Download 1 - Instructional Information Services

High School CCSS Mathematics II Curriculum Guide -Quarter 1Columbus City Schools Page 1 of 162 Table of Contents RUBRIC – IMPLEMENTING STANDARDS FOR MATHEMATICAL PRACTICE ....................... 11 Mathematical Practices: A Walk-Through Protocol .............................................................................. 16 Curriculum Timeline .............................................................................................................................. 19 Scope and Sequence ............................................................................................................................... 20 Similarity 1, 1a, 1b, 2, 3, 4, 5 ................................................................................................................. 29 Teacher Notes .......................................................................................................................................... 30 Are You Golden? ................................................................................................................................. 43 The Gumps ........................................................................................................................................... 47 The Gumps and Similar Figures .......................................................................................................... 55 Draw Similar Triangles ........................................................................................................................ 61 Similar Quilt Blocks............................................................................................................................. 63 Quilt Calculations ................................................................................................................................ 64 Investigating Triangles with Two Pairs of Congruent Angles ............................................................. 67 Similar Triangles Application .............................................................................................................. 71 Find the Scale Factor............................................................................................................................ 72 Let’s Prove the Pythagorean Theorem ................................................................................................. 76 Proving the Pythagorean Theorem, Again!.......................................................................................... 80 Trigonometric Ratios G-SRT 6, 7, 8 ...................................................................................................... 82 Teacher Notes .......................................................................................................................................... 83 Exploring Special Right Triangles (45-45-90)..................................................................................... 97 Exploring Special Right Triangles (30-60-90)..................................................................................... 99 Discovering Trigonometric Ratios ..................................................................................................... 104 Make a Model: Trigonometric Ratios ................................................................................................ 108 Let’s Measure the Height of the Flagpole .......................................................................................... 112 Applications of Trigonometry Using Indirect Measurement ............................................................. 114 Find the Missing Side or Angle ......................................................................................................... 122 Between the Uprights ......................................................................................................................... 124 Solve the Triangle .............................................................................................................................. 129 Right Triangle Park ............................................................................................................................ 135 Find the Height................................................................................................................................... 136 Find the Height Data Sheet ................................................................................................................ 137 Applications of the Pythagorean Theorem ......................................................................................... 138 Memory Match – Up .......................................................................................................................... 140 Memory Match – Up Cards ............................................................................................................... 141 Similar Right Triangles and Trigonometric Ratios ............................................................................ 147 Similar Right Triangles and Trigonometric Ratios ............................................................................ 149 Hey, All These Formulas Look Alike!............................................................................................... 155 Problem Solving: Trigonometric Ratios ............................................................................................ 157 Grids and Graphics Addendum ............................................................................................................ 162 Page 2 of 162 Math Practices Rationale CCSSM Practice 1: Make sense of problems and persevere in solving them. Why is this practice important? What does this practice look like when students are doing it? What can a teacher do to model this practice? What questions could a teacher ask to encourage the use of this practice? What does proficiency look like in this practice? What actions might the teacher make that inhibit the students’ use of this practice? Helps students to develop critical thinking skills. Teaches students to “think for themselves”. Helps students to see there are multiple approaches to solving a problem. Students immediately begin looking for methods to solve a problem based on previous knowledge instead of waiting for teacher to show them the process/algorithm. Students can explain what problem is asking as well as explain, using correct mathematical terms, the process used to solve the problem. Frame mathematical questions/challenges so they are clear and explicit. Check with students repeatedly to help them clarify their thinking and processes. “How would you go about solving this problem?” “What do you need to know in order to solve this problem?” What methods have we studied that you can use to find the information you need? Students can explain the relationships between equations, verbal descriptions, tables, and graphs. Students check their answer using a different method and continually ask themselves, “Does this make sense?” They understand others approaches to solving complex problems and can see the similarities between different approaches. Showing the students shortcuts/tricks to solve problems (without making sure the students understand why they work). Not giving students an adequate amount of think time to come up with solutions or processes to solve a problem. Giving students the answer to their questions instead of asking guiding questions to lead them to the discovery of their own question. Page 3 of 162 CCSSM Practice 2: Reason abstractly and quantitatively. Why is this practice important? What does this practice look like when students are doing it? What can a teacher do to model this practice? What questions could a teacher ask to encourage the use of this practice? What does proficiency look like in this practice? What actions might the teacher make that inhibit the students’ use of this practice? Students develop reasoning skills that help them to understand if their answers make sense and if they need to adjust the answer to a different format (i.e. rounding) Students develop different ways of seeing a problem and methods of solving it. Students are able to translate a problem situation into a number sentence or algebraic expression. Students can use symbols to represent problems. Students can visualize what a problem is asking. Ask students questions about the types of answers they should get. Use appropriate terminology when discussing types of numbers/answers. Provide story problems and real world problems for students to solve. Monitor the thinking of students. “What is your unknown in this problem? “What patterns do you see in this problem and how might that help you to solve it?” Students can recognize the connections between the elements in their mathematical sentence/expression and the original problem. Students can explain what their answer means, as well as how they arrived at it. Giving students the equation for a word or visual problem instead of letting them “figure it out” on their own. Page 4 of 162 CCSSM Practice 3: Construct viable arguments and critique the reasoning of others Why is this practice important? What does this practice look like when students are doing it? What can a teacher do to model this practice? What questions could a teacher ask to encourage the use of this practice? What does proficiency look like in this practice? What actions might the teacher make that inhibit the students’ use of this practice? Students better understand and remember concepts when they can defend and explain it to others. Students are better able to apply the concept to other situations when they understand how it works. Communicate and justify their solutions Listen to the reasoning of others and ask clarifying questions. Compare two arguments or solutions Question the reasoning of other students Explain flaws in arguments Provide an environment that encourages discussion and risk taking. Listen to students and question the clarity of arguments. Model effective questioning and appropriate ways to discuss and critique a mathematical statement. How could you prove this is always true? What parts of “Johnny’s “ solution confuses you? Can you think of an example to disprove your classmates theory? Students are able to make a mathematical statement and justify it. Students can listen, critique and compare the mathematical arguments of others. Students can analyze answers to problems by determining what answers make sense. Explain flaws in arguments of others. Not listening to students justify their solutions or giving adequate time to critique flaws in their thinking or reasoning. Page 5 of 162 CCSSM Practice 4: Model with mathematics Why is this practice important? What does this practice look like when students are doing it? What can a teacher do to model this practice? What questions could a teacher ask to encourage the use of this practice? What does proficiency look like in this practice? What actions might the teacher make that inhibit the students’ use of this practice? Helps students to see the connections between math symbols and real world problems. Write equations to go with a story problem. Apply math concepts to real world problems. Use problems that occur in everyday life and have students apply mathematics to create solutions. Connect the equation that matches the real world problem. Have students explain what different numbers and variables represent in the problem situation. Require students to make sense of the problems and determine if the solution is reasonable. How could you represent what the problem was asking? How does your equation relate to the problems? How does your strategy help you to solve the problem? Students can write an equation to represent a problem. Students can analyze their solutions and determine if their answer makes sense. Students can use assumptions and approximations to simplify complex situations. Not give students any problem with real world applications. Page 6 of 162 CCSSM Practice 5: Use appropriate tools strategically Why is this practice important? What does this practice look like when students are doing it? What can a teacher do to model this practice? What questions could a teacher ask to encourage the use of this practice? Helps students to understand the uses and limitations of different mathematical and technological tools as well as which ones can be applied to different problem situations. Students select from a variety of tools that are available without being told which to use. Students know which tools are helpful and which are not. Students understand the effects and limitations of chosen tools. Provide students with a variety of tools Facilitate discussion regarding the appropriateness of different tools. Allow students to decide which tools they will use. How is this tool helping you to understand and solve the problem? What tools have we used that might help you organize the information given in this problem? Is there a different tool that could be used to help you solve the problem? What does proficiency look like in this practice? Students are sufficiently familiar with tools appropriate for their grade or course and make sound decisions about when each of these tools might be helpful. Students recognize both the insight to be gained from the use of the selected tool and their limitations. What actions might the teacher make that inhibit the students’ use of this practice? Only allowing students to solve the problem using one method. Telling students that the solution is incorrect because it was not solved “the way I showed you”. Page 7 of 162 CCSSM Practice 6: Attend to precision. Why is this practice important? What does this practice look like when students are doing it? What can a teacher do to model this practice? What questions could a teacher ask to encourage the use of this practice? What does proficiency look like in this practice? What actions might the teacher make that inhibit the students’ use of this practice? Students are better able to understand new math concepts when they are familiar with the terminology that is being used. Students can understand how to solve real world problems. Students can express themselves to the teacher and to each other using the correct math vocabulary. Students use correct labels with word problems. Make sure to use correct vocabulary terms when speaking with students. Ask students to provide a label when describing word problems. Encourage discussions and explanations and use probing questions. How could you describe this problem in your own words? What are some non-examples of this word? What mathematical term could be used to describe this process. Students are precise in their descriptions. They use mathematical definitions in their reasoning and in discussions. They state the meaning of symbols consistently and appropriately. Teaching students “trick names” for symbols (i.e. the alligator eats the big number) Not using proper terminology in the classroom. Allowing students to use the word “it” to describe symbols or other concepts. Page 8 of 162 CCSSM Practice 7: Look for and make use of structure. Why is this practice important? What does this practice look like when students are doing it? What can a teacher do to model this practice? What questions could a teacher ask to encourage the use of this practice? What does proficiency look like in this practice? What actions might the teacher make that inhibit the students’ use of this practice? When students can see patterns or connections, they are more easily able to solve problems Students look for connections between properties. Students look for patterns in numbers, operations, attributes of figures, etc. Students apply a variety of strategies to solve the same problem. Ask students to explain or show how they solved a problem. Ask students to describe how one repeated operation relates to another (addition vs. multiplication). How could you solve the problem using a different operation? What pattern do you notice? Students look closely to discern a pattern or structure. Provide students with pattern before allowing them to discern it for themselves. Page 9 of 162 CCSSM Practice 8: Look for and express regularity in repeated reasoning Why is this practice important? What does this practice look like when students are doing it? What can a teacher do to model this practice? What questions could a teacher ask to encourage the use of this practice? What does proficiency look like in this practice? What actions might the teacher make that inhibit the students’ use of this practice? When students discover connections or algorithms on their own, they better understand why they work and are more likely to remember and be able to apply them. Students discover connections between procedures and concepts Students discover rules on their own through repeated exposures of a concept. Provide real world problems for students to discover rules and procedures through repeated exposure. Design lessons for students to make connections. Allow time for students to discover the concepts behind rules and procedures. Pose a variety of similar type problems. How would you describe your method? Why does it work? Does this method work all the time? What do you notice when…? Students notice repeated calculations. Students look for general methods and shortcuts. Providing students with formulas or algorithms instead of allowing them to discover it on their own. Not allowing students enough time to discover patterns. Page 10 of 162 RUBRIC – IMPLEMENTING STANDARDS FOR MATHEMATICAL PRACTICE Using the Rubric: Task: Is strictly procedural. Does not require students to check solutions for errors. NEEDS IMPROVEMENT Teacher: Allots too much or too little time to complete task. Encourages students to individually complete tasks, but does not ask them to evaluate the processes used. Explains the reasons behind procedural steps. Does not check errors publicly. Is overly scaffolded or procedurally “obvious”. Requires students to check answers by plugging in numbers. (teacher does thinking) Task: EMERGING Teacher: Allows ample time for all students to struggle with task. Expects students to evaluate processes implicitly. Models making sense of the task (given situation) and the proposed solution. EXEMPLARY Summer 2011 Differentiates to keep advanced students challenged during work time. Integrates time for explicit meta-cognition. Expects students to make sense of the task and the proposed solution. (teacher mostly models) (students take ownership) Task: Task: Is cognitively Allows for multiple entry points and solution paths. demanding. Requires students to Has more than one entry defend and justify their point. solution by comparing Requires a balance of multiple solution paths. procedural fluency and conceptual Teacher: understanding. Requires students to check solutions for errors usingone other solution path. PROFICIENT Review each row corresponding to a mathematical practice. Use the boxes to mark the appropriate description for your task or teacher action. The task descriptors can be used primarily as you develop your lesson to make sure your classroom tasks help cultivate the mathematical practices. The teacher descriptors, however, can be used during or after the lesson to evaluate how the task was carried out. The column titled “proficient” describes the expected norm for task and teacher action while the column titled “exemplary” includes all features of the proficient column and more. A teacher who is exemplary is meeting criteria in both the proficient and exemplary columns. PRACTICE Make sense of problems and persevere in solving them. Teacher: Does not allow for wait time; asks leading questions to rush through task. Does not encourage students to individually process the tasks. Is focused solely on answers rather than processes and reasoning. Institute for Advanced Study/Park City Mathematics Institute Secondary School Teachers Program/Visualizing Functions Page 11 of 162 PRACTICE Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Is either ambiguously stated. Does not expect students to interpret representations. Expects students to memorize procedures withno connection to meaning. Lacks context. Does not make use of multiple representations or solution paths. NEEDS IMPROVEMENT Task: Teacher: Task: Teacher: Does not ask students to present arguments or solutions. Expects students to follow a given solution path without opportunities to make conjectures. Task: EMERGING Does not help students differentiate between assumptions and logical conjectures. Asks students to present arguments but not to evaluate them. Allows students to make conjectures without justification. Is not at the appropriate level. representation. Explains connections between procedures and meaning. tasks using a single model and interpret Expects students to Is embedded in a contrived context. (teacher does thinking) Teacher: Task: Teacher: Institute for Advanced Study/Park City Mathematics Institute Secondary School Teachers Program/Visualizing Functions Page 12 of 162 PROFICIENT expressed with multiple representations. Expects students to interpret and model using multiple representations. Provides structure for students to connect algebraic procedures to contextual meaning. Links mathematical solution with a question’s answer. Avoids single steps or routine algorithms. Teacher: EXEMPLARY Helps students differentiate between assumptions and logical conjectures. Prompts students to evaluate peer arguments. Expects students to formally justify the validity of their conjectures. Expects students to interpret, model, and connect multiple representations. Prompts students to articulate connections between algebraic procedures and contextual meaning. (teacher mostly models) (students take ownership) Task: Task: Has realistic context. Has relevant realistic context. Requires students to frame solutions in a Teacher: context. Has solutions that can be Teacher: Task: Teacher: Identifies students’ assumptions. Models evaluation of student arguments. Asks students to explain their conjectures. Summer 2011 PRACTICE Model with mathematics. Use appropriate tools strategically. NEEDS IMPROVEMENT Requires students to Task: identify variables and to perform necessary computations. Teacher: Identifies appropriate variables and procedures for students. Does not discuss appropriateness of model. Does not incorporate Task: additional learning tools. Teacher: additional learning tools. Does not incorporate EMERGING (teacher does thinking) Requires students to Task: identify variables and to compute and interpret results. Teacher: Verifies that students have identified appropriate variables and procedures. Explains the appropriateness of model. Lends itself to one learning Task: PROFICIENT Requires students to (teacher mostly models) Task: identify variables, compute and interpret results, and report findings using a mixture of representations. Illustrates the relevance of the mathematics involved. Requires students to identify extraneous or missing information. Teacher: Asks questions to help students identify appropriate variables and procedures. Facilitates discussions in evaluating the appropriateness of model. Lends itself to multiple Task: learning tools. Gives students opportunity tool. Does not involve mental estimation. Models error checking by learning tools for student use. Chooses appropriate to develop fluency in mental computations. Teacher: appropriate learning tool. Demonstrates use of computations or estimation. Teacher: Institute for Advanced Study/Park City Mathematics Institute Secondary School Teachers Program/Visualizing Functions Page 13 of 162 EXEMPLARY Requires students to (students take ownership) Task: Expects students to justify identify variables, compute and interpret results, report findings, and justify the reasonableness of their results and procedures within context of the task. Teacher: their choice of variables and procedures. Gives students opportunity to evaluate the appropriateness of model. Requires multiple learning Task: tools (i.e., graph paper, calculator, manipulative). Requires students to demonstrate fluency in mental computations. Teacher: appropriate learning tools. Allows students to choose Creatively finds appropriate alternatives where tools are not available. Summer 2011 PRACTICE Attend to precision. Look for and make use of structure. Requires students to automatically apply an algorithm to a task without evaluating its appropriateness. Does not intervene when students are being imprecise. Does not point out instances when students fail to address the question completely or directly. Gives imprecise instructions. NEEDS IMPROVEMENT Task: Teacher: Task: Teacher: Does not recognize students for developing efficient approaches to the task. Requires students to apply the same algorithm to a task although there may be other approaches. Task: EMERGING Identifies individual students’ efficient approaches, but does not expand understanding to the rest of the class. Demonstrates the same algorithm to all related tasks although there may be other more effective approaches. Requires students to analyze a task before automatically applying an algorithm. Inconsistently intervenes when students are imprecise. Identifies incomplete responses but does not require student to formulate further response. Has overly detailed or wordy instructions. (teacher does thinking) Teacher: Task: Teacher: Institute for Advanced Study/Park City Mathematics Institute Secondary School Teachers Program/Visualizing Functions Page 14 of 162 PROFICIENT Requires students to analyze a task and identify more than one approach to the problem. Consistently demands precision in communication and in mathematical solutions. Identifies incomplete responses and asks student to revise their response. Teacher: Task: Teacher: EXEMPLARY Prompts students to identify mathematical structure of the task in order to identify the most effective solution path. Encourages students to justify their choice of algorithm or solution path. Requires students to identify the most efficient solution to the task. Demands and models precision in communication and in mathematical solutions. Encourages students to identify when others are not addressing the question completely. Includes assessment criteria for communication of ideas. (teacher mostly models) (students take ownership) Task: Task: Has precise instructions. Teacher: Task: Teacher: Facilitates all students in developing reasonable and efficient ways to accurately perform basic operations. Continuously questions students about the reasonableness of their intermediate results. Summer 2011 PRACTICE Look for and express regularity in repeated reasoning. Is disconnected from prior and future concepts. Has no logical progression that leads to pattern recognition. NEEDS IMPROVEMENT Task: Teacher: Does not show evidence of understanding the hierarchy within concepts. Presents or examines task in isolation. Task: EMERGING Hides or does not draw connections to prior or future concepts. Is overly repetitive or has gaps that do not allow for development of a pattern. (teacher does thinking) Teacher: Institute for Advanced Study/Park City Mathematics Institute Secondary School Teachers Program/Visualizing Functions Page 15 of 162 PROFICIENT Reviews prior knowledge and requires cumulative understanding. Lends itself to developing a pattern or structure. (teacher mostly models) Task: Teacher: Connects concept to prior and future concepts to help students develop an understanding of procedural shortcuts. Demonstrates connections between tasks. EXEMPLARY Addresses and connects to prior knowledge in a nonroutine way. Requires recognition of pattern or structure to be completed. (students take ownership) Task: Teacher: Encourages students to connect task to prior concepts and tasks. Prompts students to generate exploratory questions based on the current task. Encourages students to monitor each other’s intermediate results. Summer 2011 Mathematical Practices: A Walk-Through Protocol Mathematical Practices Observations *Note: This document should also be used by the teacher for planning and self-evaluation. MP.1. Make sense of problems and persevere in solving them Teachers are expected to______________: Provide appropriate representations of problems. Students are expected to______________: Connect quantity to numbers and symbols (decontextualize the problem) and create a logical representation of the problem at hand. Recognize that a number represents a specific quantity (contextualize the problem). Contextualize and decontextualize within the process of solving a problem. Teachers are expected to______________: Provide time for students to discuss problem solving. Students are expected to______________: Engage in solving problems. Explain the meaning of a problem and restate in it their own words. Analyze given information to develop possible strategies for solving the problem. Identify and execute appropriate strategies to solve the problem. Check their answers using a different method, and continually ask “Does this make sense?” MP.2. Reason abstractly and quantitatively. MP.3. Construct viable arguments and critique the reasoning of others. Students are expected to____________________________: Explain their thinking to others and respond to others’ thinking. Participate in mathematical discussions involving questions like “How did you get that?” and “Why is that true?” Construct arguments that utilize prior learning. Question and problem pose. Practice questioning strategies used to generate information. Analyze alternative approaches suggested by others and select better approaches. Justify conclusions, communicate them to others, and respond to the arguments of others. Compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument, explain what it is. CCSSM National Professional Development Page 16 of 162 Mathematical Practices MP.4. Model with mathematics. MP 5. Use appropriate tools strategically Observations Teachers are expected to______________: Provide opportunities for students to listen to or read the conclusions and arguments of others. Students are expected to______________: Apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. Make assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. Experiment with representing problem situations in multiple ways, including numbers, words (mathematical language), drawing pictures, using objects, acting out, making a chart or list, creating equations, etc. Identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts, and formulas. Evaluate their results in the context of the situation and reflect on whether their results make sense. Analyze mathematical relationships to draw conclusions. Teachers are expected to______________: Provide contexts for students to apply the mathematics learned. Students are expected to______________: Use tools when solving a mathematical problem and to deepen their understanding of concepts (e.g., pencil and paper, physical models, geometric construction and measurement devices, graph paper, calculators, computer-based algebra or geometry systems.) Consider available tools when solving a mathematical problem and decide when certain tools might be helpful, recognizing both the insight to be gained and their limitations. Detect possible errors by strategically using estimation and other mathematical knowledge. Teachers are expected to______________: CCSSM National Professional Development Page 17 of 162 Mathematical Practices MP.6. Attend to precision. MP.7. Look for and make use of structure. MP.8. Look for and express regularity in repeated reasoning. Observations Students are expected to______________: Use clear and precise language in their discussions with others and in their own reasoning. Use clear definitions and state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. Specify units of measure and label parts of graphs and charts. Calculate with accuracy and efficiency based on a problem’s expectation. Teachers are expected to______________: Emphasize the importance of precise communication. Students are expected to______________: Describe a pattern or structure. Look for, develop, generalize, and describe a pattern orally, symbolically, graphically and in written form. Relate numerical patterns to a rule or graphical representation Check the reasonableness of their results. Teachers are expected to______________: Use models to examine patterns and generate their own algorithms. Use models to explain calculations and describe how algorithms work. Use repeated applications to generalize properties. Look for mathematically sound shortcuts. Apply and discuss properties. Teachers are expected to______________: Provide time for applying and discussing properties. Students are expected to______________: Describe repetitive actions in computation CCSSM National Professional Development Page 18 of 162 High School Common Core Math II Curriculum Timeline Topic Intro Unit Similarity Trigonometric Ratios Other Types of Functions Comparing Functions and Different Representations of Quadratic Functions Modeling Unit and Project Quadratic Functions: Solving by Factoring Quadratic Functions: Completing the Square and the Quadratic Formula Probability Geometric Measurement Geometric Modeling Unit and Project Standards Covered G – SRT 1 G – SRT 1a G – SRT 1b G – SRT 6 G – SRT 2 G – SRT 3 G – SRT 4 G – SRT 7 G – SRT 5 Grading Period 1 1 No. of Days 5 20 G – SRT 8 1 20 A – CED 1 A – CED 4 A – REI 1 N – RN 1 N – RN 2 N – RN 3 F – IF 4 F – IF 5 F – IF 6 F – IF 7 F – IF 7a F– IF 9 F – IF 4 F – IF 7b F – IF 7e F – IF 8 F – IF 8b F– BF1 A– CED 1 A– CED 2 F– BF 1 F– BF 1a F – BF 1b F– BF 3 F – BF 1a F – BF 1b F – BF 3 A – SSE 1b N–Q2 2 15 F – LE 3 N– Q 2 S – ID 6a S – ID 6b A – REI 7 2 20 2 10 A – APR 1 A – REI 1 A – REI 4b F – IF 8a A – CED 1 A – SSE 1b A – SSE 3a 3 20 A – REI 1 A – REI 4 A – REI 4a A – REI 4b A – SSE 3b F – IF 8 F – IF 8a A – CED 1 N – CN 1 N – CN 2 N – CN 7 3 20 S – CP 1 S – CP 2 S – CP 3 G – GMD 1 S – CP 4 S – CP 5 S – CP 6 G – GMD 3 S – CP 7 4 20 4 10 G – MG 1 G – MG 2 G – MG 3 4 15 Page 19 of 162 High School Common Core Math II 1st Nine Weeks Scope and Sequence Intro Unit – IO (5 days) Topic 1 – Similarity (20 days) Geometry (G – SRT): 1) Similarity, Right Triangles, and Trigonometry: Understand similarity in terms of similarity transformations. G – SRT 1: Verify experimentally the properties of dilations given by a center and a scale factor. G – SRT 1a: A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. G – SRT 1b: The dilation of a line segment is longer or shorter in the ratio given by the scale factor. G – SRT 2: Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. G – SRT 3: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Geometry (G – SRT): 2) Similarity, Right Triangles, and Trigonometry: Prove theorems involving similarity. G – SRT 4: Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. G – SRT 5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Topic 2 – Trigonometric Ratios (20 days) Geometry (G – SRT): 3) Similarity, Right Triangles, and Trigonometry: Define trigonometric ratios and solve problems involving .right triangles G – SRT 6: Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. G – SRT 7: Explain and use the relationship between the sine and cosine of complementary angles. G – SRT 8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Page 20 of 162 High School Common Core Math II 2nd Nine Weeks Scope and Sequence Topic 3 – Other Types of Functions (15 days) Creating Equations (A – CED): 4) Create equations that describe numbers or relationships A – CED 1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. A – CED 4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R. Reasoning with Equations and Inequalities (A – REI): 5) Understand solving equations as a process of reasoning and explain the reasoning. A – REI 1: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. The Real Number System (N – RN): 6) Extend the properties of exponents to rational exponents. N – RN 1: Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5. N – RN 2: Rewrite expressions involving radicals and rational exponents using the properties of exponents. The Real Number System (N – RN): 7) Use properties of rational and irrational numbers. N – RN 3: Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Interpreting Functions (F – IF): 8) Interpret functions that arise in applications in terms of the context. F – IF 4*: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.* Interpreting Functions (F – IF): 9) Analyze functions using different representations. F – IF 7b: Graph square root, cube root, and absolute value functions. F – IF 7e: Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Page 21 of 162 F – IF 8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. F – IF 8b: Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay. Building Functions (F – BF): 10) Build a function that models a relationship between two quantities. F – BF 1: Write a function that describes a relationship between two quantities. F – BF 1a: Determine an explicit expression, a recursive process, or steps for calculation from a context. F – BF 1b: Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Building Functions (F – BF): 11) Build new functions from existing functions. F – BF 3: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Seeing Structure in Expressions (A – SSE): 12) Interpret the structure of expressions. A – SSE 1b: Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)n as the product of P and a factor not depending on P. Quantities (NQ): 13) Reason quantitatively and use units to solve problems. N – Q 2: Define appropriate quantities for the purpose of descriptive modeling. Topic 4 – Comparing Functions and Different Representations of Quadratic Functions (20 days) Interpreting Functions (F – IF): 14) Interpret functions that arise in applications in terms of the context. F – IF 4*: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.* F – IF 5*: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.* Page 22 of 162 F – IF 6: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Interpreting Functions (F – IF): 15) Analyze functions using different representations. F – IF 7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. F – IF 7a*: Graph linear and quadratic functions and show intercepts, maxima, and minima.* F – IF 9: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Creating Equations (A – CED): 16) Create equations that describe numbers of relationships. A – CED 1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. A – CED 2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Building Functions (F – BF): 17) Build a function that models a relationship between two quantities. F – BF 1: Write a function that describes a relationship between two quantities. F – BF 1a: Determine an explicit expression, a recursive process, or steps for calculation from a context. F – BF 1b: Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Building Functions (F – BF): 18) Build new functions from existing functions. F – BF 3: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Linear and Exponential Models (F – LE): 19) Construct and compare linear and exponential models and solve problems. F- LE 3: Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. Quantities (N-Q): 20) Reason quantitatively and use units to solve problems. N – Q 2: Define appropriate quantities for the purpose of descriptive modeling. Page 23 of 162 Interpreting Categorical and Quantitative Data (S – ID): 21) Summarize, represent, and interpret data on two categorical and quantitative variables. S – ID 6a: Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear and exponential models. S – ID 6b: Informally assess the fit of a function by plotting and analyzing residuals. Reasoning with Equations and Inequalities (A – REI): 22) Solve systems of equations. A – REI 7: Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = -3x and the circle x2 + y2 = 3. Modeling Unit and Project –(10 days) Page 24 of 162 High School Common Core Math II 3rd Nine Weeks Scope and Sequence Topic 5–Quadratic Functions – Solving by factoring (20 days) Arithmetic with Polynomials and Rational Expressions (A – APR): 23) Perform arithmetic operations on polynomials. A – APR 1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Reasoning with Equations and Inequalities (A – REI): 24) Understand solving equations as a process of reasoning and explain the reasoning. A – REI 1: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Reasoning with Equations and Inequalities (A – REI): 25) Solve equations and inequalities in one variable. A – REI 4b: Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Interpreting Functions (F – IF): 26) Analyze functions using different representations. F – IF 8a: Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Creating Equations (A – CED): 27) Create equations that describe numbers of relationships. A – CED 1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Seeing Structure in Expressions (A – SSE): 28) Interpret the structure of expressions. A – SSE 1b: Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)n as the product of P and factor not depending on P. Seeing Structure in Expressions (A – SSE): 29) Write expressions in equivalent forms to solve problems. A – SSE 3a: Factor a quadratic expression to reveal the zeros of the function it defines. Topic 6– Quadratic Functions – Completing the Square/Quadratic Formula (20 days) Reasoning with Equations and Inequalities (A – REI): 30) Understand solving equations as a process of reasoning and explain the reasoning. Page 25 of 162 A – REI 1: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Reasoning with Equations and Inequalities (A – REI): 31) Solve equations and inequalities in one variable. A – REI 4: Solve quadratic equations in one variable. A – REI 4a: Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. A – REI 4b: Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Seeing Structure in Expressions (A – SSE): 32) Write expressions in equivalent forms to solve problems. A – SSE 3b: Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. Interpreting Functions (F – IF): 33) Analyze functions using different representations. F – IF 8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. F – IF 8a: Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Creating Equations (A – CED): 34) Create equations that describe numbers of relationships. A – CED 1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. The Complex Number System (N – CN): 35) Perform arithmetic operations with complex numbers. N – CN 1: Know there is a complex number i such that i 2 has the form a+bi with a and b real. 1 , and every complex number N – CN 2: Use the relation i 2 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. The Complex Number System (N – CN): 36) Use complex numbers in polynomial identities and equations. N – CN 7: Solve quadratic equations with real coefficients that have complex solutions. Page 26 of 162 High School Common Core Math II 4th Nine Weeks Scope and Sequence Topic 7 –Probability (20 days) Conditional Probability and the Rules of Probability (S – CP): 37) Understand independence and conditional probability and use them to interpret data. S – CP 1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”). S – CP 2: Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. S – CP 3: Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. S – CP 4: Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from you school will favor science given that the student is in the tenth grade. Do the same for other subjects and compare the results. S – CP 5: Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. Conditional Probability and the Rules of Probability (S – CP): 38) Use the rules of probability to compute probabilities of compound events in a uniform probability model. S – CP 6: Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. S – CP 7: Apply the Addition Rule, P(A or B) = P(B) – P(A and B), and interpret the answer in terms of the model. Topic 8 – Geometric Measurement (10 days) Geometric Measurement and Dimension (G – GMD): 39) Explain volume formulas and use them to solve problems. G – GMD 1: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. G – GMD 3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Page 27 of 162 Geometric and Modeling Project-(15 days) *Modeling with Geometry (G – MG): 40) Apply geometric concepts in modeling situations. G – MG 1*: Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).* G – MG 2*: Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).* G – MG 3*: Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).* Page 28 of 162 COLUMBUS CITY SCHOOLS HIGH SCHOOL CCSS MATHEMATICS II CURRICULUM GUIDE TOPIC 1 CONCEPTUAL CATEGORY TIME RANGE 20 days Similarity 1, 1a, 1b, 2, 3, 4, Geometry 5 Domain: Geometry: Similarity, Right Triangles, and Trigonometry (G – SRT): Cluster 1) Understand similarity in terms of similarity transformations. 2) Prove theorems involving similarity. GRADING PERIOD 1 Standards 1) Understand similarity in terms of similarity transformations. G – SRT 1: Verify experimentally the properties of dilations given by a center and a scale factor. G – SRT 1a: A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. G – SRT 1b: The dilation of a line segment is longer or shorter in the ratio given by the scale factor. G – SRT 2: Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. G – SRT 3: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. 2) Prove theorems involving similarity. G – SRT 4: Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. G – SRT 5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 29 of 162 TEACHING TOOLS Vocabulary: AA, center of dilation, corresponding parts, cross product, dilation, extremes, figure, image, irregular polygon, means, midsegment, proportion, proportional, ratio, regular polygon, rotational symmetry, scale factor, similar polygons, tessellation, transformations, transversal segments, similarity Teacher Notes Dilations A dilation is a transformation that produces an image that is the same shape as the original, but is a different size. A dilation used to create an image larger than the original is called an enlargement. A dilation used to create an image smaller than the original is called a reduction. The website for the Topic index for dilations at Regents Prep is sited below. It includes lessons, practice and teacher support. http://www.regentsprep.org/Regents/math/geometry/GT3/indexGT3.htm The website below provides a lesson with a warm-up, vocabulary, and examples with solutions for dilations. http://www.chs.riverview.wednet.edu/math/aitken/Integrated-Old/int1-notes/Unit6/Int1_6-6Dilations-notes.pdf At the website below, teachers can look at a tutorial for dilations. There are four different ones: dilating a triangle; invariants in dilation; dilations in the coordinate plane; and problem solving with dilations. http://education.ti.com/en/us/professional-development/pd_onlinegeometry_free/course-outline-andtake-this-course Geometry with Cabri Jr. and the TI-84 Plus Module 9 DILATIONS Lesson 1 - dilating a triangle Lesson 2 - invariants in a dilation Lesson 3 - dilations in the coordinate plane Lesson 4 - problem solving with dilations Similarity Similar polygons are two polygons with congruent corresponding angles and proportional corresponding sides. If the cross product is equal, then the corresponding sides are proportional. Similarity of polygons can be proven in three different ways: Angle-Angle Similarity, Side-Side-Side Similarity, and Side-AngleSide Similarity. A-A Similarity is used when two pairs of corresponding angles are congruent. S-S-S Similarity is used when all three pairs of corresponding sides are proportional. S-A-S Similarity is used when two pairs of corresponding sides are proportional and their included angles are congruent. Below the website listed contains lessons, practice and teacher resources on similarity. http://www.regentsprep.org/Regents/math/geometry/GP11/indexGP11.htm A tutorial on the Pythagorean Theorem and trigonometry can be found at the website below. https://activate.illuminateed.com/playlist/resourcesview/rid/50c56098efea65b540000000/id/50c4c151 efea65fd18000003/bc0/user/bc1/playlist/bc0_id/4fff3767efea650023000698 CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 30 of 162 The website below has cliff notes on AA triangle similarity. http://www.cliffsnotes.com/study_guide/Similar-Triangles.topicArticleId-18851,articleId-18812.html The website below gives examples of SAS, AA, and SSS triangle similarity. http://www.analyzemath.com/Geometry/similar_triangles.html This website offers a teacher resource that includes the warm-up, algebraic review, lecture notes, practice, and hands on activities for similar polygons. http://teachers.henrico.k12.va.us/math/IGO/05Similarity/5_2.html This website offers a teacher resource that includes the warm-up, algebraic review, lecture notes, practice, and hands on activities for similar triangles. http://teachers.henrico.k12.va.us/math/IGO/05Similarity/5_3.html The scale factor is the ratio of lengths of two corresponding sides of similar polygons. The phrase “scale factor” is used in different ways. Example1: If the length of a side of Square A is 4 and the length of a side of Square B is 7, then the scale factor of Square A to Square B is 4/7. Example2: If the length of a side of Square A is 4 and Square A is enlarged by a scale factor of 2, then the length of a side of the new square is 8. Scale factor is used to produce dilations, which can be smaller or larger than the original figure. Real life applications include reading maps, blueprints, and varying recipe sizes. This website offers a teacher resource that includes the warm-up, algebraic review, lecture notes, practice, and hands on activities for using proportions. http://teachers.henrico.k12.va.us/math/IGO/05Similarity/5_1.html This website offers a teacher resource that includes the warm-up, algebraic review, lecture notes, practice, and hands on activities for proportional parts. http://teachers.henrico.k12.va.us/math/IGO/05Similarity/5_4.html The TI-84 and Cabri Jr. can be used for special triangles. An on-line tutorial can be found at the website below. Module 11 SPECIAL TRIANGLES - Lesson 3 - constructing a right triangle http://education.ti.com/en/us/professional-development/pd_onlinegeometry_free/course-outline-andtake-this-course' The TI-84 and Cabri Jr. can be used for special triangles. A tutorial can be found at the website below. Module 14 PROPORTIONS - Lesson 1 - similar triangles http://education.ti.com/en/us/professional-development/pd_onlinegeometry_free/course-outline-andtake-this-course This website offers a teacher resource that includes the warm-up, algebraic review, lecture notes, practice, CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 31 of 162 and hands on activities for Pythagorean Theorem. http://teachers.henrico.k12.va.us/math/IGO/07RightTriangles/7_2.html Below a website is listed for a video tutorial for solving a triangle using SAS. http://patrickjmt.com/solving-a-triangle-sas-example-1/ Below a website is listed for a video tutorial for another example of solving a triangle for SAS. http://patrickjmt.com/solving-a-triangle-sas-example-2/ The website below has a video tutorial to find the missing side and angles of triangle using SAS. http://patrickjmt.com/side-angle-side-for-triangles-finding-missing-sidesangles-example-1/ Another example of finding the missing side and angles of a triangle using SAS can be found at the website below. http://patrickjmt.com/side-angle-side-for-triangles-finding-missing-sidesangles-example-2/ Misconceptions/Challenges: Students do not match up the corresponding sides of figures, and therefore incorrectly set up proportions between similar polygons, which cause them to get the incorrect side lengths or transversal segments. Students believe that adding a particular value to all sides of a polygon will create a similar polygon. Students mix up the possible values of the scale factors for enlargements and reductions. Students do not multiply the scale factor by all sides in the polygon. Students think that all polygons of a particular shape (for example all right traingles, or all rectangles) are similar; they do not recognize that they can have different corresponding angles. CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 32 of 162 Instructional Strategies: SRT 1 Analyzing Congruence Proofs. http://map.mathshell.org.uk/materials/lessons.php?taskid=452 This lesson focuses on the concepts of congruency and similarity, including identifying corresponding sides and corresponding angles within and between triangles. Students will identify and understand the significance of a counter-example, and prove and evaluate proofs in a geometric context Key Visualizations, Geometry: http://ccsstoolbox.agilemind.com/animations/standards_content_visualizations_geometry.html This website has an animation where students can explore dilations of lines by selecting points along the line and thinking about point-by-point dilations. Students make a connection between dilations and ratios. Photocopy Faux Pas http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig_te_04091 3.pdf The website below provides a lesson on the essential features of dilation. It is Classroom Task: 6.1 found on pages 4 – 10. SRT 1a Dialations http://psdsm2ccss.pbworks.com/w/page/56495542/GSRT1%20Verify%20experimentally%20properti es%20of%20dilations Create a dilation of a line segment AB through point C with a scale factor of 2:1 to create segment EF. Find lengths of all segments, EF, AB, BC, CE and CF. Dilate and Reflect http://education.ti.com/xchange/US/Math/AlgebraII/16008/Transformations_Dilating_Functions_Teac her.pdf Students will use the Nspire Handheld to dilate and reflect different types of functions by grabbing points. Students will understand the effect of the coefficient on the vertical stretch or shrink of the function Properties of Dilations http://education.ti.com/en/us/activity/detail?id=0C732215F7EC479AB1A6350A64B161B2 Students explore the properties of dilations and the relationships between the original and image figures. Playing with Dilations http://www.cpalms.org/RESOURCES/URLresourcebar.aspx?ResourceID=SSGunMzEork=D Students explore dilations and rotations using Virtual Manipulates Dilation and Scale Factor http://www.illustrativemathematics.org/illustrations/602 Give student a copy of the picture so they can draw the points A’, B’ and C’. Provide extra space below the picture. This task enables the students to verify that a dilation takes a line that does not pass through the CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 33 of 162 center of a line parallel to the original line and the dilation of the line segment is longer or shorter by a scale factor. Rulers may be useful for duplication of lengths without formal constructions Properties of Dilations https://www.cohs.com/editor/userUploads/file/Meyn/321%20Ch%205a%20Student%20WB.pdf Lesson 5-1 page 5. Activity to investigate properties of dilations using geometry software. Properties include: dilations preserve angle measure, betweenness, collinearity, maps a lines not passing through the center of the dilation to a parallel line and leaves a line passing through the center unchanged, and a dilation of a line segment is longer or shorter in the ratio given by the scale factor. Analogy of Dilation to zoom http://www.geogebra.org/cms/ Draws an analogy of dilation to zoom-in and zoom-out of a camera, a document camera, an iPad, or using geometry software programs such as Geogebra. SRT 1b It is beneficial to use real-life data to discuss ratios with students. You can ask students to compare the number of male students to female students, the number of students in tennis shoes to students not in tennis shoes, and the number of students with homework to students without homework. Have students complete the activity “Are You Golden?” (Included in this Curriculum Guide). Divide students into groups of 3-4. This activity will allow students to discover the golden ratio by finding the ratios of various body parts. Take two triangles that are congruent. The sides can be 3, 4, and 5 units long. Set up ratios comparing 1 . Introduce similar triangles. All congruent triangles are corresponding sides. The ratios all reduce to1 = 1 similar triangles with a scale factor of 1:1. The corresponding sides of similar triangles are proportional and corresponding angles are congruent. Take two similar triangles. One has sides 6, 8, and 12. The other has sides 9, 12, and 18. Each ratio of the corresponding sides reduces to 2:3. Next, we can present situations with similar triangles where the length of one side is missing. We can demonstrate how we can set up ratios comparing corresponding sides and use properties of proportions to calculate the missing side. Similarity and Triangles. https://www.cohs.com/editor/userUploads/file/Meyn/321%20Ch%205a%20Student%20WB.pdf Practice work on applying similarity to triangles. Lesson 5-4 page 17. Students use dilations and rigid motions to map the image of triangle ABC to triangle DEF ( This lesson can also be found in SRT 3) Discuss what a blueprint is and the purpose it serves. Have students do the following activity in small 1 cooperative learning groups. Ask them to make a blueprint of the classroom. Use the scale: inch = 1 foot. 4 Use quarter-inch graph paper for this activity. Have students measure the length and width of the room. Point out those decisions that will need to be made, such as where doors and windows should be located on the scale drawing. As an extension, a scale drawing of the building or the cafeteria could be done. Ask students if the same scale should be used. Ask them to explain why or why not. Discuss options. The link below contains an explanation about dilations. http://www.frapanthers.com/teachers/zab/Geometry(H)/GeometryinaNutshell/GeometryNutshell2005/ Text/Dilations.pdf CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 34 of 162 The website below has a practice sheet for dilations. http://mathematicsburns.cmswiki.wikispaces.net/file/view/DilationsTranslations+activity+worksheet+ for+2-20.pdf Have students complete the activities “The Gumps” and “The Gumps and Similar Figures” (included in this Curriculum Guide) to lead students into discovering that mathematically similar figures have congruent angles and proportional The Gumps sides. Divide students into groups of 3-5. Each group should create one set of figures based on the coordinates given in the chart. Graph paper is required and some figures may require more than one sheet. The sample figures drawn in this Curriculum Guide use a scale factor of 2 in order for each figure to fit on one sheet of paper. Transparencies can be made of the figures to overlay them in order to show that the angles of Giggles, Higgles, and Ziggles are congruent. Are They Similar? http://www.illustrativemathematics.org/illustrations/603 The activity includes a picture of two triangles that appear to be similar but to prove similarity they need further information. Ask students to provide a sequence of similarity transformations that map one triangle to the other one. Remind students that all parts of one triangle get mapped to the corresponding parts of the other one. An additional task includes asking the students to prove or disprove that the triangles are similar in each problem using properties of parallel lines and the definition of similarity. Transformations and Similarity https://www.cohs.com/editor/userUploads/file/Meyn/321%20Ch%205a%20Student%20WB.pdf Similarity practice. Lesson 5-3. Students find that two plane figures are similar if and only if one can be obtained from the other by transformations. Geometry Problems: Circles and Triangles, http://map.mathshell.org/materials/lessons.php?taskid=222 Students solve problems by determining the lengths of the sides in right triangles. They also determine the measurements of shapes by decomposing complex shapes into simpler ones. Scale (or Grid) Drawings and Dilations. http://www.regentsprep.org/Regents/math/geometry/GT3/DActiv.htm Students work with scale (or grid) drawing to reinforce the concept of scalar factor. Angles and Similarity http://education.ti.com/calculators/downloads/US/Activities/Detail?id=13153 Students use technology (TI-Nspire or Nspire CAS) to experiment with the measures of the angles of similar triangles to determine conditions necessary for two triangles to be similar. Corresponding Parts of Similar Triangles http://education.ti.com/calculators/downloads/US/Activities/Detail?id=13150 Students use technology(TI-Nspire or Nspire CAS) to change the scale factor (r) between similar triangles, identify the corresponding parts, and establish relationships between them. Nested Similar Triangles http://education.ti.com/calculators/downloads/US/Activities/Detail?id=13152 Students use technology (TI-Nspire or Nspire CAS) to discover the conditions that make triangles similar by CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 35 of 162 moving the sides opposite the common angles. Demonstrate to students the properties of similarity. Draw a triangle and ask students “How would you draw a triangle similar to the triangle shown?” Include in your discussion that angle measures are the same and sides are proportional. Have students draw two triangles one that is similar to and larger than the original and one that is similar to and smaller than the original. Follow up this introduction to similarity with the “Draw Similar Triangles” activity (included in this Curriculum Guide). Students will need a protractor, straightedge, calculator, and a copy of the worksheet. Students can do this activity in partners or individually. Quilts are a beautiful, practical, and historically significant use of geometric shapes. Students will work with triangles in historic quilt patterns by creating triangles similar to those in a quilt block and then creating their own pattern with the new triangles as described in the activity “Similar Quilt Blocks” (included in this Curriculum Guide). Students may work individually or in groups. Students will need a copy of the “Similar Quilt Blocks” sheet, the “Quilt Calculations” sheet (included in this Curriculum Guide), the “Quilt Design #1” sheet (included in this Curriculum Guide), the “Quilt Design #2” sheet (included in this Curriculum Guide), a ruler, a protractor, and materials to make their quilt design (e.g., construction paper, scissors, etc.). After students have completed this activity, have students share their creations and any challenges they may have had in creating their new pattern. Students who express an interest in this art form may find additional information by searching the web using the keyword “quilt”. Scale Factor Area Perimeter http://education.ti.com/calculators/downloads/US/Activities/Detail?id=13154 Students use technology (TI-Nspire or Nspire CAS) to explore the relationship of perimeter and area in similar triangles when the scale factor is changed. Transformations with Lists, http://education.ti.com/calculators/downloads/US/Activities/Detail?id=10278 Students use list operations to perform reflections, rotations, translations, and dilations on a figure and graph the resulting image using a scatter plot.. Dilations. http://www.frapanthers.com/teachers/zab/Geometry(H)/ClassNotes/14.6Dilations.pdf The website below has practice for dilations. SRT 2 Triangle Similarity. https://ccgps.org/G-SRT_9DRF.html This website offers internet resources for triangle similarity. Investigating Triangles with Two Pairs of Congruent Angles (AA similarity): Have students complete the activity “Investigating Triangles with Two Pairs of Congruent Angles” (included in this Curriculum Guide). Students should discover the AA Similarity Theorem from this activity. Students will need protractors and straightedges to complete this activity. Draw a triangle on the chalkboard. Label the vertices of the triangle A, B, and C. Double the length of AB from point A. Label the resulting endpoint B'. Double the length of AC from point A. Label the resulting endpoint C'. Connect B' and C'. Compare ABC and AB'C'. Discuss with students whether or not the triangles are similar. (They are similar because of SAS for ~ ’s.) CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 36 of 162 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Have students complete the activity “Similar Triangles Application” (included in this Curriculum Guide) to use their skills with similar triangles in a real life situation. Meter sticks and mirrors are required materials. Students should stand several feet away from an object, placing the mirror on the ground between themselves and the object. The student should place themselves or the mirror in such a way that s/he can spot the top of the object in the mirror. A partner should take three measurements: the distance the student is standing from the mirror, the distance from the mirror to the base of the object,, and the distance from the students line of sight to the ground. Using proportions and similar triangles, the students should be able to indirectly calculate the height of the object. Have students do the activity “Find the Scale Factor” (included in this Curriculum Guide) for more practice in using scale factor to solve similarity problems. Before doing this activity, discuss scale factors with the students. For example, discuss with the students how the scale factor is 5:1 not 4:1 in the figure below. 24 6 Falling Down a Rabbit Hole Can Lead to a King Sized Experience - Exploring Similar Figures Using Proportions,” http://alex.state.al.us/lesson_view.php?id=30067 Students explore similarity. They simplify ratios, solve proportions using cross products, and use properties of proportions to solve real-world problems. Similarity Transformation https://www.cohs.com/editor/userUploads/file/Meyn/321%20Ch%205a%20Student%20WB.pdf: Students find that two plane figures are similar if and only if one can be obtained from the other by transformations (reflections, translations, rotations, and/or dilations Lesson 5-3, page 13 (This lesson is also found in SRT 1b) Triangle Dilations http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig_te_04091 3.pdf Students examine relationships of proportions in triangles that are known to be similar to each other based on dilations. Classroom Task 6.2 pages 11-20 (This lesson is also found in SRT 5.) Similar Triangles and Other Figures http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig_te_04091 3.pdf Students compare the definitions of similarity based on dilations and relationships between corresponding sides and angles. Classroom Task 6.3 pages 21-23 (This lesson is also found in SRT 3.) CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 37 of 162 SRT3 The website below contains lessons for SRT3. https://ccgps.org/G-SRT_AVKU.html Similarity and Triangles. https://www.cohs.com/editor/userUploads/file/Meyn/321%20Ch%205a%20Student%20WB.pdf Practice work on applying similarity to triangles. Lesson 5-4 page 17. Students use dilations and rigid motions to map the image of triangle ABC to triangle DEF. (This lesson was also provided in SRT 1b) Practice with Similarity Proofs, http://www.regentsprep.org/Regents/math/geometry/GP11/PracSimPfs.htm Eight formative assessment questions are provided. Similar Triangles and Other Figures http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig_te_04091 3.pdf Students compare the definitions of similarity based on dilations and relationships between corresponding sides and angles. Classroom Task 6.3 pages 21-23 (This lesson is also found in SRT 3.) SRT4 Pythagorean Theorem https://ccgps.org/G-SRT_G6QQ.html A power point presentation on the Pythagorean Theorem. A Proportionality Theorem. https://www.cohs.com/editor/userUploads/file/Meyn/321%20Ch%205a%20Student%20WB.pdf Students find what happens when a line that is parallel to one side of a triangle “splits” the other two sides. The sides are dived proportionally. It is known as the Side-Splitting Theorem. (This lesson is also found in SRT 5.) Proving the Pythagorean Theorem https://www.cohs.com/editor/userUploads/file/Meyn/321%20Ch%205a%20Student%20WB.pdf Students will use their knowledge about similar triangles to prove the Pythagorean Theorem. Applying Angle Theorems http://map.mathshell.org/materials/lessons.php?taskid=214 Students use geometric mean properties to solve problems using the measures of interior and exterior angles of polygons Have students complete the activity “Let’s Prove the Pythagorean Theorem” (included in this Curriculum Guide) to construct a proof of the Pythagorean Theorem. Have students complete the activity “Proving the Pythagorean Theorem, Again!”(included in this Curriculum Guide) to reinforce the proof of the Pythagorean Theorem. Proofs of the Pythagorean Theorem http://map.mathshell.org/materials/lessons.php?taskid=419&subpage=concept Below link: Students interpret diagrams, link visual and algebraic representations, and produce a mathematical argument CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 38 of 162 Cut by a Transversal http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig_te_04091 3.pdf Students examine proportional relationships of segments when two transversals intersect sets of parallel lines. Classroom Task: 6.4 pages 30-37 Measured Reasoning http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig_te_04091 3.pdf Students apply theorems about lines, angles, and proportional relationships when parallel lines are crossed by multiple transversals. Classroom Task6.5 pages 38-45. (This lesson can also been found at SRT 5.) Pythagoras by Proportions http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig_te_04091 3.pdf Students use similar triangles to prove the Pythagorean Theorem and theorems about geometric means in right triangles. Classroom Task 6.6 pages 36-52(This lesson can also been found at SRT 5.) Finding the Value of a Relationship http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig_te_04091 3.pdf Students solve for unknown values in right triangles using trigonometric ratios. Classroom Task 6.9 pages 67-74 (This lesson can also been found at SRT 5.) SRT5 Proving the Pythagorean Theorem https://www.cohs.com/editor/userUploads/file/Meyn/321%20Ch%205a%20Student%20WB.pdf Students will use their knowledge about similar triangles to prove the Pythagorean Theorem. (Lesson 5-7, page 27) Measured Reasoning, http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig_te_04091 3.pdf Students apply theorems about lines, angles, and proportional relationships when parallel lines are crossed by multiple transversals. Classroom Task 6.5 pages 38-45 (This lesson can also been found at SRT 4.) Finding the Value of a Relationship http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig_te_04091 3.pdf Students solve for unknown values in right triangles using trigonometric ratios. Classroom Task 6.9 pages 67-74 (This lesson can also been found at SRT 4.) Pythagoras by Proportions http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig_te_04091 3.pdf Students use similar triangles to prove the Pythagorean Theorem and theorems about geometric means in CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 39 of 162 right triangles. Classroom Task 6.6 pages 36-52 (This lesson can also been found at SRT 4.) How Tall is the School’s Flagpole https://ccgps.org/G-SRT_AVKU.html Students will apply math concepts concerning similar triangles and trigonometric functions to real life situations. The students will find measurements of objects when they are unable to use conventional measurement. (This lesson can also be found at SRT 3) A Proportionality Theorem. https://www.cohs.com/editor/userUploads/file/Meyn/321%20Ch%205a%20Student%20WB.pdf Students find what happens when a line that is parallel to one side of a triangle “splits” the other two sides. The sides are dived proportionally. It is known as the Side-Splitting Theorem. (This lesson is also found in SRT 4) Solving Problems Using Similarity. https://www.cohs.com/editor/userUploads/file/Meyn/321%20Ch%205a%20Student%20WB.pdf Students use proportionality of corresponding sides to find side lengths of two similar polygons. Lesson 5-5 Solving Geometry Problems: Floodlights http://map.mathshell.org/materials/lessons.php?taskid=429&subpage=problem Students make models, draw diagrams, and identify similar triangles to solve problems. .https://www.georgiastandards.org/Frameworks/GSO%20Frameworks/MathII_Unit2_%20Student_E dition_revised_8-10-09.pdf This website contains a set of lessons on right triangle trigonometry. These lessons include discovering special right triangles, discovering trigonometric ratio relationships, and determining side or angle measures using trigonometry How Far Can You Go in a New York Minute? http://illuminations.nctm.org/LessonDetail.aspx?id=L848 Students use proportions and similar figures to adjust the size of the New York City Subway Map so that it is drawn to scale. http://education.ti.com/en/us/activity/detail?id=A760474813204FBB944031327521B742&ref=/en/us/ac tivity/search/subject?d=6B854F0B5CB6499F8207E81D1F3A25E6&s=B843CE852FC5447C8DD8 8F6D1020EC61&sa=71A40A9FD9E84937B8C6A8A4B4195B58&t=3CC394B76E4347CF8C EFCADAACAE9754 Students will explore the ratio of perimeter, area, surface area, and volume of similar figures in twodimensional figures using graphing technology. Triangle Dilations http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig_te_04091 3.pdf Students examine relationships of proportions in triangles that are known to be similar to each other based on dilations. Classroom Task 6.2 pages 11-20 (This lesson is also found in SRT 2.) CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 40 of 162 Reteach: Construct ABC with sides 5, 8, and 10 units. Tell students that the scale factor of 1 2 . Ask the students to construct DEF. ABC to Construct ABC with sides 5, 8, and 10 units. Tell students that the scale factor of ABC to Ask the students to construct RST. DEF is RST is 2. Construct two equilateral triangles of different sizes on the chalkboard. Ask students to determine if they are congruent or similar. Ask the students to justify their answers. (The equilateral triangles that were drawn are not congruent because the sides do not have the same length. They are similar because the angles all have a measure of 60° and the ratios of the lengths of the corresponding sides are the same.) Extensions: Use coordinate geometry and graph paper to draw the dilation of a figure. Use construction tools to construct the dilation of a figure. Take a map of Ohio or the United States. Make a transparency of the map. Place it over a coordinate plane. Write the coordinates of many of the border points. Have groups multiply each coordinate by a 1 scale factor. Have some groups use a scale factor of 3. Have others use a scale factor of 3 . Tape pages of graph paper together. Have students graph the new image. Discussion: Are the maps proportional? Textbook References: Textbook: Geometry, Glencoe (2005): pp. 282-287, 288 Supplemental: Geometry, Glencoe (2005): Chapter 6 Resource Masters Study Guide and Intervention, pp. 295-296 Skills Practice, p. 297 Practice, p. 298 Reading to Learn Mathematics, pp. vii-viii, 299 Enrichment, p. 300 Textbook: Geometry, Glencoe (2005): pp. 289-297 Supplemental: Geometry, Glencoe (2005): Chapter 6 Resource Masters Study Guide and Intervention, pp. 301-302 Skills Practice, p. 303 Practice, p. 304 Reading to Learn Mathematics, pp. vii-viii, 305 Textbook: Geometry, Glencoe (2005): pp. 298-306, 307-315, 316-323 CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 41 of 162 Supplemental: Geometry, Glencoe (2005): Chapter 6 Resource Masters Learning to Read Mathematics, pp. ix-x Study Guide and Intervention, pp. 307-308, 313-314, 319-320 Skills Practice, pp. 309, 315, 321 Practice, pp. 310, 316, 322 Reading to Learn Mathematics, pp. 311, 317, 323 Enrichment, pp. 312, 318, 324 Textbook: Geometry, Glencoe (2005): pp. 490 – 493 Textbook: Algebra 1, Algebra 1 (2005): pp. 197 – 203 CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 42 of 162 G –SRT 1b Name ______________________________ Date ____________ Period ________ Are You Golden? Materials: meter stick or tape measure, calculator For each group members, measure the length from the shoulder to the tip of the fingers and the length from the elbow to the tip of the fingers. Record the data below. Column A Column B TABLE 1 Column C Column D Group Members’ Names Length from shoulder to tip of fingers (cm) Length from Elbow to tip of fingers (cm) Find the Ratio of Column B to Column C Column E Decimal form of Column D Round to 2 decimal places. 1. Examine Column E. What do you notice about all of the decimals? 2. Find the average of all the decimals in Column E. Round to two decimal places. Now, measure each group member’s height and the height of the navel from the ground (make sure to take off your shoes). Record the data below. Column A Column B Group Members’ Names Height (cm) TABLE 2 Column C Column D Height of Navel from the Ground (cm) Find the Ratio of Column B to Column C Column E Decimal form of Column D Round to 2 decimals places. 3. Examine Column E. What do you notice about all of the decimals? 4. Find the average of all the decimals in Column E. Round to two decimal places. CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 43 of 162 5. How do the decimal averages in #2 and #4 compare? The ratios that you found are very close to what is known as the 1 5 “Golden Ratio”, which is . The decimal approximation 2 of the “Golden Ratio” is 1.618033989… Set up proportions to answer the following questions based on the “Golden Ratio”. 6. If a person’s arm (length of shoulder to tip of fingers) is 68 cm long, what is the length of this person’s elbow to the tip of the fingers? 7. If the height of a person’s navel from the ground is 105 cm tall, how tall is this person? CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 44 of 162 G –SRT 1b Name ______________________________ Date ____________ Period ________ Are You Golden? Answer Key Materials: meter stick or tape measure, calculator For each group members, measure the length from the shoulder to the tip of the fingers and the length from the elbow to the tip of the fingers. Record the data below. Column A Column B TABLE 1 Column C Column D Group Members’ Names Length from shoulder to tip of fingers (cm) Length from elbow to tip of fingers (cm) Find the ratio of Column B to Column C Column E Decimal form of Column D Round to 2 decimal places. 1. Examine Column E. What do you notice about all of the decimals? Answers Will Vary. 2. Find the average of all the decimals in Column E. Round to two decimal places. Answers Will Vary. Now, measure each group member’s height and the height of the navel from the ground (make sure to take off your shoes). Record the data below. TABLE 2 Column A Column B Column C Column D Column E Decimal form of Height of navel Find the ratio of Group Members’ Column D Height (cm) from the ground Column B to Names Round to 2 (cm) Column C decimal places. 3. Examine Column E. What do you notice about all of the decimals? Answers Will Vary. 4. Find the average of all the decimals in Column E. Round to two decimal places. Answers Will Vary. CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 45 of 162 5. How do the decimal averages in #2 and #4 compare? Answers Will Vary Students could conclude that the decimal average is very close to 1.6. The ratios that you found are very close to what is known as the 1 5 “Golden Ratio”, which is . The decimal approximation 2 of the “Golden Ratio” is 1.618033989… Set up proportions to answer the following questions based on the “Golden Ratio”. 6. If a person’s arm (length of shoulder to tip of fingers) is 68 cm long, what is the length of this person’s elbow to the tip of the fingers? 68 1.618 x x 42.03 cm 7. If the height of a person’s navel from the ground is 105 cm tall, how tall is this person? x = 1.618 105 x 169.89 cm CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 46 of 162 G –SRT 1b Name ______________________________ Date ____________ Period ________ The Gumps There are imposters lurking among the family of Gumps. Using the following criteria, you will create a set of characters. They will all look somewhat alike but only some of them are considered to be mathematically similar. Each group should create a set of characters in order to answer the questions that follow. Every graph within the group should be drawn using the same scale in order to see the changes between the Gumps. More than one piece of graph paper may be needed for a particular character. Plot each point on graph paper. For the points in SET 1 and SET 3, connect them in order and connect the last point to the first point. For SET 2, connect the points in order but do not connect the last point to the first point. For SET 4, make a dot at each point. CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 47 of 162 G –SRT 1b Name ______________________________ Date ____________ Period ________ The Gumps Giggles (x,y) SET 1 (4,0) (4,6) (2,4) (0,4) (4,8) (2,10) (2,14) (4,16) (5,18) (6,16) (8,16) (9,18) (10,16) (12,14) (12,10) (10,8) (14,4) (12,4) (10,6) (10,0) (8,0) (8,4) (6,4) (6,0) SET 2 (4,11) (6,10) (8,10) (10,11) SET 3 (6,11) (6,12) (8,12) (8,11) SET 4 (5,14) (9,14) Higgles (2x,2y) SET 1 Wiggles (3x,y) SET 1 Ziggles (3x,3y) SET 1 Miggles (x,3y) SET 1 SET 2 SET 2 SET 2 SET 2 SET 3 SET 3 SET 3 SET 3 SET 4 SET 4 SET 4 SET 4 CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 48 of 162 The Gumps Answer Key Giggles (x,y) SET 1 (4,0) (4,6) (2,4) (0,4) (4,8) (2,10) (2,14) (4,16) (5,18) (6,16) (8,16) (9,18) (10,16) (12,14) (12,10) (10,8) (14,4) (12,4) (10,6) (10,0) (8,0) (8,4) (6,4) (6,0) SET 2 (4,11) (6,10) (8,10) (10,11) SET 3 (6,11) (6,12) (8,12) (8,11) SET 4 (5,14) (9,14) Higgles (2x,2y) SET 1 (8,0) (8,12) (4,8) (0,8) (8,16) (4,20) (4,28) (8,32) (10,36) (12,32) (16,32) (18,36) (20,32) (24,28) (24,20) (20,16) (28,8) (24,8) (20,12) (20,0) (16,0) (16,8) (12,8) (12,0) SET 2 (8,22) (12,20) (16,20) (20,22) SET 3 (12,22) (12,24) (16,24) (16,22) SET 4 (10,28) (18,28) Wiggles (3x,y) SET 1 (12,0) (12,6) (6,4) (0,4) (12,8) (6,10) (6,14) (12,16) (15,18) (18,16) (24,16) (27,18) (30,16) (36,14) (36,10) (30,8) (42,4) (36,4) (30,6) (30,0) (24,0) (24,4) (18,4) (18,0) SET 2 (12,11) (18,10) (24,10) (30,11) SET 3 (18,11) (18,12) (24,12) (24,11) SET 4 (15,14) (27,14) CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Ziggles (3x,3y) SET 1 (12,0) (12,18) (6,12) (0,12) (12,24) (6,30) (6,42) (12,48) (15,54) (18,48) (24,48) (27,54) (30,48) (36,42) (36,30) (30,24) (42,12) (36,12) (30,18) (30,0) (24,0) (24,12) (18,12) (18,0) SET 2 (12,33) (18,30) (24,30) (30,33) SET 3 (18,33) (18,36) (24,36) (24,33) SET 4 (15,42) (27,42) Miggles (x,3y) SET 1 (4,0) (4,18) (2,12) (0,12) (4,24) (2,30) (2,42) (4,48) (5,54) (6,48) (8,48) (9,54) (10,48) (12,42) (12,30) (10,24) (14,12) (12,12) (10,18) (10,0) (8,0) (8,12) (6,12) (6,0) SET 2 (4,33) (6,30) (8,30) (10,33) SET 3 (6,33) (6,36) (8,36) (8,33) SET 4 (5,42) (9,42) Columbus City Schools 6/28/13 Page 49 of 162 Giggles Higgles CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 50 of 162 CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 51 of 162 Wiggles CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 52 of 162 Ziggles CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 53 of 162 Miggles CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 54 of 162 G –SRT 1b Name ______________________________ Date ____________ Period ________ The Gumps and Similar Figures 1. Use a protractor to measure the following angles of the Gumps’ bodies. Giggles Higgles Wiggles Ziggles Miggles Top of Ear Under Arm Neck Smile Do you notice anything about the above measurements? If so, explain. Count the length of the following sides of the Gumps’ bodies. Giggles Higgles Wiggles Ziggles Miggles Width of Head Length of Leg Width of Hand Width of Waist Total Height Compare each Gump’s measurements to Giggles’ measurements. Describe any patterns that you notice. Giggles and Higgles are mathematically similar. Describe what you think it means for two figures to be mathematically similar. What other Gump(s) fit this description. Why? Complete the following table. CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 55 of 162 Nose Width Nose Length Width Length Nose Perimeter Nose Area Giggles (Gump 1) Higgles (Gump 2) Ziggles (Gump 3) Prediction for Gump 4 Prediction for Gump 5 . . . Prediction for Gump 10 Prediction for Gump 20 Prediction for Gump 100 Wiggles Miggles Make ratios using the nose perimeter for the following figures: Gump 2:Gump 1 Gump 3:Gump 1 Gump 4:Gump 1 Gump 5:Gump 1 CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 56 of 162 Make a comparison between the scale factor of objects and the ratio of their perimeters. Make ratios using the nose area for the following figures: Gump 2:Gump 1 Gump 3:Gump 1 Gump 4:Gump 1 Gump 5:Gump 1 Make a comparison between the scale factor of objects and the ratio of their areas. Look at Gump 10, Gump 20 and Gump 100. Using your answers to #9 and #11, show the relationship between scale factor of objects and the ratio of their perimeters and areas. CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 57 of 162 G –SRT 1b Name ______________________________ Date ____________ Period ________ The Gumps and Similar Figures Answer Key 1. Use a protractor to measure the following angles of the Gumps’ bodies. Giggles Higgles Wiggles Ziggles o o o Top of Ear 53 53 112 53o Under Arm 45o 45o 72o 45o o o o Neck 90 90 37 90o Smile 153o 153o 171o 153o Miggles 19o 18o 143o 124o Do you notice anything about the above measurements? If so, explain. Giggles, Higgles and Ziggles have the same angle measurements. They are the same shape just different sizes which preserves their angle measurements. The other two figures are stretched because only one of their dimensions was changed. Count the length of the following sides of the Gumps’ bodies. (Remember to count by 2 on the sample drawings since the scale is 2!) Giggles Higgles Wiggles Ziggles Width of Head 10 20 30 30 Length of Leg 4 8 4 12 Width of Hand 2 4 6 6 Width of Waist 6 12 18 18 Total Height 18 36 18 54 Miggles 10 12 2 6 54 Compare each Gump’s measurements to Giggles’ measurements. Describe any patterns that you notice. All of Higgles’ measurements are two times that of Giggles’. All of Ziggles’ measurements are three times that of Giggles’. Wiggles’ widths only are three times larger than Giggles’ widths because only the x-values were multiplied by 3. Miggles’ lengths only are three times larger than Giggles’ lengths because only the y-values were multiplied by 3. Giggles and Higgles are mathematically similar. Describe what you think it means for two figures to be mathematically similar. Two figures are mathematically similar if their angle measures are the same and all of their dimensions are proportional. What other Gump(s) fit this description. Why? Ziggles is also mathematically similar to Giggles and Higgles because they have the same angle measurements and their sides are all proportional. CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 58 of 162 Complete the following table. Nose Width Nose Length Width Length Nose Perimeter Nose Area Giggles (Gump 1) 1 cm 2 cm 1 2 6 cm 2 cm2 Higgles (Gump 2) 2 cm 4 cm 2 1 = 4 2 12 cm 8 cm2 Ziggles (Gump 3) 3 cm 6 cm 3 1 = 6 2 18 cm 18 cm2 4 cm 8 cm 4 1 = 8 2 24 cm 32 cm2 5 cm 10 cm 5 1 = 10 2 30 cm 50 cm2 10 cm 20 cm 10 1 = 20 2 60 cm 200 cm2 20 cm 40 cm 20 1 = 40 2 120 cm 800 cm2 100 cm 200 cm 100 1 = 200 2 600 cm 20,000 cm2 1 cm 6 cm 1 1 = 6 2 14 cm 6 cm2 2 cm 3 cm 2 1 = 3 2 10 cm 6 cm2 Prediction for Gump 4 Prediction for Gump 5 . . . Prediction for Gump 10 Prediction for Gump 20 Prediction for Gump 100 Wiggles Miggles Make ratios using the nose perimeter for the following figures: Gump 2:Gump 1 Gump 3:Gump 1 12 6 2 1 24 6 4 1 Gump 4:Gump 1 18 6 3 1 30 6 5 1 Gump 5:Gump 1 Make a comparison between the scale factor of objects and the ratio of their perimeters. The ratio of the perimeters of two objects is the same as the scale factor. CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 59 of 162 Make ratios using the nose area for the following figures: Gump 2:Gump 1 Gump 3:Gump 1 8 2 4 1 Gump 4:Gump 1 32 2 18 2 9 1 50 2 25 1 Gump 5:Gump 1 16 1 Make a comparison between the scale factor of objects and the ratio of their areas. The ratios of the areas of two object is equal to the square of the scale factor. 12. Look at Gump 10, Gump 20 and Gump 100. Using your answers to #9 and #11, show the relationship between scale factor of objects and the ratio of their perimeters and areas. Perimeter of Gump 10 Perimeter of Gump 1 Area of Gump 10 Area of Gump 1 102 12 Perimeter of Gump 20 Perimeter of Gump 1 Area of Gump 20 Area of Gump 1 10 1 20 1 202 12 Perimeter of Gump 100 Perimeter of Gump 1 Area of Gump 100 Area of Gump 1 1002 12 100 1 x 6 x 2 10 1 100 1 x 6 x 2 20 1 400 1 x 6 x 2 100 1 10,000 1 CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Perimeter of Gump 10 = 60 cm Area of Gump 10 = 200 cm2 Perimeter of Gump 20 = 120 cm Area of Gump 20 = 800 cm2 Perimeter of Gump 100 = 600 cm Area of Gump 100 = 20,000 cm2 Columbus City Schools 6/28/13 Page 60 of 162 G –SRT 1b Name ______________________________ Date ____________ Period ________ Draw Similar Triangles Instructions: In each problem, draw a triangle similar to the one shown. Remember, corresponding angles of similar triangles have the same measure. Sides of similar triangles are proportional. Show all calculations that verify the triangles are similar. 1. 2. 3. 4. 5. _________________________________________________________________________ CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 61 of 162 6. 7. 8. 9. 10. CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 62 of 162 G –SRT 1b Name ______________________________ Date ____________ Period ________ Similar Quilt Blocks Quilts are a beautiful, practical, and historically significant use of geometric shapes. Create a quilt block using triangles that are similar to the triangles in the quilt block you selected. Select one of the quilt blocks shown on the “Quilt Design” pages. Your quilt block may be a replica of the given quilt block or it may be of your own design. Verify that your triangles are similar and show calculations on the “Quilt Calculations” page. Draw your quilt design in the space below. CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 63 of 162 G –SRT 1b Name ______________________________ Date ____________ Period ________ Quilt Calculations Measure the sides and angles of each triangle in the quilt block. Record these values in the “Original Triangle” section of the chart. Draw a triangle similar to the one you just measured. Measure the sides and angles and record these values in the “New Triangle” section of the chart. Verify that the sides of the similar triangles are proportional and place those calculations in the “Calculations” area. Original Triangle angle A angle B angle C side a side b side c New Triangle angle A angle B angle C side a side b side c Calculations Original Triangle angle A angle B angle C side a side b side c New Triangle angle A angle B angle C side a side b side c Calculations Original Triangle angle A angle B angle C side a side b side c New Triangle angle A angle B angle C side a side b side c Calculations Original Triangle angle A angle B angle C side a side b side c New Triangle angle A angle B angle C side a side b side c Calculations CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 64 of 162 Quilt Design #1 Hopscotch Grandma’s is from the Quilt Pattern Collection of the Camden-Carrol Library, Morehead State University. CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 65 of 162 Quilt Design #2 Laced Star is from the Quilt Pattern Collection of the Camden-Carrol Library, Morehead State University. G –SRT 2 CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 66 of 162 Name ______________________________ Date ____________ Period ________ Investigating Triangles with Two Pairs of Congruent Angles Given: a triangle with one angle measure of 40o and another angle measure of 50o: 1. Construct a triangle with the given angle measures. Label the 40o angle A, the 50o angle B, and the third angle C. 2. Use a ruler to find the length of each side of triangle ABC to the nearest tenth of a centimeter. AB= BC= AC= 3. Draw a second triangle that has the same angle measurements but is not congruent to triangle ABC. Label this triangle A'B'C'. 4. Use a ruler to find the length of each side of triangle A’B’C’ to the nearest tenth of a centimeter. A'B'= B'C'= A'C'= 5. How do the sides of triangle A'B'C' compare to the sides of triangle ABC? 6. How does the measurement of angle C compare to the measurement of angle C'? CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 67 of 162 7. What conclusion can be drawn about triangle ABC compared to triangle A'B'C'? Given: a triangle with one angle measure of 80o and another angle measure of 60o: 8. Construct a triangle with the given angle measures. Label the 80o angle M, the 60o angle N, and the third angle O. 9. Use a ruler to find the length of each side of triangle MNO to the nearest tenth of a centimeter. MN= NO= MO= 10. Draw a second triangle that has the same angle measurements but is not congruent to triangle MNO. Label this triangle M'N'O'. 11. Use a ruler find the length of each side of triangle M'N'O' to the nearest tenth of a centimeter. M'N'= N'O'= 12. M'O'= How do the sides of triangle M'N'O' compare to the sides of triangle MNO? 13. How does the measurement of angle O compare to the measurement of angle O'? 14. What conclusion can be drawn about triangle MNO compared to triangle M'N'O'? 15. What can you conclude about two triangles given two pair of congruent angles? CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 68 of 162 Name ___________________________________ Date __________________ Period ________ Investigating Triangles with Two Pairs of Congruent Angles Answer Key Given a triangle with one angle measure of 40o and another angle measure of 50o. 1. Construct a triangle with the given angle measures. Label the 40o angle A, the 50o angle B, and the third angle C. Answers may vary. 2. Use a ruler to find the length of each side of triangle ABC to the nearest tenth of a centimeter. AB= Answers may vary. BC= Answers may vary. AC= Answers may vary. 3. Draw a second triangle that has the same angle measurements but is not congruent to ABC . Label this triangle A'B'C'. Answers may vary. 4. Use a ruler to find the length of each side of triangle A'B'C' to the nearest tenth of a centimeter. A'B'= Answers may vary. B'C'= Answers may vary. A'C'= Answers may vary. 5. How do the sides of triangle A'B'C' compare to the sides of triangle ABC? They are proportional. 6. How does the measurement of angle C compare to the measurement of angle C'? They are congruent. 7. What conclusion can be drawn about triangle ABC compared to triangle A'B'C'? They are similar Given a triangle with one angle measure of 80o and another angle measure of 60o. 8. Construct a triangle with the given angle measures. Label the 80o angle M, the 60o angle N, and the third angle O. Answers may vary. 9. Use a ruler to find the length of each side of triangle MNO to the nearest tenth of a centimeter. MN= Answers may vary. NO= Answers may vary. MO= Answers may vary. 10. Draw a second triangle that has the same angle measurements but is not congruent to triangle MNO. Label this triangle M'N'O'. Answers may vary. 11. Use a ruler to find the length of each side of triangle M'N'O' to the nearest tenth of a centimeter. CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 69 of 162 M'N'= Answers may vary. N'O'= Answers may vary. M'O'= Answers may vary. 12. How do the sides of triangle M'N'O' compare to the sides of triangle MNO? They are proportional. 13. How does the measurement of angle O compare to the measurement of angle O'? They are congruent. 14. What conclusion can be drawn about triangle MNO compared to triangle M'N'O'? They are similar. 15. What can you conclude about two triangles given two pair of congruent angles? They are similar. CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 70 of 162 G –SRT 2 Name ______________________________ Date ____________ Period ________ Similar Triangles Application Use a mirror, a meter stick, and similar triangles to calculate the height of three objects in the room. Think about what information will be needed and how to accurately collect it. Describe the object, its location, and the measurements taken. Description of Object Distance From Student To Mirror Distance From The Mirror To The Base Of The Object Draw a sketch of each situation and explain why this scenario involves similar Distance From Line Of Sight To Ground s. Label your picture with your measurements and use proportions or scale factor to calculate the height of each object. Record your calculated heights below. Now, measure the actual height of each object. Record the actual (measured) heights below. Describe how well the calculated height matches the actual height. If there is a significant discrepancy, explain where any error may have occurred and if it can be corrected. CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 71 of 162 G –SRT 2 Name ______________________________ Date ____________ Period ________ Find the Scale Factor For each exercise, find the scale factor of figure A to figure B and solve for x. 1. 24 A A B 6 3 x 2. 5 x 6 A x+4 B 3. A CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 72 of 162 20 B 15 x 4 4. A 8 10 B x x+1 CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 73 of 162 Name ___________________________________ Date __________________ Period ________ Find the Scale Factor Answer Key For each exercise, find the scale factor of figure A to figure B and solve for x. 1. Scale Factor = 5; x = 12 24 A 6 B 3 x 2. Scale Factor = 5 x 1 ; x = 20 5 6 x+4 A B 3. Scale Factor = 6; x = 3 CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 74 of 162 A 20 B 15 4 x 4. Scale Factor = 3; x = 4 A 8 10 B x x+1 CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 75 of 162 G –SRT 4 Name ______________________________ Date ____________ Period ________ Let’s Prove the Pythagorean Theorem Given the square below, mark a point, E, on AB (not a midpoint). Next, mark a point, F, on DA such that DF = AE. Now, mark a point, G, on CD such that CG = AE. Again, mark a point, H, on BC such that BH = AE. Once you have marked all the new points, connect them to create another square that is inscribed in square ABCD. Label each side of the new smaller square x. A B D C Examine AE and BE . Decide which segment is shorter, s, and which segment is longer, l. Label each segment either s or l accordingly. Do the same thing for DF and AF ; CG and DG ; BH and CH . How many right triangles do you see? Name all of them. In each right triangle, what are the s, l and x (i.e. is it the leg or hypotenuse of the right triangle)? Represent the area of square ABCD in terms of s and l. Simplify the expression. Represent the combined area of all the triangles in terms of s and l. Simplify the expression. Represent the area of the smaller square in terms of x. CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 76 of 162 Write an expression for the total area of all the right triangles and the smaller square. What should this total area be equal to and why? Write an equation that relates part E to Part C. Identify and eliminate any common terms on each side of the equation. Explain what each variable in the new equation represents. You have just proven the Pythagorean Theorem! CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 77 of 162 Name ___________________________________ Date __________________ Period ________ Let’s Prove the Pythagorean Theorem Answer Key Given the square below, mark a point, E, on AB (not a midpoint). Next, mark a point, F, on DA such that DF = AE. Now, mark a point, G, on CD such that CG = AE. Again, mark a point, H, on BC such that BH = AE. Once you have marked all the new points, connect them to create another square that is inscribed in square ABCD. Label each side of the new smaller square x. Examine AE and BE . Decide which segment is shorter, s, and which segment is longer, l. Label each segment either s or l accordingly. Do the same thing for DF and AF ; CG and DG ; BH and CH . How many right triangles do you see? Name all of them. Four triangles - AEF, BEH, CGH, DFG (students could label these differently) In each right triangle, what are the s, l and x (i.e. is it the leg or hypotenuse of the right triangle)? s is a leg, l is a leg and x is the hypotenuse Represent the area of square ABCD in terms of s and l. Simplify the expression. Area = (s + l)2 = s2 + 2sl + l2 Represent the combined area of all the triangles in terms of s and l. Simplify the expression. Area = 4(½)sl = 2sl Represent the area of the smaller square in terms of x. Area = x2 Write an expression for the total area of all the right triangles and the smaller square. What should this total area be equal to and why? Total area = 2sl + x2 CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 78 of 162 This total area should equal to the area of the original square ABCD because the original square consists of the 4 right triangle and the inscribed square. Write an equation that relates part E to Part C. Identify and eliminate any common terms on each side of the equation. Explain what each variable in the new equation represents. s2 + 2sl + l2 = 2sl + x2 s2 + l2 = x2 s and l are the legs of the right triangle and x is the hypotenuse. You have just proven the Pythagorean Theorem! CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 79 of 162 G –SRT 4 Name ______________________________ Date ____________ Period ________ Proving the Pythagorean Theorem, Again! a 2 b c 3 a c 1 If the formula for finding the area of a trapezoid is b (base1 base2 )height , find the area of the above 2 trapezoid. Simplify the expression. If the formula for finding the area of a triangle is base height , find the areas of each triangle in the 2 picture above. Write an equation relating #1 and #2. Using your algebra skills, try to manipulate the equation so that only the Pythagorean Theorem remains. CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 80 of 162 Name ___________________________________ Date __________________ Period ________ Proving the Pythagorean Theorem, Again! Answer Key a 2 b c 3 a c 1 If the formula for finding the area of a trapezoid is b (base1 base2 )height , find the area of the above 2 trapezoid. Simplify the expression. base1 = a Area of trapezoid = base2 = b height = a + b (a b)(a b) 2 If the formula for finding the area of a triangle is picture above. Area of triangle1 = ½ ab a 2 ab ab b 2 2 a2 2ab b 2 2 base height , find the areas of each triangle in the 2 Area of triangle2 = ½ ab Area of triangle3 = ½ c2 Write an equation relating #1 and #2. Using your algebra skills, try to manipulate the equation so that only the Pythagorean Theorem remains. Area of triangle1 + Area of triangle2 + Area of triangle3 = Area of trapezoid 2 2 1 1 1 2 a +2ab+b ab + ab + c = 2 2 2 2 2 2 ab + ab + c = a + 2ab + b2 2ab + c2 = a2 + 2ab + b2 c2 = a2 + b2 CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 81 of 162 COLUMBUS CITY SCHOOLS HIGH SCHOOL CCSS MATHEMATICS II CURRICULUM GUIDE TOPIC 2 CONCEPTUAL CATEGORY TIME RANGE GRADING 20 days Trigonometric Ratios Geometry PERIOD G-SRT 6, 7, 8 1 Domain: Similarity, Right Triangles, and Trigonometry (G – SRT): Cluster 3) Define Trigonometric ratios and solve problems involving right triangles. Standards 3) Define Trigonometric ratios and solve problems involving similarity. G – SRT 6: Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. G – SRT 7: Explain and use the relationship between the sine and cosine of complementary angles. G – SRT 8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 82 of 162 TEACHING TOOL Vocabulary: acute angle, adjacent, angle of depression, angle of elevation, complementary angles, corresponding sides, cosine, geometric mean, hypotenuse, opposite, proportion, Pythagorean Theorem, Pythagorean triple, ratio, right triangle, similar triangles, sine, solving a triangle, special right triangles, tangent, trigonometric ratios, trigonometry Teacher Notes Properties of Radicals n An expression that contains a radical sign is called a radical expression, a . The expression under the radical sign is the radicand and the numeric value, n is the index. We read this as “the nth root of 5 a. Looking at the radical expression 3x , 3x is the radicand and 5 is the index. 1. c is a square root of a, if c2 = a, e.g., 2 is a square root of 4 because 22 = 4 and -2 is a square root of 4 because (-2)2 = 4. Because there are two values that satisfy the equation x2 = 4, we take the term square root to mean the principal square root which has a non-negative value. In this case 4 2 is the principal square root. Mathematically, we express this as: a2 a 2. c is a cube root of a if c = a, e.g., 3 is a cube root of 27 because 33 = 27 and -3 is a cube root of 27 because (-3)3 = -27. The cube root of a negative number is negative. The cube root of a positive number is positive. 3. c is an nth root of a if cn = a. Note that if the index is odd and the radicand is negative then the 3 5 32 2 because (-2)5 = -32. The following are general principal root is negative. For example, rules for taking the roots of positive and negative numbers. The answer is the principal root. The answer is the opposite of the principal root. The answer is both roots, the positive and the negative root. odd number negative number even number negative number odd number positive number even number positive number The answer is a negative number. There is no real solution. The answer is the principal root. The answer is the principal root. For any value of x and any even number n, = -5, then 8 ( 5)8 5 xn x . For example, if x = 4, then 6 46 5 . For any value of x and any odd number n greater than 1, example, if x = 4, then . If x = -5, then 4. product rule: n n a n b n 9 ( 5)9 ab , e.g. 3 n 4 4 . If x xn x . For 5. 7 3 5 CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 3 7 5 3 35 , and Columbus City Schools 6/28/13 Page 83 of 162 ( x 6) ( x 6) ( x 6)( x 6) x 2 36 The product rule can be used for factoring to simplify radical expressions as shown below. 50 25 2 25 2 5 2 and 72 22233 2 22 33 23 2 6 2 19 19 72 can be written as the product of prime factors, and then simplified, but 19 is a prime number so it is already in its simplest form. 5. quotient rule: given n a and n b,b 0, n a b n a , e.g. n b x2 16 x2 16 x 4 6. principle of powers: if a = b then an = bn This website offers a teacher resource that includes a power point presentation for operations with radical expressions. http://teachers.henrico.k12.va.us/math/hcpsalgebra1/module11-3.html This website has an on-line explanation of radicals. http://www.regentsprep.org/Regents/math/algtrig/ATO3/simpradlesson.htm The website below is a teacher resource that has lessons, practice and a tutorial. http://www.regentsprep.org/Regents/math/ALGEBRA/AO1/indexAO1.htm The two following websites have practice with operations with radicals. http://www.algebralab.org/practice/practice.aspx?file=Algebra1_13-2.xml http://www.algebralab.org/practice/practice.aspx?file=Algebra1_13-3.xml Right Triangles Remind students that it is better to remember the Pythagorean Theorem as leg2 + leg2 = hypotenuse2 rather than a2 + b2 = c2, since there is no guarantee that c is always the hypotenuse. There are two special right triangles. The first is a 45-45-90 triangle. The special ratio is 1:1: 2 . The second is a 30-60-90 triangle. The special ratio is 1: 3 : 2 . Solving special right triangles http://www.youtube.com/watch?v=nVTtSE5nv7c http://www.youtube.com/watch?v=NsNaYwHtowA Trigonometry is based on similar right triangles. The sine (sin) of an angle is the ratio of the opposite side to the hypotenuse. The cosine (cos) of an angle is the ratio of the adjacent side to the hypotenuse. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 84 of 162 The cosine of an angle is the ratio of the adjacent side to the hypotenuse. The tangent (tan) of an angle is the ratio of the opposite side to the adjacent side. There are many different ways to help your students remember the sine, cosine, and tangent functions. Use the old Indian chief SOH CAH TOA. Tell a story of how a great Indian chief was also a great mathematician. And he developed sine, cosine, and tangent to match his name. SOH (sin = opp / hyp) CAH (cos = adj / hyp) TOA (tan = opp / adj) The following phrase could also be used. Some Caught Taking Old Horse Another Horse Oats Away The geometric mean is the square root of the product of two numbers. In right triangles, an altitude drawn to the hypotenuse is the geometric mean of the measures of the two segments of the hypotenuse. Each leg of a right triangle is the geometric mean of the measure of the adjacent segment of the hypotenuse and the total measure of the hypotenuse. angle of depression The angle of elevation is the angle between the line of sight and the horizontal when looking up. The angle of depression is the angle between the line of sight and the horizontal when looking down. It is helpful to remember that the angle of elevation and the angle of depression are alternate interior angles to each other. angle of elevation Real life applications are architecture and engineering. Right triangle trigonometry is one of the more practical day-to-day applications of mathematics. Used to find lengths and angles, it is a necessity in construction and home improvement. For example, if you wish to build a deck that is a regular polygon, you only need the length of one side to find the area using trigonometry and simple geometry. The three trigonometric functions, sine, cosine, and tangent are simply ratios of the sides of right triangles. These values can be found in a table, in a calculator, or in a textbook. By the Angle-Angle Similarity Theorem, if the measures of two of the angles of a pair of triangles are equal, then the triangles are similar. Since we are working with right triangles only, all triangles with a second angle of the same measure are similar and their sides are proportional. The given angle is called “theta” and is represented by the symbol . The side of the triangle across from is the “opposite side”. The side of the triangle next to is the “adjacent side”. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 85 of 162 hypotenuse opposite side adjacent side The trigonometric ratios are: sin = opposite =o hypotenuse h cos Words Symbol Trigonometric sine θ sin θ Ratios cosine θ cos θ tangent θ tan θ θ= adjacent =a hypotenuse h tan θ= opposite o = adjacent a Definition opposite sin hypotenuse adjacent cos hypotenuse opposite tan adjacent If the angle measure is 30°, 45° or 60° in a right triangle, special trigonometric relationships exist. θ sin θ 1 2 cos θ tan θ csc θ sec θ cot θ 3 3 2 3 2 3 2 3 3 2 2 45˚ 1 1 2 2 2 2 1 3 2 3 3 60˚ 2 3 2 2 3 3 2 2 Remind students that it is better to remember the Pythagorean Theorem as leg + leg = hypotenuse2 rather than a2 + b2 = c2, since there is no guarantee that c is always the hypotenuse. 30˚ There are two special right triangles. The first is a 45-45-90 triangle. The special ratio is 1 :1 : 2 . The second is a 30-60-90 triangle. The special ratio is 1: 3 : 2 . 45o 30o x 2 x 2x x 3 45 o x x x CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 60o Columbus City Schools 6/28/13 Page 86 of 162 m∠A + m∠B = 90o B sin A= opposite side of A a c sin B= opposite side of B b cos A= adjacent side of A b cos B= opposite side of B a c c hypotenuse c a hypotenuse C b a2 + b2 = c2 hypotenuse hypotenuse c A tan A= opposite side of A a tan B= opposite side of B b b a adjacent side of A adjacent side of B Students must understand that triangles with congruent angles are similar triangles. Students must understand that the ratio of two sides in one triangle is equal to the ratio of the corresponding two sides of all other similar triangles. Right Triangles Right Triangle Trigonometry http://patrickjmt.com/right-triangles-and-trigonometry/ A website video tutorial on right triangle trigonometry. Evaluating Trigonometric functions http://patrickjmt.com/evaluating-trigonometric-functions-for-an-unknown-angle-given-a-pointon-the-angle-ex-1/ Evaluating trigonometric functions for an unknown angle given a point on the angle. Right Triangle Trigonometry http://teachers.henrico.k12.va.us/math/IGO/07RightTriangles/7_4.html Teacher resource that includes the warm-up, algebraic review, lecture notes, practice, and hands on activities for right triangle trigonometry. Special Right Triangles http://teachers.henrico.k12.va.us/math/IGO/07RightTriangles/7_3.html Teacher resource that includes the warm-up, algebraic review, lecture notes, practice, and hands on activities for special right triangles. Basic Trigonometry http://education.ti.com/en/us/activity/detail?id=469426FC7D1542A9B54240E5C87A8593 Students define basic terms relating to trigonometry and use trigonometric ratios using their TI-84 calculator. Module 16 Trigonometric Ratios http://education.ti.com/en/us/professional-development/pd_onlinegeometry_free/course-outlineand-take-this-course Using TI – 84 and Cabri Jr for special triangles Sine and Cosine of Complementary Angles http://learni.st/users/60/boards/3370-sine-and-cosineCCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 87 of 162 of-complementary-angles-common-core-standard-9-12-g-srt-7#/users/60/boards/3370-sine-andcosine-of-complementary-angles-common-core-standard-9-12-g-srt-7 Tutorials to explain the relationship between the sine and cosine of complementary angles. Co-Functions http://www.regentsprep.org/Regents/math/algtrig/ATT6/cofunctions.htm Practice and warm ups to explain co-functions Finding Height Using Trigonometry http://patrickjmt.com/finding-the-height-of-an-object-using-trigonometry-example-1/ Tutorial on finding the height of an object using trigonometry example 1 Finding Height Using Trigonometry http://patrickjmt.com/finding-the-height-of-an-object-using-trigonometry-example-2/ Example 2 Finding Height Using Trigonometry http://patrickjmt.com/finding-the-height-of-an-object-using-trigonometry-example-3/ Example 3 Finding Height Using Trigonometry http://patrickjmt.com/trigonometry-word-problem-finding-the-height-of-a-building-example-1/ Word Problem 1 Finding Height Using Trigonometry http://patrickjmt.com/trigonometry-word-problem-example-2/ Word Problem 2 Misconceptions/Challenges: SRT 6 Students struggle labeling the opposite, adjacent and hypotenuse. Sometimes they use the shortest leg as the opposite leg or confuse adjacent and hypotenuse. Students get confused of where the angle of depression is located. Students confuse the difference on how to use the calculator when finding values of a missing side or missing angle. Students may apply the ratios of the special right triangles to all right triangles. Once trigonometry is taught, students like to use that instead of the ratios of special triangles. But to get exact values, they must use the ratios. SRT 8 Students may not substitute the hypotenuse in for ‘c’ in the Pythagorean Theorem. Angle of depression is often mislabeled as the angle between the vertical and hypotenuse Students incorrectly identify corresponding legs when using hypotenuse-leg congruence for right triangles. Students do not understand that equilateral triangles are also equiangular and vice versa. Students do not realize that congruent angles in an isosceles triangle are opposite the congruent sides. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 88 of 162 Instructional Strategies: This is an entire unit that covers all three standards. There are many references to everyday objects in the lessons. https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_912_AccelCoorAlgebraAnalyticGeom_Unit8SE.pdf This link has instructional strategies and sample formative assessment tasks as well as key concepts and vocabulary. http://www.schools.utah.gov/CURR/mathsec/Core/Secondary-II/II-5-G-SRT-6.aspx project ideas http://ccss.performanceassessment.org/taxonomy/term/1045 The following website has practice on simplifying radical expressions. http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Simplifying%20Radicals.pd f SRT 6 Stay in Shape http://www.teachengineering.org/view_activity.php?url=http://www.teachengineering.org/col lection/cub_/activities/cub_navigation/cub_navigation_lesson03_activity1.xml Lesson on how triangles and The Pythagorean Theorem are used in measuring distance. Fit by Design https://access.bridges.com/usa/en_US/choices/pro/content/applied/topic/aom14CX.html lesson relates actual and calculated measures of right triangles to objects created by mechanical drafters or designers Calculating Volumes of Compound Objects http://map.mathshell.org/materials/lessons.php?taskid=216 Decomposing shapes into simpler ones and using right triangles to solve real-world problems. Geometry Problems: Circles and Triangles http://map.mathshell.org/materials/lessons.php?taskid=222 Students determine the lengths of sides in right triangles to solve problems. Hopewell Geometry. http://map.mathshell.org/materials/tasks.php?taskid=127&subpage=apprentice How the Hopewell people constructed earthworks using right triangles. Have students complete the activity “Exploring Special Right Triangles 45-45-90” (included in this Curriculum Guide) to reinforce the properties of 45-45-90 triangles. Have students complete the activity “Exploring Special Right Triangles 30-60-90” (included in this Curriculum Guide) to reinforce the properties of 30-60-90 triangles. Have students complete the activity “Discovering Trigonometric Ratios” (included in this curriculum guide) to develop their understanding of trigonometry. Students will need centimeter rulers and protractors to measure the parts of the given triangles. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 89 of 162 Have students complete the activity “Make a Model: Trigonometric Ratios” (included in Curriculum Guide) to discover that the trigonometric ratios of any right triangle with specific acute angles are the same regardless of the lengths of the sides. Calculators may be helpful for this activity. Eratosthenes Finds the Circumference of the Earth https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_9-12_AnalyticGeo_Unit2SE.pdf Students examine a diagram and verify the two triangles are similar. Page 12 Discovering Special Triangles https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_9-12_AnalyticGeo_Unit2SE.pdf Students use real-world situations to discover special right triangles. Page 16 Finding Right Triangles in Your Environment https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_9-12_AnalyticGeo_Unit2SE.pdf Students find right triangles such as a ramp. Page 20 Create Your Own Triangles https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_9-12_AnalyticGeo_Unit2SE.pdf Students use paper, compass, straight edge and protractor to create right triangles and verify the measurements. Page 22 The Tangent Ratio https://www.cohs.com/editor/userUploads/file/Meyn/321%20Ch%206a%20Student%20WB. pdf Students solve real-world problems using the tangent ratio Are Relationships Predictable .http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig _te_040913.pdf Students develop and use right triangle relationships based on similar triangles. Classroom Task: 6. pages 53-59. (This strategy can also be found in SRT8.) Relationships with Meaning http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig _te_040913.pdf Students find relationships between the sine and cosine ratios for right triangles, including the Pythagorean identity. Classroom Task: 6.8 pages 60-66 (This strategy can also be found in SRT7.) Solving Right Triangles Using Trigonometric Relationships http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig _te_040913.pdf Students set up and solve right triangles modeling real world context. Classroom Task: 6.10 found on pages 75-81 (This strategy can also be found in SRT7.) CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 90 of 162 SRT 7 Relationships with Meaning http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig _te_040913.pdf Students find relationships between the sine and cosine ratios for right triangles, including the Pythagorean identity. Classroom Task: 6.8 pages 60-66 (This strategy can also be found in SRT 6.) Solving Right Triangles Using Trigonometric Relationships http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig _te_040913.pdf Students set up and solve right triangles modeling real world context. Classroom Task: 6.10 found on pages 75-81 (This strategy can also be found in SRT 6.) Create Your Own Right Triangles https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_9-12_AnalyticGeo_Unit2SE.pdf Students use paper, compass, straight edge and protractor to create right triangles and verify the measurements. Page 22. Discovering Trigonometric Ratio Relationships https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_9-12_AnalyticGeo_Unit2SE.pdf Students use degree measurements of acute angles from right triangles to determine trigonometric ratios. Page 27 Finding Right Triangles in Your Environment https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_9-12_AnalyticGeo_Unit2SE.pdf Students find right triangles such as a ramp. Page 20 The Sine and Cosine Ratios https://www.cohs.com/editor/userUploads/file/Meyn/321 Ch 6a Student WB.pdf Students use sine and cosine ratios of acute angles of right triangles to solve real-world problems. Page 7 Special Right Triangles https://www.cohs.com/editor/userUploads/file/Meyn/321 Ch 6a Student WB.pdf Students investigate special right triangles. Page 11 SRT 8 Have students complete the activity “Application of Trigonometry” (included in this Curriculum Guide) to practice using trigonometry to solve indirect measurement questions. Horizons https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_9-12_AnalyticGeo_Unit2SE.pdf Students use trigonometric ratios to determine distance to the horizon from different locations. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 91 of 162 Page 11 Finding Right Triangles in Your Environment https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_9-12_AnalyticGeo_Unit2SE.pdf Students find right triangles such as a ramp. Page 20 Create Your Own Triangles https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGP S_Math_9-12_AnalyticGeo_Unit2SE.pdf Students use paper, compass, straight edge and protractor to create right triangles and verify the measurements. Page 22. Find that Side or Angle https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_9-12_AnalyticGeo_Unit2SE.pdf Students use graphing technology to find the values of sine and cosine in real-world situations. Access Ramp http://www.achieve.org/files/CCSS-CTE-Task-AccessRamp-FINAL.pdf Students design an access ramp which complies with the Americans with Disabilities Act (ADA) requirements and includes pricing based on local costs. Land Surveying Project http://alex.state.al.us/lesson_view.php?id=25108 Students learn the basics of civil engineering in land surveying. The Clock Tower http://alex.state.al.us/lesson_view.php?id=25107 Students use the Pythagorean Theorem, and Sine, Cosine, and Tangent to find unknown heights of objects. Solving Right Triangles. https://www.cohs.com/editor/userUploads/file/Meyn/321%20Ch%206a%20Student%20WB. pdf Students determine the angle measures in right triangles. Lesson 6-4 Page 15 Determine the Missing Sides of Special Right Triangles http://www.kutasoftware.com/FreeWorksheets/GeoWorksheets/8Special%20Right%20Triangles.pdf Students practice finding missing sides of special right triangles Applied Trigonometry http://learni.st/users/60/boards/3453-trig-ratios-and-the-pythagorean-theorem-common-corestandard-9-12-g-srt-8 Several tutorials on trigonometry Solving Right Triangles https://www.cohs.com/editor/userUploads/file/Meyn/321%20Ch%206a%20Student%20WB. pdf Students determine how to find unknown angle measures of a right triangle. Page 15 CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 92 of 162 Have students complete the activity “Let’s Measure the Height of a Flagpole” (included in this Curriculum Guide) to practice solving problems involving indirect measurement. A clinometer can be built from a piece of cardboard, a drinking straw, a piece of string, a washer, and a protractor as shown on the first page of the activity record sheet. Templates for protractors are included “Grids and Graphics” (included in this Curriculum Guide). Note: Answers will vary. Have students practice finding the missing side or angle of a right triangle with the “Find the Missing Side or Angle” activity (included in this Curriculum Guide). Students will need a copy of the activity and a scientific or graphing calculator. Remind students of the legend of Soh Cah Toa, the “great trigonometry leader”. Embellish the story yourself or ask students to spin their own tall tale, write a poem or create a rap that includes a description of the trigonometric ratios: sine = opposite/hypotenuse, cosine =adjacent/hypotenuse, and tangent = opposite/adjacent. Review how the trigonometric ratios can be used to find missing angle measures or side lengths in right triangles. Arrange students in groups of three to play “paper football”. After each group has made its football by folding a sheet of paper, the group will assign duties and measure its field (the distance along the ground from where the ball is kicked to the uprights). One student will be the kicker, one will hold up their hands as the uprights, and one will measure the height of the “football” from the ground as it crosses the uprights. Each group will draw its field, record the measurements, and calculate the angle of elevation the ball makes with the ground for each kick, for at least 5 kicks. Follow up with the “Between the Uprights” activity (included in this Curriculum Guide). Remind students that the angle the goal post makes with the ground is 90 . Students will need a copy of the activity, a sheet of paper to use to make a football, and a calculator. Discuss with students the effect of a five or ten yard penalty on the results for each situation. How significantly would the angle or distances be changed? Arrange students into groups of two. Student will practice their right triangle solving skills with the exercise “Solve the Triangle” (included in this Curriculum Guide). Students will need a copy of the activity, a calculator, a ruler, and a protractor. Have students take turns explaining to their partner how they solved one of the problems on the sheet. Students design their ideal city park in the activity “Right Triangle Park” (included in this Curriculum Guide). The catch is that their “ideal” park can be made up of only right triangles. Students will then measure two parts of each triangle and calculate the remaining parts using trigonometry. Students will need a calculator, ruler, protractor, and a copy of the worksheet. Allow students time to share with the class their design. Ask students to point out several of the right triangles in their design and to explain how they calculated the lengths and/or angles for a few of the triangles. Have students complete the activity “Applications of the Pythagorean Theorem” (included in this Curriculum Guide) to explore real-life applications of the Pythagorean Theorem. Students can reinforce their similar triangle skills by using the properties of similar triangles to measure objects around school. Have the students select three objects they want to measure and use a mirror to create a pair of similar triangles as instructed in the “Find the Height” activity (included in this Curriculum Guide). Each group of two or three students will need a tape measure, mirror, and a copy of the activity instructions. Each student will need a copy of the activity data CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 93 of 162 sheet. After students have completed the activity, discuss as a class their strategies for finding the height of each object. Close Enough http://www.teachengineering.org/view_activity.php?url=http%3A%2F%2Fwww.teachengin eering.org%2Fcollection%2Fcub_%2Factivities%2Fcub_navigation%2Fcub_navigation_les son04_activity1.xml Hands-on activity shows how accurate measurement is important as students use right triangle trigonometry and angle measurements to calculate distances Six Trigonometric Ratio Values of Special Acute http://illuminations.nctm.org/LessonDetail.aspx?id=L383 A puzzle for practicing knowledge of all six trigonometric ratios. Two activities involve angle of elevation and angle of declination. Solving Problems Using Trigonometry http://education.ti.com/en/us/activity/detail?id=EB3E2581FFEC4FDA8FC94C3AA51F3D31 Students use TI-84 calculator to find the angle of elevation or the angle of depression. Basic Trigonometry http://education.ti.com/en/us/activity/detail?id=469426FC7D1542A9B54240E5C87A8593 Students define basic terms relating to trigonometry and use trigonometric ratios. Are Relationships Predictable http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig _te_040913.pdf Students develop the right triangle trigonometric relationships based on similar triangles. (This strategy can also be found in SRT8.) Reteach: Have students complete the activity “Memory Match” (included in this Curriculum Guide) to reinforce right triangle terminology. Students can work in groups of three or four. First place all cards facedown and then have each student take turns drawing two cards. If the two cards drawn go together as a pair the student will keep it as a match. Students take turns drawing. The student with the most pairs or matches wins. Students will need a scientific calculator. Additional practice in solving proportions, using the properties of similar triangles and right triangle trigonometry is available in the “Similar Right Triangles and Trigonometric Ratios” activity (included in this Curriculum Guide). Have students complete the activity “Hey, All These Formulas Look Alike” (included in this Curriculum Guide) to investigate the tangent relationship. Note: For the visual learner, the use of highlighted notes may lead to greater understanding. Highlighting (x2 – x1 ) x and ( y2 – y1) y in yellow and pink, respectively, may aide the visual learner in the formula comparisons. Extensions: Have students complete the activity “Similar Right Triangles and Trigonometric Ratios” CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 94 of 162 (included in Curriculum Guide) to make connections between similar triangles and trigonometric ratios Have students complete the activity “Problem Solving: Trigonometric Ratios” (included in Curriculum Guide) to apply their knowledge of trigonometric ratios. Given the sides of a right triangle inscribed in a circle and circumscribed about another circle, students find the radii of each circle. Also, students analyze sample solutions and compare their own solutions to those given. http://map.mathshell.org/materials/lessons.php?taskid=403#task403 Textbook References: Textbook: Geometry, Glencoe (2005): pp. 349, 350-356 Supplemental: Geometry, Glencoe (2005): Chapter 7 Resource Masters Study Guide and Intervention, pp. 357-358 Skills Practice, p. 359 Practice, p. 360 Enrichment, p. 362 Textbook: Geometry, Glencoe (2005): pp. 357-363 Supplemental: Geometry, Glencoe (2005): Chapter 7 Resource Masters Study Guide and Intervention, pp. 363-364 Skills Practice, p. 365 Practice, p. 366 Enrichment, p. 368 Textbook: Geometry, Glencoe (2005): pp. 364-370 Supplemental: Geometry, Glencoe (2005): Chapter 7 Resource Masters Study Guide and Intervention, pp. 369-370 Skills Practice, p. 371 Practice, p. 372 Enrichment, p. 374 Textbook: Geometry, Glencoe (2005): pp. 342-348 371-376 Supplemental: Geometry, Glencoe (2005): Chapter 7 Resource Masters Study Guide and Intervention, pp. 351-352, 375-376 Skills Practice, pp. 353, 377 Practice, pp. 354, 378 Enrichment, pp. 356, 380 Supplemental: Integrated Mathematics: Course 3, McDougal Littell (2002): Teacher’s Resources for Transfer Students, pp. 39-40 Supplemental: Integrated Mathematics: Course 3, McDougal Littell (2002): Skills Bank, p. 104 Overhead Visuals, folders A, 10 CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 95 of 162 Textbook: Algebra 1, Glencoe (2005): pp. 622 – 630 Textbook: Algebra 1, Glencoe (2005): pp. 698 – 708 CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 96 of 162 G-SRT 6 SRT- 6 Name ___________________________________ Date __________________ Period ________ Exploring Special Right Triangles (45-45-90) Given the isosceles right triangle below. l h l 1. What is the measure of each acute angle? Explain. 2. a) If the length of each leg is 1 unit, find the length of the hypotenuse. Leave answer exact and simplified. b) What are the side-length ratios of leg: leg: hypotenuse? 3. a) If the length of each leg is 2 units, find the length of the hypotenuse. Leave answer exact and simplified. b) How many times longer is the hypotenuse than the leg? c) What are the side-length ratios of leg: leg: hypotenuse? Simplify the ratio. 4. a) If the length of each leg is 5 units, find the length of the hypotenuse. Leave answer exact and simplified. b) How many times longer is the hypotenuse than the leg? c) What are the side-length ratios of leg: leg: hypotenuse? Simplify the ratio. 5. What can you conclude about the side-length ratios of leg: leg: hypotenuse of any isosceles right triangle? CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 97 of 162 Name ___________________________________ Date __________________ Period ________ Exploring Special Right Triangles (45-45-90) Given the isosceles right triangle below. l Answer Key h 1. What is the measurel of each acute angle? Explain how you know. Each acute angle is 45o. Since this is an isosceles right triangle, each angle opposite the legs are congruent. Since there’s a total of 90o for both acute angles and they are congruent, they must be 45o each. 2. a) If the length of each leg is 1 unit, find the length of the hypotenuse. Leave answer exact and simplified. Hypotenuse = 2 b) What are the side-length ratios of leg: leg: hypotenuse? 1: 1: 2 3. a) If the length of each leg is 2 units, find the length of the hypotenuse. Leave answer exact and simplified. Hypotenuse = 8 = 2 2 b) How many times longer is the hypotenuse than the leg? The hypotenuse is 2 times longer than the leg. c) What are the side-length ratios of leg: leg: hypotenuse? Simplify the ratio. 2: 2: 2 2 which is really 1: 1: 2 4. a) If the length of each leg is 5 units, find the length of the hypotenuse. Leave answer exact and simplified. Hypotenuse = 50 = 5 2 b) How many times longer is the hypotenuse than the leg? The hypotenuse is 2 times longer than the leg. c) What are the side-length ratios of leg: leg: hypotenuse? Simplify the ratio. 5: 5: 5 2 which is really 1: 1: 2 5. What can you conclude about the side-length ratios of leg: leg: hypotenuse of any isosceles right triangle? They will always be 1: 1: 2 . CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 98 of 162 G-SRT 6 Name ___________________________________ Date __________________ Period ________ Exploring Special Right Triangles (30-60-90) 1. Given the equilateral triangle below whose sides are 2 units long. A 2 2 B 2 a) What is the angle measure of each acute angle? C b) Fold the triangle along vertex A so that vertex B maps onto vertex C. Draw a segment along the crease. Label point D where the new segment intersects segment BC . What is special about AD ? What is the length of BD (label it in the diagram above)? Explain your reasoning. c) What does AD do to d) Examine above)? A? ABD . What is the measure of BAD and BDA (label it in the diagram e) Find the length of AD . Leave exact and simplified. f) In ABD , how many times longer is the hypotenuse than the shorter leg? How many times longer is the longer leg than the shorter leg? g) In this 30o-60o-90o triangle, what are the side length ratios of short leg: long leg: hypotenuse? CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 99 of 162 2. Given the equilateral triangle below whose sides are 4 units long. A 4 4 B C 4 a) Fold the triangle along vertex A so that vertex B maps onto vertex C. Draw a segment along the crease. Label point D where the new segment intersects BC . What is the length of BD (label it in the diagram above)? Explain your reasoning. b) Examine above)? ABD . What is the measure of BAD and BDA (label it in the diagram c) Find the length of AD . Leave exact and simplified. d) In ABD , how many times longer is the hypotenuse than the shorter leg? How many times longer is the longer leg than the shorter leg? CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 100 of 162 e) In this 30o-60o-90o triangle, what are the side length ratios of short leg: long leg: hypotenuse? 3. Repeat the steps above using a different number for the length of the side of the equilateral triangle. What can you conclude about the side-length ratios of short leg: long leg: hypotenuse for any 30o-60o-90o triangle? CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 101 of 162 Name ___________________________________ Date __________________ Period ________ Exploring Special Right Triangles (30-60-90) Answer Key 1. Given the equilateral triangle below whose sides are 2 units long. A 2 B 2 C D a) What is the angle measure of each acute angle? 2 60o b) Fold the triangle along vertex A so that vertex B maps onto vertex C. Draw a segment along the crease. Label point D where the new segment intersects BC . What is special about AD ? What is the length of BD (label it in the diagram above)? Explain your reasoning. AD is the altitude, bisector, bisector, and median of ABC . BD = 1 because AD bisects BC c) What does AD do to A ? It bisects A d) Examine ABD . What is the measure of above)? m BAD = 30o and m BDA = 90o BAD and BDA (label it in the diagram e) Find the length of AD . Leave exact and simplified. AD = 3 f) In ABD, how much longer is the hypotenuse than the shorter leg? How much longer is the longer leg than the shorter leg? The hypotenuse is twice as long as the shorter leg and the longer leg is 3 times as long as the shorter leg. g) In this 30o-60o-90o triangle, what are the side-length ratios of short leg: long leg: hypotenuse? 1: 3:2 CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 102 of 162 2. Given the equilateral triangle below whose sides are 4 units long. A 4 4 B D C 4 a) Fold the triangle along vertex A so that vertex B maps onto vertex C. Draw a segment along the crease. Label point D where the new segment intersects BC . What is the length of BD (label it in the diagram above)? Explain your reasoning. BD = 2 because AD bisects BC b) Examine ABD . What is the measure of BAD and BDA (label it in the diagram above)? m BAD = 30o and m BDA = 90o c) Find the length of AD . Leave exact and simplified. AD = 12 = 2 3 d) In ABD , how many times longer is the hypotenuse than the shorter leg? How many times longer is the longer leg than the shorter leg? The hypotenuse is twice as long as the shorter leg and the longer leg is 3 times as long as the shorter leg. e) In this 30o-60o-90o triangle, what are the side-length ratios of short leg: long leg: hypotenuse? 1: 3 : 2 3. Repeat the steps above using a different number for the length of the side of the equilateral triangle. What can you conclude about the side-length ratios of short leg: long leg: hypotenuse for any 30o-60o-90o triangle? They will always be 1: 3 : 2 . CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 103 of 162 G-SRT 6 Name ___________________________________ Date __________________ Period ________ Discovering Trigonometric Ratios For the following right triangles, find the indicated ratios. C C' A' B' B A Find each length to the nearest quarter of an inch and after division round the quotient to three decimal places. 1. Length of AB = Length of AC Length of A B = Length of A C 2. Length of BC = Length of AC Length of B / C / = Length of A/ C / 3. Length of AB = Length of BC Length of A/ B / = Length of B / C / Triangles ABC and A/B/C/ are similar triangles. 4. From the above experiment, what can you conclude about these ratios? Find the measures of 5. m C and C= C/ to the nearest tenth of a degree. 6. m C/ = Using your calculator, find the following using the value of C from above. (#5) 7. sin 9. tan C= 8. cos C= C= 10. What do you notice about the values in #7 - #9 as compares to the ratios in #1 - #3? 11. Match sine, cosine, and tangent to the three ratios in #1 - #3. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 104 of 162 Find the measures of 12. A = A and A/ . 13. A/ = Using your calculator, find the following using the value of A from above. (#12) 14. sin 16. tan A= 15. cos A= A= 17. Did this change how sine, cosine, and tangent match with the ratios in #1 - #3? If so, how and why? 18. Write a general equation for sine, cosine, and tangent that could be used with any right triangle. 19. Make your own triangles to test the above equations. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 105 of 162 G-SRT 6 Name ___________________________________ Date __________________ Period ________ Discovering Trigonometric Ratios Answer Key For the following right triangles, find the indicated ratios. C Find each length to the nearest quarter of an inch and after division round the quotient to three decimal places. 2.5 1. Length of AB 2 = 0.8 1.5 = Length of AC 2.5 2. Length of BC 1.5 B = 2.5 = 0.6 Length of AC 2 Length of A/ B / 1 = 0.8 C' = 1.25 Length of A/ C / 1.25 .75 / / Length of B C .75 = 0.6 A' = B' A 1.25 Length of A/ C / 1 3. Length of AB 2 = 1.333 = Length of BC 1.5 Length of A/ B / 1 = 1.333 = .75 Length of B / C / DUE TO HUMAN AND ROUNDING ERRORS THESE MAY BE CLOSE BUT NOT BE EXACT. Triangles ABC and A/B/C/ are similar triangles. 4. From the above experiment, what can you conclude about these ratios? Both ratios in #1 are close to being the same. Both ratios in #2 are close to being the same. Both ratios in #3 are close to being the same. Find the measures of 5. m C and C = 53.1o C/ to the nearest tenth of a degree. 6. m C/ = 53.1o Using your calculator, find the following using the value of C from above. (#5) 7. sin 9. tan C = 0.7997 8. cos C = 0.6004 C = 1.3319 10. What do you notice about the values in #7 - #9 as compares to the ratios in #1 - #3? The value in #7 is close to the value in #1 The value in #8 is close to the value in #2 The value in #9 is close to the value in #3 CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 106 of 162 11. Match sine, cosine, and tangent to the three ratios in #1 - #3. Sine matches with the ratio in #1 Cosine matches the ratio in #2 Tangent matches the ratio in #3 Find the measures of 12. A and A = 36.9o A/ . 13. A/ = 36.9o Using your calculator, find the following using the value of A from above. (#12) 14. sin 16. tan A = 0.6004 15. cos A = 0.7997 A = 0.7508 17. Did this change how sine, cosine, and tangent match with the ratios in #1 - #3? If so, how and why? Yes this changed. When looking at #14 - #16, sine matches with #2, cosine matches with #1, and tangent matches with the reciprocal of #3. 18. Write a general equation for sine, cosine, and tangent that could be used with any right triangle. sin = opposite hypotenuse cos = adjacent hypotenuse tan = opposite adjacent 19. Make your own triangles to test the above equations. Answers may vary. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 107 of 162 G-SRT 6 Name ___________________________________ Date __________________ Period ________ Make a Model: Trigonometric Ratios Materials: protractor, metric ruler, compass, plain or graph paper, and scientific calculator for each person in the group. Directions: Everyone in the group should do Steps 1 - 5 individually. Steps 6 – 8 should be done collectively. Step 1: On a sheet of graph or plain paper, use a protractor to make as large a right triangle ABC as possible with m B = 90°, m A = 20°, and m C = 70°. Label the vertices appropriately. Step 2: Use your ruler to measure sides AB, AC, and BC to the nearest millimeter. AB = ______ mm AC = ______ mm BC = ______ mm Step 3: Recall by definition: r hypotenuse y leg opposite sin = length of leg opposite θ y = length of hypotenuse r cos = length of leg adjacent θ x = length of hypotenuse r tan = length of leg opposite θ y = length of leg adjacent θ x x leg adjacent to Step 4: Use the information obtained in Step 2 to complete the following statements. Write the following ratios in fraction form and decimal form to the nearest thousandth. Fraction Decimal sin 20° = length of leg opposite A length of hypotenuse cos 20° = length of leg adjacent to A = __________ = __________ length of hypotenuse tan 20° = length of leg opposite A = __________ = __________ length of leg adjacent to A = __________ = __________ CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 108 of 162 sin 70° = length of leg opposite C = __________ = __________ length of hypotenuse cos 70° = length of leg adjacent C = __________ = __________ length of hypotenuse tan 70° = length of leg opposite C = __________ = __________ length of leg adjacent C Step 5: In the tables below, record your ratios in decimal form to the nearest thousandth in the appropriate boxes (Individual) under sin A, cos A, tan A, sin C, cos C, and tan C. Step 6: Compare the ratios obtained by the members of your group. Calculate the average of each of the ratios found by the members of your group. In the tables below, record the ratios in the appropriate boxes (Group Averages) under sin A, cos A, tan A, sin C, cos C, and tan C. Step 7: Use a calculator to check your group’s results. Calculate sin 20 , cos 20 , tan 20 , sin 70 , cos 70 , and tan 70 . Record the results in your tables. How do the trigonometric ratios that were found by measuring the sides compare with the trigonometric ratios that were found by using a calculator? ______________________________________________________________ m A = 20 Ratios (Individual) sin A cos A tan A sin C cos C tan C Ratios (Group Averages) Ratios (Calculator) m C = 70 Ratios (Individual) Ratios (Group Averages) Ratios (Calculator) CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 109 of 162 Step 8: Questions for Discussion A. Are all right triangles with acute angles measuring 20° and 70° similar? Explain. __________________________________________________________________ B. For any two right triangles with acute angles measuring 20° and 70°: The sin 20°, cos 20°, and tan 20° are ____________________________ the same. sometimes, always, or never The sin 70°, cos 70°, and tan 70° are ____________________________ the same. sometimes, always, or never C. Why are the trigonometric ratios of any right triangle with acute angles measuring 20° and 70° the same regardless of the lengths of the sides? __________________________________________________________________ CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 110 of 162 Name ___________________________________ Date __________________ Period ________ Make a Model: Trigonometric Ratios Answer Key The responses given in Steps 1-6 are based on the lengths of the sides of the different triangles that are drawn by individual students. The responses in Steps 1-6 may vary. Step 7: Use a calculator to check your group’s results. Calculate sin 20 , cos 20 , tan 20 , sin 70 , cos 70 , and tan 70 . Record the results in your tables. How do the trigonometric ratios that were found by measuring the sides compare with the trigonometric ratios that were found by using a calculator? They are the equal or approximately equal to each other. sin A cos A tan A m A = 20 Ratios (Individual) May Vary May Vary May Vary Ratios (Group Averages) May Vary May Vary May Vary Ratios (Calculator) .342 .940 .364 m C = 70 Ratios (Individual) Ratios (Group Averages) Ratios (Calculator) sin C May Vary May Vary .940 cos C May Vary May Vary .342 tan C May Vary May Vary 2.747 Step 8: Questions for Discussion A. Are all right triangles with acute angles measuring 20° and 70° similar? Explain. Yes. Two triangles are similar if their corresponding angles are congruent. B. For any two right triangles with acute angles measuring 20° and 70°: The sin 20°, cos 20°, and tan 20° are always the same. The sin 70°, cos 70°, and tan 70° are always the same. C. Why are the trigonometric ratios of any right triangle with acute angles measuring 20° and 70° the same regardless of the lengths of the sides? A trigonometric ratio is a ratio of the lengths of two sides of a right triangle. All right triangles with acute angles measuring 20° and 70° are similar; therefore, the ratio of any two sides of one triangle will equal the ratio of the corresponding two sides of another. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 111 of 162 G-SRT 8 Name ___________________________________ Date __________________ Period ________ Let’s Measure the Height of the Flagpole 50 60 40 0 13 0 12 0 14 30 0 15 20 10 0 16 0 17 The clinometer is used to measure the heights of objects. It is a simplified version of the quadrant, an important instrument in the Middle Ages, and the sextant, an instrument for locating the positions of ships. Each of these devices has arcs which are graduated in degrees for measuring angles of elevation. The arc of the clinometer is marked from 0 to 90 degrees. When an object is sighted through the straw, the number of degrees in angle BXY can be read from the arc. Angle BAC is the angle of elevation of the clinometer. Angle BXY on the clinometer is equal to the angle of elevation, angle BAC. 70 0 11 X 20 30 40 50 10 0 17 0 16 0 15 0 13 0 12 60 70 0 14 0 11 0 10 80 90 0 10 80 Drinking Straw B A Y C Objective: You will use your skills of right triangle trigonometry to measure the height of the school’s flagpole. Materials: - clinometer, meter stick, calculator CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 112 of 162 Procedures & Questions: - Pick a certain distance (in meters) that you want to stand from the flagpole. Record it below. _____________ meters - Use the clinometer and look through the straw to locate the top of the flagpole. Record the angle measure that is created from the string below. ______________ degrees - Draw a picture of this situation and label all parts clearly. - Use your knowledge of right triangle trigonometry to find the height of the flagpole. Show algebraic work. Round answer to two decimal places. - Can you think of another method to find the height of the flagpole? Explain clearly and be very specific. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 113 of 162 G-SRT 8 Name _____________________________ Date ______________ Period ________ Applications of Trigonometry Using Indirect Measurement 1. ODOT (Ohio Department of Transportation) uses an electronic measurement device to measure distances by recording the time required for a signal to reflect off the object. They use the equipment to survey a portion of the Hocking Hills as below. How much taller is the left part of the Hocking Hills than the right part? T M 950 ft 880 ft 60o 50o B C A 2. You are designing a jet plane as shown. In preparing the documentation for your design, you are required to find the measures of RPQ and PQR in the wing (triangle PQR). What are the measures? P 30 ft 12 ft R Q 3. The first flight of a biplane (doubled-winged plane) was the historic flight of the Wright brothers in 1903. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 114 of 162 A B C E D G F Use the diagram to find the measure of the indicated segment or angle. Given that ADGE is a rectangle, BFC is equilateral, AEF DGF, EF = 15, and BC = 9. Round your answers to two decimal places. a) BF b) AE c) AF d) AB e) AFE f) FAB g) ABF h) FBC In #3h, you can find m value? FBC in two ways. Describe the two ways. Do they yield the same CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 115 of 162 4. You are standing beside Alum Creek to survey the structure of Hoover Reservoir. Using an electronic measuring device, you find the angle of elevation to the top of the dam to be 55 o, and the distance to the top of the dam to be 922 feet. 922 ft 55º 500 ft x ft a) Use the diagram to find the height of the dam. b) If you are standing 500 feet from the base of the dam, find x. 5. You are standing 382.5 feet away from the center of the Eiffel Tower and the angle of elevation is 70o. Find the height of the Eiffel Tower. 70º 382.5 ft CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 116 of 162 6. A yacht is sailing toward the lighthouse and a airplane is flying toward the lighthouse as well. The lighthouse is 250 feet tall. The yacht is 400 feet from the lighthouse and the airplane is 300 feet from the lighthouse and has the same height as the top of the lighthouse. 300 ft y 250 ft x 400 ft Find the angle of elevation of the yacht and the angle of depression of the airplane. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 117 of 162 Name ___________________________________ Date __________________ Period ________ Application of Trigonometry Using Indirect Measurement Answer Key 1. ODOT (Ohio Department of Transportation) uses an electronic measurement device to measure distances by recording the time required for a signal to reflect off the object. They use the equipment to survey a portion of the Hocking Hills as below. How much taller is the left part of the Hocking Hills than the right part? T M 950 ft 880 ft 60o 50o B C A In the right triangle ∆ATB, you can use the sine ratio to find the length of TB . TB TB sin TAB = sin 60o = 950(sin 60o) = TB 822.72 TB 950 TA Use the same procedure to find the length of MC in AMC. MC MC sin MAC = sin 50o = 880(sin 50o) = MC 674.12 MC MA 880 From these two approximations, you can conclude that the difference in the heights is: 822.72 – 674.12 = 148.6 feet. 2. You are designing a jet plane as shown. In preparing the documentation for your design, you are required to find the measures of RPQ and PQR in the wing (triangle PQR). What are the measures? P 30 ft 12 ft R Q To find the measure of RPQ, you can use the tangent ratio. RQ 30 = 2.5 tan P = tan P = m P 68.2o RP 12 Because P and Q are complementary, you can determine the measure of Q to be m Q = 90o – 68.2o = 21.8o 3. The first flight of a biplane (doubled-winged plane) was the historic flight of the Wright brothers in 1903. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 118 of 162 A B C E D G F Use the diagram to find the measure of the indicated segment or angle. Given that ADGE is a rectangle, BFC is equilateral, AEF DGF, EF = 15, and BC = 9. Round your answers to two decimal places. a) BF b) AE 9 7.79 or 4.5 3 c) AF d) AB 10.5 16.90 e) AFE 27.46o f) FAB 27.46o g) ABF 120o h) FBC 60o In #3h, you can find m value? FBC in two ways. Describe the two ways. Do they yield the same Method 1: Each angle of equilateral 4.5 = 1 Method 2: cos FBC = 9 2 So m FBC = 60o; yes. FBC is 60o. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 119 of 162 4. You are standing beside Alum Creek to survey the structure of Hoover Reservoir. Using an electronic measuring device, you find the angle of elevation to the top of the dam to be 55 o, and the distance to the top of the dam to be 922 feet. 922 ft 55º 500 ft x ft a) Use the diagram to find the height of the dam. opp hyp opp sin 55o = 922 sin 55o = 922(sin 55o) = opp 922(.819) opp 755.12 opp = the height of the dam b) If you are standing 500 feet from the base of the dam, find x. adj hyp 500 + x cos 55o = 922 cos 55o = 922(cos 55o) = 500 + x 922(.573) 500 + x 528.31 500 + x 28.31 x CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 120 of 162 5. You are standing 382.5 feet away from the center of the Eiffel Tower and the angle of elevation is 70o. Find the height of the Eiffel Tower. opp adj opp tan 70o = 382.5 tan 70o = 382.5(tan 70o) = opp 382.5(2.747) opp 1050.7 ft opp = height of Eiffel Tower 70º 382.5 ft 6. A yacht is sailing toward the lighthouse and an airplane is flying toward the lighthouse as well. The lighthouse is 250 feet tall. The yacht is 400 feet from the lighthouse and the airplane is 300 feet from the lighthouse and has the same height as the top of the lighthouse. 300 ft y 250 ft x 400 ft Find the angle of elevation of the yacht and the angle of depression of the airplane. Angle of Elevation: tan x = Angle of Depression: opp adj tan y = opp adj tan x = 250 400 tan y = 250 300 tan x = .625 tan y 32o y x CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 .833 39.81o Columbus City Schools 6/28/13 Page 121 of 162 G-SRT 8 Name ___________________________________ Date __________________ Period ________ Find the Missing Side or Angle Instructions: Find the missing side or angle as indicated in each of the right triangles below. 1. 2. x = ___________ 10 = ___________ 23 28 18 x 3. 4. 9 x 45 30 x = ___________ c c = ___________ 70 5. x 55 6. = ___________ x = __________ 25 11 2 38 17 7. a 8. 8 65 a = ___________ b = __________ b 9. Describe a situation when you would use sine. Use illustrations to support your answer. 10. Describe a situation when you would use cos-1. Use illustrations to support your answer. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 122 of 162 Name ___________________________________ Date __________________ Period ________ Find the Missing Side or Angle Answer Key Instructions: Find the missing side or angle as indicated in each of the right triangles below. 1. 2. 10 x= 18.81 = 23 28 18 x 3. 4. 9 x 51.5o x= 8.46 70 5. 6. 11 = 45 30 42.42 c x 55 10.30 c= x= 14.34 25 2 7. 8. 17 a 65 a= 38 7.93 b= 6.25 8 b 9. Describe a situation when you would use sine. Use illustrations to support your answer. When you know the measure of an angle and the measure of either the opposite side or the hypotenuse. x 15 25 10. Describe a situation when you would use cos-1. Use illustrations to support your answer. When you know the measure of the adjacent side and the hypotenuse and want to find the measure of the angle. 10 x 5 CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 123 of 162 G-SRT 8 Name ___________________________________ Date __________________ Period ________ Between the Uprights Using the picture below: Find the angle of elevation the ball makes with the ground when it is kicked. Find the length of the most direct path from where the ball is kicked to where it crosses the uprights (hypotenuse). 24 feet 60 50 40 30 20 10 yards Using the picture below: Find the angle of elevation the ball makes with the ground when it is kicked. Find the length of the most direct path from where the ball is kicked to where it crosses the uprights (hypotenuse). 20 feet 40 30 20 10 yards Using the picture below: CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 124 of 162 Find the angle of elevation the ball makes with the ground when it is kicked. Find the length of the most direct path from where the ball is kicked to where it hits the uprights (hypotenuse). 10 feet 50 40 30 yards 10 20 Using the picture below: If the angle of elevation the ball makes with the ground when it is kicked is 27o, at what distance from the ground will it cross the uprights? Find the length of the most direct path from where the ball is kicked to where it crosses the uprights (hypotenuse). ? 30 CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 20 yards 10 Columbus City Schools 6/28/13 Page 125 of 162 Write your own problem for the picture below. Label all parts. Solve the problem showing all calculations. ___ feet 30 20 yards CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 10 Columbus City Schools 6/28/13 Page 126 of 162 G-SRT 8 Name ___________________________________ Date __________________ Period ________ Between the Uprights Answer Key Using the picture below: Find the angle of elevation the ball makes with the ground when it is kicked. 7.59 Find the length of the most direct path from where the ball is kicked to where it crosses the uprights (hypotenuse). 60.53 yds 24 feet 60 50 40 20 30 10 yards Using the picture below: Find the angle of elevation the ball makes with the ground when it is kicked. 9.46 Find the length of the most direct path from where the ball is kicked to where it crosses the uprights (hypotenuse). 40.55 yds 20 feet 40 30 CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 20 10 yards Columbus City Schools 6/28/13 Page 127 of 162 Using the picture below: Find the angle of elevation the ball makes with the ground when it is kicked. 3.81 Find the length of the most direct path from where the ball is kicked to where it hits the uprights (hypotenuse). 50.11 yds 10 feet 50 20 30 yards 40 10 Using the picture below: If the angle of elevation the ball makes with the ground when it is kicked is 27o, at what distance from the ground will it cross the uprights? 45.86 ft = 15.29 yds Find the length of the most direct path from where the ball is kicked to where it crosses the uprights. 33.67 yds ? 30 20 yards 10 Write your own problem for the picture below. Label all parts. Solve the problem showing all calculations. Answers will vary. ___ feet 30 20 yards CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 10 Columbus City Schools 6/28/13 Page 128 of 162 G-SRT 8 Name ___________________________________ Date __________________ Period ________ Solve the Triangle Instructions: Measure the sides (centimeters) and/or angle (degrees) listed in the given column for the right triangles shown below. Example: A b C c a B Given Side or Angle A a Measure Calculated Side or Angle B b c Measure Once you have completed your measurements, solve each triangle (find all missing sides and angles), placing values in the table. Trade papers with your partner and check each other's completed triangles using the following checklist: ____ all calculations are correct ____ the sum of all angles of each triangle is 180o, accuracy within 1o ____ the Pythagorean Theorem holds true for your values of the legs and hypotenuse, i.e., a2 + b2 = c2 1. A c b C a B Given Side or Angle B c CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Measure Calculated Side or Angle Measure Columbus City Schools 6/28/13 Page 129 of 162 2. A c b a C 3. B A Given Side or Angle b c c b C Measure Measure Calculated Side or Angle Calculated Side or Angle Measure Measure B a 4. A Given Side or Angle A b c b B Given Side or Angle A c C a Measure Calculated Side or Angle Measure 5. C b a B c A Given Side or Angle B b CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Measure Calculated Side or Angle Measure Columbus City Schools 6/28/13 Page 130 of 162 6. b A C a c Given Side or Angle A a Measure Calculated Side or Angle Measure B 7. A c b C 8. B a C b B a c Given Side or Angle B a Given Side or Angle a b Measure Measure Calculated Side or Angle Calculated Side or Angle Measure Measure A CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 131 of 162 Name ___________________________________ Date __________________ Period ________ Solve the Triangle Answer Key Instructions: Measure the sides (centimeters) and/or angle (degrees) listed in the given column for the right triangles shown below. Example: A c b C B a Given Side or Angle A a Measure o 30 1.5 cm Calculated Side or Angle B b c Measure 60o 2.6 cm 3 cm Once you have completed your measurements, solve each triangle (find all missing sides and angles), placing values in the table. Trade papers with your partner and check each other’s completed triangles using the following checklist: ____ all calculations are correct ____ the sum of all angles of each triangle is 180o, accuracy within 1o ____ the Pythagorean Theorem holds true for your values of the legs and hypotenuse, i.e., a2 + b2 = c2 1. A c b C a B Given Side or Angle B c CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Measure 27o 2.8 cm Calculated Side or Angle A a b Measure 63o 2.50 cm 1.27 cm Columbus City Schools 6/28/13 Page 132 of 162 A 2. a C 3. Given Side or Angle A c c b B Measure o 50 3.3 cm Calculated Side or Angle B a b Measure 40o 2.53 cm 2.12 cm A Given Side or Angle b c c b C B a Measure 2.5 cm 2.7 cm Calculated Side or Angle A B a Measure 22.19o 67.81o 1.02 cm A 4. c b B C a Given Side or Angle A b Measure o 65 1.9 cm Calculated Side or Angle B a c Measure 25o 4.07 cm 1.72 cm 5. a C b B c Given Side or Angle B b Measure 21o 1.6 cm A Calculated Side or Angle A a c Measure 69o 4.17 cm 4.46 cm 6. b A c C a Given Side or Angle A a Measure 42o 2.5 cm Calculated Side or Angle B b c Measure 48o 2.78 cm 3.74 cm B CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 133 of 162 7. A b 8. c C a B C a B b c Given Side or Angle B a Given Side or Angle a b Measure 29o 2.6 cm Measure 2.5 cm 4 cm Calculated Side or Angle A b c Calculated Side or Angle A B c Measure 61o 1.44 cm 2.97 cm Measure 32o 58o 4.72 cm A CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 134 of 162 G-SRT 8 Name ___________________________________ Date __________________ Period ________ Right Triangle Park Because of your reputation for drawing and your keen mathematical ability, you have been selected to design a very special park for your neighborhood! This park will be designed using only right triangles! Instructions: Design a city park using only right triangles. Your park must include at least 4 different components such as picnic tables, swing sets, slides, gardens, skating ramps, etc. Draw your design in the area provided below. Measure one side and one acute angle of each triangle in your design. Solve and label each triangle, using trigonometry to find the missing sides and angles. Right Triangle Park CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 135 of 162 G-SRT 8 Name ___________________________________ Date __________________ Period ________ Find the Height When you see an image in a mirror, the angle your line of sight makes with the ground is the same as the angle the top of the object being reflected makes with the ground as shown below. You and your work group will use this fact and your knowledge of similar triangles to find the heights of structures in your school yard. Instructions: Select 3 tall objects you wish to measure (tree, flagpole, smokestack, school, goalpost, etc). Place a mirror on the ground between yourself and the object whose height you are calculating. Stand so you can see the top of the object in the mirror. While you stand, your partner will measure the ground distance from you to the mirror and from the mirror to the object. Record the measurements on the “Find the Height” data sheet. Record the mirror watcher's height on the data sheet. Set up your proportion and find the height of the object. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 136 of 162 G-SRT 8 Name ___________________________________ Date __________________ Period ________ Find the Height Data Sheet Object measured Sketch your reflection experiment in the box below. Label all measurements. Proportion _____________________________ Height of Object _________________________ Object measured Sketch your reflection experiment in the box below. Label all measurements. Proportion _____________________________ Height of Object _________________________ Object measured Sketch your reflection experiment in the box below. Label all measurements. Proportion _____________________________ Height of Object _________________________ CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 137 of 162 G-SRT 8 Name ___________________________________ Date __________________ Period ________ Applications of the Pythagorean Theorem For each of the following word problems, draw a picture to represent the situation, write an equation and solve for the missing parts. A 25-ft ladder leans against the side of a house. If you place the ladder 15 ft from the base of the house, how high up will the ladder reach? A broadcast antenna needs a support wire replaced. If the support wire is attached to the ground 58 ft from the antenna base and is attached to the antenna 125 ft from the ground, how long is the support wire? Ralph purchased a 7 m slide and it covers a 4.3 m distance on the ground. How tall is the slide’s ladder? The bases on a baseball diamond are 90 ft apart. If the catcher stands at home plate and throws to second base, how far does the catcher throw? CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 138 of 162 Name ___________________________________ Date __________________ Period ________ Applications of the Pythagorean Theorem Answer Key For each of the following word problems, draw a picture to represent the situation, write an equation and solve for the missing parts. A 25-ft ladder leans against the side of a house. If you place the ladder 15 ft from the base of the house, how high up will the ladder reach? x2 + 152 = 252 x = 20 ft 25 ft x ft 15 ft A broadcast antenna needs a support wire replaced. If the support wire is attached to the ground 58 ft from the antenna base and is attached to the antenna 125 ft from the ground, how long is the support wire? 2 2 125 ft 2 125 + 58 = x x = 137.8 x ft 58 ft ft Ralph purchased a 7 m slide and it covers a 4.3 m distance on the ground. How tall is the slide’s ladder? 4.32 + x 2 = 72 x = 5.5 xm 7m 4.3 m The bases on a baseball diamond are 90 ft apart. If the catcher stands at home plate and throws to second base, how far does the catcher throw? 902 + 902 = x 2 x = 127.3 ft 90 ft x ft 90 ft CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 139 of 162 Reteach Name ___________________________________ Date __________________ Period ________ Memory Match – Up Students can be put into groups of 3 – 4. First place all cards face down and have each student take turns drawing two cards. If the two cards drawn go together as a pair, then the student will keep it as a match. The student with the most matches wins. Note: There are 3 cards that say “1”. There are 2 cards that say “ 3 ”. There are 2 cards that say 3 ”. There are 2 cards that say “ 1 ”. Make sure that students know that two cards with the exact “ 2 2 same expression on them are not considered a match. For example: A card with a “1” on it does not match a card with a “1” on it. A card with a “1” on it is a match with a card that has “tan 45º” on it. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 140 of 162 Memory Match – Up Cards Pythagorean Theorem 2 1 45o ? 45o 30o 45o ? ? 60o 45o ? 1 3 CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 141 of 162 3 1 2 2 2 1 tan 45º sin 45º sin 30º cos 30º CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 142 of 162 30o 30o ? 60o 60o ? It can be used to solve for an acute angle in a right triangle. 3 2 sin 1 2 3 2 3 3 -1 CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 143 of 162 sin cos opposite hypotenuse tan 3 adjacent hypotenuse opposite adjacent 1 2 leg2 + leg2 = hypotenuse2 o tan 45 o sin 30 CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Page 144 of 162 Columbus City Schools 6/28/13 oo 45 sin 30 cos tan 30º sin 60º cos 60º tan 60º CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 145 of 162 Memory Match-Up Answer Key leg2 + leg2 = hypotenuse2 Pythagorean Theorem opposite hypotenuse adjacent hypotenuse opposite adjacent sin cos tan 1 2 3 2 3 3 2 2 sin 30º cos 30º tan 30º sin 45º tan 45º 1 3 2 1 2 sin 60º cos 60º 3 tan 60º It can be used to solve for an acute angle in a right triangle. sin -1 ? 45 º 1 45º ? ? 45º 2 45º ? 30º 2 60º 1 30º 60º ? ? 3 30º 60º CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 146 of 162 Reteach Name ___________________________________ Date __________________ Period ________ Similar Right Triangles and Trigonometric Ratios Draw a right triangle. Label it ABC, with C being the right angle. Measure the sides in centimeters and the angles in degrees. Complete this chart by filling in the measurements for each angle and each side. Side or Angle Measure A B A B C Remember that the trigonometric ratios are defined as shown below. sin = length of opposite leg length of hypotenuse cos = length of adjacent leg length of hypotenuse tan = length of opposite leg length of adjacent leg Complete the chart from your measurements. Use the trigonometric functions on your calculator to find the values. If the two sets of values are not about the same, measure and compute again. Trigonometric Value From Measurement From Calculator sin A cos A tan A sin B cos B tan B CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 147 of 162 Draw a right triangle, DEF, whose angles are the same as those in triangle ABC, but whose sides are twice as long. Complete the chart as you did for triangle ABC. Side or Measure Angle D E d e f Trigonometric Value From Measurement From Calculator sin D cos D tan D sin E cos E tan E Make a triangle GHI, that is similar to the other two triangles, with side GH measuring 20 cm long. Show how you find the length of the other two sides. What do you know about the sine, cosine, and tangents of angles G and H? CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 148 of 162 Extension Name ___________________________________ Date __________________ Period ________ Similar Right Triangles and Trigonometric Ratios Given: rt. ABC ~ rt. DEF A D b e c f d E a F B the fact that the triangles are similar C to find the missing term (?). Write the missing term Part A: Use in the space provided. __________ 1. a b d ? __________ 2. ? b f e __________ 3. e f ? c __________ 4. d ? f c __________ 5. a c ? f __________ 6. b ? e f __________ 7. ? d b a __________ 8. f d c ? CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 149 of 162 Part B: Describe two ways that similarity proportions can be formed. ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ Part C: Each of the proportions below are true for the two similar triangles given. The ratios that form the proportions can be written as trigonometric ratios. Complete the statements below that correspond to the given proportions to make them true. A D b e c f d E a 1. 3. 5. Bd a = b e sin A = sin _____ C c f 2. = b e sin C = sin _____ c f = b e cos A = cos _____ a d = c f tan A = tan _____ F 4. 6. a d = b e cos C = cos _____ c f = a d tan C = tan _____ CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 150 of 162 Part D: Use the information in Part C to complete the following statements. 1. sin C = cos _____ 2. cos A = sin _____ 3. sin A = cos _____ 4. cos C = sin _____ 5. Describe the relationship that exists between angles A and C? ________________________________________________________________________ 6. sin D = cos _____ 7. cos F = sin _____ 8. sin F = cos _____ 9. cos D = sin _____ 10. Describe the relationship that exists between angles D and F? ________________________________________________________________________ Conclusion: ________________________________________________________________________ ________________________________________________________________________ Part E: Complete the following statements. 1. sin 20° = cos _____ 2. cos 35° = sin _____ 3. sin 10° = cos _____ 4. cos 45° = sin _____ 5. sin 30° = cos _____ 6. cos 60° = sin _____ 7. sin x° = cos _____ 8. cos y° = sin _____ CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 151 of 162 Name ___________________________________ Date __________________ Period ________ Similar Right Triangles and Trigonometric Ratios Answer Key Given: rt. ABC ~ rt. DEF A D b e c f d E a B F C Part A: Use the fact that the triangles are similar to find the missing term (?). Write the missing term in the space provided. e 1. a b d ? c 2. ? b f e b 3. e f ? c a 4. d ? f c d 5. a c ? f c 6. b ? e f e 7. ? d b a a 8. f d c ? CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 152 of 162 Part B: Describe two ways that similarity proportions can be formed. 1. One way: Each ratio of the proportion compares a side of one triangle with the corresponding side of the other triangle. 2. Another way: Each ratio of the proportion compares two sides from the same triangle with two corresponding sides of the other triangle. Part C: Each of the proportions below are true for the two similar triangles given. The ratios that form the proportions can be written as trigonometric ratios. Complete the statements below that correspond to the given proportions to make them true. A D b e c f d E a B a d 1. = b e sin A = sin D C 2. c f = b e cos A = cos D 3. 5. 4. a d = c f tan A = tan D 6. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 F c f = b e sin C = sin F a d = b e cos C = cos F c f = a d tan C = tan F Columbus City Schools 6/28/13 Page 153 of 162 Part D: Use the information in Part C to complete the following statements. 1. sin C = cos A 2. cos A = sin C 3. sin A = cos C 4. cos C = sin A 5. Describe the relationship that exists between angles A and C? Angles A and C are complementary angles. The sum of their measures equals 90°. 6. sin D = cos F 7. cos F = sin D 8. sin F = cos D 9. cos D = sin F 10. Describe the relationship that exists between angles D and F? Angles D and F are complementary angles. The sum of their measures equals 90°. Conclusion: In a right triangle, the sine of one of the acute angles equals the cosine of the other acute angle (complement of the angle). Part E: Complete the following statements. 1. sin 20° = cos 70° 2. cos 35° = sin 55° 3. sin 10° = cos 80° 4. cos 45° = sin 45° 5. sin 30° = cos 60° 6. cos 60° = sin 30° 7. sin x° = cos (90 – x)° 8. cos y° = sin (90 – y)° CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 154 of 162 Reteach Name ___________________________________ Date __________________ Period ________ Hey, All These Formulas Look Alike! Show your work. Include formulas in your explanations. Consider the ABC. Show that CBA is a right angle. C (3, 4) A (0, 0) B (3, 0) Complete the chart. Slope, AC Distance, AC Pythagorean Theorem, ABC Tan CAB Write Formulas Substitute Values and Simplify Compare the expressions and the values for the slope of AC and tan CAB. Are the formulas the same? Are the values equal? Support your answer by showing your work. This exercise utilized the first quadrant only. Predict if your conclusions will hold if the triangle is rotated 90 , about the origin, counterclockwise. Test your prediction. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 155 of 162 Name ___________________________________ Date __________________ Period ________ Hey, All These Formulas Look Alike! Answer Key Show your work. Include formulas in your explanations. Consider the ABC. Show that CBA is a right angle. Solution: If CBA is a right angle, then CB BA . If CB BA , then m1 * m2 = -1. This is a special case in which the slope of one of the perpendicular lines is undefined and the slope of the other line is zero. 4 0 4 0 0 0 Slope BC = Slope BA = 3 3 0 3 0 3 C (3, 4) A (0, 0) B (3, 0) 2. Complete the chart. Slope, AC Write Formulas Substitute Values and Simplif m y2 x2 4 0 3 0 y1 x1 ( y2 y1 )2 (3 0)2 (4 0)2 AC ( x2 3 4 3 Pythagorean Theorem, ABC (x2-x1)2+(y2-y1)2=AC2 Distance, AC 2 x1 )2 4 2 AC 9 16 AC 25 AC Tan CAB length of opp. side length of adj. side (3-0)2+(4-0)2=(AC)2 32 + 42 = (AC)2 9 + 16 = (AC)2 25 = (AC)2 4 0 3 0 4 3 25 ( AC )2 5 = AC 5 = AC 3. Compare the expressions and the values for the slope of AC and tan CAB. Are the formulas the same? Are the values equal? Support your answer by showing your work. The formula for the slope of AC and the equation for tan CAB are the same. Slope of AC = y2 x2 y1 x1 Tan CAB = The values are the same. The slope AC of and tan length of opposite side length of adjacent side CAB both equal 4 0 3 0 y2 x2 y1 x1 4 3 4. This exercise utilized the first quadrant only. Predict if your conclusions will hold if the triangle is rotated 90 , about the origin, counterclockwise. Test your prediction. Answers will vary. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 156 of 162 Extension Name ___________________________________ Date __________________ Period ________ Problem Solving: Trigonometric Ratios Materials: scientific calculator Use the information given in the figure below to determine the sine, cosine, and tangent of Explain your answer. Sin = _______ Cos = _______ Tan . = _______ (0,5) B A C (3,4) (5,0) Use the information given in the figure below to determine the perimeter of rectangle ABCD. Support your answer by showing your work. B C 100 cm 35 A D Perimeter = ____________ 3. Which of the following trigonometric ratios: sine, cosine, or tangent of an angle can have a value greater than 1? Why is it that the values of the other two trigonometric ratios can never be greater than 1? Explain. 4. John, an employee of the U.S. Forestry Service has been asked to determine the height of a tall tree in Wayne National Forest. He uses an angle measuring device to determine the angle of CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 157 of 162 elevation (angle formed by the line of sight to the top of the tree and a horizontal) to be about 33 . He walks off 40 paces to the base of the tree. If each pace is .6 meters, how tall is the tree to the nearest meter? Support your answer by showing your work and including a diagram. 5. Determine the perimeter to the nearest centimeter and the area to the nearest square centimeter of the triangle shown below. Support your answer by showing your work and giving an explanation. B 10 cm A C 6. Use what you know about the side lengths of special right triangles to complete the following table. Express your answers in simplified radical form. 30 45 45 30 45 60 60 Sin Cos Tan CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 158 of 162 Name ___________________________________ Date __________________ Period ________ Problem Solving: Trigonometric Ratios Answer Key Use the information given in the figure below to determine the sine, cosine, and tangent of . 4 3 4 Explain your answer. Sin = Cos = Tan = (0,5) 5 5 3 B (3,4) Solution: The lengths of AC and BC can be determined by using the coordinates of point B(3,4). The length of AB can be determined by using the fact that it is a radius of a circle. AB = 5, BC = 4, and AC = 3. By definition: A C (5,0) BC 4 AC 3 BC 4 sin θ = = ;; cos θ = = ;; and tan θ = = AB 5 AB 5 AC 3 Use the information given in the figure below to determine the perimeter of rectangle ABCD. Support your answer by showing your work. Solution: By definition: 57.36 cm B C AB AB sin 35° = ; .5736 ; AB 57.36 100 100 BD BD cos 35° = ; .8192 ; BD 81.92 100 100 81.92 cm 100 cm 35 A D The perimeter of the rectangle = 2(57.36) +2(81.92) = 278.56 cm.; Perimeter = 278.56 cm Which of the following trigonometric ratios: sine, cosine, or tangent of an angle can have a value greater than 1? Why is it that the values of the other two trigonometric ratios can never be greater than 1? Explain. Solution: The tangent of an angle can be greater than 1. The sine of an acute angle of a right triangle length of leg opposite the angle is defined as and the cosine of an acute angle of a right triangle is length of hypotenuse length of leg adjacent to the angle defined as . The length of a leg of a right triangle will always length of hypotenuse be less than the length of the hypotenuse. If the numerator of a fraction is less than the denominator, the fraction is always less than 1. Therefore, the sine and cosine of an angle will never be greater than 1 by definition of the sine and cosine ratios. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 159 of 162 4. John, an employee of the U.S. Forestry Service has been asked to determine the height of a tall tree in Wayne National Forest. He uses an angle measuring device to determine the angle of elevation (angle formed by the line of sight to the top of the tree and a horizontal) to be about 33 . He walks off 40 paces to the base of the tree. If each pace is .6 meters, how tall is the tree to the nearest meter? Support your answer by showing your work and including a diagram. Solution: h tan 33° = 24 h h .6494 24 h 16 m 33 24 m 5. Determine the perimeter to the nearest centimeter and the area to the nearest square centimeter of the triangle shown below. Support your answer by showing your work and giving an explanation. Sample Solution: Triangle ABC is an isosceles right triangle. The legs have equal lengths, therefore the acute angles each have a measure of 45 . The ratio of the sides of a 45 - 45 - 90 triangle is 10 10 2 B 1:1: 2 . The length of each leg is = = 5 2. 2 2 The perimeter of the triangle is 45 10 + 5 2 + 5 2 = 10 + 10 2 24 cm. 10 cm The area of the triangle is 1 1 • 5 2 • 5 2 = • 25 • 2 2 2 45 25 cm 2 . A C Sample Solution: Triangle ABC is an isosceles right triangle. The legs have equal lengths, therefore the acute angles each have a measure of 45 . The lengths of the legs can be found by using the sine and cosine ratios. B AC 10 AC = sin 45° •10 .707 •10 7.07 cm AB cos B = cos 45° = 10 AB = cos 45° •10 .707 •10 7.07 cm sin B = sin 45° = 45 10 cm 45 A C The perimeter of the triangle is 7.07 + 7.07 + 10 = 24.14 24 cm. The area of the triangle is (.5)(7.07)(7.07) = 24.99 25 cm2 CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 160 of 162 6. Use what you know about the side lengths of special right triangles to complete the following table. Express your answers in simplified radical form. 30 45 45 60 30 Sin 1 2 Cos Tan 3 2 1 3 = 3 3 45 1 2 = 2 2 1 2 = 2 2 1 =1 1 CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 60 3 2 1 2 3 = 3 1 Columbus City Schools 6/28/13 Page 161 of 162 Grids and Graphics Addendum CCSSM II Comparing Functions and Different Representations of Quadratic Functions FIF 3, 4, 5, 6, 7, 7a, 9, F-BF 1, 1a, 1b, A-CED 1, 2, , F-LE 3, , N-NQ 2, , S-ID 6a, 6b, A-REI 7 Quarter 2 Page 162 of 162 Columbus City Schools 6/28/13 Algebra Tiles Template Grids and Graphics Page 1 of 18 Columbus Public Schools 6/27/13 10 by 10 Grids Grids and Graphics Page 2 of 18 Columbus Public Schools 6/27/13 20 by 20 Grids Grids and Graphics Page 3 of 18 Columbus Public Schools 6/27/13 Small Coordinate Grids Grids and Graphics Page 4 of 18 Columbus Public Schools 6/27/13 Dot Paper Grids and Graphics Page 5 of 18 Columbus Public Schools 6/27/13 Isometric Dot Paper Grids and Graphics Page 6 of 18 Columbus Public Schools 6/27/13 Quarter-Inch Grid Grids and Graphics Page 7 of 18 Columbus Public Schools 6/27/13 Half-Inch Graph Paper Grids and Graphics Page 8 of 18 Columbus Public Schools 6/27/13 One-Inch Grid Paper Grids and Graphics Page 9 of 18 Columbus Public Schools 6/27/13 Centimeter Grid Grids and Graphics Page 10 of 18 Columbus Public Schools 6/27/13 Pascal’s Triangle Template Grids and Graphics Page 11 of 18 Columbus Public Schools 6/27/13 Probability Spinners Grids and Graphics Page 12 of 18 Columbus Public Schools 6/27/13 Protractor 60 80 70 50 120 130 140 30 150 20 160 10 170 110 100 90 100 110 80 70 40 60 120 130 140 30 150 20 160 10 170 30 120 40 130 140 30 150 20 160 10 170 60 50 110 80 100 90 100 80 110 70 Grids and Graphics 60 120 130 60 40 30 120 130 140 30 150 20 160 10 170 160 10 170 110 80 100 90 100 80 110 70 60 120 130 60 40 30 120 40 130 140 30 150 20 160 10 170 150 20 160 10 170 110 80 100 90 100 80 110 70 60 120 130 60 50 40 30 40 150 20 110 80 100 90 100 80 110 70 160 10 170 Page 13 of 18 120 130 60 140 50 40 30 150 20 160 10 170 110 80 100 90 100 80 110 70 120 130 60 140 50 40 30 150 20 160 10 170 70 120 130 140 30 150 20 160 10 170 140 50 110 120 80 70 130 60 140 50 40 150 30 160 20 10 170 70 50 140 50 100 40 150 20 110 70 50 140 50 70 120 130 140 30 150 20 160 10 170 40 160 10 170 70 50 120 130 140 30 150 20 160 10 170 100 90 40 150 20 80 70 50 140 40 40 60 130 60 50 70 50 60 120 110 80 100 90 100 80 110 70 120 130 60 140 50 40 30 150 20 160 10 170 Columbus Public Schools 6/27/13 Tangram Template Grids and Graphics Page 14 of 18 Columbus Public Schools 6/27/13 Blank 11 Grids and Graphics 11 Geoboards Page 15 of 18 Columbus Public Schools 6/27/13 Blank Number Lines Grids and Graphics Page 16 of 18 Columbus Public Schools 6/27/13 Rulers mm 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 mm mm mm mm mm Grids and Graphics Page 17 of 18 Columbus Public Schools 6/27/13 Websites for Graph Paper and More! Below you will find great web sites to visit for graph paper and other things to use in your math activities. http://www.mathematicshelpcentral.com/graph_paper.htm (requires Adobe Acrobat Reader version 5.0 or higher to view or print graphs) This is a wonderful collection of all different kinds of graphs from full-page format to several per page for multiple problems. You will also find a page set up specifically for proofs and graph paper for 3-space, polar coordinates, and logarithms. http://mathpc04.plymouth.edu/gpaper.html At this site you will find several versions of coordinate, semi-logarithmic, full logarithmic, polar, and triangular graph paper. http://mason.gmu.edu/~mmankus/Handson/manipulatives.htm This is site to go to if you need to make math manipulatives. Cutouts are available for pattern blocks, geometric shapes, base-ten and base-five blocks, xy blocks, attribute blocks, rods, and color tiles. Graph paper can be printed as well. http://www.handygraph.com/free_graphs.htm Several forms of coordinate graphs and number lines sized just right for homework and tests. http://donnayoung.org/frm/spepaper.htm Not only does this site have graph paper it contains notebook paper, Lego design paper, music paper, and award certificates. http://www.lib.utexas.edu/maps/map_sites/outline_sites.html#W Outline maps for states, countries, regions, and the world. Grids and Graphics Page 18 of 18 Columbus Public Schools 6/27/13 High School CCSS Mathematics II Curriculum Guide -Quarter 2Columbus City Schools Page 1 of 399 Table of Contents RUBRIC – IMPLEMENTING STANDARDS FOR MATHEMATICAL PRACTICE ....................... 12 Mathematical Practices: A Walk-Through Protocol .............................................................................. 17 Curriculum Timeline .............................................................................................................................. 20 Scope and Sequence ............................................................................................................................... 21 Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7E, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-NQ 2...................................................................................................................... 30 Teacher Notes: ......................................................................................................................................... 33 Equation of a Circle with Center (0, 0) ................................................................................................ 66 Equation of a Circle with Center (h, k) ................................................................................................ 70 Using Formulas .................................................................................................................................... 74 Literal Madness .................................................................................................................................... 78 Adding and Subtracting Radicals Cards .............................................................................................. 81 Multiplying and Dividing Radicals Cards ........................................................................................... 83 Match Me Cards ................................................................................................................................... 87 Radicals Rule! ...................................................................................................................................... 89 Addition and Subtraction of Radicals .................................................................................................. 93 Multiplication and Division of Radicals .............................................................................................. 95 Radical Expressions ............................................................................................................................. 97 Radical and Exponent Matching .......................................................................................................... 99 Find Two Ways .................................................................................................................................. 101 Graphing Square Roots ...................................................................................................................... 103 Graphing Cube Roots ......................................................................................................................... 111 Graphing Cube and Square Roots ...................................................................................................... 119 Real Number System – Classification ............................................................................................... 132 Real Number System Cards ............................................................................................................... 133 Real Numbers – Perimeter Investigation ........................................................................................... 134 Real Numbers – Perimeter Homework .............................................................................................. 138 Real Numbers – Circumference & Area Investigation ...................................................................... 139 Real Numbers –Area Homework ....................................................................................................... 143 Real Numbers – Extension ................................................................................................................. 144 Real Number System Cards ............................................................................................................... 146 Graphing? Absolutely! ....................................................................................................................... 158 Absolute Value Equation and Graph Cards ....................................................................................... 166 2’s Are Wild ....................................................................................................................................... 168 Connecting Functions ........................................................................................................................ 176 Different = Same ................................................................................................................................ 190 Three Different Exponential Functions.............................................................................................. 194 Practice ............................................................................................................................................... 197 Absolute Value Graphs ...................................................................................................................... 204 Absolute Value Graphs ...................................................................................................................... 209 Investigate Compound Interest .......................................................................................................... 220 Exponent Properties Exploration ....................................................................................................... 225 Multiplying Binomials ....................................................................................................................... 235 A Number Called e ............................................................................................................................ 237 Comparing Functions and Different Representations of Quadratic Functions F-IF 3, 4, 5, 6, 7, 7a, 9, F-BF 1, 1a, 1b, A-CED 1, 2, , F-LE 3, , N-NQ 2, , S-ID 6a, 6b, A-REI 7........................................... 239 Page 2 of 399 Teacher Notes: ....................................................................................................................................... 242 Quadratics Inquiry Project ................................................................................................................. 269 Quadratics Inquiry.............................................................................................................................. 275 Math - Problem Solving : Quadratics ................................................................................................ 278 Graphs ................................................................................................................................................ 279 Families of Graphs ............................................................................................................................. 283 Tables Graphs Equations ............................................................................................................. 293 Vertex Form and Transformations ..................................................................................................... 299 Graphing Quadratic Functions ........................................................................................................... 307 What Do You Need for the Graph? ................................................................................................... 324 Linear, Exponential and Quadratic Functions.................................................................................... 326 Zeros of Quadratic Functions ............................................................................................................. 331 Calculator Discovery.......................................................................................................................... 331 Quadratic Qualities ............................................................................................................................ 335 Match the Graphs –Equations ............................................................................................................ 347 Match the Graph-Graphs .................................................................................................................... 348 Graph It! ............................................................................................................................................. 350 Quadratic Functions and Rates of Change ......................................................................................... 358 Linear or Quadratic? .......................................................................................................................... 364 Don’t Change That Perimeter! ........................................................................................................... 368 Toothpicks and Models ...................................................................................................................... 372 Patterns with Triangles....................................................................................................................... 376 Patterns with Stacking Pennies .......................................................................................................... 378 Leap Frog Investigation ..................................................................................................................... 380 Area Application ................................................................................................................................ 384 Toothpick Patterns ............................................................................................................................. 389 Ball Bounce Activity.......................................................................................................................... 392 Water Fountain Activity..................................................................................................................... 398 Page 3 of 399 Math Practices Rationale CCSSM Practice 1: Make sense of problems and persevere in solving them. Why is this practice important? What does this practice look like when students are doing it? What can a teacher do to model this practice? What questions could a teacher ask to encourage the use of this practice? What does proficiency look like in this practice? What actions might the teacher make that inhibit the students’ use of this practice? Helps students to develop critical thinking skills. Teaches students to “think for themselves”. Helps students to see there are multiple approaches to solving a problem. Students immediately begin looking for methods to solve a problem based on previous knowledge instead of waiting for teacher to show them the process/algorithm. Students can explain what problem is asking as well as explain, using correct mathematical terms, the process used to solve the problem. Frame mathematical questions/challenges so they are clear and explicit. Check with students repeatedly to help them clarify their thinking and processes. “How would you go about solving this problem?” “What do you need to know in order to solve this problem?” What methods have we studied that you can use to find the information you need? Students can explain the relationships between equations, verbal descriptions, tables, and graphs. Students check their answer using a different method and continually ask themselves, “Does this make sense?” They understand others approaches to solving complex problems and can see the similarities between different approaches. Showing the students shortcuts/tricks to solve problems (without making sure the students understand why they work). Not giving students an adequate amount of think time to come up with solutions or processes to solve a problem. Giving students the answer to their questions instead of asking guiding questions to lead them to the discovery of their own question. Page 4 of 399 CCSSM Practice 2: Reason abstractly and quantitatively. Why is this practice important? What does this practice look like when students are doing it? What can a teacher do to model this practice? What questions could a teacher ask to encourage the use of this practice? What does proficiency look like in this practice? What actions might the teacher make that inhibit the students’ use of this practice? Students develop reasoning skills that help them to understand if their answers make sense and if they need to adjust the answer to a different format (i.e. rounding) Students develop different ways of seeing a problem and methods of solving it. Students are able to translate a problem situation into a number sentence or algebraic expression. Students can use symbols to represent problems. Students can visualize what a problem is asking. Ask students questions about the types of answers they should get. Use appropriate terminology when discussing types of numbers/answers. Provide story problems and real world problems for students to solve. Monitor the thinking of students. “What is your unknown in this problem? “What patterns do you see in this problem and how might that help you to solve it?” Students can recognize the connections between the elements in their mathematical sentence/expression and the original problem. Students can explain what their answer means, as well as how they arrived at it. Giving students the equation for a word or visual problem instead of letting them “figure it out” on their own. Page 5 of 399 CCSSM Practice 3: Construct viable arguments and critique the reasoning of others Why is this practice important? What does this practice look like when students are doing it? What can a teacher do to model this practice? What questions could a teacher ask to encourage the use of this practice? What does proficiency look like in this practice? What actions might the teacher make that inhibit the students’ use of this practice? Students better understand and remember concepts when they can defend and explain it to others. Students are better able to apply the concept to other situations when they understand how it works. Communicate and justify their solutions Listen to the reasoning of others and ask clarifying questions. Compare two arguments or solutions Question the reasoning of other students Explain flaws in arguments Provide an environment that encourages discussion and risk taking. Listen to students and question the clarity of arguments. Model effective questioning and appropriate ways to discuss and critique a mathematical statement. How could you prove this is always true? What parts of “Johnny’s “ solution confuses you? Can you think of an example to disprove your classmates theory? Students are able to make a mathematical statement and justify it. Students can listen, critique and compare the mathematical arguments of others. Students can analyze answers to problems by determining what answers make sense. Explain flaws in arguments of others. Not listening to students justify their solutions or giving adequate time to critique flaws in their thinking or reasoning. Page 6 of 399 CCSSM Practice 4: Model with mathematics Why is this practice important? What does this practice look like when students are doing it? What can a teacher do to model this practice? What questions could a teacher ask to encourage the use of this practice? What does proficiency look like in this practice? What actions might the teacher make that inhibit the students’ use of this practice? Helps students to see the connections between math symbols and real world problems. Write equations to go with a story problem. Apply math concepts to real world problems. Use problems that occur in everyday life and have students apply mathematics to create solutions. Connect the equation that matches the real world problem. Have students explain what different numbers and variables represent in the problem situation. Require students to make sense of the problems and determine if the solution is reasonable. How could you represent what the problem was asking? How does your equation relate to the problems? How does your strategy help you to solve the problem? Students can write an equation to represent a problem. Students can analyze their solutions and determine if their answer makes sense. Students can use assumptions and approximations to simplify complex situations. Not give students any problem with real world applications. Page 7 of 399 CCSSM Practice 5: Use appropriate tools strategically Why is this practice important? What does this practice look like when students are doing it? What can a teacher do to model this practice? What questions could a teacher ask to encourage the use of this practice? Helps students to understand the uses and limitations of different mathematical and technological tools as well as which ones can be applied to different problem situations. Students select from a variety of tools that are available without being told which to use. Students know which tools are helpful and which are not. Students understand the effects and limitations of chosen tools. Provide students with a variety of tools Facilitate discussion regarding the appropriateness of different tools. Allow students to decide which tools they will use. How is this tool helping you to understand and solve the problem? What tools have we used that might help you organize the information given in this problem? Is there a different tool that could be used to help you solve the problem? What does proficiency look like in this practice? Students are sufficiently familiar with tools appropriate for their grade or course and make sound decisions about when each of these tools might be helpful. Students recognize both the insight to be gained from the use of the selected tool and their limitations. What actions might the teacher make that inhibit the students’ use of this practice? Only allowing students to solve the problem using one method. Telling students that the solution is incorrect because it was not solved “the way I showed you”. Page 8 of 399 CCSSM Practice 6: Attend to precision. Why is this practice important? What does this practice look like when students are doing it? What can a teacher do to model this practice? What questions could a teacher ask to encourage the use of this practice? What does proficiency look like in this practice? What actions might the teacher make that inhibit the students’ use of this practice? Students are better able to understand new math concepts when they are familiar with the terminology that is being used. Students can understand how to solve real world problems. Students can express themselves to the teacher and to each other using the correct math vocabulary. Students use correct labels with word problems. Make sure to use correct vocabulary terms when speaking with students. Ask students to provide a label when describing word problems. Encourage discussions and explanations and use probing questions. How could you describe this problem in your own words? What are some non-examples of this word? What mathematical term could be used to describe this process. Students are precise in their descriptions. They use mathematical definitions in their reasoning and in discussions. They state the meaning of symbols consistently and appropriately. Teaching students “trick names” for symbols (i.e. the alligator eats the big number) Not using proper terminology in the classroom. Allowing students to use the word “it” to describe symbols or other concepts. Page 9 of 399 CCSSM Practice 7: Look for and make use of structure. Why is this practice important? What does this practice look like when students are doing it? What can a teacher do to model this practice? What questions could a teacher ask to encourage the use of this practice? What does proficiency look like in this practice? What actions might the teacher make that inhibit the students’ use of this practice? When students can see patterns or connections, they are more easily able to solve problems Students look for connections between properties. Students look for patterns in numbers, operations, attributes of figures, etc. Students apply a variety of strategies to solve the same problem. Ask students to explain or show how they solved a problem. Ask students to describe how one repeated operation relates to another (addition vs. multiplication). How could you solve the problem using a different operation? What pattern do you notice? Students look closely to discern a pattern or structure. Provide students with pattern before allowing them to discern it for themselves. Page 10 of 399 CCSSM Practice 8: Look for and express regularity in repeated reasoning Why is this practice important? What does this practice look like when students are doing it? What can a teacher do to model this practice? What questions could a teacher ask to encourage the use of this practice? What does proficiency look like in this practice? What actions might the teacher make that inhibit the students’ use of this practice? When students discover connections or algorithms on their own, they better understand why they work and are more likely to remember and be able to apply them. Students discover connections between procedures and concepts Students discover rules on their own through repeated exposures of a concept. Provide real world problems for students to discover rules and procedures through repeated exposure. Design lessons for students to make connections. Allow time for students to discover the concepts behind rules and procedures. Pose a variety of similar type problems. How would you describe your method? Why does it work? Does this method work all the time? What do you notice when…? Students notice repeated calculations. Students look for general methods and shortcuts. Providing students with formulas or algorithms instead of allowing them to discover it on their own. Not allowing students enough time to discover patterns. Page 11 of 399 RUBRIC – IMPLEMENTING STANDARDS FOR MATHEMATICAL PRACTICE Using the Rubric: Task: Is strictly procedural. Does not require students to check solutions for errors. NEEDS IMPROVEMENT Teacher: Allots too much or too little time to complete task. Encourages students to individually complete tasks, but does not ask them to evaluate the processes used. Explains the reasons behind procedural steps. Does not check errors publicly. Is overly scaffolded or procedurally “obvious”. Requires students to check answers by plugging in numbers. (teacher does thinking) Task: EMERGING Teacher: Allows ample time for all students to struggle with task. Expects students to evaluate processes implicitly. Models making sense of the task (given situation) and the proposed solution. EXEMPLARY Summer 2011 Differentiates to keep advanced students challenged during work time. Integrates time for explicit meta-cognition. Expects students to make sense of the task and the proposed solution. (teacher mostly models) (students take ownership) Task: Task: Is cognitively Allows for multiple entry points and solution paths. demanding. Requires students to Has more than one entry defend and justify their point. solution by comparing Requires a balance of multiple solution paths. procedural fluency and conceptual Teacher: understanding. Requires students to check solutions for errors usingone other solution path. PROFICIENT Review each row corresponding to a mathematical practice. Use the boxes to mark the appropriate description for your task or teacher action. The task descriptors can be used primarily as you develop your lesson to make sure your classroom tasks help cultivate the mathematical practices. The teacher descriptors, however, can be used during or after the lesson to evaluate how the task was carried out. The column titled “proficient” describes the expected norm for task and teacher action while the column titled “exemplary” includes all features of the proficient column and more. A teacher who is exemplary is meeting criteria in both the proficient and exemplary columns. PRACTICE Make sense of problems and persevere in solving them. Teacher: Does not allow for wait time; asks leading questions to rush through task. Does not encourage students to individually process the tasks. Is focused solely on answers rather than processes and reasoning. Institute for Advanced Study/Park City Mathematics Institute Secondary School Teachers Program/Visualizing Functions Page 12 of 399 PRACTICE Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Is either ambiguously stated. Does not expect students to interpret representations. Expects students to memorize procedures withno connection to meaning. Lacks context. Does not make use of multiple representations or solution paths. NEEDS IMPROVEMENT Task: Teacher: Task: Teacher: Does not ask students to present arguments or solutions. Expects students to follow a given solution path without opportunities to make conjectures. Task: EMERGING Does not help students differentiate between assumptions and logical conjectures. Asks students to present arguments but not to evaluate them. Allows students to make conjectures without justification. Is not at the appropriate level. representation. Explains connections between procedures and meaning. tasks using a single model and interpret Expects students to Is embedded in a contrived context. (teacher does thinking) Teacher: Task: Teacher: Institute for Advanced Study/Park City Mathematics Institute Secondary School Teachers Program/Visualizing Functions Page 13 of 399 PROFICIENT expressed with multiple representations. Expects students to interpret and model using multiple representations. Provides structure for students to connect algebraic procedures to contextual meaning. Links mathematical solution with a question’s answer. Avoids single steps or routine algorithms. Teacher: EXEMPLARY Helps students differentiate between assumptions and logical conjectures. Prompts students to evaluate peer arguments. Expects students to formally justify the validity of their conjectures. Expects students to interpret, model, and connect multiple representations. Prompts students to articulate connections between algebraic procedures and contextual meaning. (teacher mostly models) (students take ownership) Task: Task: Has realistic context. Has relevant realistic context. Requires students to frame solutions in a Teacher: context. Has solutions that can be Teacher: Task: Teacher: Identifies students’ assumptions. Models evaluation of student arguments. Asks students to explain their conjectures. Summer 2011 PRACTICE Model with mathematics. Use appropriate tools strategically. NEEDS IMPROVEMENT Requires students to Task: identify variables and to perform necessary computations. Teacher: Identifies appropriate variables and procedures for students. Does not discuss appropriateness of model. Does not incorporate Task: additional learning tools. Teacher: additional learning tools. Does not incorporate EMERGING (teacher does thinking) Requires students to Task: identify variables and to compute and interpret results. Teacher: Verifies that students have identified appropriate variables and procedures. Explains the appropriateness of model. Lends itself to one learning Task: PROFICIENT Requires students to (teacher mostly models) Task: identify variables, compute and interpret results, and report findings using a mixture of representations. Illustrates the relevance of the mathematics involved. Requires students to identify extraneous or missing information. Teacher: Asks questions to help students identify appropriate variables and procedures. Facilitates discussions in evaluating the appropriateness of model. Lends itself to multiple Task: learning tools. Gives students opportunity tool. Does not involve mental estimation. Models error checking by learning tools for student use. Chooses appropriate to develop fluency in mental computations. Teacher: Demonstrates use of computations or estimation. Teacher: appropriate learning tool. Page 14 of 399 EXEMPLARY Requires students to (students take ownership) Task: Expects students to justify identify variables, compute and interpret results, report findings, and justify the reasonableness of their results and procedures within context of the task. Teacher: their choice of variables and procedures. Gives students opportunity to evaluate the appropriateness of model. Requires multiple learning Task: tools (i.e., graph paper, calculator, manipulative). Requires students to demonstrate fluency in mental computations. Teacher: appropriate learning tools. Allows students to choose appropriate alternatives Creatively finds where tools are not available. PRACTICE Attend to precision. Look for and make use of structure. Requires students to automatically apply an algorithm to a task without evaluating its appropriateness. Does not intervene when students are being imprecise. Does not point out instances when students fail to address the question completely or directly. Gives imprecise instructions. NEEDS IMPROVEMENT Task: Teacher: Task: Teacher: Does not recognize students for developing efficient approaches to the task. Requires students to apply the same algorithm to a task although there may be other approaches. Task: EMERGING Identifies individual students’ efficient approaches, but does not expand understanding to the rest of the class. Demonstrates the same algorithm to all related tasks although there may be other more effective approaches. Requires students to analyze a task before automatically applying an algorithm. Inconsistently intervenes when students are imprecise. Identifies incomplete responses but does not require student to formulate further response. Has overly detailed or wordy instructions. (teacher does thinking) Teacher: Task: Teacher: Institute for Advanced Study/Park City Mathematics Institute Secondary School Teachers Program/Visualizing Functions Page 15 of 399 PROFICIENT Requires students to analyze a task and identify more than one approach to the problem. Consistently demands precision in communication and in mathematical solutions. Identifies incomplete responses and asks student to revise their response. Teacher: Task: Teacher: EXEMPLARY Prompts students to identify mathematical structure of the task in order to identify the most effective solution path. Encourages students to justify their choice of algorithm or solution path. Requires students to identify the most efficient solution to the task. Demands and models precision in communication and in mathematical solutions. Encourages students to identify when others are not addressing the question completely. Includes assessment criteria for communication of ideas. (teacher mostly models) (students take ownership) Task: Task: Has precise instructions. Teacher: Task: Teacher: Facilitates all students in developing reasonable and efficient ways to accurately perform basic operations. Continuously questions students about the reasonableness of their intermediate results. Summer 2011 PRACTICE Look for and express regularity in repeated reasoning. Is disconnected from prior and future concepts. Has no logical progression that leads to pattern recognition. NEEDS IMPROVEMENT Task: Teacher: Does not show evidence of understanding the hierarchy within concepts. Presents or examines task in isolation. Task: EMERGING Hides or does not draw connections to prior or future concepts. Is overly repetitive or has gaps that do not allow for development of a pattern. (teacher does thinking) Teacher: Institute for Advanced Study/Park City Mathematics Institute Secondary School Teachers Program/Visualizing Functions Page 16 of 399 PROFICIENT Reviews prior knowledge and requires cumulative understanding. Lends itself to developing a pattern or structure. (teacher mostly models) Task: Teacher: Connects concept to prior and future concepts to help students develop an understanding of procedural shortcuts. Demonstrates connections between tasks. EXEMPLARY Addresses and connects to prior knowledge in a nonroutine way. Requires recognition of pattern or structure to be completed. (students take ownership) Task: Teacher: Encourages students to connect task to prior concepts and tasks. Prompts students to generate exploratory questions based on the current task. Encourages students to monitor each other’s intermediate results. Summer 2011 Mathematical Practices: A Walk-Through Protocol Mathematical Practices Observations *Note: This document should also be used by the teacher for planning and self-evaluation. MP.1. Make sense of problems and persevere in solving them Teachers are expected to______________: Provide appropriate representations of problems. Students are expected to______________: Connect quantity to numbers and symbols (decontextualize the problem) and create a logical representation of the problem at hand. Recognize that a number represents a specific quantity (contextualize the problem). Contextualize and decontextualize within the process of solving a problem. Teachers are expected to______________: Provide time for students to discuss problem solving. Students are expected to______________: Engage in solving problems. Explain the meaning of a problem and restate in it their own words. Analyze given information to develop possible strategies for solving the problem. Identify and execute appropriate strategies to solve the problem. Check their answers using a different method, and continually ask “Does this make sense?” MP.2. Reason abstractly and quantitatively. MP.3. Construct viable arguments and critique the reasoning of others. Students are expected to____________________________: Explain their thinking to others and respond to others’ thinking. Participate in mathematical discussions involving questions like “How did you get that?” and “Why is that true?” Construct arguments that utilize prior learning. Question and problem pose. Practice questioning strategies used to generate information. Analyze alternative approaches suggested by others and select better approaches. Justify conclusions, communicate them to others, and respond to the arguments of others. Compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument, explain what it is. CCSSM National Professional Development Page 17 of 399 Mathematical Practices MP.4. Model with mathematics. MP 5. Use appropriate tools strategically Observations Teachers are expected to______________: Provide opportunities for students to listen to or read the conclusions and arguments of others. Students are expected to______________: Apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. Make assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. Experiment with representing problem situations in multiple ways, including numbers, words (mathematical language), drawing pictures, using objects, acting out, making a chart or list, creating equations, etc. Identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts, and formulas. Evaluate their results in the context of the situation and reflect on whether their results make sense. Analyze mathematical relationships to draw conclusions. Teachers are expected to______________: Provide contexts for students to apply the mathematics learned. Students are expected to______________: Use tools when solving a mathematical problem and to deepen their understanding of concepts (e.g., pencil and paper, physical models, geometric construction and measurement devices, graph paper, calculators, computer-based algebra or geometry systems.) Consider available tools when solving a mathematical problem and decide when certain tools might be helpful, recognizing both the insight to be gained and their limitations. Detect possible errors by strategically using estimation and other mathematical knowledge. Teachers are expected to______________: CCSSM National Professional Development Page 18 of 399 Mathematical Practices MP.6. Attend to precision. MP.7. Look for and make use of structure. MP.8. Look for and express regularity in repeated reasoning. Observations Students are expected to______________: Use clear and precise language in their discussions with others and in their own reasoning. Use clear definitions and state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. Specify units of measure and label parts of graphs and charts. Calculate with accuracy and efficiency based on a problem’s expectation. Teachers are expected to______________: Emphasize the importance of precise communication. Students are expected to______________: Describe a pattern or structure. Look for, develop, generalize, and describe a pattern orally, symbolically, graphically and in written form. Relate numerical patterns to a rule or graphical representation Check the reasonableness of their results. Teachers are expected to______________: dents to look for and discuss regularity in reasoning. Use models to examine patterns and generate their own algorithms. Use models to explain calculations and describe how algorithms work. Use repeated applications to generalize properties. Look for mathematically sound shortcuts. Apply and discuss properties. Teachers are expected to______________: Provide time for applying and discussing properties. Students are expected to______________: Describe repetitive actions in computation CCSSM National Professional Development Page 19 of 399 High School Common Core Math II Curriculum Timeline Topic Intro Unit Similarity Trigonometric Ratios Other Types of Functions Comparing Functions and Different Representations of Quadratic Functions Modeling Unit and Project Quadratic Functions: Solving by Factoring Quadratic Functions: Completing the Square and the Quadratic Formula Probability Geometric Measurement Geometric Modeling Unit and Project Standards Covered G – SRT 1 G – SRT 1a G – SRT 1b G – SRT 6 G – SRT 2 G – SRT 3 G – SRT 4 G – SRT 7 G – SRT 5 Grading Period 1 1 No. of Days 5 20 G – SRT 8 1 20 A – CED 1 A – CED 4 A – REI 1 N – RN 1 N – RN 2 N – RN 3 F – IF 4 F – IF 5 F – IF 6 F – IF 7 F – IF 7a F– IF 9 F – IF 4 F – IF 7b F – IF 7e F – IF 8 F – IF 8b F– BF1 A– CED 1 A– CED 2 F– BF 1 F– BF 1a F – BF 1b F– BF 3 F – BF 1a F – BF 1b F – BF 3 A – SSE 1b N–Q2 2 15 F – LE 3 N– Q 2 S – ID 6a S – ID 6b A – REI 7 2 20 2 10 A – APR 1 A – REI 1 A – REI 4b F – IF 8a A – CED 1 A – SSE 1b A – SSE 3a 3 20 A – REI 1 A – REI 4 A – REI 4a A – REI 4b A – SSE 3b F – IF 8 F – IF 8a A – CED 1 N – CN 1 N – CN 2 N – CN 7 3 20 S – CP 1 S – CP 2 S – CP 3 G – GMD 1 S – CP 4 S – CP 5 S – CP 6 G – GMD 3 S – CP 7 4 20 4 10 G – MG 1 G – MG 2 G – MG 3 4 15 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 20 of 399 Columbus City Schools 6/28/13 High School Common Core Math II 1st Nine Weeks Scope and Sequence Intro Unit – IO (5 days) Topic 1 – Similarity (20 days) Geometry (G – SRT): 1) Similarity, Right Triangles, and Trigonometry: Understand similarity in terms of similarity transformations. G – SRT 1: Verify experimentally the properties of dilations given by a center and a scale factor. G – SRT 1a: A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. G – SRT 1b: The dilation of a line segment is longer or shorter in the ratio given by the scale factor. G – SRT 2: Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. G – SRT 3: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Geometry (G – SRT): 2) Similarity, Right Triangles, and Trigonometry: Prove theorems involving similarity. G – SRT 4: Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. G – SRT 5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Topic 2 – Trigonometric Ratios (20 days) Geometry (G – SRT): 3) Similarity, Right Triangles, and Trigonometry: Define trigonometric ratios and solve problems involving .right triangles G – SRT 6: Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. G – SRT 7: Explain and use the relationship between the sine and cosine of complementary angles. G – SRT 8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Page 21 of 399 High School Common Core Math II 2nd Nine Weeks Scope and Sequence Topic 3 – Other Types of Functions (15 days) Creating Equations (A – CED): 4) Create equations that describe numbers or relationships A – CED 1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. A – CED 4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R. Reasoning with Equations and Inequalities (A – REI): 5) Understand solving equations as a process of reasoning and explain the reasoning. A – REI 1: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. The Real Number System (N – RN): 6) Extend the properties of exponents to rational exponents. N – RN 1: Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5. N – RN 2: Rewrite expressions involving radicals and rational exponents using the properties of exponents. The Real Number System (N – RN): 7) Use properties of rational and irrational numbers. N – RN 3: Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Interpreting Functions (F – IF): 8) Interpret functions that arise in applications in terms of the context. F – IF 4*: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.* Interpreting Functions (F – IF): 9) Analyze functions using different representations. F – IF 7b: Graph square root, cube root, and absolute value functions. F – IF 7e: Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Page 22 of 399 F – IF 8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. F – IF 8b: Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay. Building Functions (F – BF): 10) Build a function that models a relationship between two quantities. F – BF 1: Write a function that describes a relationship between two quantities. F – BF 1a: Determine an explicit expression, a recursive process, or steps for calculation from a context. F – BF 1b: Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Building Functions (F – BF): 11) Build new functions from existing functions. F – BF 3: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Seeing Structure in Expressions (A – SSE): 12) Interpret the structure of expressions. A – SSE 1b: Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)n as the product of P and a factor not depending on P. Quantities (NQ): 13) Reason quantitatively and use units to solve problems. N – Q 2: Define appropriate quantities for the purpose of descriptive modeling. Topic 4 – Comparing Functions and Different Representations of Quadratic Functions (20 days) Interpreting Functions (F – IF): 14) Interpret functions that arise in applications in terms of the context. F – IF 4*: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.* F – IF 5*: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.* Page 23 of 399 F – IF 6: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Interpreting Functions (F – IF): 15) Analyze functions using different representations. F – IF 7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. F – IF 7a*: Graph linear and quadratic functions and show intercepts, maxima, and minima.* F – IF 9: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Creating Equations (A – CED): 16) Create equations that describe numbers or relationships. A – CED 1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. A – CED 2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Building Functions (F – BF): 17) Build a function that models a relationship between two quantities. F – BF 1: Write a function that describes a relationship between two quantities. F – BF 1a: Determine an explicit expression, a recursive process, or steps for calculation from a context. F – BF 1b: Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Building Functions (F – BF): 18) Build new functions from existing functions. F – BF 3: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Linear and Exponential Models (F – LE): 19) Construct and compare linear and exponential models and solve problems. F- LE 3: Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. Quantities (N-Q): 20) Reason quantitatively and use units to solve problems. N – Q 2: Define appropriate quantities for the purpose of descriptive modeling. Page 24 of 399 Interpreting Categorical and Quantitative Data (S – ID): 21) Summarize, represent, and interpret data on two categorical and quantitative variables. S – ID 6a: Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear and exponential models. S – ID 6b: Informally assess the fit of a function by plotting and analyzing residuals. Reasoning with Equations and Inequalities (A – REI): 22) Solve systems of equations. A – REI 7: Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = -3x and the circle x2 + y2 = 3. Modeling Unit and Project –(10 days) Page 25 of 399 High School Common Core Math II 3rd Nine Weeks Scope and Sequence Topic 5–Quadratic Functions – Solving by factoring (20 days) Arithmetic with Polynomials and Rational Expressions (A – APR): 23) Perform arithmetic operations on polynomials. A – APR 1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Reasoning with Equations and Inequalities (A – REI): 24) Understand solving equations as a process of reasoning and explain the reasoning. A – REI 1: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Reasoning with Equations and Inequalities (A – REI): 25) Solve equations and inequalities in one variable. A – REI 4b: Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Interpreting Functions (F – IF): 26) Analyze functions using different representations. F – IF 8a: Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Creating Equations (A – CED): 27) Create equations that describe numbers of relationships. A – CED 1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Seeing Structure in Expressions (A – SSE): 28) Interpret the structure of expressions. A – SSE 1b: Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)n as the product of P and factor not depending on P. Seeing Structure in Expressions (A – SSE): 29) Write expressions in equivalent forms to solve problems. A – SSE 3a: Factor a quadratic expression to reveal the zeros of the function it defines. Topic 6– Quadratic Functions – Completing the Square/Quadratic Formula (20 days) Reasoning with Equations and Inequalities (A – REI): 30) Understand solving equations as a process of reasoning and explain the reasoning. Page 26 of 399 A – REI 1: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Reasoning with Equations and Inequalities (A – REI): 31) Solve equations and inequalities in one variable. A – REI 4: Solve quadratic equations in one variable. A – REI 4a: Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. A – REI 4b: Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Seeing Structure in Expressions (A – SSE): 32) Write expressions in equivalent forms to solve problems. A – SSE 3b: Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. Interpreting Functions (F – IF): 33) Analyze functions using different representations. F – IF 8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. F – IF 8a: Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Creating Equations (A – CED): 34) Create equations that describe numbers or relationships. A – CED 1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. The Complex Number System (N – CN): 35) Perform arithmetic operations with complex numbers. N – CN 1: Know there is a complex number i such that i 2 has the form a+bi with a and b real. 1 , and every complex number N – CN 2: Use the relation i 2 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. The Complex Number System (N – CN): 36) Use complex numbers in polynomial identities and equations. N – CN 7: Solve quadratic equations with real coefficients that have complex solutions. Page 27 of 399 High School Common Core Math II 4th Nine Weeks Scope and Sequence Topic 7 –Probability (20 days) Conditional Probability and the Rules of Probability (S – CP): 37) Understand independence and conditional probability and use them to interpret data. S – CP 1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”). S – CP 2: Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. S – CP 3: Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. S – CP 4: Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from you school will favor science given that the student is in the tenth grade. Do the same for other subjects and compare the results. S – CP 5: Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. Conditional Probability and the Rules of Probability (S – CP): 38) Use the rules of probability to compute probabilities of compound events in a uniform probability model. S – CP 6: Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. S – CP 7: Apply the Addition Rule, P(A or B) = P(B) – P(A and B), and interpret the answer in terms of the model. Topic 8 – Geometric Measurement (10 days) Geometric Measurement and Dimension (G – GMD): 39) Explain volume formulas and use them to solve problems. G – GMD 1: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. G – GMD 3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Page 28 of 399 Geometric and Modeling Project-(15 days) *Modeling with Geometry (G – MG): 40) Apply geometric concepts in modeling situations. G – MG 1*: Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).* G – MG 2*: Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).* G – MG 3*: Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).* Page 29 of 399 COLUMBUS PUBLIC SCHOOLS HIGH SCHOOL CCSSM MATHEMATICS II CURRICULUM GUIDE TOPIC 3 CONCEPTUAL CATEGORY Other Types of Functions AFunctions, Algebra, Number CED 1, 4, A-REI 1, N-RN 1, 2, and Quantity 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Domain: Creating Equations (A – CED) Cluster 4) Create equations that describe numbers of relationships. TIME RANGE 20 days GRADING PERIOD 2 Domain: Reasoning with Equations and Inequalities (A – REI) Cluster 5) Understand solving equations as a process of reasoning and explain the reasoning. Domain: The Real Number System (N – RN) Cluster 6) Extend the properties of exponents to rational exponents. 7) Use properties of rational and irrational numbers. Domain: Interpreting Functions (F – IF) Cluster 8) Interpret functions that arise in applications in terms of the context. 9) Analyze functions using different representations. Domain: Building Functions (F – BF) Cluster 10) Build a function that models a relationship between two quantities. 11) Build new functions from existing functions. Domain: Seeing Structure in Expressions (A – SSE) Cluster 12) Interpret the structure of expressions. Domain: Quantities (NQ) Cluster 13) Reason quantitatively and use units to solve problems. Standards 4) Create equations that describe numbers of relationships A – CED 1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 30 of 399 Columbus City Schools 6/28/13 rational and exponential functions. A – CED 4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R. 5) Understand solving equations as a process of reasoning and explain the reasoning. A – REI 1: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. 6) Extend the properties of exponents to rational exponents. N – RN 1: Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5. N – RN 2: Rewrite expressions involving radicals and rational exponents using the properties of exponents. 7) Use properties of rational and irrational numbers. N – RN 3: Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. 8) Interpret functions that arise in applications in terms of the context. F – IF 4*: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.* 9) Analyze functions using different representations. F – IF 7b: Graph square root, cube root, and absolute value functions. F – IF 7e: Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. F – IF 8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. F – IF 8b: Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay. 10) Build a function that models a relationship between two quantities. F – BF 1: Write a function that describes a relationship between two quantities. F – BF 1a: Determine an explicit expression, a recursive process, or steps for calculation CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 31 of 399 Columbus City Schools 6/28/13 from a context. F – BF 1b: Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. 11) Build new functions from existing functions. F – BF 3: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. 12) Interpret the structure of expressions. A – SSE 1b: Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)n as the product of P and factor not depending on P. 13) Reason quantitatively and use units to solve problems. N – Q 2: Define appropriate quantities for the purpose of descriptive modeling. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 32 of 399 Columbus City Schools 6/28/13 TEACHING TOOLS Vocabulary: absolute value function, base, binomial, compound interest, conjugate, cube root function, decay factor, decay rate, decreasing function, dimensional analysis, distributive property, domain, exponent, exponential decay, exponential function, exponential growth, extraneous root, first difference, function, global behavior, growth factor, growth rate, horizontal compression, horizontal stretch, increasing function, index, intercepts, interval, like term, linear function, local behavior, monomial, nth root, piecewise function, polynomial degree, power, principal root, radical function, radical exponent, radicand, range, rate of change, rational exponent, rational root, rationalizing denominator, restricted domain, root, step function, square root function, transformation, translation, vertical compression, vertical stretch, zeros. Teacher Notes: Circles The standard form for the equation of a circle, where the center is the origin and length of the radius is r is x2 + y2 = r2. If the center is not the origin, then the equation is (x – h)2 + (y – k)2 = r2, where (h, k) is the center of the circle and r is the length of the radius of the circle. If the equation given is (x + 7)2 + (y – 4)2 = 92, then the center of the circle is (- 7, 4) and the length of the radius is 9 units. Properties of Radicals An expression that contains a radical sign is called a radical expression, n a . The expression under the radical sign is the radicand and the numeric value, n is the index. We read this as “the nth root of a. Looking at the radical expression 5 3x , 3x is the radicand and 5 is the index. 1) c is a square root of a, if c2 = a, e.g., 2 is a square root of 4 because 22 = 4 and -2 is a square root of 4 because (-2)2 = 4. Because there are two values that satisfy the equation x2 = 4, we take the term square root to mean the principal square root which has a non-negative value. In this case 4 2 is the principal square root. Mathematically, we express this as: a2 a 2) c is a cube root of a if c3 = a, e.g., 3 is a cube root of 27 because 33 = 27 and -3 is a cube root of -27 because (-3)3 = -27. The cube root of a negative number is negative. 3) The cube root of a positive number is positive. c is an nth root of a if cn = a. Note that if the index is odd and the radicand is negative then the principal square root is negative. For CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 33 of 399 Columbus City Schools 6/28/13 example, 5 32 2 because (-2)5 = -32. The following are general rules for taking the roots of positive and negative numbers. The answer is the principal root. The answer is the opposite of the principal root. The answer is both roots, the positive and the negative root. odd number negative number even number negative number odd number positive number even number positive number The answer is a negative number. There is no real solution. The answer is the principal root. The answer is the principal root. For any value of x and any even number n, If x = -5, then 8 5 8 5 For example, if x = 4, then 4) product rule: n a n x . For example, if x = 4, then xn 6 46 4 5 . For any value of x and any odd number n greater than 1, 5 45 n b 4 . If x = -5, then n ( x 6) ab , e.g. ( x 6) 3 7 3 5 9 5 3 9 7 5 ( x 6)( x 6) 4. n xn . 5. 3 35 , and x 2 36 The product rule can be used for factoring to simplify radical expressions as shown below. 50 5) quotient rule: given n 25 2 a and n b,b 25 0, 2 5 2 n a b n a n b , e.g. x2 16 x2 16 x 4 6) principle of powers: if a = b then an = bn CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 34 of 399 Columbus City Schools 6/28/13 Convert from Radical to Rational Exponents Solving and simplifying radical expressions or equations sometimes requires conversion from one form to another. The properties for converting from radical to exponent form are shown below. 1 an a x n n n 1 a1 a x or read: “the nth root of a to the first power” e.g.,9 2 n a x 2 91 read: “the nth root of a to the x power” e.g.,8 2 3 3. 3 82 4 When multiplying or dividing radical expressions with different indices, convert to exponent form, multiply or divide and convert back to radical form. 5 4 For example: x5 x 1 5 x4 x2 2 7 x4 x4 x4 4 x7 Rationalizing the Denominator When calculations warrant a radical in the denominator, this can be accomplished by multiplying by a factor of one in a form that will make the denominator a perfect square as shown in the example below. 1 a 1 a a a a a 2 a a Adding and Subtracting Radicals Adding and subtracting radicals follows a process very similar to adding and subtracting like terms. In this case, the radicands and the indices must both be the same. In the example, 3 5 17 6 5 17 , notice that the radicand, 17, is the same in both terms as is the index, 5. These terms can be added, giving 3 5 17 6 5 17 9 5 17 . Initially it appears that the expression 5 2 2 8 cannot be simplified because the radicands are different from each other. By simplifying the second term we see that 2 8 can be simplified to 4 2 , thus they are like radicals and can be subtracted. The simplification is as follows: 5 2 2 8 5 2 2 4 2 5 2 2 4 2 5 2 4 2 2 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 35 of 399 Columbus City Schools 6/28/13 Multiplying and Dividing Radicals One can multiply radicals using the distributive property (FOIL). When dividing radicals, rationalize the denominator by multiplying both the numerator and denominator by the conjugate. Anything of the form a b c d and a b c d are conjugates. For example, to 2 simplify 5 3 3: 2 3 5 3 10 2 3 5 3 ( 3) 2 10 7 3 3 25 3 5 3 5 3 25 ( 3) 2 Graphs of the Square Root and Cube Root Function The graph of the square root function, f ( x) shown below. f ( x) x 0 1 4 9 16 f(x) 0 1 2 3 4 x -8 -1 0 1 8 3 x , has a domain of all non-negative numbers, as x (x, f(x)) (0, 0) (1, 1) (4, 2) (9, 3) (16, 4) The graph of the cube root function, f ( x) below. f ( x) 7 7 3 22 3 x , has a domain of all real numbers, as shown x f(x) -2 -1 0 1 2 (x, f(x)) (-8, -2) (-1, -1) (0, 0) (1, 1) (8, 2) CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 36 of 399 Columbus City Schools 6/28/13 Polynomials A monomial is a single algebraic expression such as: x, x2y, 16, 3x4y. A polynomial is a monomial by itself or a sum of monomials. Each monomial expression is simplified when there are no duplicate bases and no parentheses or operators. The degree of a monomial can be found by taking the sum of the exponents of all the variables. (Example: 4x3y2 has a degree of five.) The degree of a polynomial is the greatest degree of any term in the polynomial. (Example: 4x2y4 has a degree of 6 and 2x3y5 has a degree of eight, therefore the polynomial 4x2 y 4 2x3 y5 has a degree of eight.) While polynomials with a degree of one represent linear relationships (x + y = 7), polynomials with a degree greater than one represent non-linear relationships (x2 + y2 = 16 or y = x2 + x + 4). Polynomial expressions can be simplified by using the distributive property to combine like terms such as 2(x + 3) + 4(x – 1) = 2x + 6 + 4x – 4 = 6x + 2. The associative property is used to add and subtract polynomials. Subtracting one polynomial from another is the same as adding the opposite of the expression that is to be subtracted. We also use the distributive property to multiply each term of one polynomial by each term of another polynomial when multiplying polynomials. When multiplying polynomials, we must use the laws of exponents (This is covered in depth in the textbook). The study of properties: closure, identity, inverse, commutative, and associative, will build upon students’ prior knowledge of these and will require them to use these properties to simplify algebraic as well as numerical expressions. The distributive property will also be introduced. Likewise, students will use either single properties or a combination of them to evaluate algebraic expressions, add and subtract monomials and polynomials, and use the concept of combining like terms to simplify these expressions. Distributive Property: 3x x 2 5 x 2 3x3 15 x 2 6 x Multiply each term inside the parentheses by the factor 3x. FOIL method for multiplication of two binomials. (Students should understand they are simply using the distributive property twice in this situation, so that they can use the same process when one or both of the polynomials contain more than two terms.) Outside Inside x 2 x 5 First Last CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 37 of 399 Columbus City Schools 6/28/13 F x2 O I L 5x - 2x - 10 2 x + 3x – 10 Exponential Functions There are two types of exponential functions, exponential growth and decay. In the form f x ab x , if b > 1 this is an exponential growth function and if 0 < b < 1 this is an exponential decay function. If b =1 this will be a constant function. The a value represents the initial value or y-intercept. The use of the compound interest formula should be emphasized in this topic. nt r The compound interest formula is A P 1 where A is the balance if P dollars is invested n into an account paying an annual interest of r percent compounding n times per year for t years. When teaching transformations, students should understand that in the form f x ab x k,a describes whether the graph is steeper than y = bx and the graph is reflected about the x-axis if a is negative. The constant h describes the horizontal translation and the constant k describes the vertical translation. In addition, students should be able to graph f x ab x k k based on some reference points on f(x) = bx. For example: Given (0, 1), (1, 2) and (2, 4) on y = 2x, graph f x reference points. Answer: Since f x 2x 1 2x 1 4 using the three given points as 4 is a shift left 1 and down 4 from y = 2x, (0, 1) is now at (-1, -3); (1, 2) is now at (0, -2); (2, 4) is now at (1, 0). Radical and Rational Functions To simplify radicals by adding and subtracting, you must first have like radicals, that is, you need like terms. In order to get like radicals, the radicands may need to be simplified by using a factor tree to break down the square-root term. Example: 24 6 4 3 2 2 2 2 6 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 38 of 399 Columbus City Schools 6/28/13 To multiply radicals together, you multiply the radical coefficients together, and you multiply the radicands together. (The same way you multiply variables together.) You may then need to simplify the radicand. Example: 3 6 4 10 12 60 12 2 15 24 15 When dividing square roots, you must rationalize the denominator; that is you cannot have an answer with a square-root in the denominator. 2 5 8 3 Example: 2 5 3 2 15 2 15 15 24 12 8 3 3 8 9 ***Students may need to do this when solving simple quadratic equations, in the form ax2+c, algebraically. multiply the top and the bottom by the same number To solve simple quadratic equations in the form ax2 + c = 0, isolate the variable, and then take the square-root of both sides. (Don’t forget you will have two solutions.) Example: Example: 2 2 x2 4 5 (x + 5) = 100 ( x 5) 2 100 x + 5 = ±10 x + 5 = 10 or x=5 or 2 x2 1 1 x2 2 x + 5 = -10 x = -15 x2 1 2 x 2 2 1 2 or 1 2 2 or 2 Exponential Graphs The graphs of exponential functions of the form f ( x) upon the value of b and x. b x have typical behaviors depending CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 39 of 399 Columbus City Schools 6/28/13 For f ( x) bx Where 0 b 1 and x 1 An example is f ( x) 2 0 , the graph exhibits exponential decay. x : In the graph above, as the x values increase, the y values decrease and get closer to 0, approaching the x-axis, with the x-axis acting as a horizontal asymptote. For f ( x) bx Where b > 1 and x > 0, the graph exhibits exponential growth. An example is f ( x) 2x : In the graph above as the x values increase, the y values increase toward infinity, but as the x values decrease the corresponding y values get closer to 0, approaching the x-axis with the x-axis acting as a horizontal asymptote. Definition: A line is an “asymptote” for a curve if the distance between the line and the curve approaches zero as we move farther and farther out along the line. An interesting curiosity of a study of the two graphs above is that they are reflections of each other over the y-axis, and the function f x and 2 1 2 1 x x is equivalent to f x 1 2 2 x Since in f x 2 x, 2 x can be written as 2 1 x , x . A table that summarizes the characteristics of these two functions follows. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 40 of 399 Columbus City Schools 6/28/13 Function f ( x) 2x f ( x) 1 2 Domain Range Y-intercept X-intercept All real numbers All real numbers (0,1) None y>0 x All real numbers All real numbers (0,1) y>0 None x For the more general form of an exponential function, (i.e., f ( x) ab ), the factor a represents a vertical stretch or compression of the basic exponential graph. If a < 0, the graph will be reflected over the x-axis. Teacher Notes for A-CED 4 Written notes on solving literal equations. http://www.purplemath.com/modules/solvelit.htm Patterns and Functions A function is a relationship in which every value of x has a unique corresponding value of y. When a function is graphed it will pass the vertical line test. In other words, when a vertical line is drawn anywhere on the graph, it only intersects the graph at one point. A function can be represented with an equation (e.g., y = 3x – 2) or in function notation (e.g., f(x) = 3x – 2). The notation f(x) can be translated as “f of x” or “the function of x”. Before determining the type of function, it is important to identify which is the independent variable (x) and which is the dependent variable (y). The dependent variable is a result of the independent variable when a function has been applied to it. The value of the dependent variable is determined by the value of the related independent variable. Linear and Non-Linear 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 Linear Relationships Non-Linear Relationships In a linear relationship there will be a common difference between consecutive terms. The difference between consecutive y-values can also be referred to as the finite difference. When the x-values are given as consecutive integers, the equation for a linear relationship can be easily developed. First find the finite difference between the y-values. This number is then multiplied CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 41 of 399 Columbus City Schools 6/28/13 by the independent variable (x). If this product is not equal to the dependent variable (y) then add or subtract whatever value will result in the dependent variable. For example, in the table below a linear relationship (function) is represented. The finite difference between terms is 4 (4 is added each time to find the next y-value); therefore the independent variable must be multiplied by 4 as part of the process to determine the dependent variable. When 1 is multiplied by 4 the result is not equal to 6, we must add 2 more in order to equal 6. Multiplying by 4 and then adding 2 will also work for all of the other pairs of values in the relationship. Therefore, the function y = 4x + 2 can be used to represent this relationship. The finite difference, 4, is the slope and the amount added, 2, is the y-intercept. x y 1 6 2 10 3 14 4 18 5 22 x 4x + 2 A function that is non-linear will have finite differences that are not the same. There will, however, be a pattern to the differences. In a quadratic relationship the finite differences will increase or decrease by the same amount. In other words, the difference between each succeeding terms is constant. When you have to find two “levels” of differences to arrive at a constant finite difference, the relationship is a quadratic one. The related equation for a quadratic function has a degree of two (i.e., it has a variable that is squared) and all exponents are positive. If it requires going to three levels of differences to arrive at a constant value, the equation would have a degree of three (i.e., it has a variable that is cubed) and would not be quadratic but cubic. Four levels would indicate a degree of four, and so on. Not all relationships will eventually have a constant value for the finite differences, no matter how many levels are found. When the relationship is exponential (e.g., y = 2x) the finite differences will never all be the same value. For example, in the table below, the first level of finite differences would be 3, 6, 12, 24, . . . and so on. The second level of finite differences would also be 3, 6, 12, 24, . . . and so on. This relationship will never result in finite differences that are the same value. x y 0 3 +3 +3 1 6 +6 +6 2 12 +12 +12 3 24 +24 4 48 x 3(2x) Students can use finite differences to quickly determine if a relationship is linear, quadratic, or something else. If the first level of finite differences is all the same value, then the relationship is linear (i.e., it will be a first-degree equation). If the second level of finite differences is all the CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 42 of 399 Columbus City Schools 6/28/13 same value, then the relationship is quadratic (i.e., it will be a second-degree equation). Examples are shown below of linear relationships and quadratic relationships. Linear Relationships x y x y 9 1 -2.5 2 -2 3 -1.5 –2 4 -1 –2 5 -0.5 x 1 x–3 2 0 7 1 –2 5 2 3 3 1 4 x –2 -2x + 9 +½ +½ +½ +½ Quadratic Relationships x y 0 -4 1 -3 2 3 0 5 4 12 x x2 – 4 +1 +3 +5 +7 +2 +2 x y 1 1 2 2 2 3 4 +2 4 5 x 1 2 8 12 1 2 1 2 x 2 Graphing Absolute-Value Functions: The Purplemath website has a description of graphing absolute value functions. http://www.purplemath.com/modules/graphabs.htm CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 43 of 399 Columbus City Schools 6/28/13 Graphing Absolute Value Functions: Descriptions of graphing absolute value functions from tables can be found at this site. http://www.purplemath.com/modules/graphabs2.htm Misconceptions/Challenges: Students move the graph the opposite direction on horizontal shifts, because they confuse the fact that “h” is positive in the portion of the equation which says (x-h). Students move the graph up and down instead of left and right. Students confuse the horizontal and vertical shifts. Students confuse vertical stretches and compressions, often assuming that the effect of the equation is on the horizontal; for example they think the graph is simply expanding (getting wider) or shrinking (getting narrower) horizontally, rather than actually expanding or shrinking vertically. Students do not understand the different inequality signs such as and ; they do not understand why one includes the stated value and the other does not. Students confuse interval notation with inequality symbols. Students do not understand the concept of absolute value; they cannot visualize why there are two solutions to an absolute value equation, because they do not understand the definition of absolute value. Instructional Strategies: A – CED 1 1) http://www.montereyinstitute.org/courses/Algebra1/U08L2T1_RESOURCE/index.html Polynomials The website provides a warm up, video presentation, worked problems, practice and review on graphing equations in slope intercept form. 2) http://www.montereyinstitute.org/courses/Algebra1/U03L2T5_RESOURCE/index_tabless.ht ml?tabless=true&activetab=pres Non-linear Functions: The link below links to a video that introduces various non-linear functions including exponential and quadratic functions. 3) http://education.ti.com/en/timath/us/detail?id=92645D7C429D4E32A32A271DAED110A5&s CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 44 of 399 Columbus City Schools 6/28/13 a=71A40A9FD9E84937B8C6A8A4B4195B58 Exploring Circle Equations: Students explore the equation of a circle by connecting the coordinates of the center of the circle and the length of the radius to the corresponding parts of the equation. 4) http://map.mathshell.org.uk/materials/lessons.php?taskid=406 Equations of Circles 1: In this lesson students use the Pythagorean theorem to determine the equation of a circle and translate between geometric features of circles and their equations. 5) http://map.mathshell.org.uk/materials/lessons.php?taskid=425 Equations of Circles 2: In this lesson students translate between the equations of circles and their geometric features and sketch a circle from its equation. 6) http://www.kutasoftware.com/FreeWorksheets/GeoWorksheets/11Equations%20of%20Circles.pdf Equation of Circle: Practice sheet identifying radius and center to write equation and vice versa. 7) http://www.regentsprep.org/Regents/math/algtrig/ATC1/indexATC1.htm Equation of a Circle: This website offers a lesson, practice and teacher resources for the equation of a circle. 8) http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod3_quadfun_062 213.pdf Experimenting with Exponents: In this lesson (pp. 3-8), students look at values on continuous exponential functions that come between integers. 9) From the activity “Equation of a Circle with Center (0, 0)” (included in this Curriculum Guide), students will be able to come up with the equation for a circle with the center at (0,0) is x2 + y2 = r2. 10) From the activity “Equation of a circle with Center (h, k)” (included in this Curriculum Guide), students will be able to come up with the equation for a circle with the center at (h, k); that is (x – h)2 + (y – k)2 = r2. Students will be able to describe how a change in a constant affects the graph of a conic (transformation). 11) http://learnzillion.com/lessonsets/120-create-equations-and-inequalities-in-one-variable-anduse-them-to-solve-problems Create Equations and Inequalities: This website contains four lessons. Students create CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 45 of 399 Columbus City Schools 6/28/13 linear equations to solve problems by identifying relationships between important information; create compound linear inequalities by identifying relationships between important information; create an equation that models geometric change by visualizing an extending a pattern; and compare linear and geometric growth by creating and solving equations. 12) http://illuminations.nctm.org/LessonDetail.aspx?id=L606 Light It Up: In this activity students will conduct an investigation and develop rational functions that model three specific forms of a rational function. Students will determine the relationship between the graph, equation and problem context. A – CED 4 1) http://anakamura.weebly.com/uploads/4/9/1/4/4914438/alg1x_3-8_lecture_notes.pdf Solving Equations and Formulas: This website has lessons on solving and using literal equations. 2) http://www.montereyinstitute.org/courses/Algebra1/U02L1T4_RESOURCE/index.html Solving Literal Equations: This website provides a video presentation, worked problems, practice and test questions for solving a specific variable. 3) http://www.khanacademy.org/math/algebra/solving-linear-equations-andinequalities/solving_for_variable/v/rearrange-formulas-to-isolate-specific-variables Rearrange Formulas to Isolate Specific Variables: This link provides a tutorial for solving for a specific variable. 4) http://alex.state.al.us/lesson_view.php?id=23922 Solving Literal Equations: This website provides a power point presentation and practice for students on solving literal equations. 5) http://www.purplemath.com/modules/solvelit.htm Solving Literal Equations: A video tutorial and written explanation for solving equations for a variable can be found at this site. 6) http://www.regentsprep.org/Regents/math/ALGEBRA/AE4/indexAE4.htm Equations: A lesson, practice problems and teacher resources can be found at this site. 7) http://www.slideshare.net/crainsberg/solving-literal-equations Literal Equations: This website offers a set of slides on solving literal equations. 8) http://www.mcckc.edu/common/services/BR_Tutoring/files/math/equat_inequ/Practice_Solvin g_Literal_Equations.pdf CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 46 of 399 Columbus City Schools 6/28/13 Solving Literal Equations: Sample worked problems and practice problems are found at this site. 9) Have the students do the “Using Formulas” activity (included in this Curriculum Guide). 10) Have the students complete the “Literal Madness” activity (included in this Curriculum Guide). A – REI 1 1) http://map.mathshell.org.uk/materials/lessons.php?taskid=218 Sorting Equations and Identities: In this lesson students will be able to recognize the differences between equations and identities. They will test validity of special cases by substituting numbers into algebraic statements. Students will also note common errors when manipulating expressions such as use of the distributive property and squares of binomials. 2) http://map.mathshell.org/materials/tasks.php?taskid=293&subpage=novice Reasoning with Equations and Inequalities: This task includes a set of 6 short questions to solve. N – RN 1 1) http://www.regentsprep.org/Regents/math/algtrig/ATO1/indexATO1.htm Negative and Fractional Exponents: This website has lessons, practice and teacher resources for positive, negative and zero exponents, fractional exponents, and evaluating rational exponents. 2) http://www.kutasoftware.com/FreeWorksheets/Alg2Worksheets/Simplifying%20Rational%20 Exponents.pdf Simplify Rational Exponents: This lesson allows students to practice simplifying rational exponents. 3) http://www.montereyinstitute.org/courses/Algebra1/U07L3T4_RESOURCE/index.html Fractional Exponents: The website provides a warm up, video presentation, worked problems, practice and review on fractional exponents and writing them as a radical expression. 4) http://www.montereyinstitute.org/courses/Algebra1/U07L3T4_RESOURCE/index_tabless.ht ml?tabless=true&activetab=pres Fractional Exponents Video: The website provides a video presentation on fractional exponents and writing them as a radical expression. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 47 of 399 Columbus City Schools 6/28/13 5) http://www.khanacademy.org/math/arithmetic/exponents-radicals/world-ofexponents/e/exponents_3?exid=exponents_3 Fractional Exponents: This website has practice on fractional and integer bases raised to positive and negative fractional exponents 6) http://www.youtube.com/watch?feature=player_embedded&v=aYE26a5E1iU Level 3 Exponents: This website has a tutorial on converting bases with fractional exponents to radical expressions. 7) http://www.youtube.com/watch?feature=player_embedded&v=jO4wOQQiVZg Radical Equivalent to Rational Exponents: A tutorial is provided at this site on changing expressions from rational exponent to radical form. 8) http://www.slideshare.net/jessicagarcia62/simplifying-radical-expressions-rational-exponentsradical-equations Simplifying Radicals: This website provides a series of slides on simplifying radicals. Slides 1-21pertain to simplifying radicals. 9) http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod3_quadfun_062 213.pdf Half Interested: In this lesson (pp. 9-17), students use the definition of radicals and rules of exponents to attach meaning to fractional exponents. 10) Discuss with students that sometimes there are one, two, or no real solutions when finding the square root of a number. Using actual numbers, have students determine the square root of 4, 5, 0, and –1. Students should be guided, if necessary, to determine that 4 would have two rational solutions (2, and –2), 5 would have two irrational solutions 5 , 0 would have one 1 . Students can summarize rational solution (0), and –1 would have no real solutions their findings using variables and record this in their notes. Discussion should include why negative numbers do not have real solutions for their square roots. Students should develop the understanding that because multiplying two numbers that are the same (and therefore have the same sign) will always result in a positive product, it is impossible to have a real square root for a negative number. A sample summary for these findings is shown below. i. If d > 0, then x2 = d has two solutions: x = d . ii. If d = 0, then x2 = d has one solution: x = 0. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 48 of 399 Columbus City Schools 6/28/13 11) Discuss like square roots with students and draw parallels between adding and subtracting like square roots with like terms of polynomials. 12) Have students practice adding and subtracting radicals, using “Adding and Subtracting Radical Cards” (included in this Curriculum Guide). 13) Students should be given instruction on the requirements for the simplest form of radicals, and practice simplifying radicals including getting rid of fractions under the radical sign and radicals that appear in the denominator of a fraction. 14) Discuss how to multiply two square-root terms together. Compare multiplying the radical coefficients together, and radicands together, with multiplying coefficients together, and variables together. 15) Have students practice multiplication and division of radicals using “Multiplying and Dividing Radical Cards” (included in this Curriculum Guide) 16) Have students practice radicals by using “Match Me Cards” (included in this Curriculum Guide). 17) “Radicals Rule!” (included in this Curriculum Guide) can be used to introduce students to the rules which are involved in the simplification of radial expressions. 18) Remind students how to add and subtract like radicals. Include examples of radicals that must be simplified before they can be added or subtracted. Students will practice the addition and subtraction of radicals with the worksheet “Addition and Subtraction of Radicals” (included in this Curriculum Guide). 19) Students will have simplified multiplication and division of radicals with numbers. This idea should be expanded to include variables in the simplification of radicals with the worksheet “Multiplication and Division of Radicals” (included in this Curriculum Guide). 20) In “Radical Expressions” (included in this Curriculum Guide), students will practice simplifying radicals. 21) http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod3_quadfun_062 213.pdf More Interesting: In this lesson (pp. 18-23), students verify properties and rules for exponents hold true for rational exponents. 22) http://www.montereyinstitute.org/courses/Algebra1/U07L3T4_RESOURCE/index_tabless.ht ml?tabless=true&activetab=pres Radical Equivalent to Rational Exponents 2: At this site a tutorial is provided on changing CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 49 of 399 Columbus City Schools 6/28/13 radicals to expressions with power in rational form. 23) http://www.youtube.com/watch?feature=player_embedded&v=NuccqpiUHrk Rational Exponents and Exponent Laws: In this tutorial radical expressions are changed to expressions with rational exponents. 24) http://www.youtube.com/watch?feature=player_embedded&v=rco7DMcy-oE More Rational Exponents and Exponent Laws: In this tutorial radical expressions are changed to expressions with rational exponents. 25) http://patrickjmt.com/exponents-multiplying-variables-with-rational-exponents-basic-ex-1/ Exponents: Multiplying Variables with Rational Exponents – Basic Ex 1: This website provides a video presentation on multiplying expressions with rational exponents. 26) http://patrickjmt.com/exponents-multiplying-variables-with-rational-exponents-basic-ex-2/ Exponents: Multiplying Variables with Rational Exponents – Basic Ex 2: This website provides an additional video presentation on multiplying expressions with rational exponents. 27) https://www.khanacademy.org/math/algebra/exponent-equations/exponent-propertiesalgebra/v/fractional-exponent-expressions-1 Fractional Exponents, Example 1: This website provides a video tutorial for simplifying an expression with rational exponents. N – RN 2 1) http://www.ixl.com/promo?partner=google&phrase=common%20core%20strands&redirect= %2Fmath%2Fstandards%2Fcommon-core%2Fhighschool&gclid=CJO_8sme37cCFac7MgoduRIATg IXL State Standards: The website below links to interactive practice with radicals. 2) http://www.kutasoftware.com/FreeWorksheets/Alg2Worksheets/Radicals%20and%20Rational %20Exponents.pdf Radical and Rational Exponents: This practice sheet allows students to write rational expressions as rational expressions and vice versa. 3) http://www.regentsprep.org/Regents/math/algtrig/ATO1/RatPowersTeacher.HTM Exploration of Rational Exponents: This website offers a practice sheet to simplify expressions with rational exponents and radical expressions using technology. 4) http://www.regentsprep.org/Regents/math/algtrig/ATO1/indexATO1.htm Negative and Fractional Exponents: This website offers lessons, practice, and teacher resources on negative and fractional exponents. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 50 of 399 Columbus City Schools 6/28/13 5) http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod3_quadfun_062 213.pdf Radical Ideas: In this lesson (pp. 24-31), students become fluent converting between exponential and radical notation. 6) http://www.slideshare.net/jessicagarcia62/simplifying-radical-expressions-rational-exponentsradical-equations Simplifying Radicals: This website provides a series of slides with notes on expressions with rational exponents. Slides 12-29 pertain to converting between radical expressions to expressions with rational exponents. 7) http://www.khanacademy.org/math/algebra/exponent-equations/exponent-propertiesalgebra/v/radical-equivalent-to-rational-exponents-2 Radical Equivalent to Rational Exponents 2: This website has a video tutorial showing how to change a radical to its equivalent rational exponent. 8) https://www.khanacademy.org/math/algebra/exponent-equations/exponent-propertiesalgebra/v/fractional-exponent-expressions-2 Fractional Exponents, Example 2: This website provides a video tutorial for simplifying an expression with exponents and a radical expression. 9) https://www.khanacademy.org/math/algebra/exponent-equations/exponent-propertiesalgebra/v/fractional-exponent-expressions-3 Fractional Exponents, Example 3: This website provides a video tutorial with an example of simplifying an expression with rational exponents and a radical expression. N – RN 3 1) http://map.mathshell.org.uk/materials/lessons.php?taskid=424 Rational and Irrational Numbers 1: In this lesson students will classify numbers as rational or irrational and change between different representation of rational and irrational numbers. 2) http://map.mathshell.org.uk/materials/lessons.php?taskid=434 Rational and Irrational Numbers 2: In this lesson students will demonstrate understanding of properties of rational and irrational numbers. The will find irrational and rational numbers to exemplify general statements and reason with properties of rational and irrational numbers. 3) http://educ.jmu.edu/~taalmala/235_2000post/235contradiction.pdf Proof: Why is the sum of a rational and irrational number irrational?: This website provides proofs for this standard. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 51 of 399 Columbus City Schools 6/28/13 4) http://map.mathshell.org/materials/tasks.php?taskid=289&subpage=novice The Real Number System: This task consists of a set of 4 short questions on operations with radicals. 5) By using the lesson, “Rational and Irrational” (included in this Curriculum Guide) students will be able to: classify real numbers as either rational or irrational numbers, graph real numbers accurately and precisely on a number line, calculate the exact perimeter, circumference, and area of shapes that model mathematical situations, based on observations of the sum and product of real numbers, students will be able to determine: if the sum of the product of rational numbers is rational, if the sum of a rational number and an irrational number is irrational, and if the product of a nonzero rational number and an irrational number is irrational. (Note that this lesson includes the following activities: “Real Number System-Classification”, “Real Number System Cards”, “Real Numbers Perimeter Investigation”, “Real Numbers Perimeter Homework”, “Real Numbers-Circumference and Area Investigation”, “Real Numbers Area Homework”, and “Real Numbers-Extension.” Also not that all activities appear in this order, followed by all of the corresponding answer keys at the end. 6) http://www.shmoop.com/common-core-standards/ccss-hs-n-rn-3.html Math.N-RN.3: This site provides an explanation of the standard and sample problems and solutions. F – IF 4* 1) http://map.mathshell.org.uk/materials/lessons.php?taskid=430 Functions and Everyday Situations: In this lesson students will translate between different representation of linear, exponential and quadratic functions. Use graphing technology for this lesson. Ask students to determine the key features such as intercepts, intervals where the function is increasing, positive, or negative; relative maximums or minimums; symmetries; end behavior; and periodicity. 2) http://www.ixl.com/math/algebra-1/identify-linear-quadratic-and-exponential-functions-fromtables IXL Identify Linear, Exponential or Quadratic Functions: This website offers interactive practice for identifying linear, exponential or quadratic functions from tables. 3) http://www.ixl.com/math/algebra-1/identify-linear-quadratic-and-exponential-functions-fromgraphs IXL Identify Linear, Exponential or Quadratic Functions: This website offers interactive practice for identifying linear, exponential or quadratic functions from graphs. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 52 of 399 Columbus City Schools 6/28/13 4) http://map.mathshell.org/materials/lessons.php?taskid=426&subpage=concept Comparing Investments: In this lesson students will translate between descriptive, algebraic, tabular data, and graphical representation of exponential and linear functions. 5) https://commoncorealgebra1.wikispaces.hcpss.org/Unit+2 Lacrosse Tournament: This lesson is provided for the standard F-IF 4at the website provided below Students are instructed to use a graphing utility to complete this assignment. Students will determine an algebraic representation of the data and answer questions on this real-world situation. F – IF 7b 1) http://betterlesson.com/lesson/307202/graphing-the-absolute-value-function-using-y-a-x-h-k Graphing the Absolute Value Function: This website has a lesson and lesson resources for graphing absolute value functions. 2) http://map.mathshell.org/materials/tasks.php?taskid=264&subpage=apprentice Sorting Functions: At this website students are given four graphs, four equations, four tables, and four rules. Their task is to match each graph with an equation, a table and a rule. 3) http://www.montereyinstitute.org/courses/Algebra1/U11L1T1_RESOURCE/index_tabless.ht ml?tabless=true&activetab=pres Simplifying Rational Expressions: The website below provides a video presentation of simplifying rational expression. 4) http://www.ixl.com/math/algebra-1/graph-an-absolute-value-function IXL Graph Absolute Value Functions: This website offers interactive practice for graphing absolute value functions. 5) http://www.montereyinstitute.org/courses/Algebra1/U02L2T1_RESOURCE/index.html Absolute Value: This website offers a warm up, video tutorial, practice and review for the meaning of absolute value and how to determine it. 6) http://education.ti.com/en/timath/us/detail?id=3FE56C10EF57478CB386AE374A96BEDD&s a=291B0ACD31104D178C0EA77ABC7FB53A Radical Transformations: Students will use the Transformational Graphing application to examine how the square root function is transformed on the coordinate plane. 7) http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Graphing%20Abs%20Value. pdf Graphing Absolute Value Functions: An assignment is provided at this link for practice graphing absolute value functions CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 53 of 399 Columbus City Schools 6/28/13 8) http://education.ti.com/en/us/activity/detail?id=15A3E986F08A463882927F0208FB6817 Exploring Power Functions 2: Students will describe the shape, end behavior, and key points for radical functions. 9) http://www.youtube.com/watch?v=kAQ-CPyKzq8 Absolute Value Graphs: Video tutorial on graphing absolute value functions 10) http://education.ti.com/en/us/activity/detail?id=29EC5E10B5AC441DAF8387226A13DCA1 Absolute Value: Handheld activity on graphing absolute value functions. This lesson involves the family of absolute value functions of the form f(x) = a |x + c| + b. Students will explore the family of absolute value functions of the form f(x) = a |x + c| + b. and discover the effect of each parameter on the graph of y = f(x). 11) http://education.ti.com/en/us/activity/detail?id=8B9C1960BC04457498DAF0BD1BFF70A9 Exploring Transformations: Students will explore transformations of absolute functions and examine the effect of stretching and translating the coordinates of the graph. 12) http://teachers2.wcs.edu/high/fhs/jamesa/Lists/Calendar/Attachments/859/Activity%20%20Exploring%20Transformations%20of%20Abs%20Value%20Fn.pdf Activity: Exploring Graphs of Absolute Value Functions: Graphing calculator activity to investigate absolute value functions. 13) http://www.dlt.ncssm.edu/algebra/HTML/05.htm Piecewise Defined Functions as Models: In this lesson students develop a piecewise-defined linear function using domain restrictions and the linear regression line. This model provides specific information in the slopes to compare the data of the two trends. 14) http://www.dlt.ncssm.edu/AFM/turnpike.htm Tolls on the New Jersey Turnpike: Students develop a model for the toll structure of the New Jersey turnpike using a piecewise defined linear function developed using data analysis. 15) http://education.ti.com/en/us/activity/detail?id=919012C8691F44EA8E22DC388D5612EB Transformations of Functions 1: Students investigate vertical and horizontal translations of a function and will be able to recognize the effect of a horizontal and vertical translation on the graph of a function. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 54 of 399 Columbus City Schools 6/28/13 16) http://education.ti.com/en/us/activity/detail?id=9591A4F69AF249D4BEB093204DA8624D Transformations of Functions 2: Students investigate vertical stretches and reflections through the x-axis of a function. Students will recognize the effect of a vertical stretch, vertical compression, and reflection through the x-axis on the graph of a function. 17) http://www.regentsprep.org/Regents/math/algtrig/ATE1/indexATE1.htm Absolute Value Equations: This website has a lesson, practice and teacher resources on absolute value. 18) http://education.ti.com/en/us/activity/detail?id=911D4E08E86E44678874752A4E0433F3&ref =/en/us/activity/search/subject?d=3AF8B2EA285D41F2A983D320C8A3A0B6&s=B843 CE852FC5447C8DD88F6D1020EC61&sa=2D7AB06424004125A392EB9A075CABC0 &t=2151DCE714B04646B61A0C77DC042FED Introducing the Absolute Value Function: In this activity students will examine data and investigate the absolute value function. 19) Students can practice converting from radical to exponent form and from exponent to radical form with the activity “Radical and Exponent Matching” (included in this Curriculum Guide). With a partner, students will cut out either the radical expression cards or the exponential expression cards. Students will place exponent and radical cards face down. The partners take turns drawing a pair of cards and determining if they have equivalent expressions in exponent and radical form, i.e. a match. The player with the most matches at the end wins. 20) Recognizing the various forms in which equivalent radical and exponential expressions can be written is useful in problem solving and test taking. In the activity “Find Two Ways” (included in this Curriculum Guide), students will rewrite a radical or an exponential expression in two additional equivalent ways. Answers can vary. 21) At this point, students should have done many graphs that involve transformations. Supply the students with the worksheets “Graphing Square Roots” (included in this Curriculum Guide) and a graphing calculator. Students should spend about half of the period completing the worksheets by themselves. When students have them completed or about half-way through the period, have the students pair-share their answers. Graphs that have not been completed in the classroom should be assigned for homework. Debrief with a summary of the properties of the graphs of square root functions including the algebraic method of finding the domain and range of the square root function and a reminder of the exponential form of the equations. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 55 of 399 Columbus City Schools 6/28/13 22) Since students have just completed the transformations with square root functions, the worksheet “Graphing Cube Roots” (included in this Curriculum Guide) could be done the same way as “Graphing Square Roots” or it could be done as a homework assignment. 23) The worksheet “Graphing Cube and Square Root Functions” (included in this Curriculum Guide) can be done as a review or as reinforcement to these concepts. 24) Use a number line taped on the floor to model absolute value equations. A student would be asked to stand on a number, for example 3, and then another student would identify the points that are a given distance from the first student, for example two units. The solutions would be 1 and 5. Students would then write the absolute value equation that represents the equation, |x – 3| = 2. This could also be used to model absolute value inequalities by having students use a rope or ruler with an arrow on the end to model the solution set. 25) Explore graphing absolute value equations on the coordinate graph with the activity “Graphing? Absolutely!” (included in this Curriculum Guide). Students start by graphing y = |x|, by making a t-table, and using graph paper or a graphing calculator. Students should discuss with a partner any patterns or unique characteristics they notice about this graph compared to the other linear graphing they have done in the past. Students will then graph several other equations, each time discussing any patterns or characteristics they notice. 26) Discuss how to determine the vertex for an absolute value equation without graphing the equation. Students should study the equations from the “Graphing? Absolutely!” (included in this Curriculum Guide) activity and discuss the relationships they notice between the vertex of the graph and the equation. Assuming an absolute value equation is of the form y = a|x + b| + c, students should be guided to notice that the x coordinate of the vertex can be found by solving the following equation for x: x + b = 0. The y-coordinate would then be found by substituting the value for x and then solving for y. Students should practice finding the vertex when given an equation, then make a table of values that they will use with the vertex to graph the equation. When making the table use x-coordinates that are to the left and right of the vertex. Examples are shown below. Find the vertex: y = 3|x – 2| – 1 x–2=0 x=2 y = 3|2 – 2| – 1 y = -1 The vertex = (2, -1) y = |x| + 3 x=0 y = |0| + 3 y=3 The vertex = (0, 3) CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 56 of 399 Columbus City Schools 6/28/13 x y 0 5 1 2 2 -1 3 2 4 5 x -2 -1 0 1 2 y 5 4 3 4 5 27) Have students play a memory game in which they match the equation with the graph using the “Absolute Value Equation and Graph Cards” (included in this Curriculum Guide). 28) Have students complete the “2’s Are Wild” activity (included in this Curriculum Guide). Students will generate sets of points for equations based on y = |x| and variations of this equation, then graph the points to determine patterns in the graphs. 29) Through this calculator discovery activity, students will see that vertically shrinking/stretching with absolute values is very similar to vertically shrinking/stretching with quadratic functions. x in y1 with a standard window. Then have Have students graph the basic function f x students graph each function below on the same window. Students should discuss how each of the following functions compared to the basic function, that is, is it a vertical shrink (wider) or vertical stretch (narrower). 1 1 x y5 x 2 4 Students should look at the table for all five functions and compare the y-values in each. They should notice that, for example, the y-values in y2 are doubled that of the y-values in y1 for the same x-values. In addition, the y-values in y5 are a fourth of the y-values in y1 for the same xvalues. y2 2x y3 5x y4 30) Let students do the activity “Connecting Functions” (included in this Curriculum Guide) in order for them to write absolute value functions as piecewise functions. Student will make the connection that absolute value functions are just the union of linear functions with a restricted domain. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 57 of 399 Columbus City Schools 6/28/13 31) http://www.slideshare.net/tdimella/absolute-value-functions-graphs Absolute Value Functions and Graphs: This site has a set of slides showing how to graph absolute value functions with a table and graphing technology. 32) http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Graphing%20Abs%20Value. pdf Graphing Absolute Value Functions: A practice sheet for graphing absolute value functions can be found at this site. 33) http://teachers.henrico.k12.va.us/math/HCPSAlgebra2/Documents/1-5/1_5CW.pdf Absolute Value Functions: This lesson allows students to graph the functions and determine the vertex. 34) http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod4_morefeatures _062213.pdf Some of This, Some of That: In this lesson (pp. 3-6), students build on prerequisite skills to develop an understanding of piece-wise functions. 35) http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod4_morefeatures _062213.pdf Bike Lovers: Students build on their understanding of piece-wise functions with additional practice in this lesson (pp. 7-12). F – IF 7e 1) http://illuminations.nctm.org/LessonDetail.aspx?id=L829 Drug Filtering: In this lesson, students observe a model of exponential decay, and how kidneys filter their blood. They will calculate the amount of a drug in the body over a period of time. Then, they will make and analyze the graphical representation of this exponential function. 2) Using different representations of exponential functions, have the students use interval notation (or symbols of inequality) to communicate key features of the graphs. 3) https://commoncorealgebra1.wikispaces.hcpss.org/Unit+2 Analyzing Residuals: This website offers a lesson plan and student practice on analyzing residuals. F – IF 8 1) Have students complete “Different = Same” (Included in this Curriculum Guide). In this activity students will change exponential equations from the form y = a(b)x into y = a(1 + r)x or CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 58 of 399 Columbus City Schools 6/28/13 y = a( 1 - r)x and vice versa. 2) Have students complete “Three Different Exponential Functions” (included in this Curriculum Guide). By graphing exponential growth and decay functions, students determine the initial value and the growth or decay factor. F – IF 8b 1) http://www.regentsprep.org/Regents/math/ALGEBRA/AE7/ExpDecayL.htm Exponential Growth and Decay: This website has examples of growth and decay functions. 2) http://hotmath.com/help/gt/genericalg1/section_9_6.html Hot Math Practice Problems: A series of practice problems can be found at this site. 3) http://www.pkwy.k12.mo.us/homepage/nhsalgebra1/file/6.3_Homework_Day1.pdf Homework Practice #1: Practice sheet on determining growth or decay of a function. 4) http://www.pkwy.k12.mo.us/homepage/nhsalgebra1/file/6.3_ExtraPractice_Day1.pdf Homework Practice #2: Practice sheet on determining growth or decay given a graph or function. 5) http://www.pkwy.k12.mo.us/homepage/nhsalgebra1/file/Exponential%20Growth%20&%20D ecay/6.3_Homework_Day2.pdf Homework Practice #3: Practice sheet on determining growth or decay and the initial value given an equation. 6) http://www.pkwy.k12.mo.us/homepage/nhsalgebra1/file/Exponential%20Growth%20&%20D ecay/Day%202%20-%20HW(1).pdf Writing Exponential Functions: Students determine the growth or decay factor and initial value given a table or function. F – BF 1 1) http://www.montereyinstitute.org/courses/Algebra1/U08L2T1_RESOURCE/index.html Polynomials: The website below provides a warm up, video presentation, worked problems, practice and review on polynomials. 2) http://www.ixl.com/math/algebra-1/write-linear-quadratic-and-exponential-functions IXL Write Linear, Exponential and Quadratic Functions: This website offers interactive practice for writing linear, exponential and quadratic functions. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 59 of 399 Columbus City Schools 6/28/13 3) http://map.mathshell.org.uk/materials/lessons.php?taskid=215 Generalizing Patterns: Table Tiles: In this lesson students will examine data and look for patterns to identify linear and quadratic relationships. F – BF 1a 1) http://www.ixl.com/math/algebra-1/write-linear-quadratic-and-exponential-functions Linear, Quadratic and Exponential Functions: At this site there are interactive problems to write linear, quadratic and exponential functions. F – BF 1b 1) http://map.mathshell.org/materials/tasks.php?taskid=295&subpage=novice Building Functions: In this short task, students determine which graphs represent the equations. 2) http://map.mathshell.org/materials/tasks.php?taskid=279&subpage=expert Skeleton Tower: In this task student determine a rule for calculating the total number of cubes needed to build towers of different heights. F – BF 3 1) http://betterlesson.com/lesson/307202/graphing-the-absolute-value-function-using-y-a-x-h-k Graphing the Absolute Value Function: The following website has a lesson and lesson resources for graphing absolute value functions. 2) http://alex.state.al.us/lesson_view.php?id=23782 Explore the Transformations of Linear and Absolute Value Functions Using Graphing Utilities: This inquiry lessons allows students to explore the rigid and non-rigid transformations of linear and absolute value functions using a graphing utility. 3) https://commoncorealgebra1.wikispaces.hcpss.org/Unit+5 4) Piecewise Functions Lesson: This website has a lesson on piecewise functions. Search under the standard F-IF 7. 5) Have students complete “Absolute Value Graphs” (Included in this Curriculum Guide). In this activity students will investigate the graphs and determine the effects of a, h, and k. 6) https://commoncorealgebra1.wikispaces.hcpss.org/Unit+2 Vertical Shifts of Functions: At this site you can download a task given matching of different representations of linear and exponential functions. 7) http://education.ti.com/en/us/activity/detail?id=B15F536614A0428E8C072248C23C76EE 8) Exploration of Absolute Value Functions: Using graphing technology students explore the CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 60 of 399 Columbus City Schools 6/28/13 transformations of absolute value functions. A – SSE 1b 1) http://www.montereyinstitute.org/courses/Algebra1/U08L2T2_RESOURCE/index.html Adding and Subtracting Polynomials: The website provides a warm up, video presentation, worked problems, practice and review on adding and subtracting polynomials. 2) http://www.montereyinstitute.org/courses/Algebra1/U08L2T2_RESOURCE/index_tabless.ht ml?tabless=true&activetab=pres Adding and Subtracting Polynomials Video: The website below provides a video presentation of adding and subtracting polynomials. 3) http://www.montereyinstitute.org/courses/Algebra1/U08L2T3_RESOURCE/index.html Multiplying Polynomials: The website below provides a warm up, video presentation, worked problems, practice and review on multiplying polynomials. 4) http://www.montereyinstitute.org/courses/Algebra1/U08L2T3_RESOURCE/index_tabless.ht ml?tabless=true&activetab=pres Multiplying Polynomials Video: The website links to a video introduces and explains the concept. 5) http://www.montereyinstitute.org/courses/Algebra1/U08L1T1_RESOURCE/index.html Multiplying and Dividing Monomials: This site provides a warm up, video presentation, worked problems and practice on multiplying and dividing monomials. 6) http://www.montereyinstitute.org/courses/Algebra1/U08L2T4_RESOURCE/index.html Special Products of Polynomials: The website provides a warm up, video presentation, worked problems, practice and review on special products of polynomials. 7) http://www.montereyinstitute.org/courses/Algebra1/U08L2T4_RESOURCE/index_tabless.ht ml?tabless=true&activetab=pres Special Products of Polynomials Video: The website links to a video introduces and explains the concept. 8) http://www.uen.org/core/math/downloads/proving_patterns.pdf Proving Patterns: Students will analyze quadratic patterns related to the difference of squares and use patterns with the number line. They will then prove the general rule with a sequence of calculations that model inductive reasoning. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 61 of 399 Columbus City Schools 6/28/13 9) http://map.mathshell.org/materials/lessons.php?taskid=426&subpage=concept Comparing Investments: In this lesson students will translate between descriptive, algebraic, and tabular data, and graphical representation of exponential and linear functions. 10) http://www.montereyinstitute.org/courses/Algebra1/U11L1T3_RESOURCE/index.html Adding and Subtracting Rational Expressions: This site provides a warm up, video presentation, worked problems and practice on adding and subtracting rational expression. 11) http://www.montereyinstitute.org/courses/Algebra1/U11L1T2_RESOURCE/index.html Multiplying and Dividing Rational Expressions: This site provides a video presentation explaining the topic, practice exercises, worked examples, practice problems, and a review. 12) http://www.montereyinstitute.org/courses/Algebra1/U07L3T1_RESOURCE/index.html Simplifying Radical Expressions: This site provides a video presentation, practice exercises, worked examples, and a review for simplifying radical expressions. 13) Through the activity “Investigate Compound Interest” (included in this Curriculum Guide), students should be able to come up with a formula to calculate compound interest. Students will also have a deeper understanding of the difference between simple and compound interest. N –Q 2 1) http://illuminations.nctm.org/ActivityDetail.aspx?ID=16 Flowing Through Mathematics: At the website a simulation of water flowing from a tube through a hole in the bottom is provided. The diameter of the hole can be adjusted and data can be gathered for the height or volume of water in the tube at any time. 2) http://www.dlt.ncssm.edu/algebra/HTML/05.htm Piecewise Defined Functions as Models: In this lesson students develop a piecewise-defined linear function using domain restrictions and the linear regression line. This model provides specific information in the slopes to compare the data of the two trends. 3) https://commoncorealgebra2.wikispaces.hcpss.org/file/detail/F.IF.B.4%20Lesson%20Exponen tial%20Graph%20Characteristics.doc Exponential Graph Characteristics: This website contains a lesson plan on exponential graph characteristics. 4) https://commoncorealgebra2.wikispaces.hcpss.org/Unit+2 Exponential Characteristics: On this website, locate the standard and click on Exponential Graph Characteristics. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 62 of 399 Columbus City Schools 6/28/13 5) https://commoncorealgebra2.wikispaces.hcpss.org/Unit+5 Radical and Rational Exponents: This website contains a lesson plan on radical and rational exponents. 6) http://www.ixl.com/math/algebra-1/interpret-a-scatter-plot Interpret Scatter plots: Interactive practice problems on interpreting scatter plots can be found at this website. Reteach: 1) Have students work on “Exponent Properties Exploration” (included in this curriculum guide) to review the exponent properties, and why they work. 2) http://www.montereyinstitute.org/courses/Algebra1/U07L1T1_RESOURCE/index.html Rules of Exponents: The website below provides a warm up, video presentation, worked problems, practice and review on rules of exponents. 3) http://www.montereyinstitute.org/courses/Algebra1/U07L1T1_RESOURCE/index_tabless.htm l?tabless=true&activetab=pres Rules of Exponents Video: The website below provides a video presentation on rules of exponents. 4) Have the students use the “Multiplying Binomials” (Curriculum Guide) for additional practice with multiplying binomials together. Extensions: 1) http://education.ti.com/en/timath/us/detail?id=FABF80DD572743E89EBE4A8BB2BB9202&s a=291B0ACD31104D178C0EA77ABC7FB53A Exponential Growth: Students will find an approximation for the value of the mathematical constant e and to apply it to exponential growth and decay problems 2) Students will use the internet and/or the library resource center to find a direct application of a radical function. The student will make a visual representation of their application and share this information with the remainder of the class. 3) Students can explore the concept of radical inequalities. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 63 of 399 Columbus City Schools 6/28/13 4) From the activity “A Number Called e” (included in this Curriculum Guide), students will discover Euler’s number by evaluating a compounded interest formula and examine what happens to A, the balance, as it is compounded more frequently. The principal used is $1, the interest rate is 100% and the time is 1 year. Students will see that the expression 1 1 1 n n (1) will go towards e as n gets larger 5) http://teachers.henrico.k12.va.us/math/hcpsalgebra1/module7-7.html 6) Multiplying Binomials: Use the Project Graduation Multiplying Polynomials Puzzle. Students will cut out the squares and match the multiplication of polynomials problems with their products. 7) http://enlvm.usu.edu/ma/nav/activity.jsp?sid=nlvm&cid=4_2&lid=189 Interactive Web-site for Multiplying Binomials: Binomials can be multiplied using interactive tiles. 8) http://www.regentsprep.org/Regents/math/ALGEBRA/AV3/Smul_bin.htm Multiplying Binomials: This websites offers explanations of using different methods of multiplying binomials. 9) https://commoncorealgebra1.wikispaces.hcpss.org/file/view/F.BF.1+Discounting+Tickets.pdf Students are given the task, “Discounting Tickets” by Maryland CCRG Algebra Task Project. They will determine which promotional plan to use to increase ticket sales the most. Students will combine two functions, linear and exponential, to determine the best plan. Textbook References: Textbook: Algebra I, Glencoe (2005): pp. 554-560, 561-565, 566, 586-592, 593-597, 648654, 655-659, 660-664, 838-839, 843 Supplemental: Algebra I, Glencoe (2005): Chapter 10 Resource Masters Reading to Learn Mathematics, pp. 607, 613 Study Guide and Intervention, pp. 603-604, 609-610 Skills Practice, pp. 605, 611 Practice, pp. 606, 612 Chapter 11 Resource Masters CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 64 of 399 Columbus City Schools 6/28/13 Reading to Learn Mathematics, pp. 647, 653, Study Guide and Intervention, pp. 643-644, 649-650 Skills Practice, pp. 645, 651 Practice, pp. 646, 652 Enrichment, pp. 648, 654 Chapter 12 Resource Masters Reading to Learn Mathematics, pp. 715, 721, 727 Study Guide and Intervention, pp. 711-712, 717-718, 723-724 Practice, pp. 713, 719, 725 Skills Practice, pp. 714, 720, 726 Enrichment pp. 716, 722, 728 Textbook: Algebra 2,Glencoe (2003): pp. 89-95, 245-249, 250-256, 257-262, 522, 523–525, 527 – 528, 838, 849 Supplemental: Algebra 2,Glencoe (2003): Chapter 5 Resource Masters Reading to Learn Mathematics, p. 267, 273, 279 Study Guide and Intervention, pp. 263-264, 269-270, 275-276 Skills Practice, p. 265, 271, 277 Practice, p. 266, 272, 278 Enrichment, p. 268, 274, 280 Chapter 10 Resource Masters Study Guide and Intervention, pp. 573-574 Skills Practice, pp. 575 Practice, pp. 576 Textbook: Integrated Mathematics: Course 3, McDougal Littell (2002): pp. 85-91, 114-117, 281-287, 489-491, 638-640, 642 Textbook: Advanced Mathematical Concepts, Glencoe (2004): pp. 45-51, 137-145, 159-168, , 169-170 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 65 of 399 Columbus City Schools 6/28/13 A – CED 1 Name ___________________________________ Date __________________ Period ________ Equation of a Circle with Center (0, 0) 1. Given the graph of a circle whose center is (0, 0) and some points on the circle, complete the table below. (2, 5 ) (-1, 8) Ordered Pairs (x-coordinate)2 (y-coordinate)2 x2 + y2 a) What is the length of the radius of this circle? b) What is the length of the radius squared? c) Do you see any connection between the length of the radius squared and the x2 + y2 column in the table above? CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 66 of 399 Columbus City Schools 6/28/13 2. Given the graph of a circle whose center is (0, 0) and some points on the circle, complete the table below. (-2, 21 ) Ordered Pairs (x-coordinate)2 (y-coordinate)2 x2 + y2 a. What is the length of the radius of this circle? b. What is the length of the radius squared? a. Do you see any connection between the length of the radius squared and the x2 + y2 column in the table above? 3. Based on #1c and #2c, write a general equation to represent all the points (x, y) of a circle whose center is (0, 0) and whose radius is r. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 67 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Equation of a Circle with Center (0, 0) Answer Key 1. Given the graph of a circle whose center is (0, 0) and some points on the circle, complete the table below. (2, 5 ) 8) (-1, Ordered Pairs (-3, 0) (0, 3) (2, 5 ) (3, 0) (0, -3) (-1, 8 ) (x-coordinate)2 9 0 4 9 0 1 (y-coordinate)2 0 9 5 0 9 8 x2 + y2 9 9 9 9 9 9 b. What is the length of the radius of this circle? 3 c. What is the length of the radius squared? 9 d. Do you see any connection between the length of the radius squared and the x2 + y2 column in the table above? They are the same value. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 68 of 399 Columbus City Schools 6/28/13 2. Given the graph of a circle whose center is (0, 0) and some points on the circle, complete the table below. (-2, 21 ) Ordered Pairs (-5, 0) (-2, 21 ) (0, 5) (5, 0) (3, -4) (0, -5) (x-coordinate)2 25 4 0 25 9 0 (y-coordinate)2 0 21 25 0 16 25 x2 + y2 25 25 25 25 25 25 a. What is the length of the radius of this circle? 5 b. What is the length of the radius squared? 25 c. Do you see any connection between the length of the radius squared and the x2 + y2 column in the table above? They are the same value. 3. Based on #1c and #2c, write a general equation to represent all the points (x, y) of a circle whose center is (0, 0) and whose radius is r. x2 + y2 = r2 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 69 of 399 Columbus City Schools 6/28/13 A – CED 1 Name _______________________________ Date __________________ Period ________ Equation of a Circle with Center (h, k) 1. Given the graph of a circle whose center is (2, 3) and some points on the circle, complete the table below. Ordered Pairs (x-coordinate – 2)2 (y-coordinate – 3)2 (x – 2)2 + (y – 3)2 Column 2 + Column 3 a. What is the length of the radius of this circle? b. What is the length of the radius squared? c. Do you see any connection between the length of the radius squared and column 4 in the table above? d. How is this circle shifted from a circle with the same radius length but centered at (0, 0)? CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 70 of 399 Columbus City Schools 6/28/13 2. Given the graph of a circle whose center is (-2, 1) and some points on the circle, complete the table below. (1, 7 1) (-4, - 12 1 ) Ordered Pairs (x-coordinate – (-2))2 (y-coordinate – 1)2 (x – (-2))2 + (y – 1)2 Column 2 + Column 3 a. What is the length of the radius of this circle? b. What is the length of the radius squared? c. Do you see any connection between the length of the radius squared and column 4 in the table above? d. How is this circle shifted from a circle with the same radius length but centered at (0, 0)? 3. Based on #1c and #2c, write a general equation to represent all the points (x, y) of a circle whose center is (h, k) and whose radius is r. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 71 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Equation of a Circle with Center (h, k) Answer Key 1. Given the graph of a circle whose center is (2, 3) and some points on the circle, complete the table below. Ordered Pairs (x-coordinate – 2)2 (y-coordinate – 3)2 (x – 2)2 + (y – 3)2 Column 2 + Column 3 (2, 8) (5, 7) (7, 3) (2, -2) (-2, 0) (-3, 3) 0 9 25 0 16 25 25 16 0 25 9 0 25 25 25 25 25 25 a. What is the length of the radius of this circle? 5 b. What is the length of the radius squared? 25 c. Do you see any connection between the length of the radius squared and column 4 in the table above? They both are the same value. d. How is this circle shifted from a circle with the same radius length but centered at (0, 0)? This circle is shifted right 2 and up 3 from a circle with radius 5 centered at (0, 0). CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 72 of 399 Columbus City Schools 6/28/13 2. Given the graph of a circle whose center is (-2, 1) and some points on the circle, complete the table below. (1, 7 1) (-4, - 12 1 ) Ordered Pairs (x-coordinate – (-2))2 (y-coordinate – 1)2 (x – (-2))2 + (y – 1)2 Column 2 + Column 3 (-6, 1) (-2, 5) (1, 7 1) (2, 1) (-2, -3) (-4, 12 1 ) 16 0 9 16 0 4 0 16 7 0 16 12 16 16 16 16 16 16 a. What is the length of the radius of this circle? 4 b. What is the length of the radius squared? 16 c. Do you see any connection between the length of the radius squared and column 4 in the table above? They both are the same value. d. How is this circle shifted from a circle with the same radius length but centered at (0, 0)? This circle is shifted left 2 and up 1 from a circle with the radius 4 centered at (0, 0). 3. Based on #1c and #2c, write a general equation to represent all the points (x, y) of a circle whose center is (h, k) and whose radius is r. (x – h)2 + (y – k)2 = r2 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 73 of 399 Columbus City Schools 6/28/13 A – CED 4 Name ___________________________________ Date __________________ Period ________ Using Formulas There are many formulas used in everyday life and in science. Depending on what information you know and what you wish to find, you may need to use a formula in a different way. Look at the following examples. Example 1 The formula for finding the perimeter of a rectangle is P = 2l + 2w, where P represents perimeter, l represents length, and w represents width. A. Find the perimeter of a rectangle with a length of 20 inches and a width of 14 inches. Substitute the value of each variable into the formula and solve the equation. In part A, you solved the equation by evaluating the right side of the equation because you knew the value of every variable on the right hand side. B. Find the length of the rectangle if the perimeter is 240 inches and the width is 50 inches. Substitute the value of each variable into the formula and solve the equation. In part B, you knew some information from each side of the equation and you had to solve the equation by performing operations on both sides of the equation. Example 2 Find the width of the rectangle if: A) the length is 10 feet and the perimeter is 50 feet. B) the length is 25 cm and the perimeter is 75 cm. C) the length is 145 inches and the perimeter is 540 inches. A. B. C. Although this worked well enough, it was inefficient because you repeated the same operations three times with a different set of numbers. By solving the formula for the width, and then substituting the different values for length and perimeter, you would not be repeating steps. Second method: Solve the formula for width, and then substitute the values for length and perimeter. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 74 of 399 Columbus City Schools 6/28/13 P = 2l + 2w To solve for w, treat w as the variable which you want to isolate, and deal with P and l the way you normally use numbers in solving equations. P – 2l = 2l + 2w – 2l P – 2l = 2w First, subtract 2l from both sides of the equation. P 2l 2w 2 2 P 2l w 2 A. Next, divide both sides of the equation by 2. Now you can find the width of each rectangle by substituting the values for the perimeter and the length of each rectangle into the new expression. B. C. As you can see, the second method is more efficient if you need to find the value of the width in more than one situation. Extra Practice h b1 b2 . Solve the formula for the height. 2 (You will need to get the variable h on one side, by itself.) 1. The formula for the area of a trapezoid is A Find the height of the trapezoid if: A) the area is 400 in2 and the bases are 10 inches and 30 inches. B) the area is 1500 cm2 and the bases are 50 cm and 150 cm. C) the area is 20 yd2 and the bases are 5 yd and 10 yd. 2. Using the formula for a trapezoid, solve the formula for b2. Find b2 if: A) the area is 400 ft2, the height is 4 ft, and one base is 30 ft. B) the area is 1500 cm2, the height is 10 cm, and one base is 100 cm. C) the area is 20 yd2, the height is 1 yd, and one base is 4 yd. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 75 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Using Formulas Answer Key There are many formulas used in everyday life and in science. Depending on what information you know and what you wish to find, you may need to use a formula in different ways. Look at the following examples. Example 1 The formula for finding the perimeter of a rectangle is P = 2l + 2w, where P represents perimeter, l represents length, and w represents width. A. Find the perimeter of a rectangle with a length of 20 inches and a width of 14 inches. Substitute the value of each variable into the formula and solve the equation. P = 2(20) + 2(14) P = 40 + 28 P = 68 Perimeter = 68 inches B. Find the length of the rectangle if the perimeter is 240 inches and the width is 50 inches. Substitute the value of each variable into the formula and solve the equation. 240 = 2l + 2(50) 240 = 2l + 100 140 = 2l 70 = l, length = 70 inches In part A, you solved the equation by evaluating the right side of the equation because you knew the value of every variable on the right hand side. In part B, you knew some information from each side of the equation and you had to solve the equation by performing operations on both sides of the equation. Example 2 Find the width of the rectangle if A) the length is 10 feet and the perimeter is 50 feet. B) the length is 25 cm and the perimeter is 75 cm. C) the length is 145 inches and the perimeter is 540 inches. A. P = 2l + 2w 50 = 2(10) + 2w 50 = 20 + 2w 50 - 20= 20 + 2w- 20 30 = 2w 15 = w width = 15 feet B. P = 2l + 2w 75 = 2(25) + 2w 75 = 50 + 2w 75 - 50 = 50 + 2w - 50 25 = 2w 12.5 = w width = 12.5 cm C. P = 2l + 2w 540 = 2(145) + 2w 540 = 290 + 2w 540 - 290= 290 + 2w - 290 250 = 2w 125 = w width = 125 inches Although this worked well enough, it was inefficient because you repeated the same operations three times with a different set of numbers. By solving the formula for the width, and then substituting the different values for length and perimeter, you would not be repeating steps. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 76 of 399 Columbus City Schools 6/28/13 Second method: Solve the formula for width, and then substitute the values for length and perimeter. P = 2l + 2w To solve for w, treat w as the variable which you want to isolate, and deal with P and l the way you normally use numbers in solving equations. P – 2l = 2l + 2w – 2l First, subtract 2l from both sides of the equation. P – 2l = 2w Next, divide both sides of the equation by 2. P 2l 2 2w 2 P 2l 2 w 50 - 2(10) 2 50 - 20 w= 2 30 w= = 15 2 width = 15 feet A. w = Now you can find the width of each rectangle by substituting the values for the perimeter and the length of each rectangle into the new expression. 75 - 2(25) 2 75 - 50 w= 2 25 w= = 12.5 2 width = 12.5 cm B. w = 540 - 2(145) 2 540 - 290 w= 2 250 w= = 125 2 width = 125 inches C. w = As you can see, the second method is more efficient if you need to find the value of the width in more than one situation. Extra Practice h b1 b2 1. The formula for the area of a trapezoid is A . Solve the formula for the height. 2 2A = h b1 + b2 2A =h b1 + b2 Find the height of the trapezoid if: A) the area is 400 in2 and the bases are 10 inches and 30 inches. B) the area is 1500 cm2 and the bases are 50 cm and 150 cm. C) the area is 20 yd2 and the bases are 5 yd and 10 yd. A. height = 20 inches B. height = 15 cm C. height = 2.67 yds 2. Using the formula for a trapezoid, solve the formula for b2. 2A = b1 + b2 h 2A - b1 = b2 h Find b2 if: A) the area is 400 ft2, the height is 4 ft, and one base is 30 ft. b2 = 170 ft B) the area is 1500 cm2, the height is 10 cm, and one base is 100 cm. b2 = 200 cm C) the area is 20 yd2, the height is 1 yd, and one base is 4 yd. b2 = 36 yd CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 77 of 399 Columbus City Schools 6/28/13 A – CED 4 Name ___________________________________ Date __________________ Period ________ Literal Madness Solve the following literal equations for the indicated variable: 1. Solve for x: x d 3 2. Solve for z: a z b 2 3. Solve for a: s 4. Solve for s: v2 u 2 2as 5. Solve for n: s n 2 180 6. Solve for P: A P Prt 7. Solve for h: T 8. Solve for d: c 2dh 3bh 9. Solve for s: T c c 1 2 at 2 2 r 2 2 rh sk a s 1 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 78 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Literal Madness Answer Key Solve the following literal equations for the indicated variable: x d 1. Solve for x: x + d = 3c c, 3 x = 3c - d a z b 2 2. Solve for z: c, a + z – b = 2c z = 2c – a + b 3. Solve for a: 1 2 at , 2 s 2s = at2 2s t2 v2 4. Solve for s: 5. Solve for n: u 2 2as , a v 2 u2 v 2 u2 2a 2as s n 2 180 , s s = 180n - 360 s + 360 = 180n or s = n-2 180 s +2= n 180 s + 360 =n 180 6. Solve for P: A P Prt , A P 1 rt A 1 rt 7. Solve for h: T 2 r 2 2 rh , P T 2 r2 2 rh CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 79 of 399 Columbus City Schools 6/28/13 2 r2 h 2 r c + 3bh = 2dh c 3bh d 2h T 8. Solve for d: c 2dh 3bh , 9. Solve for s: T sk a , s 1 T ( s 1) sk a Ts T sk a Ts sk T a s T k s T a T a T k CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 80 of 399 Columbus City Schools 6/28/13 Adding and Subtracting Radicals Cards 2 3 27 5 3 8 72 4 2 125 20 7 5 24 54 5 6 3 18 19 2 2 50 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 81 of 399 Columbus City Schools 6/28/13 Adding and Subtracting Radicals Cards 80 45 32 128 27 48 5 12 2 75 6 3 2 98 3 72 32 2 40 250 3 10 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 82 of 399 Columbus City Schools 6/28/13 Multiplying and Dividing Radicals Cards 2 6 2 3 3 12 6 5 10 5 2 (2 6)(3 2) 2 (5 2) 12 3 50 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 83 of 399 Columbus City Schools 6/28/13 Multiplying and Dividing Radicals Cards 8 27 6 6 2 3( 18 32) 14 6 2(2 3 48) 2 6 (2 3 3)(2 3 3) 3 3 3 2 2 2 2 3 2 2 -23 9 6 2 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 84 of 399 Columbus City Schools 6/28/13 Multiplying and Dividing Radicals Cards 2 3 6 3 1 2 2 2 50 18 5 3 15 3 5 3 12 2 6 2 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 85 of 399 Columbus City Schools 6/28/13 Multiplying and Dividing Radicals Cards 7 12 21 6 4 3 8 6 5 24 30 12 3 48 3 4 8 27 2 6 9 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 86 of 399 Columbus City Schools 6/28/13 Match Me Cards (2 3)(5 6) 24 48 30 2 2 6 75 9 3 2 3 6 3 8 2 4 2 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 87 of 399 Columbus City Schools 6/28/13 Match Me Cards (5 3)(5 20 3) 80 5 50 4 32 45 22 3 5 9 2 (3 2)2 18 ( 3)( 20) 2 15 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 88 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Radicals Rule! Read each rule and complete the examples that apply to the rule. Rule 1: If two numbers are multiplied under a radical sign, you can rewrite them under two different radical signs that are multiplied together. The first example is done for you. 9 36 = 3∙6 = 18 1. (9)(36) 2. (4)(16)(25) = 3. 49x = 4. (4)(11)x 4 y 3 = Rule 2: If two radical expressions are multiplied together, you rewrite them as products under the same radical sign. 5. 3 5 = (3)(5) = 15 6. 8 2 = 7. 7 x3 21x 2 = 8. 12 3y 8 = Rule 3: You can factor the number under a radical and take the square root of the factors that are perfect squares. 9. 12 10. 20x = 4•3 = 2 3 11. 125x3 y 5 z = 12. 27h12 = Rule 4: If two numbers are divided under a radical sign, you can rewrite them under two different radial signs separated by a division sign. 13. 64 36 64 8 4 = = 36 6 3 14. x4 = 4 y6 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 89 of 399 Columbus City Schools 6/28/13 N-RN 1 15. 147x 4 = z10 16. 3 y8 = 25 Rule 5: You can multiply the numerator and denominator of a radical expression by the same number and not change the value of the expression. This will help you eliminate radical expressions in the denominator. 17. 3 5 19. 5 = 7 3 5 15 • = 5 5 5 18. 3y2 = 2 20. 6 x3 = 15 y 5 Rule 6: You can add and subtract radical expressions if each index is the same and each radicand is the same. When you add radical expressions, add the coefficients. 21. 2 3 + 3 3 = 5 3 22. 3 5 5 3 5 = 23. 14 x - 9x = 24. 5 2x3 - 7 8x 3 = 25. The square root of a number plus two times the square root of the same number is twelve. What is the number? 26. Seth lives in Bethesda. Sarah lives nine miles east of Seth. Tara lives twelve miles north of Sarah. How far does Seth live from Tara? CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 90 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Radicals Rule! Answer Key Read each rule and complete the examples that apply to the rule. Rule 1: If two numbers are multiplied under a radical sign, you can rewrite them under two different radical signs that are multiplied together. The first example is done for you. 1. (9)(36) 3. 49x = 9 36 = 3∙6 = 18 49 • x = 7 x 4 • 16 • 25 = 40 2. (4)(16)(25) = 4. (4)(11)x 4 y 3 = 4 • 11 • x 4 • y 3 =2x2y 11y Rule 2: If two radical expressions are multiplied together, you rewrite them as products under the same radical sign. 5. 3 5 = 7. 7 x3 (3)(5) = 15 21x 2 = 7 •7 • 3 • x 5 =7x2 3x 6. 8 2 = 8. 12 8 • 2 = 16 = 4 3y 8 = 36y 8 = 6y4 Rule 3: You can factor the number under a radical and take the square root of the factors that are perfect squares. 9. 11. 12 4•3 = 2 3 125x3 y 5 z = 5 • 25 • x 3 y 5 z 10. 20x = 12. 27h12 = 4 • 5 • x = 2 5x 9 • 3h12 = 3h6 3 = 5xy2 5xyz Rule 4: If two numbers are divided under a radical sign, you can rewrite them under two different radial signs separated by a division sign. 13. 64 36 64 8 4 = = 36 6 3 14. x4 = 4 y6 x4 4y 6 = x2 2y 3 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 91 of 399 Columbus City Schools 6/28/13 3y 8 7x 2 3 147x 4 3 y8 y4 3 7 •7 • 3x 4 15. = = 16. = = z10 25 z5 5 25 z 10 Rule 5: You can multiply the numerator and denominator of a radical expression by the same number and not change the value of the expression. This will help you eliminate radical expressions in the denominator. 17. 3 5 19. 5 7 5 7 5 • = = 7 7 7 7 3 5 15 • = 5 5 5 18. 3y2 y 3 2 y 6 • = = 2 2 2 2 20. 3x 10xy 15y x 6x 6 x3 • = 2 = 5 15y 3 15 y y 15y 15y Rule 6: You can add and subtract radical expressions if each index is the same and each radicand is the same. When you add radical expressions, add the coefficients. 21. 2 3 + 3 3 = 5 3 23. 14 x - 9x = 14 x - 3 x = 11 x 22. 3 5 53 5 = 6 3 5 24. 5 2x3 - 7 8x 3 = 5x 2x - 14x 2x = -11x 2x 25. The square root of a number plus two times the square root of the same number is twelve. What is the number? x +2 x = 12 3 x = 12 x =4 x = 16 26. Seth lives in Bethesda. Sarah lives nine miles east of Seth. Tara lives twelve miles north of Sarah. How far does Seth live from Tara? 92 + 122 = c2 81 + 144 = c2 225 = c2 15 = c CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 92 of 399 Columbus City Schools 6/28/13 N-RN 1 Name ___________________________________ Date __________________ Period ________ Addition and Subtraction of Radicals Use the properties of exponents and radicals to complete the given operation. 1. -3 144 + 2 81 = 2. 8 196 - 3. 8 13 + 8 13 = 4. 5 6 + 7 54 = 5. 7 18 - 8 = 7. -8 32 + 5 72 = 9. 2 5 98 72 = 5 6 11. 5 432 + 6 147 = 25 = 6. -6 10 - 360 = 8. 9 225 + 2 144 = 3 10. 6 384 - 9 216 = 12. 2 72 - 5 512 = 5 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 93 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Addition and Subtraction of Radicals Answer Key Use the properties of exponents and radicals to complete the given operation. 1. -3 144 + 2 81 = -18 2. 8 196 - 3. 8 13 + 8 13 = 16 13 4. 5 6 + 7 54 = 26 6 5. 7 18 - 8 = 19 2 7. -8 32 + 5 72 = -2 2 9. 2 5 -11 98 72 = 2 5 6 5 11. 5 432 + 6 147 = 102 3 25 = 107 6. -6 10 - 360 = -12 10 8. 9 225 + 2 144 = 143 3 10. 6 384 - 9 216 = -6 6 12. 2 -388 72 - 5 512 = 2 = -77.6 2 5 5 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 94 of 399 Columbus City Schools 6/28/13 N-RN 1 Name ___________________________________ Date __________________ Period ________ Multiplication and Division of Radicals Use the properties of exponents and radicals to complete the given operation. 1. (-4 49 )( 9 ) = 2. 16 = 7 196 3. (12 7 )2 = 4. 2 50 = 5 2 5. (11 5 )(7 10 ) = 6. 160 = 810 7. 1200 = 588 9. ( 250 )(-8 180 ) = 8. (8 .49 )(-5 7 ) = 10. 70 138 = 14 23 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 95 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Multiplication and Division of Radicals Answer Key Use the properties of exponents and radicals to complete the given operation. 1. (-4 49 )( 9 ) = -84 2. 16 2 = 49 7 196 3. (12 7 )2 = 1008 4. 2 50 =2 5 2 5. (11 5 )(7 10 ) = 385 2 6. 160 4 = 9 810 7. 1200 10 = 7 588 9. ( 250 )(-8 180 ) = -1200 2 8. (8 .49 )(-5 7 ) = -28 7 10. 70 138 =5 6 14 23 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 96 of 399 Columbus City Schools 6/28/13 N-RN 1 Name ___________________________________ Date __________________ Period ________ Radical Expressions Simplify the Expression. 5 32 = 2. 4 3. 17 3 1728 = 4. 63 64 = 7 1. 5. 7. 9. 4 625 = 494 - 2 = 6. -8 3 8k 9 c3 g 4 = 20736e9 f 10 j 8 d 6 = 8. 20e5 h3 = 10. 375g 4 = 3 324 f 5 3 12 f 5 = CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 97 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Radical Expressions Answer Key Simplify the Expression. 5 32 = -2 2. 4 3. 17 3 1728 = -204 4. 63 24 3 64 = =3 7 7 7 1. 625 = 5 4 5. 7. 9. 4 494 - 2 = 2399 6. -8 3 8k 9 c3 g 4 = -16k3cg 3 g or -16k3c g 3 20736e9 f 10 j 8 d 8 = 12e2f2j2d2 4 ef 2 8. 20e5 h3 = 2e2h 5eh 10. 375g 4 = 5g2 15 3 324 f 5 3 12 f 5 =3 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 98 of 399 Columbus City Schools 6/28/13 Radical and Exponent Matching Exponential Expression Cards 8 1 3 25 1 17 4 3 2 3 x y 2a 1 3 2 x c5 27 2 1 3 1 2 81 4 3 7 x 3 8 5y 1 abc 7 3 a 1 3 2 x5 64x 5 3 1 2 32 5 6 2 7 1 5 y2 16x 1 2 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 99 of 399 a5 1 2 2 3 4 1 2 2 1 9 Columbus City Schools 6/28/13 Radical and Exponent Matching Radical Expression Cards 3 4 8 253 2a 5 c 7 8 125y 3 7 3 x 5 3 5 x2 64x 2 x7 5 2 9 6 a 2b 2 c 2 3 x3 y 3 3 23 2 17 y 4 y2 4x a CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 100 of 399 9 a5 Columbus City Schools 6/28/13 N-RN 1 Name ___________________________________ Date __________________ Period ________ Find Two Ways Find two equivalent expressions for each radical or exponential expression given below. Given Expression Equivalent Expression #1 Equivalent Expression #2 1 24 1 25 a 5 4 x x 3 x 81 9 3 x5 a2 a4 x3 x x x 2 3 1 6 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 101 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Find Two Ways Answer Key Find two equivalent expressions for each radical or exponential expression given below. Student answers will vary. Given Expression Equivalent Expression #1 Equivalent Expression #2 1 24 1 42 1 16 1 25 1 1 5 25 5 4 a4 x x 3 x x a5 a4 a x3 x2 x a2 x 5 3 x 3 x2 1 4 a2 x3 1 6 3 2 2 a2 a4 x 1 x6 32 x5 2 3 1 2 1 3 3 x 3 4 81 9 3 1 2 x 1 2 x x2 3 2 3 x2 6 x CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 102 of 399 6 x3 Columbus City Schools 6/28/13 F-IF 7b Name ___________________________________ Date __________________ Period ________ Graphing Square Roots Using your graphing calculator, complete the following table: Note: The Parent Function has been done for you. Equation Domain x and/or y Transformations Sketch of the Graph with Three Exact (in radical and intercept(s) from the Points Labeled and Range Parent exponential Function form) y= x Domain: xNone – This is [0,∞) intercept(s): the Parent (0,0) Square Root Function 1 y = x2 Range: [0,∞) y= yintercept(s): (0,0) x +2 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 103 of 399 Columbus City Schools 6/28/13 Equation (in radical and exponential form) y= x -3 y= x 3 y= x 2 Equation Domain x and/or y Transformations and intercept(s) from the Range Parent Function Sketch of the Graph with Three Exact Points Labeled Domain Sketch of the Graph with Three Exact x and/or y Transformations CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 104 of 399 Columbus City Schools 6/28/13 (in radical and exponential form) and Range intercept(s) from the Parent Function Points Labeled y=- x y= 1 x 2 y=3 x CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 105 of 399 Columbus City Schools 6/28/13 Equation (in radical and exponential form) y= x y= 5 x Domain x and/or y Transformations and intercept(s) from the Range Parent Function Sketch of the Graph with Three Exact Points Labeled y= -2 x 2 + 3 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 106 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Graphing Square Roots Answer Key Using your graphing calculator, complete the following table: Note: The Parent Function has been done for you. Equation Domain x and/or y Transformations Sketch of the Graph with Three Exact (in radical and intercept(s) from the Points Labeled and Range Parent exponential Function form) y= x Domain: xNone – This is [0,∞) intercept(s): the Parent (0,0) Square Root Function 1 y = x2 Range: [0,∞) y= yintercept(s): (0,0) Domain: x-intercept: none x + 2 [0, ∞) 1 2 y= x +2 Range: [2, ∞) Vertical shift up 2 units y-intercept: (0, 2) CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 107 of 399 Columbus City Schools 6/28/13 Equation (in radical and exponential form) y= x -3 1 Domain and Range x and/or y Transformations intercept(s) from the Parent Function Domain: x[0, ∞) intercept: (9, 0) Range: [-3, ∞) yintercept: (0, -3) Vertical shift down 3 units Domain: x[-3, ∞) intercept: (-3, 0) Range: [0, ∞) yintercept: (0, 3 ) Horizontal shift left 3 units Domain: x[2, ∞) intercept: (2, 0) Range: [0, ∞) yintercept: none Horizontal shift right 2 units Sketch of the Graph with Three Exact Points Labeled y = x2 - 3 y= x 3 y= x+ 3 y= 1 2 x 2 y= x-2 1 2 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 108 of 399 Columbus City Schools 6/28/13 Equation and Equation in Exponential Form Domain and Range x and/or y intercept(s) Transformations from the Parent Function Sketch of the Graph with Three Exact Points Labeled Domain: xReflection about the [0, ∞) intercept: x-axis (0, 0) Range: [0, - ∞) y-intercept: (0, 0) y=- x 1 y = - x2 y= 1 x 2 y= 1 x 2 y=3 x Domain: xVertical shrink by a [0, ∞) intercept: 1 factor of (0, 0) 2 Range: [0, ∞) y-intercept: (0, 0) 1 2 Domain: x[0, ∞) intercept: Vertical stretch by a (0, 0) factor of 3 Range: [0, ∞) y-intercept: (0, 0) 1 y = 3 x2 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 109 of 399 Columbus City Schools 6/28/13 Equation and Equation in Exponential Form y= x and/or y intercept(s) Transformations from the Parent Function Domain: xReflection about the (- ∞, 0] intercept: y-axis (0, 0) Range: [0, ∞) y-intercept: (0, 0) x y = -x Domain and Range 1 2 Sketch of the Graph with Three Exact Points Labeled 10 8 6 (-9., 3.) 4 (-4., 2.) 2 -10 -8 -6 -4 -2 (0., 0.) 2 4 6 8 10 -2 -4 -6 -8 -10 y= 5 x y= 5-x 1 2 y= -2 x 2 +3 y= -2 x - 2 1 2 Domain: xReflection about the (- ∞, 5] intercept: y-axis and horizontal (5, 0) shift right 5 units Range: [0, ∞) y-intercept: (0, 5 ) Domain: x[2, ∞) intercept: (4.25, 0) Range: (-∞, 3] y-intercept: none Reflection about the x-axis, vertical stretch by a factor of 2, horizontal shift right 2 units, and vertical shift up 3 units +3 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 110 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Graphing Cube Roots Using your graphing calculator, complete the following table: Note: The Parent Function has been done for you. Equation Domain x and/or y Transformations Sketch of the Graph with Three Exact and and intercept(s) from the Points Labeled Equation in Range Parent Exponential Function Form y= 3 x y= x y= 3 1 3 Domain: x(- ∞, ∞) intercept: (0, 0) Range: (-∞, ∞) yintercept: (0, 0) None – This is the Parent Cube Root Function x+3 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 111 of 399 Columbus City Schools 6/28/13 F-IF 7b Equation Domain x and/or y Transformations and and intercept(s) from the Equation in Range Parent Exponential Function Form y= 3 x-2 y= 3 x-4 y= 3 x+3 Sketch of the Graph with Three Exact Points Labeled CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 112 of 399 Columbus City Schools 6/28/13 Equation Domain x and/or y Transformations and and intercept(s) from the Equation in Range Parent Exponential Function Form y=- y= 3 Sketch of the Graph with Three Exact Points Labeled x 13 x 2 y=33 x CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 113 of 399 Columbus City Schools 6/28/13 Equation and Equation in Exponential Form y= 3 -x y= 3 5- x Domain x and/or y Transformations and intercept(s) from the Range Parent Function Sketch of the Graph with Three Exact Points Labeled y= 13 x 1 +5 2 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 114 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Graphing Cube Roots Answer Key Using your graphing calculator, complete the following table: Note: The Parent Function has been done for you. Equation Domain x and/or y Transformations Sketch of the Graph with Three Exact and and intercept(s) from the Points Labeled Equation in Range Parent Exponential Function Form y= 3 x y= x y= 3 Domain: x(- ∞, ∞) intercept: (0, 0) Range: (-∞, ∞) yintercept: (0, 0) 1 3 x+3 y= x 1 3 xDomain: intercept: (- ∞, ∞) (-27, 0) +3 Range: (-∞, ∞) None – This is the Parent Cube Root Function Vertical shift up 3 units yintercept: (0, 3) CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 115 of 399 Columbus City Schools 6/28/13 Equation and Equation in Exponential Form y= 3 x-2 y= x y= 3 1 3 -2 x-4 y = x-4 1 3 Domain and Range x and/or y Transformations intercept(s) from the Parent Function Domain: x(- ∞, ∞) intercept: (8, 0) Range: (-∞, ∞) yintercept: (0, -2) Domain: x(- ∞, ∞) intercept: (4, 0) Range: (-∞, ∞) yintercept: (0, 3 -4 ) Sketch of the Graph with Three Exact Points Labeled Vertical shift down 2 units Horizontal shift right 4 units 10 y= 3 x+3 y= x+ 3 1 3 Domain: x(- ∞, ∞) intercept: (-3, 0) Range: (-∞, ∞) yintercept: (0, 3 3 ) 8 Horizontal shift left 3 units 6 (8., 6.) 4 2 -10 -8 -6 -4 -2 (-1., -3.) -2 (1., 3.) (0.,20.) 4 6 8 10 -4 -6 (-8., -6.) -8 -10 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 116 of 399 Columbus City Schools 6/28/13 Equation and Equation in Exponential Form y=- 3 x and/or y Transformations intercept(s) from the Parent Function Domain: x(- ∞, ∞) intercept: (0, 0) Range: (-∞, ∞) yintercept: (0, 0) x y=- x Domain and Range 1 3 Sketch of the Graph with Three Exact Points Labeled Reflection about the xaxis 10 y= y= 13 x 2 1 3 1 x 2 Domain: x(- ∞, ∞) intercept: (0, 0) Range: (-∞, ∞) yintercept: (0, 0) Vertical shrink by a factor of 1 2 8 6 4 (8., 1.) 2 (1., .5) (-1., -.5) -10 -8 -6 -4 -2 (-8., -1.) -2 (0.,20.) 4 6 8 10 -4 -6 -8 -10 y=33 x y=3 x 1 3 Domain: x(- ∞, ∞) intercept: (0, 0) Range: (-∞, ∞) yintercept: (0, 0) Vertical stretch by a factor of 3 10 8 6 (8., 6.) 4 2 -10 -8 -6 -4 -2 (-1., -3.) -2 (1., 3.) (0.,20.) 4 6 8 10 -4 -6 (-8., -6.) -8 -10 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 117 of 399 Columbus City Schools 6/28/13 Equation and Equation in Exponential Form y= 3 Domain: (- ∞, ∞) -x y = -x Domain and Range Range: (-∞, ∞) 1 3 x and/or y intercept(s) Transformations from the Parent Function x-intercept: Reflection (0, 0) about the yaxis y-intercept: (0, 0) Sketch of the Graph with Three Exact Points Labeled 10 8 6 (-8., 2.) 4 (-1., 1.) 2 (0., 0.) 2 4 -10 -8 -6 -4 -2 -2 -4 (1., -1.) 6 8 10 (8., -2.) -6 -8 -10 y= 3 Domain: (- ∞, ∞) 5- x y= 5-x 1 3 y= 13 x 1 +5 2 y= 1 1 - x - 1 3 +5 2 Range: (-∞, ∞) Domain: (- ∞, ∞) Range: (-∞, ∞) x-intercept: Reflection (5, 0) about the yaxis and a y-intercept: horizontal shift right 5 (0, 3 5 ) x-intercept: Reflection (1001, 0) about the xaxis, vertical y-intercept: shrink by a (0, 5.5) 1 factor of , 2 vertical shift up 5 units, and horizontal shift right 1 unit CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 118 of 399 Columbus City Schools 6/28/13 F-IF 7b Name ___________________________________ Date __________________ Period ________ Graphing Cube and Square Roots Use your graphing calculator to complete the table. Equation Domain x and/or y Transformations and and intercept(s) from the Equation in Range Parent Exponential Function Form Sketch of the Graph with Three Exact Points Labeled 1 y = x2 y= 3 x CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 119 of 399 Columbus City Schools 6/28/13 Equation and Equation in Exponential Form y = ( x 5) Domain and Range x and/or y intercept(s) Transformations from the Parent Function Sketch of the Graph with Three Exact Points Labeled 1 3 y= 3 x y= 3 5 x CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 120 of 399 Columbus City Schools 6/28/13 Equation and Equation in Exponential Form Domain and Range x and/or y intercept(s) Transformations from the Parent Function Sketch of the Graph with Three Exact Points Labeled 1 y = ( x 4) 2 y= x 1 2 y= 3 4 x 2 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 121 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Graphing Cube and Square Roots Answer Key Use your graphing calculator to complete the table. Equation and Equation in Exponential Form 1 y = x2 x y= y= 3 x 1 y = x3 Domain and Range x and/or y Transformations intercept(s) from the Parent Function Domain: x[0, ∞) intercept: (0, 0) Range: [0, ∞) yintercept: (0, 0) Domain: x(- ∞, ∞) intercept: (0, 0) Range: (-∞, ∞) yintercept: (0, 0) Sketch of the Graph with Three Exact Points Labeled None – this is the parent function None – this is the parent function CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 122 of 399 Columbus City Schools 6/28/13 Equation and Equation in Exponential Form 1 0 8 6 4 ( 2 - - - - - - ( - ( 2 4 6 8 1 1 ( ( 1 4 1 8 6 4 2 5 2 - - 4 2 . 0 2 3 0 , . , . 8 1 0 y = ( x 5) y= 3 . 1 3 x-5 Domain and Range , , . 4 6 - - 0 - 2 . 1 , 1 . ) . - ) 2 . x and/or y intercept(s) ) . ) ) Domain: x(- ∞, ∞) intercept: (5, 0) Range: (-∞, ∞) yintercept: (0, - 3 5 ) Transformations from the Parent Function Sketch of the Graph with Three Exact Points Labeled Horizontal shift right 5 units 10 y= 3 x y = 3- x 1 2 Domain: x(- ∞, 3] intercept: (3, 0) Range: [0, ∞) yintercept: (0, 3 ) Reflection about the yaxis and horizontal shift right 3 units 8 6 (-6., 3.) 4 2 (-1., 2.) -10 -8 -6 -4 -2 -2 (2., 1.) 2 4 (3., 0.) 6 8 10 -4 -6 -8 -10 y= 3 5 x y= - 5-x 1 3 Domain: x(- ∞, ∞) intercept: (5, 0) Range: (-∞, ∞) yintercept: (0, - 3 5 ) Reflection about the xaxis and horizontal shift right 5 units CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 123 of 399 Columbus City Schools 6/28/13 Equation and Equation in Exponential Form 1 y = ( x 4) 2 y= x+4 y= x 1 2 y= 1 - x + 1 2 -2 y= 3 4 x 2 y= 1 4 - x 3 +2 Domain and Range x and/or y intercept(s) Transformations from the Parent Function Domain: x[-4, ∞) intercept: (-4, 0) Range: [0, ∞) yintercept: (0, 2) Horizontal shift left 4 units Domain: x[-1, ∞) intercept: none Range: (- ∞, -2] yintercept: (0, -3) Reflection about the xaxis, horizontal shift right 1 unit, and a vertical shift down 2 units Domain: x(- ∞, ∞) intercept: (12, 0) Range: (-∞, ∞) yintercept: (0, 3 4 + 2 ) Sketch of the Graph with Three Exact Points Labeled Reflection about the yaxis and a horizontal shift right 2 units CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 124 of 399 Columbus City Schools 6/28/13 Conceptual Category: Number & Quantity Course: Math 2 Quarter: 2 Domain: The Real Number System (N-RN3) Time Frame: 3 days Cluster: Use properties of rational and irrational numbers Common Core Content Standards Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational Mathematical Practices Model with mathematics Attend to precision Look for and express regularity in repeated reasoning Expectations for Learning Students will be able to: classify real numbers as either rational or irrational numbers graph real numbers accurately and precisely on a number line calculate the exact perimeter, circumference, and area of shapes that model mathematical situations based on observations of the sum and product of real numbers, students will be able to determine: The sum of the product of rational numbers is rational. The sum of a rational number and an irrational number is irrational. The product of a nonzero rational number and an irrational number is irrational. Prerequisite Skills Students must be able to: describe differences between rational and irrational numbers use technology to show that some numbers (rational) can be expressed as terminating or repeating decimals and others (irrational) as non-terminating and non-repeating decimals round numbers to a specific place value and comprehend the concept of non-terminating and nonrepeating decimals CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 125 of 399 Columbus City Schools 6/28/13 recognize that natural numbers, whole numbers, integers are subsets of rational numbers; and that rational numbers and irrational numbers are subsets of the real number system recognize and identify perfect squares and their roots find the square root of perfect squares, and approximate the square root of non-perfect squares as consecutive integers between which the root lies (a radicand of 76 is between perfect squares 64 and 81) determine the appropriate formula used to calculate the perimeter, circumference, or area for situations modeled by certain shapes identify the correct measurements to be substituted into the formulas based on the situation use the order of operations correctly to determine the perimeter, circumference, or area translate geometric symbols for congruence to its correct numerical value for use in calculations Misconceptions/Challenges Students do not understand that natural numbers, whole numbers, and integers are examples of rational numbers. Students do not understand that repeating decimals are classified as rational numbers. Students use “infinity” to describe non-terminating decimals. Students use the wrong formula to represent a mathematical situation or substitute the wrong values into the formula. For example, given an isosceles triangle, students will sometimes substitute the slant height for the height in the formula rather than the altitude. Students are challenged using problem solving skills to find a missing measurement when calculating the area of irregular figures. Strategies to Address Misconceptions/Challenges Use a calculator to compare and contrast the attributes of given rational and irrational numbers. Define a rational number and use the definition to define an irrational number. Use a website (see Technology & Other Resources) that demonstrates methods for changing repeating decimals to fractions. Discussions about (for example, ask students to explain the differences between infinity and ). Warm-ups at the beginning of Days 2 and 3 to discuss matching formulas to shapes and CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 126 of 399 Columbus City Schools 6/28/13 identifying what numerical values in the image correlate with the variables in the formula. Students can work collaboratively with a partner on activities; pair/share reasoning and communications should facilitate understanding. Instructional Strategies Guided Discussion – Warm-up-- Day 1: classify real numbers (definitions of rational and irrational numbers); Day 2: derivation of perimeter formula and geometric symbols for equality; Day 3: matching shapes to the formulas used to determine circumference or area. Independent Practice – Students work individually to classify and graph real numbers on a number line; calculating the perimeter, circumference, or area of specific shapes. Think-Pair-Share – Students think, classify, and calculate individually; then pair up with a partner to share the results after each problem. Students should discuss their results or help each other. Guided Practice – While the students are working independently or in groups of two, the teacher circulates to provide guidance and support to foster critical thinking and understanding through the use of quality questioning (When graphing, where would lie on number line in relationship to 22/7?). Day 1 – Warm-up with a class discussion on classifying real numbers (definitions of rational and irrational numbers). Students can complete the real number classification cards activity and then graph the numbers on a number line. Students can work in pairs to check their answers. Teachers may laminate and cut out the cards to create a class set of real number cards to save paper and to be able to use in subsequent years. Day 2 – Warm-up with derivation of perimeter formula and geometric symbols for equality. For each situation, students will determine the formula necessary to calculate the perimeter of the shape represented in each situation. Fill in the appropriate quantities (numbers) that represent the lengths of each side, and then calculate the exact perimeter. Classify each of the quantities substituted into the formula as rational or irrational numbers; then students should determine if the resulting CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 127 of 399 Columbus City Schools 6/28/13 perimeter is a rational or irrational number. Students should conclude that the sum of rational numbers is a rational number, and the sum of a rational number and an irrational number is an irrational number. Homework Day 2 – Students will create two polygons, one where the sides are all rational, and one where at least one side is irrational and the remaining sides are rational. They will also calculate the perimeter of each polygon and remember to classify each length and the final area as rational or irrational. Day 3 – For each situation, students will determine the formula necessary to calculate the area or circumference of the shape represented in each situation. Fill in the appropriate quantities (numbers) that represent the measurements needed to find the area, and then calculate the exact area. Classify each of the quantities substituted into the formula as rational or irrational numbers, and then determine if the resulting area is a rational or irrational number. Students should be able to conclude that the product of rational numbers is a rational number, and the product of a nonzero rational number and an irrational number is an irrational number. Homework Day 3 – Students will create two polygons, one where the measurements used to calculate the area are all rational, and one where at least one length used to calculate the area is irrational and the remaining lengths are rational. They will also calculate the area of each polygon and remember to classify each length and the final area as rational or irrational. Extension Discussion – Ideas for group discussions: What if a situation required you to find the sum or product of two irrational numbers? Would the sum or product be equal to a rational number or an irrational number? In groups of 3, justify your answer. Technology & Other Resources http://easycalculations.com/recursive-fraction.php On-line calculator to change repeating decimals to equivalent fractions. Scientific calculators (any model) can be used for decimal approximations of real numbers. http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fractions/FractionsCalc.html Online calculator to turn a fraction into a decimal where you decide how many decimal places are shown; tool will indicate whether the decimal repeats or terminates; will also indicate how many decimal places until it terminates, or the period of the recurring decimal. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 128 of 399 Columbus City Schools 6/28/13 Universal Skills (Relationships through 21st Century 4C Skills-Collaboration, Communication, Creativity, Critical Thinking) Students will: apply critical thinking and problem solving strategies during structured learning experiences present resources and data in a format that effectively communicates the meaning of the data and implications for solving the problem(s) using different perspectives assume a leadership position by guiding the thinking of peers in a direction that leads to successful completion of a challenging task or project demonstrate a positive work ethic in various settings, including the classroom and during structured learning experiences Tasks and Assessments Real Number System Activity Sheets (attached) Real Number System Activity Sheets Key (attached) Content Elaborations (Rigor & Relevancy) Can we predict whether the type of number (irrational or rational) that results from the sum or product of rational or irrational numbers? Real Number System Definitions: Rational Numbers versus Irrational Numbers o Rational Number: a real number that can be expressed as the quotient or fraction of two integers where the denominator cannot equal zero. Any repeating or terminating decimal can also be written as a fraction, and therefore can also be classified as a rational number. o Irrational Number: a real number that is NOT rational. Any real number that cannot be expressed as the exact ratio of two integers. Any non-repeating AND non-terminating decimal represents an irrational number. Use a calculator to determine decimal approximations out to five decimal places (the hundredthousandths place value) o Convert square roots to decimals CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 129 of 399 Columbus City Schools 6/28/13 o Convert fractions to decimals (including mixed numbers and improper fractions) o Convert to a decimal o Expand repeating decimals to five decimal places o Round decimals precisely to the fifth decimal place Place Value: Tens Ones . Tenths Hundredths Thousandths Ten-Thousandths Hundred-Thousandths Graph a real number on a number line, ordering them based on place value, ensuring that they are ordered correctly from left to right, points are labeled clearly, and student attends to precision when deciding where to place a point when it falls between two consecutive integers. 22 22 22 ; does not 3.14159 and 3.14286 (Please note that 7 7 7 begin repeating until the seventh decimal place and repeats the first six decimal places, 22 22 both will be graphed 3.142857 ) The irrational number and the rational number 7 7 between 3 and 4 on a number line, and both will lie very close to 3 because the tenths place is a 1. The two real numbers do not vary until the third decimal place (or thousandths 22 place). In the third decimal place, has the number 2 and has the number 1, therefore, 7 22 should be graphed on the left side of . The graph will look like this: 7 o Example: versus Select the proper formula for each situation or figure (shape or drawing) to calculate perimeter, circumference, or area. See answer key for examples. Find perimeter or area of irregular figures and use problem solving skills to figure out missing measurements needed for those calculations. Recognize geometric symbols for congruency (measures are equal) for particular sides marked with the “same” number of segment markers. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 130 of 399 Columbus City Schools 6/28/13 = + m Quantities used in calculations should be simplified including equivalent fractions and square roots 30 of perfect squares. For example, 5 and 25 5 . 6 Understand that “exact values” means that decimals are never rounded. For rational decimals, they are either repeating and are marked appropriately with a repeating bar over the digits that repeat, or terminating decimals where all decimal places are written out to the last decimal place. For irrational numbers that are square roots of non-perfect squares or , those values should be left in their radical (square root) or symbol form in the answer. Vocabulary: real number system, rational numbers, irrational numbers, sum, product, triangle, parallelogram, rectangle, square, trapezoid, heptagon, semi-circle, circle, square root, perfect square, area, perimeter, circumference, diameter, radius CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 131 of 399 Columbus City Schools 6/28/13 F-1F 8 Name ___________________________________ Date __________________ Period ________ Real Number System – Classification Give a definition in your own words for rational numbers and irrational numbers. Classify each card as a rational or irrational number and write them below under the correct column (make sure you label each number by the letter written on the card). Give a decimal approximation for each number and if the number is longer than five decimal places, write the number to the hundredthousandths place, but remember to place the “…” after any non-terminating decimal (repeating and irrational). Do not round the fifth decimal place. Rational Numbers Irrational Numbers Definition: Definition: Example Cards: Example Cards: Finally, graph each number on the number line below by marking a point where each number will be located and labeling each point with its corresponding letter. Make sure that your scale is consistent, all your points will fit on the number line, and the placement of a point between integers appropriately reflects the correct decimal approximations for each number. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 132 of 399 Columbus City Schools 6/28/13 Real Number System Cards A B 4.5 E 1.74 I C 6 F 0.25 4 3 4 D 0.523109786… G H 4 3.4 J K L 0.3 9 16 2.754 M N O P 8 9 4.23 Q R 23 9 S 0 T 2 3 4 U V W X 22 7 0.171171117... 0.30 4 5 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 133 of 399 2 5 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Real Numbers – Perimeter Investigation Perimeter of any polygon is the distance around the outside of the polygon. Perimeter = the sum of the distances around the outside of the polygon Circumference of a Circle Formula: C 2 r For each situation, determine the formula necessary to calculate the perimeter of the shape represented in each situation. Fill in the appropriate quantities (numbers) that represent the lengths of each side, and then calculate the exact perimeter. Classify each of the quantities you substituted into the formula as rational or irrational numbers, and then determine if the resulting perimeter is a rational or irrational number. Results that are irrational can be written out to five decimal places followed by the “…” indicating the decimal is non-terminating. Example: Jane baked a cake in a trapezoidal shape, as seen in the image below. She wants to decorate it by wrapping a ribbon around the sides of the cake so the ends of the ribbon just touch with no overlap. Find the exact length of the ribbon that Jane needs. Formula: Perimeter of a trapezoid (4 sides) _______ + _______ + _______ + _______ = _______ Fill in the quantity for each side: 11.23 in + 5 in + 5.23 in + 5 in = 26.46 in Classify each quantity in the real number system: rational + rational + rational + rational = rational 1. Benjamin has a watch with a square face that has a side length of 1.6 inches. Determine the perimeter of Ben’s watch face. Formula: Fill in the quantity for each side: Classify each quantity in the real number system: _______ + _______ + _______ + _______ = _______ CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 134 of 399 Columbus City Schools 6/28/13 2. Haley is getting little triangles (striped like candy corn) painted on her fingernails for Halloween. The nail technician must first outline the triangle in black before she fills in the yellow, white, and orange stripes. Determine the total distance that the nail technician must outline for one of Haley’s nails. Formula: Fill in the quantity for each side: Classify each quantity in the real number system: _______ + _______ + _______ = _______ 3. Fred works for ODOT painting road lines and arrows like the one seen below. Fred was recruited to paint a large scale arrow, with dimensions shown below, on a billboard that can be seen from the highway. Fred always outlines the arrow before painting it, and he wants to know what the total distance he is outlining for the billboard. Formula: Fill in the quantity for each side: Classify each quantity in the real number system: ____ + ____ + ____ + ____ + ____ + ____ + ____ = ____ CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 135 of 399 Columbus City Schools 6/28/13 4. For numbers 1-3, summarize any conclusions you can make about the sum of rational numbers. 5. Melanie is planting a garden in the shape of a parallelogram. She wants to buy pebbles to line her garden, but she needs to know how many total feet it will take to surround her garden. Formula: Fill in the quantity for each side: Classify each quantity in the real number system: _______ + _______ + _______ + _______ = _______ 6. Sarah has a mirror in the shape of a triangle, as seen below. Her mirror has a thin wood trim around it. Find the total length of the wood trim. Formula: Fill in the quantity for each side: Classify each quantity in the real number system: _______ + _______ + _______ = _______ CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 136 of 399 Columbus City Schools 6/28/13 7. Brian is creating a large scale skateboard, with dimensions shown in the figure below, for a parade float featuring extreme sports. He is going to put a trim around the edge of the skateboard and needs to know how much trim to buy. Formula: {Remember that 2 of the sides are semi-circle arc lengths} Fill in the quantity for each variable: Classify each quantity in the real number system: _______ + _______ + _______ + _______ = _______ 8. For numbers 5-7, summarize any conclusions you can make about the sum of rational numbers and irrational numbers. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 137 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Real Numbers – Perimeter Homework On the centimeter dot paper below, create TWO polygons, one where the sides are all rational, and one where at least one side is irrational and the remaining sides are rational. Calculate the perimeter of each polygon and remember to classify each length and the final area as rational or irrational. Calculations: CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 138 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Real Numbers – Circumference & Area Investigation Area Formulas Formula Circumference 1 bh 2 Triangle: A Circle: C 2 r Parallelogram: A bh (includes rectangles and squares) Trapezoid: A Circle: A 1 h b1 b2 2 r 2 (remember: r 2 r r) For each situation, determine the formula necessary to calculate the area or circumference of the shape represented in each situation. Fill in the appropriate quantities (numbers) that represent the measurements needed to find the area, and then calculate the exact area. Classify each of the quantities you substituted into the formula as rational or irrational numbers, and then determine if the resulting area is a rational or irrational number. Results that are irrational can be written out to five decimal places followed by the “…” indicating the decimal is non-terminating. Example: Sam wants to paint a rocket on his wall, and the base of the rocket is the shape of a trapezoid as seen in the figure below. Determine the exact area that Same needs to paint. 1 Formula: Area of a trapezoid ( A h b1 b2 ) 2 _______ _______ _______ = _______ Fill in the quantity for each side: 1 2 4 in ( 5.23 in + 11.23 in) = 32.92 in Classify each quantity in the real number system: rational rational (rational) = rational CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 139 of 399 Columbus City Schools 6/28/13 1. Benjamin wears his watch with a square face all the time, so it is inevitable that he will get a tan line from his watch. Determine the area of Ben’s wrist that will stay pale (or not get tan) because he is wearing his watch. Formula: Fill in the quantity for each variable: Classify each quantity in the real number system: _______ _______ = _______ 2. Melanie now needs to lay a layer of mulch down on the soil in her, and she needs to know the area of her garden so she knows how much mulch to purchase. Formula: Fill in the quantity for each variable: Classify each quantity in the real number system: _______ _______ = _______ 3. Fred needs to paint the arrow and needs to know the area of the arrow so he can purchase paint for the billboard. Formula: {Find the sum of the areas of a triangle and a rectangle.} Fill in the quantity for each variable: CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 140 of 399 Columbus City Schools 6/28/13 Classify each quantity in the real number system: ______ ______ ______ + ______ ______ = ______ 4. For numbers 1-3, summarize any conclusions you can make about the product of rational numbers. 5. Chris walks around a circular path for exercise four days per week. Determine the distance of one lap around the path. Formula: Fill in the quantity for each variable: Classify each quantity in the real number system: _______ _______ _______ = _______ 6. The path that Chris walks surrounds a vegetable garden in his backyard. Find the maximum area of the garden that can be planted with vegetables. Formula: Fill in the quantity for each variable: Classify each quantity in the real number system: _______ _______ _______ = _______ CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 141 of 399 Columbus City Schools 6/28/13 7. Determine how much area on Sarah’s wall her triangular mirror takes up. Formula: Fill in the quantity for each variable: Classify each quantity in the real number system: _______ _______ _______ = _______ 8. Haley has had nine finger nails painted and only has one left, but the nail technician is low on paint. The technician needs to determine the total area for one of the candy corn paintings so she knows if she has enough nail polish to finish Haley’s nails . Formula: Fill in the quantity for each variable: Classify each quantity in the real number system: _______ _______ _______ = _______ 9. For numbers 5-8, summarize any conclusions you can make about the product of a nonzero rational numbers and irrational numbers. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 142 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Real Numbers –Area Homework On the centimeter dot paper below, create TWO polygons, one where the measurements used to calculate the area are all rational, and one where at least one length used to calculate the area is irrational and the remaining lengths are rational. Calculate the area of each polygon and remember to classify each length and the final area as rational or irrational. Remember that your units are in centimeters and your polygons should be to scale. Calculations: CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 143 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Real Numbers – Extension What if a situation called for you to find the sum or product of two irrational numbers? Would the sum or product be equal to a rational number or an irrational number? Discuss in groups of 3 and share out about your decision with justification. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 144 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Real Number System – Classification Answer Key Give a definition in your own words for rational numbers and irrational numbers. Classify each card as a rational or irrational number and write them below under the correct column (make sure you label each number by the letter written on the card). Give a decimal approximation for each number and if the number is longer than five decimal places, write the number to the hundred-thousandths place, but remember to place the “…” after any non-terminating decimal (repeating and irrational). Do not round the fifth decimal place. Rational Numbers Irrational Numbers Definition: A real number that can be Definition: A real number that is NOT expressed as the quotient or fraction of two rational. Any real number that cannot be integers where the denominator cannot expressed as the exact ratio of two equal zero. Any repeating or terminating integers. Any non-repeating AND nondecimal can also be written as a fraction, terminating decimal represents an and therefore can also be classified as a irrational number. rational number. Example Cards: 8 M 9 A 4.5 3 4.75 4 E 1.74 1.74747... F 0.25 C 4 G 4 H 3.4 2 0.8 0.88888... N 4.23 4.23232... O 9 3 P0 3 S 0.75 4 22 U 3.14285... 7 W 0.30 J 0.3 0.33333... 9 3 4 0.75 K X 5 16 4 L 2.754 2.75444... Example Cards: B 6 2.44948... D 0.523109786... 0.52310... I 23 4.79583... Q 3.14159... R 2 1.41421... T 2 5 4.47213... V 0.171171117... 0.17117... 0.8 Finally, graph each number on the number line below by marking a point where each number will be located and labeling each point with its corresponding letter. Make sure that your scale is consistent, all your points will fit on the number line, and the placement of a point between integers appropriately reflects the correct decimal approximations for each number. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 145 of 399 Columbus City Schools 6/28/13 Real Number System Cards A B 4.5 E 1.74 I C 6 F 0.25 4 3 4 D 0.523109786… G H 4 3.4 J K L 0.3 9 16 2.754 M N O P 8 9 4.23 Q R 23 9 S 0 T 2 3 4 U V W X 22 7 0.171171117... 0.30 4 5 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 146 of 399 2 5 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Real Numbers – Perimeter Investigation Answer Key Perimeter of any polygon is the distance around the outside of the polygon. Perimeter = the sum of the distances around the outside of the polygon Circumference of a Circle Formula: C 2 r For each situation, determine the formula necessary to calculate the perimeter of the shape represented in each situation. Fill in the appropriate quantities (numbers) that represent the lengths of each side, and then calculate the exact perimeter. Classify each of the quantities you substituted into the formula as rational or irrational numbers, and then determine if the resulting perimeter is a rational or irrational number. Results that are irrational can be written out to five decimal places followed by the “…” indicating the decimal is non-terminating. Example: Jane baked a cake in a trapezoidal shape, as seen in the image below. She wants to decorate it by wrapping a ribbon around the sides of the cake so the ends of the ribbon just touch with no overlap. Find the exact length of the ribbon that Jane needs. Formula: Perimeter of a trapezoid (4 sides) _______ + _______ + _______ + _______ = _______ Fill in the quantity for each side: 11.23 in + 5 in + 5.23 in + 5 in = 26.46 in Classify each quantity in the real number system: rational + rational + rational + rational = rational Benjamin has a watch with a square face that has a side length of 1.6 inches. Determine the perimeter of Ben’s watch face. Formula: Perimeter of a square (4 sides) Fill in the quantity for each side: 1.6 in + 1.6 in + 1.6 in + 1.6 in = 6.6 in Classify each quantity in the real number system: rational + rational + rational + rational = rational CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 147 of 399 Columbus City Schools 6/28/13 1. Haley is getting little triangles (striped like candy corn) painted on her fingernails for Halloween. The nail technician must first outline the triangle in black before she fills in the yellow, white, and orange stripes. Determine the total distance that the nail technician must outline for one of Haley’s nails. Formula: Perimeter of a triangle (3 sides) Fill in the quantity for each side: 3 3 cm + cm + 0.5 cm 4 4 0.75 cm + 0.75 cm + 0.5 cm = 2 cm Classify each quantity in the real number system: rational + rational + rational = rational 2. Fred works for ODOT painting road lines and arrows like the one seen below. Fred was recruited to paint a large scale arrow, with dimensions shown below, on a billboard that can be seen from the highway. Fred always outlines the arrow before painting it, and he wants to know what the total distance he is outlining for the billboard. Formula: Perimeter of a heptagon (7 sides) Fill in the quantity for each side: 15 30 m+ m+2m+ 3 6 25 m + 4 + 25 m + 2 m 5 m + 5 m + 2 m + 5 m + 4 m + 5 m + 2 m = 28 m Classify each quantity in the real number system: rational + rational + rational + rational + rational + rational + rational = rational CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 148 of 399 Columbus City Schools 6/28/13 3. For numbers 1-3, summarize any conclusions you can make about the sum of rational numbers. The sum of any rational numbers is always rational. 4. Melanie is planting a garden in the shape of a parallelogram. She wants to buy pebbles to line her garden, but she needs to know how many total feet it will take to surround her garden. Formula: Perimeter of a parallelogram (4 sides) Fill in the quantity for each side: 6 ft + 3 ft + 6 ft + 3 ft = (6 + 2 6 ) ft 10.89897… ft Classify each quantity in the real number system: irrational + rational + irrational + rational = irrational 5. Sarah has a mirror in the shape of a triangle, as seen below. Her mirror has a thin wood trim around it. Find the total length of the wood trim. Formula: Perimeter of triangle (3 sides) Fill in the quantity for each side: 4 2 yd + 2.64 yd + yd 5 2 yd + 2.64 yd + 0.8 yd 3.83902… yd Classify each quantity in the real number system: irrational + irrational + rational = irrational CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 149 of 399 Columbus City Schools 6/28/13 6. Brian is creating a large scale skateboard, with dimensions shown in the figure below, for a parade float featuring extreme sports. He is going to put a trim around the edge of the skateboard and needs to know how much trim to buy. Formula: Perimeter of 2 sides and 2 arc lengths {Remember that 2 of the sides are semi-circle arc lengths} 1 2 2 Fill in the quantity for each variable: 1 1 1 1.2 ft + 5 ft + 2 1.2 ft + 5 ft 5 2 5 1.2 ft + 5.2 ft + 1.2 ft + 5.2 ft 17.93982… ft Classify each quantity in the real number system: irrational + rational + irrational + rational = irrational 8. For numbers 5-7, summarize any conclusions you can make about the sum of rational numbers and irrational numbers. The sum of a rational number and an irrational number is always irrational. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 150 of 399 Columbus City Schools 6/28/13 Real Numbers – Perimeter Homework Answer Key On the centimeter dot paper below, create TWO polygons, one where the sides are all rational, and one where at least one side is irrational and the remaining sides are rational. Calculate the perimeter of each polygon and remember to classify each length and the final area as rational or irrational. Calculations: Answers will vary. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 151 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Real Numbers – Circumference & Area Investigation Answer Key Area Formulas Circumference Formula 1 bh 2 Triangle: A Circle: C 2 r Parallelogram: A bh (includes rectangles and squares) Trapezoid: A Circle: A 1 h b1 b2 2 r 2 (remember: r 2 r r) For each situation, determine the formula necessary to calculate the area or circumference of the shape represented in each situation. Fill in the appropriate quantities (numbers) that represent the measurements needed to find the area, and then calculate the exact area. Classify each of the quantities you substituted into the formula as rational or irrational numbers, and then determine if the resulting area is a rational or irrational number. Results that are irrational can be written out to five decimal places followed by the “…” indicating the decimal is non-terminating. Example: Sam wants to paint a rocket on his wall, and the base of the rocket is the shape of a trapezoid as seen in the figure below. Determine the exact area that Same needs to paint. 1 Formula: Area of a trapezoid ( A h b1 b2 ) 2 _______ _______ _______ = _______ Fill in the quantity for each side: 1 2 4 in ( 5.23 in + 11.23 in) = 32.92 in Classify each quantity in the real number system: rational rational (rational) = rational CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 152 of 399 Columbus City Schools 6/28/13 1. Benjamin wears his watch with a square face all the time, so it is inevitable that he will get a tan line from his watch. Determine the area of Ben’s wrist that will stay pale (or not get tan) because he is wearing his watch. Formula: A bh Fill in the quantity for each variable: 1.6 in 1.6 in = 25 2 in = 2.7 in2 3 Classify each quantity in the real number system: rational rational = rational 2. Melanie now needs to lay a layer of mulch down on the soil in her, and she needs to know the area of her garden so she knows how much mulch to purchase. Formula: A bh Fill in the quantity for each variable: 3 ft 5 ft = 15 ft2 Classify each quantity in the real number system: rational rational = rational 3. Fred needs to paint the arrow and needs to know the area of the arrow so he can purchase paint for the billboard. 1 Formula: A ( bh) (bh) 2 {Find the sum of the areas of a triangle and a rectangle.} Fill in the quantity for each variable: ( 1 m 2 6m 3 m) + (4 m 5 m) = 29 m2 Classify each quantity in the real number system: ( rational rational rational ) + ( rational rational ) = rational CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 153 of 399 Columbus City Schools 6/28/13 4. For numbers 1-3, summarize any conclusions you can make about the product of rational numbers. The product of any rational numbers is always rational. 5. Chris walks around a circular path for exercise four days per week. Determine the distance of one lap around the path. Formula: C 2 r Fill in the quantity for each variable: 2 3.4 m = 6.8 m 21.36283… m Classify each quantity in the real number system: rational irrational rational = irrational 6. The path that Chris walks surrounds a vegetable garden in his backyard. Find the maximum area of the garden that can be planted with vegetables. Formula: A A r2 r r Fill in the quantity for each variable: 3.4 m 3.4 m = 11.56 m2 36.31681… m2 Classify each quantity in the real number system: irrational rational rational = irrational CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 154 of 399 Columbus City Schools 6/28/13 7. Determine how much area on Sarah’s wall her triangular mirror takes up. 1 Formula: A bh 2 Fill in the quantity for each variable: 1 4 2 yd yd 2 5 2 yd = 0.4 2 yd2 0.56568… yd2 0.5 0.8 yd Classify each quantity in the real number system: rational rational irrational = irrational 8. Haley has had nine finger nails painted and only has one left, but the nail technician is low on paint. The technician needs to determine the total area for one of the candy corn paintings so she knows if she has enough nail polish to finish Haley’s nails . Formula: A 1 bh 2 Fill in the quantity for each variable: 2 1 cm (0.25 0.25) cm 2 2 0.5 0.5 cm 0.5 2 cm = 0.125 2 cm2 0.17677…cm2 Classify each quantity in the real number system: rational rational irrational = irrational 9. For numbers 5-8, summarize any conclusions you can make about the product of a nonzero rational numbers and irrational numbers. The product of a nonzero rational number and an irrational number is always irrational. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 155 of 399 Columbus City Schools 6/28/13 Real Numbers –Area Homework Answer Key On the centimeter dot paper below, create TWO polygons, one where the measurements used to calculate the area are all rational, and one where at least one length used to calculate the area is irrational and the remaining lengths are rational. Calculate the area of each polygon and remember to classify each length and the final area as rational or irrational. Remember that your units are in centimeters and your polygons should be to scale. Calculations: Answers will vary. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 156 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Real Numbers – Extension Answer Key What if a situation called for you to find the sum or product of two irrational numbers? Would the sum or product be equal to a rational number or an irrational number? Discuss in groups of 3 and share out about your decision with justification. Answers will vary, however the reason for this discussion is because there is no result that is true ALL the time. See the four examples below. Example 1: irrational + irrational = irrational 2 7 4.05996... Example 2: irrational + irrational = rational 5 5 0 Example 3: irrational irrational = irrational 3 11 33 5.74456... Example 4: irrational irrational = rational 6 6 36 6 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 157 of 399 Columbus City Schools 6/28/13 F-IF 7B Name ___________________________________ Date __________________ Period ________ Graphing? Absolutely! Complete the table of values for each equation and then graph them on the grid. Discuss with your partner anything you notice about each graph (compare the graph to its equation). 1. y = |x| x y -2 -1 0 1 2 2. y = |x – 2| x y 0 1 2 3 4 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 158 of 399 Columbus City Schools 6/28/13 3. y = |x + 2| x y -4 -3 -2 -1 0 4. y = |x| – 2 x y -2 -1 0 1 2 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 159 of 399 Columbus City Schools 6/28/13 5. y = 2|x| x y -2 -1 0 1 2 6. y = -|x| x y -2 -1 0 1 2 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 160 of 399 Columbus City Schools 6/28/13 7. y = 2|x + 2| x 2 y -4 -3 -2 -1 0 8. y = -2|x – 2| + 2 x y 0 1 2 3 4 Discuss: Write a brief paragraph about anything you noticed about how the graph is related to the equation. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 161 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Graphing? Absolutely! Answer Key Complete the table of values for each equation and then graph them on the grid. Discuss with your partner anything you notice about each graph (compare the graph to its equation). 1. 2. y = |x| x y -2 2 -1 1 0 0 1 1 2 2 y = |x – 2| x y 0 2 1 1 2 0 3 1 4 2 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 162 of 399 Columbus City Schools 6/28/13 3. 4. y = |x + 2| x y -4 2 -3 1 -2 0 -1 1 0 2 y = |x| – 2 x y -2 0 -1 -1 0 -2 1 -1 2 0 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 163 of 399 Columbus City Schools 6/28/13 5. 6. y = 2|x| x y -2 4 -1 2 0 0 1 2 2 4 y = -|x| x y -2 -2 -1 -1 0 0 1 -1 2 -2 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 164 of 399 Columbus City Schools 6/28/13 7. y = 2|x + 2| 8. 2 x y -4 2 -3 0 -2 -2 -1 0 0 2 y = -2|x – 2| + 2 x y 0 -2 1 0 2 2 3 0 4 -2 Discuss: Write a brief paragraph about anything you noticed about how the graph is related to the equation. When a value is added to the x inside the absolute value symbol (|x + 2|), the graph is moved to the left on the x-axis. When a value is subtracted from x inside the absolute value symbol (|x – 2|), the graph is moved to the right on the x-axis. The number outside tells the y–coordinate of the vertex. If the number outside is added to the absolute value, shift up that number of units. If the number outside is subtracted from the absolute value, shift down that number of units. The number that multiplies the absolute value determines how steep the lines are and how wide the “V” opens up. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 165 of 399 Columbus City Schools 6/28/13 Absolute Value Equation and Graph Cards y = |x| + 3 y = |x + 5| y = |x – 3| 7 y = -2|x – 6| + 3 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 166 of 399 Columbus City Schools 6/28/13 y = |x| |x 5 4| = y y = 3|x + 2| 9 y = -½|x + 4| + 2 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 167 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ 2’s Are Wild Directions: Make a table of values for each equation, then graph the equation on the grid. After you have graphed all of the equations, study them and write about the patterns that you notice. 1. y = |x| x -2 -1 0 1 2 2. y = 2|x| x -2 -1 0 1 2 3. y y= y 1 |x| 2 x -4 -2 0 2 4 y CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 168 of 399 Columbus City Schools 6/28/13 4. y = |x + 2| x -4 -3 -2 -1 0 5. y = |x – 2| x 0 1 2 3 4 6. y y y = |x| + 2 x -2 -1 0 1 2 y CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 169 of 399 Columbus City Schools 6/28/13 7. y = |x| x -2 -1 0 1 2 8. y y = -2|x| x -2 -1 0 1 2 9. 2 y= y 1 |x + 2| 2 x -6 -4 -2 0 2 2 y CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 170 of 399 Columbus City Schools 6/28/13 10. y = -2|x – 2| + 2 x 0 1 2 3 4 y What patterns do you notice in the graphs? CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 171 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ 2’s Are Wild Answer Key Directions: Make a table of values for each equation, then graph the equation on the grid. After you have graphed all the equations, study them and write about the patterns that you notice. 1. y = |x| x -2 -1 0 1 2 2. y = 2|x| x -2 -1 0 1 2 3. y 2 1 0 1 2 y= y 4 2 0 2 4 1 |x| 2 x -4 -2 0 2 4 y 2 1 0 1 2 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 172 of 399 Columbus City Schools 6/28/13 4. y = |x + 2| x -4 -3 -2 -1 0 5. y = |x – 2| x 0 1 2 3 4 6. y 2 1 0 1 2 y 2 1 0 1 2 y = |x| + 2 x -2 -1 0 1 2 y 4 3 2 3 4 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 173 of 399 Columbus City Schools 6/28/13 7. y = |x| x -2 -1 0 1 2 8. y 0 -1 -2 -1 0 y = -2|x| x -2 -1 0 1 2 9. 2 y= y -4 -2 0 -2 -4 1 |x + 2| 2 x -6 -4 -2 0 2 2 y 0 -1 -2 -1 0 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 174 of 399 Columbus City Schools 6/28/13 10. y = -2|x – 2| + 2 x 0 1 2 3 4 y -2 0 2 0 -2 What patterns do you notice in the graphs? Students should observe that when the absolute value of x is multiplied by a whole number, the graph is narrower. When the absolute value of x is multiplied by a fraction, the graph is wider. When a value is added to the x inside the absolute value symbol (|x + 2|), the graph is moved to the left on the x-axis. When a value is subtracted from x inside the absolute value symbol (|x – 2|), the graph is moved to the right on the x-axis. When a value is added to the absolute value of x (|x| + 3) the graph moves up on the y-axis. When a value is subtracted from the absolute value of x (|x| - 3) the graph moves down on the y-axis. When the absolute value of x is multiplied by a negative value the graph opens down instead of up. The graph can be modified in one or more of these ways if the equation is modified in one or more of these ways. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 175 of 399 Columbus City Schools 6/28/13 F-IF 7B Name ___________________________________ Date __________________ Period ________ Connecting Functions 1. 2. Equation:____________________ Equation:____________________ 10 10 8 3. 8 4. 6 6 4 4 2 -10 -8 -6 -4 -2 -2 2 2 4 6 8 10 -4 -6 -8 -10 Domain:____________________ -10 -8 -6 -4 -2 -2 2 4 6 8 10 -4 -6 -8 -10 Domain:____________________ CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 176 of 399 Columbus City Schools 6/28/13 5. 10 8 6 4 2 -10 -8 -6 -4 -2 -2 2 4 6 8 10 -4 -6 -8 -10 We can represent this graph by combining the above equations and each of their domains. Write these equations with restrictions as a piecewise function. f(x) = 6. Range of the function in #5:_______________________ 7. Coordinates of the vertex of the graph of #5:________________ Is this a minimum or maximum? CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 177 of 399 Columbus City Schools 6/28/13 Let’s examine another function that allows you to input the types of numbers included in either domain in #3 and #4, and the outputs are the type of numbers included in the range in #5. 8. Graph f ( x) x either from a table of values or on your graphing calculator. 10 8 6 4 2 -10 -8 -6 -4 -2 -2 2 4 6 8 10 -4 -6 -8 -10 9. What do you notice about the graphs of #5 and #8? Explain why this is true. 10 10. 10 11. 8 8 6 6 4 4 2 2 -10 -8 -6 -4 -2 -2 2 4 6 8 10 -4 -6 -8 -10 Equation:____________________ -10 -8 -6 -4 -2 -2 2 4 6 8 10 -4 -6 -8 -10 Equation:____________________ CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 178 of 399 Columbus City Schools 6/28/13 12. 13. 10 10 8 8 6 6 4 4 2 -10 -8 -6 -4 -2 -2 2 2 4 6 8 10 -10 -8 -6 -4 -2 -4 -2 2 4 6 8 10 -4 -6 -6 -8 -8 -10 -10 Domain:____________________ Domain:____________________ 14. 10 8 6 4 2 -10 -8 -6 -4 -2 -2 2 4 6 8 10 -4 -6 -8 -10 We can represent this graph by combining the above equations and each of their domains. Write these equations with restrictions as a piecewise function. f(x) = 15. Range of the function in #14:_______________________ 16. Coordinates of the vertex of the graph of #14:________________ Is this a minimum or maximum? CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 179 of 399 Columbus City Schools 6/28/13 17. -Describe the shifts that occurred from the graph of #5 to obtain the graph of #14. 18. Determine an absolute value equation that matches the graph of #14. Verify with either your graphing calculator or create a table of values and graph them below. 10 8 6 4 2 -10 -8 -6 -4 -2 -2 2 4 6 8 10 Equation: _______________________ -4 -6 -8 -10 10 10 8 19. 8 20. 6 6 4 4 2 -10 -8 -6 -4 -2 -2 2 2 4 6 8 10 -10 -8 -6 -4 -2 -2 -4 -4 -6 -6 -8 -8 -10 -10 Equation:____________________ 2 4 6 8 10 Equation:____________________ 21. Where do the lines in #19 and #20 intersect? 22. Combine the two equations to create an absolute value function that opens upward. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 180 of 399 Columbus City Schools 6/28/13 10 8 6 4 2 -10 -8 -6 -4 -2 -2 2 4 6 8 10 -4 -6 -8 -10 23. Write the equations with restrictions as a piecewise function. f(x) = Range:____________________ Vertex:__________________ Is this a minimum or maximum? 24. Describe the shifts that occurred from the graph of #5 to obtain the graph of #22. 25. Determine an absolute value equation that matches the graph of #22. Verify with either your graphing calculator or create a table of values and graph them below. 10 8 6 4 Equation: _____________________ 2 -10 -8 -6 -4 -2 -2 2 4 6 8 10 -4 -6 -8 -10 26. Combine the two equations from #19 and #20 to create an absolute function that opens downward. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 181 of 399 Columbus City Schools 6/28/13 10 8 6 4 2 -10 -8 -6 -4 -2 -2 2 4 6 8 10 -4 -6 -8 -10 27. Write the equations with restrictions as a piecewise function. f(x) = Range:____________________ Vertex:__________________ Is this a minimum or maximum? 28. Describe the shifts that occurred from the graph of #5 to obtain the graph of #26. 29. Determine an absolute value equation that matches the graph of #26. Verify with either your graphing calculator or create a table of values and graph them below. 10 8 6 4 2 -10 -8 -6 -4 -2 -2 2 4 6 8 10 -4 -6 -8 -10 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 182 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Connecting Functions Answer Key 10 10 8 1. 8 2. 6 6 4 4 2 2 -10 -8 Equation: -6 -4 -2 -2 2 4 6 8 -10 -8 10 -6 -4 -2 -4 -4 -6 -6 -8 -8 -10 -10 Equation: f(x) = x 4 2 -2 2 2 4 6 8 10 -10 -8 -4 -6 -4 -2 -2 2 4 6 8 10 -4 -6 -6 -8 -8 -10 Domain: 10 6 4 -2 8 8 4. 6 -6 -4 6 10 8 -10 -8 4 f(x) = -x 10 3. 2 -2 -10 x 0 OR [0, ) Domain: 5. x 0 OR ( ,0) 10 8 6 4 2 -10 -8 -6 -4 -2 -2 2 4 6 8 10 -4 -6 -8 -10 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 183 of 399 Columbus City Schools 6/28/13 We can represent this graph by combining the above equations and each of their domains. Write these equations with restrictions as a piecewise function. f(x) = x, x 0 x, x 0 0 OR [0, ) 6. Range of the function in #5: y 7. Coordinates of the vertex of the graph of #5: (0,0) Is this a minimum or maximum? minimum Let’s examine another function that allows you to input the types of numbers included in either domain in #3 and #4, and the outputs are the type of numbers included in the range in #5. 8. Graph f ( x) x either from a table of values or on your graphing calculator. y x 10 8 6 4 2 -10 -8 -6 -4 -2 -2 -4 -6 -8 -10 2 4 6 8 10 -2 2 -1 1 0 0 1 1 2 2 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 184 of 399 Columbus City Schools 6/28/13 9. What do you notice about the graphs of #5 and #8? Explain why this is true. They are the same. Explanations may vary. All x-values are possible, but only positive y-values. 10 10. 10 11. 8 8 6 6 4 4 2 2 -10 -8 -6 -4 -2 -2 2 4 6 8 -10 -8 10 -6 -4 -2 f(x) = x – 1 Equation: f(x) = -x – 5 10 10 8 8 13. 6 6 4 4 2 -2 -2 2 2 4 6 8 10 -10 -8 -4 -2 -2 2 4 6 8 10 -6 -8 -8 -10 2 OR [ 2, ) -6 -4 -4 -6 Domain: x 10 -10 -10 -6 -4 8 -8 -8 -10 -8 6 -6 -6 12. 4 -4 -4 Equation: 2 -2 -10 Domain: x 2 OR ( CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 185 of 399 , 2) Columbus City Schools 6/28/13 14. 10 8 6 4 2 -10 -8 -6 -4 -2 -2 2 4 6 8 10 -4 -6 -8 -10 We can represent this graph by combining the above equations and each of their domains. Write these equations with restrictions as a piecewise function. f(x) = x 1, x 2 x 5, x 2 15. Range of the function in #14: y 3 OR [ 3, ) 16. Coordinates of the vertex of the graph of #14: (-2,-3) Is this a minimum or maximum? minimum 17. Describe the shifts that occurred from the graph of #5 to obtain the graph of #14. Left 2 units, down 3 units 18. Determine an absolute value equation that matches the graph of #14. Verify with either your graphing calculator or create a table of values and graph them below. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 186 of 399 Columbus City Schools 6/28/13 10 8 6 4 2 -10 -8 -6 -4 -2 2 -2 4 6 8 10 f ( x) -4 x 2 3 -6 -8 -10 10 19. 20. 8 10 8 6 6 4 4 2 -10 -8 -6 -4 Equation: -2 2 2 -2 4 6 8 10 -10 -8 -6 -4 -2 -2 -4 -4 -6 -6 -8 -8 -10 -10 f(x) = -x + 6 Equation: 2 4 6 8 10 f(x)= x – 2 21. Where do the lines in #19 and #20 intersect? (4,2) 22. Combine the two equations to create the graph of an absolute function that opens upward. 10 8 6 4 2 -10 -8 -6 -4 -2 -2 2 4 6 8 10 -4 -6 -8 -10 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 187 of 399 Columbus City Schools 6/28/13 23. Write the equations with restrictions as a piecewise function. f(x) = Range: x 6, x 4 x 2, x 4 2 OR [2, ) y Vertex: (4,2) Is this a minimum or maximum? minimum 24. Describe the shifts that occurred from the graph of #5 to obtain the graph of #22. Right 4 units, up 2 units 25. Determine an absolute value equation that matches the graph of #22. Verify with either your graphing calculator or create a table of values and graph them below. 10 8 6 4 f ( x) 2 -10 -8 -6 -4 -2 -2 2 4 6 8 x 4 2 10 -4 -6 -8 -10 26. Combine the two equations from #19 and #20 to create an absolute function that opens downward. 10 8 6 4 2 -10 -8 -6 -4 -2 -2 2 4 6 8 10 -4 -6 -8 -10 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 188 of 399 Columbus City Schools 6/28/13 27. Write the equations with restrictions as a piecewise function. f(x) = Range: x 6, x 4 x 2, x 4 2 OR ( y , 2] Vertex: (4,2) Is this a minimum or maximum? maximum 28. Describe the shifts that occurred from the graph of #5 to obtain the graph of #26. Right 4 units, up 2 units, flip upside down 29. Determine an absolute value equation that matches the graph of #26. Verify with either your graphing calculator or create a table of values and graph them below. 10 8 6 4 f ( x) 2 -10 -8 -6 -4 -2 -2 2 4 6 8 x 4 2 10 -4 -6 -8 -10 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 189 of 399 Columbus City Schools 6/28/13 F-IF 8 Name ___________________________________ Date __________________ Period ________ Different = Same Luke deposited $500 into his savings account that earns 4.5% each year. He wants to determine how much money he will have in his account after 3 years. This following equation represents this situation: y = $500(1 + 0.045)3 Comparing this equation, y = P(1 + r)x with y = $500(1 + 0.045)3 and complete the table below: P r t Verbal description Numerical value y = P(1 + r)x equals y = a(b)x Change y = $500(1 + 0.045)3 into the form y = a(b)x. Write the following exponential equation into the form y = a(b)x. 1. y = 600(1 + 0.6)x _________________________ 2. y = 600(1 - 0.6)x _________________________ 3. y = 1000(1 + 0.6)x _________________________ 4. y = 1000(1 - 0.6)x _________________________ Write the following equations into the form y = P(1 + r)x or y = P(1 + r)x. 1. y = 3(1.6)x _________________________ 2. y = 5(0.4)x _________________________ 3. y = 2(3.4)x _________________________ 4. y = 8(.02)x _________________________ CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 190 of 399 Columbus City Schools 6/28/13 Write the following equations into another form of an exponential equation. Describe the functions using the terms growth or decay. Explain your reasoning. 1. y = 300(1.08)x _________________________ 2. y = 450(1 + 3)x _________________________ 3. y = 90(7/5)x _________________________ 4. y = 50(3/5)x _________________________ 5. y = 1(1.11)x _________________________ 6. y = 8(.69)x _________________________ CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 191 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Different = Same Answer Key Luke deposited $500 into his savings account that earns 4.5% each year. He wants to determine how much money he will have in his account after 3 years. This following equation represents this situation: y = $500(1 + 0.045)3 Comparing this equation, y = P(1 + r)x with y = $500(1 + 0.045)3 and complete the table below: P Verbal description Numerical value r t Initial value or deposit Rate (%) Time (in years) 500 4.5% 3 y = P(1 + r)x equals y = a(b)x Change y = $500(1 + 0.045)3 into the form y = a(b)x. y = 500 (1.045)3 Write the following exponential equation into the form y = a(b)x. 1. y = 600(1 + 0.6)x ___y = 600(1.06)x__________ 2. y = 600(1 - 0.6)x ___y = 600 (0.4)x___________ 3. y = 1000(1 + 0.6)x ___y = 1000(1.6)x __________ 4. y = 1000(1 - 0.6)x ___y = 1000(0.4)x___________ Write the following equations into the form y = P(1 + r)x or y = P(1 + r)x. 1. y = 3(1.6)x ___y = 3(1 + 0.6)x__________ 2. y = 5(0.4)x ___y = 5(1 – 0.6)x__________ 3. y = 2(3.4)x ___y = 2(1 + 2.4)x__________ 4. y = 8(.02)x ___y = 8(1 – 0.98)x_________ CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 192 of 399 Columbus City Schools 6/28/13 Write the following equations into another form of an exponential equation. Describe the functions using the terms growth or decay. Explain your reasoning. 1. y = 300(1.08)x y = 300(1 + .08)x Growth; y = a(1 + r)x a = 300, r = 8% 2. y = 450(1 + 3)x y = 450(4)x Growth; y = a(b)x a = 450, r = 300% 3. y = 90(7/5)x y = 90(1 + 2/5)x Growth; y = a(1 + r)x a = 90, r = 40% 4. y = 50(3/5)x y = 50(1 – 2/5)x Decay; y = a(1 – r)x a = 50, r = 40% 5. y = 1(1.11)x y = 1(1 + .11)x Growth; y = a(1 + r)x a = 1, r = 11% 6. y = 8(.69)x y = 8(1 - .31)x Decay; y = a(1 – r)x a = 8, r = 31% CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 193 of 399 Columbus City Schools 6/28/13 F-IF 8 Name ___________________________________ Date __________________ Period ________ Three Different Exponential Functions y = a(b)x y = a( 1 – r)x y = a(1 + r)x This is a partner activity. Discuss your findings with your partner. Use graphing technology to complete this assignment. 1. Examine the three formulas above. Explain the meaning of each based on your knowledge at this time. 2. Determine the similarities and differences between the three exponential formulas. 3. Make a table and graph y = 10(1.5)x 0 1 x y 2 3 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 194 of 399 4 Columbus City Schools 6/28/13 4. Using the table or graph, what do a and b represent in the equation? Numerical value a: _____ b: ____ Effect on graph a: _____ b: ____ Explain your reasoning. 5. The function, y = 10(1.5)x is written in the form, y = a(b)x, rewrite another way using the appropriate formula, y = a(1 + r)x or y = a( 1 – r)x. Explain the reason you chose the specific formula. 6. Make a table and graph for the functions below: a. y = 8(.4)x x y -2 -1 0 1 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 195 of 399 2 Columbus City Schools 6/28/13 What is the change in y for the function above? What is the initial value? What is the relationship between these values and the formula? Describe the graph. b. y = -4(2)x x y -2 -1 0 1 2 What is the change in y for the function above? What is the initial value? What is the relationship between these values and the formula? Describe the graph. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 196 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Practice Determine the initial value and growth factor using different representation 1. y = 15(5/4)x Initial value: __________ -2 x y 2. y = 5(3)x Initial value: __________ -5 x y Growth factor: _________ -1 0 1 2 Growth factor: _________ -3 0 1 2 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 197 of 399 Columbus City Schools 6/28/13 3. y = 3(2)x Initial value: __________ -2 x y Growth factor: _________ -1 0 1 2 Growth factor: _________ -1 0 1 2 4. y = -6(.75)x Initial value: __________ -2 x y CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 198 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Three Different Exponential Functions Answer Key y = a(b)x y = a( 1 – r)x y = a(1 + r)x This is a partner activity. Discuss your findings with your partner. Use graphing technology to complete this assignment. 1. Examine the three formulas above. Explain the meaning of each based on your knowledge at this time. Answers will vary based on understanding of exponential functions. 2. Determine the similarities and differences between the three exponential formulas. Each formula has an “a.” Each is raised to the power of “x.” Each formula represents an exponential function. Two formulas, y = a(1 + r)x and y = a( 1 – r)x, are the same except for the sign of operation within the parenthesis. 3. Make a table and graph y = 10(1.5)x. 0 1 x y 10 15 2 22.5 3 33.75 4 50.625 24 22 20 18 16 14 12 10 8 6 4 2 -5 -4 -3 -2 -1 1 2 3 4 5 4. Using the table or graph, what do a and b represent in the equation? CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 199 of 399 Columbus City Schools 6/28/13 Numerical value a: __10 b: _1.5 Effect on graph a: y-intercept b: Change factor Explain your reasoning. The y-intercept is (0, 10) and the change factor (multiply by 1.5) for y-values. 5. The function, y = 10(1.5)x is written in the form, y = a(b)x, rewrite another way using the appropriate formula, y = a(1 + r)x or y = a( 1 – r)x. Explain the reason you chose the specific formula. Since “b” is greater than 1, y = a(1 + r)x is the appropriate formula. 6. Make a table and graph for the functions below b. y = 8(.4)x -2 50 x y -1 20 0 8 1 3.2 2 1.28 52 48 44 40 36 32 28 24 20 16 12 8 4 -4 -3 -2 -1 -4 1 2 3 4 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 200 of 399 Columbus City Schools 6/28/13 What is the change in y for the function above? Multiply by .4 What is the initial value? 8 What is the relationship between these values and the formula? Substituted in the formula, “a” is 8 and b is .4 Describe the graph. The graph is decreasing. b. y = -4(2)x -2 -1 x y -3 -2 -1 1 2 -1 -2 0 -4 1 -8 2 -16 3 -2 -4 -6 -8 -10 -12 -14 -16 -18 -20 What is the change in y for the function above? 2 What is the initial value? -4 What is the relationship between these values and the formula? Substituted in the formula, “a” is -4 and b is 2. Describe the graph. It is decreasing from left to right and it has flipped across the y-axis. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 201 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Practice Answer Key Determine the initial value and growth factor using different representation 1. y = 15(5/4)x Initial value: ___15__ -2 x y 9.6 Growth factor: ___5/4____ -1 0 1 12 15 18.75 2 23.438 24 22 20 18 16 14 12 10 8 6 4 2 -5 -4 -3 -2 -1 1 2 3 4 5 2. y = 5(3)x Initial value: ____5______ -5 x y .02058 Growth factor: _____3____ -3 0 1 .18519 5 15 2 45 50 45 40 35 30 25 20 15 10 5 -5 -4 -3 -2 -1 1 2 3 4 5 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 202 of 399 Columbus City Schools 6/28/13 3. y = 3(2)x Initial value: ____3______ -2 x y .75 Growth factor: ___2______ -1 0 1 1.5 3 6 2 12 16 14 12 10 8 6 4 2 -4 -3 -2 -1 1 2 3 4 -2 -4 4. y = -6(.75)x Initial value: _____-6_____ -2 x y -10.67 Growth factor: _____.75____ -1 0 1 -8 -6 -4.5 2 -3.375 4 2 -4 -3 -2 -1 1 2 3 4 -2 -4 -6 -8 -10 -12 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 203 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Absolute Value Graphs y Part 1 ax h k I can graph an absolute value function with graphing technology. I can determine the vertex. I can determine the slope of the left and right sides. I can determine a, h and k. I can explain the effects of a, h and k on the graphs. Materials: Colored pencils, graphing calculator Choose 5 colored pencils. Use the graphing calculator and points (x, y) to graph each function. In general, there are 4 steps: 1. Enter the formula for the function you wish to graph using the Y= editor. 2. Set the viewing window. (Zoom 6) 3. Graph the function. 4. Adjust the viewing window, if necessary. 5. Use table for values to plot for graph. Example: Graph the absolute value function. This function is “built-in” and we can use it by entering its name. Step 1: Press [Y=] to display the Y= editor. Use the method 1 above to insert absolute value function. You use the [x]for the variable x. Close with ). Step 2: We assume the Standard window has been selected. (Zoom 6) Step 3: Press [GRAPH] to obtain the graph of the absolute value function. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 204 of 399 Columbus City Schools 6/28/13 Step 4 is not necessary in this case. Step 5: Use `% for table for function. Use this data to plot for the function. 1. Graph y x on the grid below. Compare each graph below to this parent graph. 10 8 6 4 2 -10 -8 -6 -4 -2 -2 2 4 6 8 10 -4 -6 -8 -10 What is the slope of the left side? _____ What is the slope of the right side?_____ State the vertex _____ 2. Graph each of the following functions on the grid below. Function Color Vertex Slope Slope What What of (x, y) of of is the is the graph left right value value side side of of “a” “h” y x State the color used for each graph. What Graph How is the did value graph of change? “k” 10 2 8 6 4 y x 5 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 y x 4 -4 x 8 -10 -6 -8 y CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 205 of 399 Columbus City Schools 6/28/13 3. Graph each of the following functions on the grid below. Function Color Vertex Slope Slope What What of (x, y) of of is the is the graph left right value value side side of of “a” “h” State the color used for each graph. What Graph How is the did value graph of change? “k” 10 y 2 x 8 6 4 2 -10 y -8 -6 -4 -2 5 x 2 4 6 8 10 -2 -4 -6 -8 -10 y x 4 y x 8 4. Graph each of the following functions on the grid below. State the color used for each graph. Function Color Vertex Slope Slope What What What of (x, y) of of is the is the is the graph left right value value value side side of of of “a” “h” “k” Graph How did graph change? 10 y x 2 8 6 4 y x 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 y x 2 -6 -8 -10 y x 5 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 206 of 399 Columbus City Schools 6/28/13 5. Graph each of the following functions on the grid below. State the color used for each graph. Function Color Vertex Slope Slope What What What of (x, y) of of is the is the is the graph left right value value value side side of of of “a” “h” “k” Graph How did graph change? 10 y x 3 8 6 4 y x 5 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 y x 1 -6 -8 -10 y x 4 6. Graph each of the following functions on the grid below. Function Color Vertex Slope Slope What What of (x, y) of of is the is the graph left right value value side side of of “a” “h” State the color used for each graph. What Graph is the value of “k” 10 y x 3 8 6 4 y x 5 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 y x 3 -6 -8 -10 y x 5 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 207 of 399 Columbus City Schools 6/28/13 How did graph change? 7. Graph each of the following functions on the grid below. State the color used for each graph. Function Color Vertex Slope Slope What What What of (x, y) of of is the is the is the graph left right value value value side side of of of “a” “h” “k” Graph How did graph change ? 10 y x 1 2 8 6 4 y x 1 2 4 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 y x 3 2 -6 -8 -10 y x 2 5 8. Summary: Explain the effects of a, h and k. 2 x 1 3 on the grid below and describe graph. 9. Graph y 1 x 3 4 10. Graph y -10 -8 -6 -4 2 on the grid below and describe the graph. 10 10 8 8 6 6 4 4 2 2 -2 2 4 6 8 10 -10 -8 -6 -4 -2 2 -2 -2 -4 -4 -6 -6 -8 -8 -10 -10 4 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 208 of 399 6 8 10 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Absolute Value Graphs y Part 2 ax h k Complete the assignment without graphing technology. I can graph an absolute value function without technology. I can determine the key features of an absolute value graph. I can determine the values of a, h and k. I can describe the effects of a, h and k on the graphs. I can write an absolute value function given a graph. I can write an absolute value function given a verbal description. 1. Graph each of the following functions on the grid below. State the color used for each graph. Function Vertex Slope Slope What What What (x, y) of of is the is the is the left right value value value side side of of of “a” “h” “k” Graph How did graph change? 10 y x 7 4 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 2 4 6 8 10 -2 -4 -6 -8 -10 10 y x 4 2 8 6 4 2 -10 -8 -6 -4 -2 -2 -4 -6 -8 -10 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 209 of 399 Columbus City Schools 6/28/13 Function Vertex Slope Slope What What What (x, y) of of is the is the is the left right value value value side side of of of “a” “h” “k” Graph How did graph change? 10 y x 1 3 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 2 4 6 8 10 -2 -4 -6 -8 -10 10 y x 2 5 8 6 4 2 -10 -8 -6 -4 -2 -2 -4 -6 -8 -10 2. Determine the equations of the functions below and explain your reasoning. a. b. -10 -8 -6 -4 10 10 8 8 6 6 4 4 2 2 -2 2 -2 4 6 8 10 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -4 -6 -6 -8 -8 -10 -10 3. Write an equation with the vertex at the origin with a slope of 3 for the left side. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 210 of 399 Columbus City Schools 6/28/13 4. Write an equation with the vertex at (2, -4) with the slope of -1 for the left side. 5. Write an equation with the vertex at (-1, 0) and a is 6. Write an equation when a is 2 . 3 1 , h is -4, and k is -9. 3 7. Given the general equation for an absolute value function, y = a|x – h| + k, explain the effects of a, h and k on the graphs and the key features. 8. Describe the changes from the parents of graph of: a. y 5x 3 2 b. y 0 .5 x 7 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 211 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Absolute Value Graphs Part 1 Answer Key y ax h k I can graph an absolute value function with graphing technology. I can determine the vertex. I can determine the slope of the left and right sides. I can determine a, h and k. I can explain the effects of a, h and k on the graphs. Materials: Colored pencils, graphing calculator Choose 5 colored pencils. Use the graphing calculator and points (x, y) to graph each function. In general, there are 4 steps: 1. Enter the formula for the function you wish to graph using the Y= editor. 2. Set the viewing window. (Zoom 6) 3. Graph the function. 4. Adjust the viewing window, if necessary. 5. Use table for values to plot for graph. Example: Graph the absolute value function. This function is “built-in” and we can use it by entering its name. Step 1: Press [Y=] to display the Y= editor. Use the method 1 above to insert absolute value function. You use the [x]for the variable x. Close with ). Step 2: We assume the Standard window has been selected. (Zoom 6) Step 3: Press [GRAPH] to obtain the graph of the absolute value function shown in Figure 2. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 212 of 399 Columbus City Schools 6/28/13 Step 4 is not necessary in this case. Step 5: Use `% for table for function. Use this data to plot for the function. 1. Graph y x on the grid below. Compare each graph below to this parent graph. 10 8 6 4 2 -10 -8 -6 -4 -2 -2 2 4 6 8 10 -4 -6 -8 -10 What is the slope of the left side? _-1__ State the vertex _(0, 0)____ What is the slope of the right side?__1_ 2. Graph each of the following functions on the grid below. Function Color Vertex Slope Slope What What of (x, y) of of is the is the graph left right value value side side of of “a” “h” y x 2 Blue State the color used for each graph. What Graph is the value of “k” How did graph change? 10 (0, 2) -1 1 1 0 2 Up 2 8 6 4 y x 5 Red (0, 5) -1 1 1 0 2 5 -10 -8 -6 -4 -2 Up 5 2 4 6 8 10 -2 y x 4 Black -4 (0, 4) -1 1 1 0 4 -6 Up 4 -8 -10 y x 8 Green (0, 8) -1 1 1 0 8 Up 8 3. Graph each of the following functions on the grid below. State the color used for each graph. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 213 of 399 Columbus City Schools 6/28/13 Function y Color Vertex Slope Slope What What What of (x, y) of of is the is the is the graph left right value value value side side of of of “a” “h” “k” Blue 2 x Graph How did graph change? 10 (0, -2) -1 1 1 0 -2 Down 2 8 6 4 y Red 5 x (0, -5) -1 1 1 0 2 -5 -10 -8 -6 -4 -2 Down 5 2 4 6 8 10 -2 y Black 4 x -4 (0, -4) -1 1 1 0 -4 Down 4 -6 -8 -10 y Green (0, -8) 8 x -1 1 1 0 4. Graph each of the following functions on the grid below. Function Color Vertex Slope Slope What What of (x, y) of of is the is the graph left right value value side side of of “a” “h” y x 2 Blue -8 Down 8 State the color used for each graph. What Graph is the value of “k” How did graph change? 10 (0, 2) 1 -1 -1 0 2 8 6 4 2 y x 4 Red (0, 4) 1 -1 -1 0 4 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 y x 2 Black y x 5 Green (0, -2) 1 -1 -1 0 -2 (0, -5) 1 -1 -1 0 -5 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 214 of 399 -10 Columbus City Schools 6/28/13 Turns down Up 2 Turns down Up 4 Turns down Down 2 Turns down Down 5 5. Graph each of the following functions on the grid below. Function Color Vertex Slope Slope What What of (x, y) of of is the is the graph left right value value side side of of “a” “h” y x 3 Blue State the color used for each graph. What Graph is the value of “k” How did graph change? 10 (-3, 0) -1 1 1 -3 0 Left 3 8 6 4 y x 5 Red (-5, 0) -1 1 1 -5 2 0 -10 -8 -6 -4 -2 2 4 6 8 10 Left 5 -2 -4 y x 1 Black (1, 0) -1 1 1 1 -6 0 Right 1 -8 -10 y x 4 Green (4, 0) -1 1 1 4 0 6. Graph each of the following functions on the grid below. Function Color Vertex Slope Slope What What of (x, y) of of is the is the graph left right value value side side of of “a” “h” Right 4 State the color used for each graph. What Graph is the value of “k” 10 y x 3 Blue (-3, 0) 1 -1 -1 -3 0 8 6 4 2 y x 5 Red (-5, 0) 1 -1 -1 -5 0 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 y x 3 Black (3, 0) 1 -1 -1 3 0 y x 5 Green (5, 0) 1 -1 -1 5 0 -10 7. Graph each of the following functions on the grid below. State the color used for each graph. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 215 of 399 Columbus City Schools 6/28/13 How did graph change? Turns down Left 3 Turns down Left 5 Turns down Right 3 Turns down Right 5 Function Color Vertex Slope Slope What What What of (x, y) of of is the is the is the graph left right value value value side side of of of “a” “h” “k” Graph How did graph change ? Left 1 Up 2 10 y x 1 Blue 2 (-1, 2) -1 1 1 -1 2 8 6 4 y x 1 2 Red 4 (1, 4) -1 1 1 1 4 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 x 3 y x 2 y 8. a: h: k: 2 Black 5 Green (3, -2) (-2, -5) 1 1 -1 -1 -1 3 -1 -2 -2 -8 10. Graph -10 -8 -6 2x 1 3 y Left 2 Down 5 -5 -4 on the grid below and describe graph. 1 x 3 4 y 2 on the grid below and describe the graph. 10 10 8 8 6 6 4 4 2 2 -2 2 4 6 8 10 -10 -8 -6 -4 -2 2 -2 -2 -4 -4 -6 -6 -8 -8 -10 -10 Right 3 Down 2 -10 Summary: Explain the effects of a, h and k. Absolute value of slope of sides Moves vertex of graph left or right Moves vertex of graph up or down 9. Graph Right 1 Up 4 4 6 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 216 of 399 8 10 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Absolute Value Graphs Part 2 Answer Key y ax h k Complete the assignment without graphing technology. I can graph an absolute value function without technology. I can determine the key features of an absolute value graph. I can determine the values of a, h and k. I can describe the effects of a, h and k on the graphs. I can write an absolute value function given a graph. I can write an absolute value function given a verbal description. 1. Graph each of the following functions on the grid below. State the color used for each graph. Function Vertex Slope Slope What What What Graph How (x, y) of of is the is the is the did left right value value value graph side side of of of change? “a” “h” “k” 10 y x 7 4 8 (-7, -4) -1 1 1 -7 Left 7 Down 4 6 -4 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 10 y x 4 2 (4, 2) -1 1 1 4 2 8 6 Right 4 Up 2 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 217 of 399 Columbus City Schools 6/28/13 Function Vertex Slope Slope (x, y) of of left right side side What is the value of “a” What is the value of “h” What Graph is the value of “k” How did graph change? 10 y x 1 3 8 (-1, 3) 1 -1 -1 -1 6 3 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 Turns down Left 1 Up 3 -2 -4 -6 -8 -10 10 y x 2 5 (2, -5) 1 1 -1 2 -5 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 6 8 10 8 10 Turns down Right 2 Down 5 -2 -4 -6 -8 -10 2. Determine the equations or the functions below and explain your reasoning. a. b. -10 -8 -6 -4 10 10 8 8 6 6 4 4 2 2 -2 2 4 6 8 10 -2 -10 -8 -6 -4 -2 2 -4 -4 -6 -6 -8 -8 -10 -10 Vertex (1, 2) and the slope is -1 on the left and 1 on the right; (h, k) is the vertex so y x 1 2 4 -2 Vertex (-6, -4) and the slope is -1 on the left and 1 on the right: (h, k) is the vertex so y x 6 4 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 218 of 399 Columbus City Schools 6/28/13 3. Write an equation with the vertex at the origin with a slope of 3 for the left side. y 3x 1 3 4. Write an equation with the vertex at (2, -4) with the slope of -1 for the left side. y x 2 4 2 5. Write an equation with the vertex at (-1, 0) and a is 3 . y 2 x 1 3 6. Write an equation when a is y 1 x 4 3 1 3 , h is -4, and k is -9. 9 7. Given the general equation for an absolute value function, y = a|x – h| + k, explain the effects of a, h and k on the graphs and the key features. a changes the shape of the V, if –a it turns down. h causes the vertex to move left or right on the graph; x h moves left and x h moves right on the x-axis. k causes the vertex to move up or down on the graph; + k moves up and – k moves down on the y-axis. 8. Describe the changes from the parents of graph of: a. y 5x 3 2 Turns down, more narrow, left 3 and up 2 b. y 0 .5 x 7 Wider, down 7 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 219 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Investigate Compound Interest In this investigation, you will explore how compound interest is different than simple interest. Simple interest is paid on the initial principal where as compound interest is paid on the initial principal and from previously earned interest. In addition, you will be able to come up with the compound interest formula. 1. You deposited $1000 into a savings account paying 6% annual interest. a) If the interest is compounded once a year, how much will you have in your account at the end of the first year (i.e. what is the balance)? Round your final answer to 2 decimal places. b) What is your balance at the end of the 2nd year? At the end of the 3rd year? At the end of the 4th year? Show work. Round your final answer to 2 decimal places. End of 2nd year __________ End of 3rd year ____________ End of 4th year ___________ CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 220 of 399 Columbus City Schools 6/28/13 c) Think of a formula to represent how much you have at the end of the tth year. Many savings institutions offer compounding intervals other than annual (yearly) compounding. For example, a bank that offers quarterly compounding computes interest on an account every quarter, that is, every 3 months. Thus instead of compounding interest once each year, the interest will be compounded 4 times each year. If a bank advertises that it is offering 6% annual interest compounded quarterly, it does not use 6% to determine interest each quarter. Instead, it will use 6%/4 = 1.5% each quarter. In this example, 6% is known as the nominal interest rate and 1.5% as the quarterly interest rate. 2. Suppose you invested your $1000 into an account that offered a nominal interest rate of 6% compounded quarterly, how much would you have in your account after: (round to 2 decimal places & show work) 3 months _______________ 6 months _______________ 9 months _______________ 1 year _________________ 4 years _______________ t years ____________________ 3. Suppose you invested your $1000 into an account that offered a nominal interest rate of 6% compounded monthly, how much would you have in your account after one year? 4. What can you conclude about how the compounding periods affect the balance? 5. Come up with a formula to represent the balance, A, if you invested P dollars at a rate of r compounded n times a year for t years. 6. Which option would you rather have? I. Investing $1000 into an account paying 5% interest compounded yearly for a year OR CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 221 of 399 Columbus City Schools 6/28/13 II. Investing $1000 into an account paying 4.75% interest compounded monthly for a year CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 222 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Investigate Compound Interest Answer Key In this investigation, you will explore how compound interest is different than simple interest. Simple interest is paid on the initial principal where as compound interest is paid on the initial principal and from previously earned interest. In addition, you will be able to come up with the compound interest formula. 1. You deposited $1000 into a savings account paying 6% annual interest. a) If the interest is compounded once a year, how much will you have in your account at the end of the first year (i.e. what is the balance)? Round your final answer to 2 decimal places. Balance = 1000 + 1000(.06)(1) = $1060 OR Balance = 1000(1 + 0.06) = 1000(1.06) = $1060 b) What is your balance at the end of the 2nd year? At the end of the 3rd year? At the end of the 4th year? Show work. Round your final answer to 2 decimal places. 1060 + 1060(0.06) = 1060(1 + 0.06) = 1060(1.06) = 1000(1.06)(1.06) = 1000(1.06)2 End of 2nd year __$1123.60___ 1123.60 + 1123.60(0.06) = 1123.60(1 + 0.06) = 1123.60(1.06) = 1000(1.06)(1.06) (1.06) = 1000(1.06)3 rd End of 3 year _ $1191.02 __ 1191.02 + 1191.02(0.06) = 1191.02(1 + 0.06) = 1191.02(1.06) = 1000(1.06)(1.06) (1.06)(1.06) = 1000(1.06)4 th End of 4 year __$1262.48__ c) Think of a formula to represent how much you have at the end of the tth year. 1000(1+ 0.6)t = 1000(1.06)t CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 223 of 399 Columbus City Schools 6/28/13 Many savings institutions offer compounding intervals other than annual (yearly) compounding. For example, a bank that offers quarterly compounding computes interest on an account every quarter, that is, every 3 months. Thus instead of compounding interest once each year, the interest will be compounded 4 times each year. If a bank advertises that it is offering 6% annual interest compounded quarterly, it does not use 6% to determine interest each quarter. Instead, it will use 6%/4 = 1.5% each quarter. In this example, 6% is known as the nominal interest rate and 1.5% as the quarterly interest rate. 2. Suppose you invested your $1000 into an account that offered a nominal interest rate of 6% compounded quarterly, how much would you have in your account after: (round to 2 decimal places & show work) 3 months ____$1015_____ 6 months ____$1030.23___ 9 months ____$1045.68___ 1 year ____$1061.37_____ 4 years ____$1268.99____ t years __1000(1+0.06/4)4t___ 3. Suppose you invested your $1000 into an account that offered a nominal interest rate of 6% compounded monthly, how much would you have in your account after one year? 12(1) 0.06 = $1061.68 12 4. What can you conclude about how the compounding periods affect the balance? The more the compound periods occur, the higher the balance will be. 1000 1+ 5. Come up with a formula to represent the balance, A, if you invested P dollars at a rate of r compounded n times a year for t years. A = P 1+ r n nt 6. Which option would you rather have? I. Investing $1000 into an account paying 5% interest compounded yearly for a year OR II. Investing $1000 into an account paying 4.75% interest compounded monthly for a year The best option is option I CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 224 of 399 Columbus City Schools 6/28/13 A-SSE 1b Name ___________________________________ Date __________________ Period ________ Exponent Properties Exploration Directions: Use a calculator if necessary to evaluate the numerical values; apply these results as you respond to the questions involving variables. A. 1. 5 5 2. 52 (The two in this problem is called a power.) 3. Explain in words what question #2 is asking you to compute mathematically. 4. Based on your explanation in question #3, expand x 8 using multiplication. 5. Expand x 4 using multiplication. 6. Expand x 8 x 4 using multiplication. 7. Write the result from question #6 as x to a power. 8. How could you take x 8 x 4 and get the result from question #7? (Create a short-cut, so you would not have to expand each portion of the problem.) CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 225 of 399 Columbus City Schools 6/28/13 B. 1. Expand x 6 using multiplication. 2. Expand x 2 using multiplication. 3. Write the expanded forms from questions #1 and #2 in the following fraction: x6 x2 4. What does any number divided by itself equal? 5. Based on your answer from question #4, cancel any terms you can from your answer in question #3, what is the result in expanded form? 6. Re-write your answer from question #5 as x to a power. 7. How can you take the problem presented in question #3 and get the final result from question #6? (Write a short-cut.) CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 226 of 399 Columbus City Schools 6/28/13 C. 1. Expand x 3 using multiplication. 2 2. Expand x 3 using multiplication. (Hint, expand x 3 using multiplication inside of the parenthesis and then think about how you would expand something to the second power.) 3. If you remove all parentheses from your answer in question #2, how could you write the result as x to a power? 4. How can you take the problem presented in question #2 and get the result from question #3? (Create a short-cut.) D. 1. Expand x 6 using multiplication. 2. Using the short-cut you created in B. x6 is equivalent to what? (Hint: write x to a power.) x6 3. Anything divided by itself is equal to? 4. Based on your results from questions #2 and #3, x 0 must equal what number? CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 227 of 399 Columbus City Schools 6/28/13 E. 1. Expand x 3 using multiplication. 2. Expand x 6 using multiplication. 3. Write the expanded form of x3 x6 4. Any number divided by itself is equal to what number? 5. Using your answer from question #4, cancel any terms you can from your expanded form of question #3. (Don’t forget to use a place holder of 1 if necessary.) 6. Write your expanded answer from question #5 using a power. x3 7. Based on your short-cut from B. 6 is equal to? (Hint, write the result as x to a power.) x 8. Based on your results from questions #6 and #7, how can you write a negative exponent with a positive exponent? CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 228 of 399 Columbus City Schools 6/28/13 F. 1. Expand x 2 using multiplication. 2. Expand y 2 using multiplication. 3. Expand x y 2 using multiplication. 4. The associative property of multiplication says that x y x y = x x __ __ 5. Write the result from question #4 using powers. 6. How could you go directly from the problem in question #3 to the result in question #5? What previously learned property is this similar to? 7. Using the same concept, write x y 2 without parenthesis. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 229 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Exponent Properties Exploration Answer Key Directions: Use a calculator if necessary to evaluate the numerical values; apply these results as you respond to the questions involving variables. A. 1. 5 5 2. 52 25 25 (The two in this problem is called a power.) 3. Explain in words what question #2 is asking you to compute mathematically. A number times itself. 4. Based on your explanation in question #3, expand x 8 using multiplication. (x)(x)(x)(x)(x)(x)(x)(x) 5. Expand x 4 using multiplication. (x)(x)(x)(x) 6. Expand x 8 x 4 using multiplication. (x)(x)(x)(x)(x)(x)(x)(x)(x)(x)(x)(x) 7. Write the result from question #6 as x to a power. (x)12 8. How could you take x 8 x 4 and get the result from question #7? (Create a short-cut, so you would not have to expand each portion of the problem.) Add the exponents together (8 plus is 12) CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 230 of 399 Columbus City Schools 6/28/13 B. 1. Expand x 6 using multiplication. (x)(x)(x)(x)(x)(x) 2. Expand x 2 using multiplication. (x)(x) 3. Write the expanded forms from questions #1 and #2 in the following fraction: x6 x2 ( x)( x)( x)( x)( x)( x) ( x)( x) 4. What does any number divided by itself equal? 1 5. Based on your answer from question #4, cancel any terms you can from your answer in question #3, what is the result in expanded form? (x)(x)(x)(x) 6. Re-write your answer from question #5 as x to a power. (x) 4 7. How can you take the problem presented in question #3 and get the final result from question #6? (Write a short-cut.) Subtract the exponents. (6 minus 2 equals 4) CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 231 of 399 Columbus City Schools 6/28/13 C. 1. Expand x 3 using multiplication. (x)(x)(x) 2 2. Expand x 3 using multiplication. (Hint, expand x 3 using multiplication inside of the parenthesis and then think about how you would expand something to the second power.) [(x)(x)(x)][(x)(x)(x)] 3. If you remove all parentheses from your answer in question #2, how could you write the result as x to a power? (x) 6 4. How can you take the problem presented in question #2 and get the result from question #3? (Create a short-cut.) Multiply the exponents. (Three times two is six) D. 1. Expand x 6 using multiplication. (x)(x)(x)(x)(x)(x) x6 2. Using the short-cut you created in B. 6 is equivalent to what? (Hint: write x to a power.) x x0 3. Anything divided by itself is equal to? 1 4. Based on your results from questions #2 and #3, x 0 must equal what number? 1 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 232 of 399 Columbus City Schools 6/28/13 E. 1. Expand x 3 using multiplication. (x)(x)(x) 2. Expand x 6 using multiplication. (x)(x)(x)(x)(x)(x) 3. Write the expanded form of x3 x6 ( x)( x)( x) ( x)( x)( x)( x)( x)( x) 4. Any number divided by itself is equal to what number? 1 5. Using your answer from question #4, cancel any terms you can from your expanded form of question #3. (Don’t forget to use a place holder of 1 if necessary.) 1 ( x)( x)( x) 6. Write your expanded answer from question #5 using a power. 1 x3 x3 7. Based on your short-cut from B. 6 is equal to? (Hint, write the result as x to a power.) x x 3 8. Based on your results from questions #6 and #7, how can you write a negative exponent with a positive exponent? One divided by the positive power is equivalent to the same negative power. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 233 of 399 Columbus City Schools 6/28/13 F. 1. Expand x 2 using multiplication. (x)(x) 2. Expand y 2 using multiplication. (y)(y) 3. Expand x y 2 using multiplication. (xy)(xy) 4. The associative property of multiplication says that x y x y = x x y y 5. Write the result from question #4 using powers. ( x) 2 ( y) 2 6. How could you go directly from the problem in question #3 to the result in question #5? What previously learned property is this similar to? Distribute the exponent. (The Distributive Property) 7. Using the same concept, write x y 2 without parenthesis. x2 y2 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 234 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Multiplying Binomials Write each product as a polynomial. 1. x 2 x 7 2. 3x 1 2 x 1 3. 4. 7x 5 7x 5 5. 3x 2 3x 2 6. - 4 x 5 3 x 7 7. 4x 5 2 10. x 4 x 7 2 x 1 11. 13. 3 x 5r 2 x 7 r 8. 8 x 9 2 5x2 3 5x2 3 14. x 8 2 9. 12. 2x 1 x 1 2 7x 1 x 2 x2 9 y3 15. 2 x 8 x 8 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 235 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Multiplying Binomials Answer Key Write each product as a polynomial. 1. x 2 x 7 x 2 7 x 2 x 14 x 2 9 x 14 4. 2. 6 x2 3 x 2 x 1 6x 7x 5 7x 5 3. 3x 1 2 x 1 2 5. 2x 1 x 1 2 x2 2 x x 1 2x 2 x 1 3x 1 6. - 4 x 5 3 x 7 3x 2 3x 2 - 12 x 2 28 x 15 x 35 49 x 2 35 x 35 x 25 49 x 2 7. 4x 5 25 9 x2 6 x 6 x 4 9x 2 2 - 12 x 2 43 x 35 4 8. 8 x 9 - 12 x 2 43 x 35 2 9. 2 7x 1 x 16 x 2 20 x 20 x 25 64 x 2 72 x 72 x 81 2 2 x 7 x 7 x2 16 x 2 40 x 25 64 x 2 144 x 81 2 5 x 7 x2 10. x 4 x 7 2 x 1 11. 5x2 3 5x2 3 12. 2 x2 9 y3 2 x 8 x 2 4 x 14 x 7 x 8 x 2 10 x 7 8x 13. 3 10 x 2 7x 3 x 5r 2 x 7 r 25 x 4 15 x 2 15 x 2 9 4 x 4 18 x 2 y 3 18 x 2 y 3 81 y 6 25 x 4 9 4 x 4 36 x 2 y 3 81 y 6 14. x 8 2 15. x 8 x 8 6 x 2 21rx 10rx 35r 2 x 2 8 x 8 x 64 x 2 8 x 8 x 64 6 x 2 11rx 35r 2 x 2 16 x 64 x 2 64 CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 236 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ A Number Called e nt r , we will examine what happens to A as n, n the compounding period, increases. In this problem we will let P = $1, the interest rate is 100% and the time is 1 year. Complete the table below by finding the balance given each specific compounding period. Round the balance value to three decimal places. Using the compound interest formula A n P 1 A P 1 r n nt A 1 (annually) 2 (semi-annually) 4 (quarterly) 12 (monthly) 52 (weekly) 365 (daily) 8760 (hourly) 525,600 (minutely) 31,536,000 (every second) Do you see a pattern in the balance? Explain. CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 237 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ A Number Called e Answer Key nt r Using the compound interest formula A P 1 , we will examine what happens to A as n, n the compounding period, increases. In this problem we will let P = $1, the interest rate is 100% and the time is 1 year. Complete the table below by finding the balance given each specific compounding period. Round the balance value to three decimal places. n 1 (annually) 2 (semi-annually) 4 (quarterly) 12 (monthly) 52 (weekly) 365 (daily) 8760 (hourly) 525,600 (minutely) 31,536,000 (every second) A 1 A 1 1 1 1(1) 2 2(1) 1 A 1 1 2 1 A 1 1 4 2.25 4(1) 1 A 1 1 12 1 A 1 1 52 A 1 1 1 365 2.441 12(1) 2.613 52(1) 2.693 365(1) 1 A 1 1 8760 A 1 1 A nt r P 1 n 1 525, 600 1 A 1 1 31, 536, 000 2.715 8760(1) 2.718 525,600(1) 2.718 31,536,000(1) 2.718 Do you see a pattern in the balance? Explain. The balance goes toward the number 2.718 as n gets larger and larger CCSSM II Quadratic Functions: Solving A-CED 1, 4, A-REI 1, N-RN 1, 2, 3, 4, F-IF 4, 7b, 7e, 8, 8b, F-BF 1, 1a, 1b. 3, A-SSE, 1b, N-Q 2 Quarter 2 Page 238 of 399 Columbus City Schools 6/28/13 COLUMBUS PUBLIC SCHOOLS HIGH SCHOOL CCSS MATHEMATICS II CURRICULUM GUIDE TOPIC 4 CONCEPTUAL CATEGORY TIME Comparing Functions and Functions, Algebra, Number and RANGE Different Representations of 20 days Quantity, Statistics and Quadratic Functions F – IF 4, 5, Probability 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Domain: Interpreting Functions (F – IF) Cluster 14) Interpret functions that arise in applications in terms of the context. 15) Analyze functions using different representations. GRADING PERIOD 2 Domain: Building Functions (F – BF) Cluster 16) Build a function that models a relationship between two quantities. 17) Build new functions from existing functions. Domain: Creating Equations (A – CED) Cluster 18) Create equations that describe numbers of relationships. Domain: Linear and Exponential Models (F – LE) Cluster 19) Construct and compare linear and exponential models and solve problems. Domain: Quantities (N - NQ) Cluster 20) Reason quantitatively and use units to solve problems. Domain: Interpreting Categorical and Quantitative Data (S – ID) Cluster 21) Summarize, represent, and interpret data on two categorical and quantitative variables. Domain: Reasoning with Equations and Inequalities (A – REI) Cluster 22) Solve systems of equations. Standards 14) Interpret functions that arise in applications in terms of the context. F – IF 4*: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 239 of 399 Columbus City Schools 6/28/13 graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.* F – IF 5*: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.* F – IF 6: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. 15) Analyze functions using different representations. F – IF 7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. F – IF 7a*: Graph quadratic functions and show intercepts, maxima, and minima.* 16) Create equations that describe numbers of relationships A – CED 1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. A – CED 2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 17) Build a function that models a relationship between two quantities. F – BF 1: Write a quadratic function that describes a relationship between two quantities. F – BF 1a: Determine an explicit expression, a recursive process, or steps for calculation from a context. F – BF 1b: Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model 18) Build new functions from existing functions. F – BF 3: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 240 of 399 Columbus City Schools 6/28/13 effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. 19) Construct and compare linear and exponential models and solve problems. F- LE 3: Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. 20) Reason quantitatively and use units to solve problems. N – NQ 2: Define appropriate quantities for the purpose of descriptive modeling. 21) Summarize, represent, and interpret data on two categorical and quantitative variables. S – ID 6a: Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear and exponential models. S – ID 6b: Informally assess the fit of a function by plotting and analyzing residuals. 22) Solve systems of equations. A – REI 7: Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = -3x and the circle x2 + y2 = 3. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 241 of 399 Columbus City Schools 6/28/13 TEACHING TOOLS Vocabulary: Axis of symmetry, average rate of change, coefficient, decreasing, decreasing functions, domain, end behavior, extrema, function, horizontal compression, horizontal stretch, horizontal translations, increasing, increasing functions, inflection point, intercepts, leading coefficient, parabola, parent graph, quadratic equation, range, rate of change, maximums, minimums, relative maximums, relative minimums, restricted domain, roots, second difference, solutions, standard form, transformation, translation, vertex, vertex form, vertical compression, vertical stretch, vertical translations, x-intercept, y-intercept, zeros Teacher Notes: The graph of a quadratic equation either intersects the x-axis in one point, two points or no points. If the graph does not intersect the x-axis then the roots of the equation are not real; they are imaginary. If the graph intersects the axis in one or two points then the x values of these points are the real roots of the equation. These roots can be identified from the graph or algebraically. The solutions, also called the roots of the equation ax2 + bx + c = 0, are the values of x where the graph of y = ax2 + bx + c crosses the x-axis. Recall that an x-intercept of a graph is the xcoordinate of any point where the graph crosses the x-axis. The graph of a quadratic equation, a parabola, may have no x-intercepts, exactly one x-intercept, or two distinct x-intercepts. The values of x at the x-intercepts are solutions, roots, or zeros. Quadratic equations in two variables can be graphed on a coordinate plane. The graph is a visual model of the relationship between the two variables and is useful in determining the solutions. That is, the graph of y = ax2 + bx + c can be used to find solutions of ax2 + bx + c = 0, where a 0. Graphing can be used to solve any quadratic equation, but gives only approximate solutions if the root(s) are irrational, and does not show any imaginary solutions. A function f given by the equation f(x) = ax2 + bx + c, where a, b, and c are real numbers, and a 0, is a quadratic function. The curve of f(x) = ax2 + bx + c is a parabola. A parabola is a graph of a quadratic function. Emphasis should be placed on questions relating to the characteristics of the graphs of quadratic functions. Emphasize their parabolic shape, the line of symmetry, the location of the vertex, the location and interpretation of the x-intercepts, and the location of maximum or minimum points. Stress that in a quadratic relationship the independent variable is raised to the second power. You can use the zero feature of a graphing calculator to find the x-intercepts of a quadratic function. In the equation f(x) = ax2 + bx + c, if the a value is positive, the parabola opens upward and the vertex is a minimum point. If the a value is negative, the parabola opens downward and the vertex is a maximum point. 10 Line of symmetry x intercept -10 x intercept 10 Vertex -10 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 242 of 399 Columbus City Schools 6/28/13 Quadratic equations should be written in a vertex form in order to describe the transformations. (In some cases in order to get an equation in this form, completing the square will need to be applied; this will be covered in topic 6.) It is important that the significance of each term be explained. The vertex form for a quadratic equation is: f(x) = a (x - h)2+ k, where a determines whether the graph is reflected about the x-axis, as well as stretched or shrunk vertically, h determines whether the graph is shifted horizontally left or right, and k determines whether the graph is shifted vertically up or down. For example, if given the function f(x) = -3(x – 2)2 – 5, a = - 3, h = 2, and k = - 5. This will transform the graph of y = x2, by reflecting it about the xaxis, stretch the graph vertically by a factor of 3, shift the graph to the right 2 units and shift the graph down 5 units. It is important to emphasis that the template is f(x) = a(x – h)2 +k. If given the function f(x) = 4(x + 1)2+ 7, then a = 4, h = - 1 and k = 7. Remember that f(x) = 4(x + 1)2 + 7 = 4 (x – (-1)) + 7. The transformations that will take place to y = x2 will be a vertical stretch by a factor of 4, a horizontal shift to the left 1 and a vertical shift up 7. When modeling, depending on the amount of information given one can use the vertex form of a quadratic equation or one can create a system to find the quadratic equation. If given the vertex and another point, the vertex form can be used; if three points are given, then a system can be created and solved. For example, assume that the vertex of a parabolic curve is known (5, - 9) and another point is known on the curve (2, 18). Using the points and the equation y = a(x – h)2+ k, one would obtain the equation: 18 a 2 5 18 a 3 2 2 9 9 18 9a 9 27 9a 3 a The equation for the quadratic would be: y = 3(x - 5)2 - 9 or y = 3x2- 30x + 66. Assume three points are given: (3, 3), (0, 66), (- 3, 183) using the standard form of the quadratic equation y = ax2 + bx + c, one would obtain the following three equations from the given points: (3, 3) (0, 66) (- 3, 183) 3 a 3 66 2 a 0 b(3) c 2 183 a -3 b 0 2 3 9a 3b c c b -3 66 c c 183 9a 3b c Solving the system: 3 9a 3b 66 183 9a 3b 66 9a 3b 63 0 9a 3b 117 0 18a 54 0 18a 54 a 3 9(3) 3b 63 0 3b 90 0 3b 90 b 30 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 243 of 399 Columbus City Schools 6/28/13 y 3x 2 30x 66 Symbolically, quadratics are identified by their degree of two, in other words the largest exponent in the equation is two. The standard form of a quadratic function is written in the form f(x) = ax2 + bx + c. x y 0 1 2 3 4 5 6 1 2 9 22 41 66 97 1st 2nd difference difference 1 7 13 19 25 31 6 6 6 6 6 Quadratic data exhibits a constant second difference. For example, if the relationship between x and y were linear, the first differences would be constant and the first difference would be the slope of the linear relation if the x values increase by 1. Because the relationship between x and y is quadratic, the second differences are constant. The constant, 6, indicates that a in the standard form of the quadratic equation is 6/2 =3. In general, a is the 2nd difference divided by 2 if the x values increase by 1. The graphs of quadratic functions are called parabolas. Every parabola opens up if a > 0 and opens down if a < 0 and has a minimum value if opening up and a maximum value if opening down. The domain of a quadratic function is the set of real numbers, while the range is y > k for a > 0 and y < k for a < 0. The point where the maximum or minimum occurs is called the vertex. In a quadratic equation written in standard form, the vertex can be found graphically by using the maximum and minimum functions on the graphing calculator, or by using a and b from the standard form equation to find the x-coordinate of the vertex, where x = -b/(2a). The y-coordinate of the vertex, is found by substituting for x into the quadratic equation. For example, in the graph of f(x) = 2x2 – 12x + 3, the x-coordinate of the vertex is 12/4=3, and the y-coordinate of the vertex is f(3) = 2(3)2 – 12(3) + 3 = -15. The vertex of the parabola is (3, -15). While changes in a change the steepness and orientation, changes in b affect the x and y coordinates of the vertex. Changes in c cause a vertical shift in the graph of the parabola, and represent the yintercept. one real zero no real zeros Quadratic functions, f(x) = ax2 + bx + c, may have one real zero, two real zeros or no real two real zeros zeros. Graphically this corresponds to the graph CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 244 of 399 Columbus City Schools 6/28/13 y = ax2 + bx + c having one x-intercept, two x-intercepts, or no x-intercepts and the quadratic equation ax2 + bx + c = 0 having one, two, or no solutions. Quadratic functions with no real zeros have two non-real zeros. The domain and range should be represented using interval notation. It is helpful to have students look at a graph from left to right (smallest value to largest value) for the domain and from bottom to top (smallest value to largest value) for the range. Parentheses indicate that values are not included in the domain or range and brackets indicate that values are included in the domain and range. If specific values are to be listed, braces should be used. If the domain is the set of numbers such that - 3 < x 5 then the interval notation that could be used is (-3, 5]. If the range is the set of numbers such that y < 0, then the interval notation that could be used is (- , 0). If the domain or range decrease or increase without boundary then - or are used respectively. Parentheses are used with the symbols - or . If the function is y = 5, the domain is the set of all real numbers, (- , ). The range contains one value which is 5 therefore the range is {5}. Getting Ready for Quadratics At this website, directions are found explaining how to use the Nspire for investigating quadratic functions. http://education.ti.com/xchange/US/Math/AlgebraII/9147/Getting%20Ready%20for%20Quadratic s_Student.pdf Creating Equations in one variable: One of the algebra common core clusters is to create equations that describe numbers or relationships. In this topic, students will be able to create equations and inequalities in one variable and use them to solve problem. At Purple Math you can find the following problem. This problem illustrates this standard by writing an equation for this situation (196Ä + 60Ä = Area of swimming pool and deck) in one variable. http://www.purplemath.com/modules/perimetr2.htm A circular swimming pool with a diameter of 28 feet has a deck of uniform width built around it. If the area of the deck is 60(pi) square feet, find its width. I have this situation: A pool is surrounded by a deck. The pool has radius 14, and the deck has width "d ". If the diameter of the pool is 28, then the radius is 14. The area of the pool is then: CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 245 of 399 Columbus City Schools 6/28/13 (Ä)r2 = (Ä)(14)2 = 196(Ä) Then the total area of the pool plus the surrounding decking is: 196(Ä) + 60(Ä) = 256(Ä) Working backwards from the area formula, I can find the radius of the whole pool-plus-deck area: 256(Ä) = (Ä)r2 256 = r2 16 = r Since I already know that the pool has a radius of 14 feet, and I now know that the whole area has a radius of 16, then clearly: the deck is two feet wide. Creating Equations in two variables: For the second standard, students will create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. The following problem can be found at Purple Math. http://www.purplemath.com/modules/quadprob3.htm A student can use graphing technology to plot the data in the table below (price hikes, total income) to find the equation it represents. You run a canoe-rental business on a small river in Ohio. You currently charge $12 per canoe and average 36 rentals a day. An industry journal says that, for every fiftycent increase in rental price, the average business can expect to lose two rentals a day. Use this information to attempt to maximize your income. What should you charge? Let's say I have no idea how to set this problem up. Instead of going straight to an equation, I'll need to put in some real numbers, see what I do when I know what the values are, and then follow the pattern to get my formula. Here is my reasoning, neatly laid out in a table: price hikes price per rental number of rentals total income / revenue none $12.00 36 $12.00×36 = $432.00 1 price hike $12.00 + 1(0.50) 36 – 1(2) $12.50×34 = $425.00 2 price hikes $12.00 + 2(0.50) 36 – 2(2) $13.00×32 = $416.00 3 price hikes $12.00 + 3(0.50) 36 – 3(2) $13.50×30 = $405.00 x price hikes $12.00 + x(0.50) 36 – x(2) (12 + 0.5x)(36 – 2x) Then my formula for my revenues R after x fifty-cent price hikes is: R(x) = (12 + 0.5x)(36 – 2x) = 432 – 6x – x2 = –x2 – 6x + 432 The maximum income will occur at the vertex of this quadratic's parabola, and the vertex is at (–3, 441) Teacher Notes for A-CED 1 http://www.purplemath.com/modules/ineqquad.htm Written notes on solving quadratic inequalities can be found on this website. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 246 of 399 Columbus City Schools 6/28/13 Teacher Notes for A-CED 2 http://www.youtube.com/watch?v=YiJC--c0Etc At this website, a tutorial can be found for creating equations in two or more variables to represent relationships between quantities and graphing equations on coordinate axes with labels and scales. Teacher Notes for A-CED 4 http://www.purplemath.com/modules/solvelit.htm Written notes on solving literal equations. Definition of Descriptive Modeling http://whatis.techtarget.com/definition/descriptive-modeling At this website a description of descriptive modeling can be found. Building Functions http://www.purplemath.com/modules/fcntrans2.htm Build new functions from existing functions is a standard for the cluster building functions in the common core state standards. Students will be able to identify the effect on the graph using transformation rules. Below are the rules listed at Purple Math. The transformations so far follow these rules: f(x) + a is f(x) shifted upward a units f(x) – a is f(x) shifted downward a units f(x + a) is f(x) shifted left a units f(x – a) is f(x) shifted right a units –f(x) is f(x) flipped upside down ("reflected about the x-axis") f(–x) is the mirror of f(x) ("reflected about the y-axis") There are two other transformations, but they're harder to "see" with any degree of accuracy. Compare the graphs of 2x2, x2, and ( 1/2 )x2 to see what is meant. 2x2 x2 _1/2 x2_ CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 247 of 399 Columbus City Schools 6/28/13 Interactive Practice: Students can experiment with different cases using the interactive parabola found at the website listed below. http://www.mathwarehouse.com/quadratic/parabola/interactive-parabola.php Lesson Plan for Quadratics (web based): http://hs-mathematics.wikispaces.com/Quadratic+Functions The site has information, including, but not limited to, the history of quadratics, which can be used in your lesson planning with students. There are also interactive pieces, which the students can manipulate during the lesson. Analyzing Residuals http://www.originlab.com/www/helponline/Origin/en/UserGuide/Graphic_Residual_Analysis.htm l http://www.opexresources.com/index.php/free-resources/articles/analysis-of-residuals-explained Notes on analyzing residuals. Exponential and Quadratic Models For this standard, students will observe graphs and tables of exponential and quadratic functions and conclude that a quantity increasing exponentially eventually exceeds a quantity increasing quadratically. For example, ask students to graph y = x2 and y = 2x on the same coordinate grid. Instruct them to compare the values of the functions at various intervals. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 248 of 399 Columbus City Schools 6/28/13 200 180 160 140 120 100 80 60 40 20 -10 -8 -6 -4 -2 2 4 6 8 10 Descriptive Modeling The common core standard is to define appropriate quantities for the purpose of descriptive modeling. Students should be able to: select and use appropriate units of measurement for problems choose appropriate scales to create quadratic graphs determine from the labels on the graph what the units of the rate of change are Definition of Descriptive Modeling At this website a description of descriptive modeling can be found. http://whatis.techtarget.com/definition/descriptive-modeling What is descriptive modeling? An explanation of descriptive modeling versus predictive modeling can be found at the following website. http://www.accudata.com/wp-content/uploads/WP-Descriptive_v.-Predictive.pdf CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 249 of 399 Columbus City Schools 6/28/13 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 250 of 399 Columbus City Schools 6/28/13 Misconceptions/Challenges: Students do not understand the meaning of the vertex and how to get two points, which are reflections of each other. Students do not make a connection between the x-value of the vertex and the axis of symmetry. Students confuse the x-intercept and y-intercept. Students confuse h and k, which are the vertex coordinates, with the x-intercept and yintercept. Students make mistakes when determining the vertex due to their lack of understanding the difference between (-3)2 and -32. Students make mistakes plotting points, because they confuse the x-axis and the y-axis. Students incorrectly assign attributes to different forms of quadratic functions; for example they incorrectly identify the k value of a quadratic equation in vertex form as the yintercept because they confuse it with the c value from a quadratic function in standard form. Students confuse the x-intercepts of the factored form regarding the positive or negative sign. Students assume the rate of change is the same for all intervals of a quadratic function. Students think an equation with a higher y-intercept has a higher maximum value. Students get confused by the concept that the “h” value comes after a subtraction sign, and therefore move the graph the opposite direction on horizontal shifts. Students confuse the horizontal and vertical shifts. Students confuse vertical stretches and compressions; they tend to see it as a horizontal change rather than a vertical change. They believe the graph gets wider horizontally, but do not understand that this is actually due to a vertical compression. Similarly students often see the graph as getting more narrow, but again do not recognize that this is due to a vertical stretch. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 251 of 399 Columbus City Schools 6/28/13 Instructional Strategies: F-BF3 1. Students will work with a group of peers to complete the “Quadratics Inquiry Project” (included in this Curriculum Guide). With and without technology students will investigate how manipulating a quadratic equation in vertex form affects the graph. 2. Provide students with “Graphs” (included in this Curriculum Guide). Students will graph groups of functions on the same grid and compare them. 3. With the use of technology students will graph quadratic graphs and describe relationships to y = x2. Provide students with “Families of Graphs” (included in this Curriculum Guide). 4. The activity, “Quadratics: Tables Graphs Equations” (included in this Curriculum Guide), will allow students to connect a table of points to the vertex form of a quadratic equation. Students will explore more on the concept of symmetric points and use this to not only graph the data but also write an equation for the data. 5. Give students quadratic equations written in vertex form. Instruct them to explore the transformations of these functions, y = a(x – h)2 + k using graphing technology. Discuss the effects of a, h and k on the graphs. 6. Have students complete “Vertex Form and Transformations” (included in this Curriculum Guide). Students will work with a partner to examine tables, graphs and equations to determine the key features of the graphs. 7. http://www.geogebra.org/cms/en/ Using graphing technology transformations that preserve characteristics of graphs of functions and which do not. Resources available are Geogebra sliders and TI Transform App. 8. http://map.mathshell.org/materials/tasks.php?taskid=295&subpage=novice Give the students the task, “Building Functions”. Students are given three quadratic graphs and asked to label them with the appropriate equation. Expand on this task by asking the students to explain their reasoning. 9. http://www.pbs.org/teachers/mathline/lessonplans/pdf/hsmp/toothpicks.pdf “Toothpicks and Transformations”. Students will review transformations of quadratics and then apply these skills in determining patterns created by a toothpick pattern. 10. http://www.illustrativemathematics.org/illustrations/741 Building a Quadratic Function From f(x) = x2: In this activity, students can graph the functions and examine the impact of the different transformations or the students could be given the graphs and lists of functions and asked to match them and explain their reasoning. 11. http://www.dlt.ncssm.edu/algebra/HTML/09.htm CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 252 of 399 Columbus City Schools 6/28/13 Toothpicks and Transformations: In this lesson students will investigate quadratic functions using toothpicks. 12. http://education.ti.com/en/us/activity/detail?id=B1A4D0199C2648109657FF99CBB00406 Transformations of a Quadratic Function: In this handheld activity students will explore transformations of a quadratic function. This "create your own" activity is designed to be student-centered, with the teacher acting as a facilitator while students work cooperatively. The time varies for this activity depending on whether the TI-Nspire document (.tns file) is provided or created by the students. 13. http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod2_structure_06 2213.pdf Shifty y’s: In this lesson (pp. 3-9), students will connect transformations to quadratic functions and parabolas. 14. http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod2_structure_06 2213.pdf Transformers: More Than Meets the y’s: Students work with the vertex form of a quadratic to connect the components to the transformations (pp. 10-14). 15. http://secondaryiiinutah.wikispaces.com/Transformations+of+Quadratic+and+Absolute+Value +Graphs http://secondaryiiinutah.wikispaces.com/Functions+and+Modeling Transformations of Quadratics: This website contains various lessons for transformations of quadratic functions. 16. http://secondaryiiinutah.wikispaces.com/Quadratic+Transformations+Exploration Quadratic Transformations Exploration: This website contains a group activity to explore transformations of quadratic functions. 17. http://secondaryiiinutah.wikispaces.com/Functions+and+Modeling Transformations of Quadratic: This website contains various lessons for transformations of quadratic functions. 18. http://secondaryiiinutah.wikispaces.com/Quadratic+Transformations+Exploration Quadratic Transformations Exploration: This website contains a group activity to explore transformations of quadratic functions. 19. http://map.mathshell.org.uk/materials/tasks.php?taskid=295 Building Functions: In this task, given three graphs students determine the equations of the quadratic functions. Students will demonstrate their knowledge of the effects of a and k in ax2 + k. F-IF 7 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 253 of 399 Columbus City Schools 6/28/13 1. Have students complete the activity “Graphing Quadratic Functions” (included in this Curriculum Guide) and analyze the x-intercepts for each graph. Students should notice that some functions cross the x-axis at one point, some cross it at two points, and some do not cross the x-axis at all. 2. Using technology students will determine the vertex and zeros of quadratic equations using “Graph in a Flash” (included in this Curriculum Guide). 3. Have the students use the “Properties of a Parabola” worksheet (included in this Curriculum Guide). Students will have looked at lines of symmetry, the relationship between the vertex and maximum or minimum, and the domain and range of a parabola. 4. The activity, “What Will My Parabola Look Like?” (included in this Curriculum Guide), allows students to see how changing coefficients, a, b and c one at a time while fixing the other two coefficients affects the graph. Students will examine the vertex and make conjectures based on the ordered pairs and the graph. Students should work in groups on this activity. 5. The students should learn how to make a table and graph the quadratic function. The students should begin by finding the x-coordinate of the vertex. Students should then make a table of values (the table should include at least two values greater than the x-coordinate for the vertex and at least two values less than the x-coordinate for the vertex). The students should plot the points and sketch the graph of the parabola. 6. Students should be able to enter any quadratic equation into a graphing calculator and obtain the graph of a parabola. Students should be able to set an appropriate window so that they can see a complete graph (x-intercepts, y-intercept, and vertex). 7. Have students use graphing calculators, and enter a variety of equations into the y= menu, changing the a and c values only. Help students compare the graph and the table of each equation to y = x2. For example, have students enter y1 = x2 and y2 = x2 + 3 into their calculators. When comparing the graphs students will see that y2 was shifted up 3 and by looking at the table they can see that all of the values in the y2 column are 3 more than those in the y1 column. Include horizontal shifts in vertex form as well. 8. http://insidemathematics.org/common-core-math-tasks/high-school/HS-F2008%20Functions.pdf In this activity students must correctly identify which points on a graph are from a quadratic function, and which points on the same graph are from a linear function, and they must then write equations for both functions. 9. Have the students complete “What Do You Need for the Graph?” (Curriculum Guide). Students will graph, by hand, quadratic functions expressed symbolically and show the key features for the graph. The key features should include the vertex, y-intercept, and xintercept(s). 10. Have the students complete “Linear, Exponential, and Quadratic Functions” (Curriculum CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 254 of 399 Columbus City Schools 6/28/13 Guide). Students will compare the key features of linear, exponential and quadratic graphs to note the effects of a, h and k on the graphs. F-IF 7a 1. http://www.kutasoftware.com/FreeWorksheets/Alg2Worksheets/Properties%20of%20Parabola s.pdf Properties of Parabolas: The sheet at this site provides practice for students on determining the vertex, minimum or maximum point, and axis of symmetry for quadratic functions. Instruct the students, after graphing the parabolas, to determine the solutions by inspection. 2. http://www.wccusd.net/cms/lib03/CA01001466/Centricity/domain/60/lessons/algebra%20i%2 0lessons/ExploringQuadraticGraphsV3.pdf Comparing Graphs: This site has a lesson on comparing graphs. 3. The activity, “Zeros of Quadratic Functions: Calculator Discovery” (included in this Curriculum Guide), allows students to connect factors of quadratics to the x-intercepts using the graphing calculator. This activity will lead into a discussion of the Zero Product Property, as well as factoring and solving quadratic equations using factoring. Emphasize that the reason it is called the Zero Product Property is because the y-value is zero when one is finding the x-intercept. Conclude with the fact that for example, if you graphed y = (x + 1) (x – 3) and y = x2 – 2x – 3 on the same screen, you should only see one graph. Hence, x2 – 2x – 3 can be factored into (x + 1) (x – 3). Once students master this concept, the teacher can have students solve equations such as 3 = x2 – 5x + 9 using factoring. Again, once students rewrite the equation 3 = x2 – 5x + 9 as 0 = x2 – 5x + 6, emphasize that they are ultimately looking for the zeros of the function f(x) = x2 – 5x + 6. Teachers should explain to students the differences between the words zeros, x-intercepts, roots, and solutions by giving the following description: a) zeros of functions, b) x-intercepts of graphs, c) roots or solutions of equations. 4. http://neaportal.k12.ar.us/index.php/2012/01/find-zeros-of-a-function-given-the-graph/ Find the Zeros of a Function Given the Graph: At this site there is a video tutorial for determining the zeros given a graph. F-IF 4 1. http://www.nctm.org/uploadedFiles/Max_min%20problem_2.pdf Give students the “Minimum Problem” activity. Students will interpret a function given a realworld scenario. Students will determine where a stake can be placed between two wires to use the least amount of wire. 2. Give students key features of a quadratic function including y-intercepts, x-intercepts, maximums, minimums, the line of symmetry, and intervals functions is increasing and decreasing. Ask them to sketch a graph given these features without technology. 3. Have students use a graphing calculator or handheld to identify the key features of a quadratic function. These key features should include y-intercepts, x-intercepts, maximums, minimums, lines of symmetry, and intervals where the functions are increasing and decreasing. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 255 of 399 Columbus City Schools 6/28/13 4. Using different representations, give the students key features of linear, exponential and quadratic graphs. These key features should include y-intercepts, x-intercepts, maximums, minimums, lines of symmetry, and intervals where the functions are increasing and decreasing. Ask students to compare these graphs for the features listed. 5. Give students a variety of quadratic equations with a table of x-values for each equation. Students are to complete the table to find the y-values and then graph the functions. Students work with a partner to identify characteristics they observe that all of their graphs have in common. Give students the activity Quadratic Qualities” (included in this Curriculum Guide). Characteristics that they should be guided to observe include: all graphs have a line of symmetry, all graphs are parabolas (u-shaped, opening either up or down), all graphs have one maximum or minimum point (vertex), and all graphs will cross the x-axis at zero, one or two points. Once students have had time to make their own observations, discuss as a class what they discovered. The terms parabola, line of symmetry, vertex, minimum, maximum, and intercepts should be defined and discussed at this point. 6. Give students the “Quadratic Qualities II” activity (Curriculum Guide). This activity is an extension of “Quadratic Qualities”. Students will graph a quadratic function given a verbal description and write verbal descriptions for the graphs in “Quadratic Qualities.” 7. Use the “Match the Graph-Equations” and “Match the Graph-Graphs” (included in this Curriculum Guide) to play a matching game. Students will match equations that are in vertex form with the appropriate graph. Make sure that students state the transformations before trying to match the graphs. This can be played like the old concentration game. 8. For an extension for “Match the Graph-Equations,” ask the students to: “Write a verbal description of each function, including the vertex, domain, range, x-intercept(s), y-intercept, minimum or maximum value. Include the intervals where the functions are decreasing and increasing. 9. Give students a graph of a line that goes up to the right and a graph of a parabola that opens up on the board and have students write down a list of similarities and differences. Do not tell students anything and have them think of this list for 3-5 minutes. Once time is up, put a list of the following words on the board and have them check if they have the following words in their list: domain, range, x-intercepts, y-intercepts, minimum, maximum, increasing, decreasing, symmetry, rate of change, etc. If they do not have these words, then have them use these words to finish their similarities and differences list. Next, give students a variety of lines and parabolas and see if their lists still hold true for these new sets of lines and parabolas (e.g., compare a line that goes down to the right to an open down or up parabola). Have students compare different types of parabolas as well. Once students are done with this comparison and contrast activity, ask students the following set of questions: 1) Does a parabola that opens up have a maximum or minimum value? 2) How can you create a parabola that always has a maximum value? 3) Can a quadratic function ever have the same range as a linear function? Why or why not? CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 256 of 399 Columbus City Schools 6/28/13 10. Have students complete “Graph It!” (included in this Curriculum Guide.) Instruct students to graph the quadratic functions given the verbal descriptions. As a culminating question students will be asked to write a description of a graph and include key features. 11. http://www.ixl.com/math/algebra-1/characteristics-of-quadratic-functions IXL Characteristics of Quadratics: This website offers interactive practice for characteristics of quadratics. F-IF 6 1. Have the students use the “Quadratic Functions and Rates of Change” activity (included in this Curriculum Guide). By the end of this activity, students will be able to see the relationship between quadratic functions and second differences. 2. Have students complete the activity “Linear or Quadratic?” (included in this Curriculum Guide) using first and second differences. 3. Students will analyze the data and look for patterns in the activity “Don’t Change that Perimeter” (included in this Curriculum Guide). 4. To investigate some properties of parabolas, have students choose 3 points on one side of a parabola, the vertex, and the 3 image points on the other side of the parabola that are reflections of the original points. Have students calculate the slopes between these points in order to recognize that the rate of change is not constant and that the parabola is symmetrical. F-IF 5 1. Give students contexts to relate the domain of a function to its graph. The domain should be limited to the subset of integers, positive or negative values. 2. http://insidemathematics.org/common-core-math-tasks/high-school/HS-F2007%20Graphs2007.pdf Graphs (2007): In this task students will determine the intersection of graphs to solve problems. 3. http://illuminations.nctm.org/LessonDetail.aspx?ID=L621 Domain Representations: In this lesson, students use graphs, tables, number lines, verbal descriptions, and symbols to represent the domain of various functions. Instruct the students to use graphing technology to graph the given functions (using tables from calculator). 4. http://www.uen.org/core/math/downloads/sec1_floating_down_river.pdf Features of Functions: On page 11 of this lesson, students can describe features of a function from its graphical representation. F-IF 9 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 257 of 399 Columbus City Schools 6/28/13 1. Give students quadratic functions using different representations (algebraically, graphically, numerically in tables and by verbal descriptions). Use technology and instruct them to compare the properties of two different functions. 2. Give students functions expressing using different representations. Instruct them to match the functions that have the same properties. 3. Give the students sets of functions expressed in different representations. Ask them which are growing at a faster rate, which have a higher initial value, and why they increases faster than the other. Instruct students to explain their reasoning. 4. http://insidemathematics.org/common-core-math-tasks/high-school/HS-F2007%20Graphs2007.pdf Use the task, “Graphs (2007)”, found at the Inside Mathematics web-site. Students should use their previous knowledge about linear and quadratic functions to predict the shapes of the graphs given equations. Students will solve two equations graphically and symbolically. This activity has students compare various linear graphs with one quadratic graph, and identify the points of intersection between the linear and quadratic graphs. F-BF 1 1. Have students use the “Toothpicks and Models” activity (included in this Curriculum Guide) to collect data and try to come up with a quadratic model for the data set. 2. The activity, “Patterns with Triangles” (included in this Curriculum Guide), introduces the concept of how to generalize patterns using quadratic functions. Students should have already been introduced to the topic of first and second differences. In addition, students should be familiar with solving equations of the form x2 = a by taking the square root of both sides. Students should be able to recognize that the number of equal triangles is the square of the figure number. Then, students should come up with the equation that relates the number of equal triangles to the figure number. 3. The activity, “Patterns with Stacking Pennies” (included in this Curriculum Guide), also allows students to generalize patterns using quadratic functions. Pass out about 30 pennies (or chips) per student. They can use this to create more figures based on the pattern they see with the first three figures or they can just draw more figures in order to complete the table. Coming up with an equation that relates the number of pennies to the figure number will take the students some time. Allow them to try every possibility and have them work in groups to come up with an equation. Let students use the graphing calculator to graph the equation that they found and view the table feature to compare with the table that they have completed on the activity sheet. 4. The “Leap Frog Investigation” (included in this Curriculum Guide) is an opportunity for students to create a table of values from a situation that can be modeled by a quadratic equation. Each student needs two groups of objects to use as a manipulative for this activity, such as two different colors of chips or cubes, nickels and pennies, etc to distinguish the two groups of frogs that are changing places. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 258 of 399 Columbus City Schools 6/28/13 5. Students should complete the “Area Application” activity (included in this Curriculum Guide) in order to see the relationship between a situation, its table and its graph. Students will discover what type of “rectangle” will maximize area if they have a fixed perimeter. This activity was written to use pipe cleaners as a manipulative to form rectangles but other objects can be used as well, and the key was written as if 12” pipe cleaners were used (which are available at the warehouse). 6. The activity, “Toothpick Patterns” (included in this Curriculum Guide), will let students make a connection between a table of values and its graph. Students will compare what type of patterns will create a linear function and what type of patterns will create a quadratic function. Students will revisit the concepts of 1st and 2nd differences to distinguish between a linear and quadratic function. Provide students with plenty of toothpicks so they can create more figures if necessary. 7. Give students tables containing linear and quadratic data. Discuss the first and second differences in the tables. Instruct the students to graph the data points to determine the connections between the differences and the family of graphs. 8. http://map.mathshell.org.uk/materials/tasks.php?taskid=295 Building Functions: This is a quick task designed to have students quickly identify which of three graphs should be matched up with which of the tree given equations. 9. http://map.mathshell.org.uk/materials/lessons.php?taskid=215 Generalizing Patterns: Table Tiles: In this lesson students will examine data and look for patterns to identify linear and quadratic relationships. F-BF 1a 1. http://schools.nyc.gov/NR/rdonlyres/48D7F470-FDD4-477FB108_E02F7D969E93/0/NYCDOEHSAlgebraAussieFirTree_Final.pdf Have students use the performance task . “Aussie Fir Tree,”. Students will investigate patterns to describe a real world relationship 2. http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod1_quadratic_06 2213.pdf Scott’s Macho March: In this lesson (pp. 14-17), students focus on changes between values in a quadratic being linear. 3. http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec1_mod2_sequences_tn _91812.pdf Don’t Break the Chain: In this lesson students determine geometric sequences by determining the constant ratio between consecutive terms (pp. 29-34). 4. http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec1_mod2_sequences_tn CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 259 of 399 Columbus City Schools 6/28/13 _91812.pdf What Comes Next? What Comes Later? In this lesson students determine recursive and explicit equations for arithmetic and geometric sequences (pp. 50-57). 5. http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec1_mod2_sequences_tn _91812.pdf Geometric Meanies: In this lesson students use a constant ratio to find missing terms in a geometric sequence (pp. 64-70). 6. http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec1_mod2_sequences_tn _91812.pdf I Know… What Do You Know? In this lesson, students develop fluency with geometric and arithmetic sequences (pp. 71-82). 7. http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec1_mod2_sequences_tn _91812.pdf Growing, Growing Dots: In this lesson students represent geometric sequences with equations, tables, graphs and story context (pp. 14-21). F-BF 1b 1. http://map.mathshell.org/materials/tasks.php?taskid=279&subpage=expert Skeleton Tower: In this activity a tower is made by stacking cubes in a particular way. Students will determine a rule for calculating the total number of cubes needed to build towers of different heights. 2. http://map.mathshell.org/materials/tasks.php?taskid=283&subpage=expert Table Tiling: In this activity students are instructed to determine the tiles needed to cover the tops of the tables of different sizes. Students must work out the number of whole, half and quarter tiles needed to cover the tables. 3. http://map.mathshell.org/materials/tasks.php?taskid=285&subpage=expert Sidewalk Stones: In Czech Republic there are sidewalks of small square blocks of stone. These stones are found in different shades. They are used to make patterns of different sizes. In this task, instruct students to look for rules to determine the number of blocks of different colors needed to make the patterns. 4. http://map.mathshell.org/materials/tasks.php?taskid=254&subpage=apprentice Sidewalk Patterns: This task is similar to Sidewalk Stones. The sidewalks in Prague are made of small square stones of different shades. These are used to make patterns of various sizes. In this task, students will determine the rules for finding the number of different colored blocks needed to make the patterns. F-LE 3 1. http://www.shmoop.com/common-core-standards/ccss-hs-f-le-3.html CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 260 of 399 Columbus City Schools 6/28/13 At this site there is an explanation of the standard and several quiz problems. Students will observe graphs and tables to determine a quantity increasing exponentially will exceed a quantity increasing linearly, quadratically, or as a polynomial function. 2. http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod1_quadratic _062213.pdf Tortoise and Hare: In this lesson (pp. 31-35), students compare quadratic and exponential functions to distinguish between types of growth in each case. 3. http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod1_quadratic _062213.pdf How Does it Grow: In this lesson (pp. 36-43), students combine quadratics with their understanding of linear and exponential functions. A-CED 1 1. https://commoncorealgebra1.wikispaces.hcpss.org/file/view/A.CED.1+Babysit+Task.pdf To Babysit or Not To Babysit?: Students are given a problem situation where a girl needs to decide which family to babysit for during the summer months. Students are asked use different problem solving techniques (graphical, numerical, algebraically and a written description). 2. http://learnzillion.com/lessons/656-create-and-solve-quadratic-equations Create and Solve Quadratic Equations : In this lesson, student will learn how to create and solve equations by modeling a situation with a quadratic relationship. This website offers a video tutorial. 3. http://learnzillion.com/lessons/657-create-and-solve-quadratic-inequalities Create and Solve Quadratic Inequalities: In this lesson, students will learn how to create and solve inequalities by using a quadratic relationship. A-CED2 1. http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod1_quadratic_06 2213.pdf Something to Talk About: This lesson (pp. 3-7), is an introduction to quadratic functions where students determine type of pattern and change. 2. http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod1_quadratic_06 2213.pdf I Rule: In this lesson (pp. 8-13), students examine patterns in multiple representations and contrast them with linear relationships. 3. http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod1_quadratic_06 2213.pdf Rabbit Run: In this lesson (pp. 18-23), students focus on maximum and minimumm points, CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 261 of 399 Columbus City Schools 6/28/13 and the domain and range for quadratics. 4. http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod1_quadratic_06 2213.pdf Look Out Below: In this lesson (pp. 24-30), students examine quadratic functions on various intervals to determine average rates of change. 5. http://learnzillion.com/lessons/662-create-and-graph-quadratic-functions Create and Graph Quadratic Functions: In this lesson students will create and graph quadratic functions. 6. http://learnzillion.com/lessons/774-model-and-solve-problems-involving-quadratic-functionsby-using-a-table-of-values Model and Graph Quadratic Functions: In this lesson students will solve problems using quadratic functions using a table of values. 7. http://learnzillion.com/lessons/253-model-quadratic-functions-drawing-graphs-and-writingequations Model Quadratic Functions: Drawing Graphs and Writing Equations : In this lesson students will learn how to model quadratic functions by drawing graphs and writing equations. 8. http://www.ixl.com/math/algebra-1/characteristics-of-quadratic-functions Characteristics of Quadratic Functions: This website offers interactive questions on characteristics of quadratic functions. This site will provide practice also for the on-line end of course test. 9. http://education.ti.com/en/us/activity/detail?id=8FF6872578B64BD9A4B016A82F5894F5 This Ti-Interactive is for radical and quadratic families: Families of Functions: In this handheld activity students will change sliders and observe the effects on the graphs of the functions. 10. http://education.ti.com/en/us/activity/detail?id=A5BC6EE7E6304770A3808B2C69A88033 Parabolic Paths: In this handheld activity students will manipulate the equation of a quadratic function so that its graph passes through a particular point. They will be able to identify the effect of changing h and k on the graph of the quadratic function in the vertex form y = a(x h)2 + k and in the standard form y = ax2 + bx + c. 11. http://education.ti.com/en/us/activity/detail?id=108E4CC31691401FAD66A3420419A904 Parametric Ball Toss: This handheld lesson involves determining the height of a ball at a given time and determining the time at which the ball is at a certain height. 12. http://education.ti.com/en/us/activity/detail?id=E9C63B78A29F47DFAA53DE57B74E212C Zeros of a Quadratic Function: In this handheld activity students will merge graphical and algebraic representations of a quadratic function and its linear factors. 13. http://www.dlt.ncssm.edu/algebra/HTML/09.htm Pig Problem: Writing and Solving Quadratic Equations: Given several problem settings, CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 262 of 399 Columbus City Schools 6/28/13 students develop quadratic functions for which they investigate maximum values, zeros, and specific values to answer specific questions about the settings. 14. http://learnzillion.com/lessons/773-model-quadratic-functions-by-drawing-graphs-and-writingequations Model Quadratic Functions by Drawing Graphs and Writing Equations: In this lesson students will learn how to represent solutions and constraints to systems of linear inequalities by graphing. 15. http://insidemathematics.org/common-core-math-tasks/high-school/HS-F2008%20Functions.pdf Functions: A performance task on quadratic functions is provided at this site. Students work with graphs and equations of linear and nonlinear functions. 16. http://illuminations.nctm.org/LessonDetail.aspx?ID=L282 Building Connections, Lessons 1 and 2: In this lesson students make connections between different classes of polynomial functions by exploring graphs. 17. http://education.ti.com/en/us/activity/detail?id=579C674EC9F646BBAD61D99B6AA209D7 Exploring Power Functions 1: In this handheld lesson, students examine the graphs of power functions with even and odd positive integer exponents. 18. http://education.ti.com/en/us/activity/detail?id=C94C136B85784562A10DEEB98E1B81D2 Standard Form of Quadratic Functions : In this Nspire lesson student use sliders to determine the effect the parameters have upon a quadratic function in standard form. N-Q 2 1. http://learni.st/users/S33572/boards/1876-choosing-units-for-modeling-real-world-situationscommon-core-standard-9-12-n-q-2 Choosing Units for Modeling Real World Situations: This website contains videos and practice problems for descriptive modeling. 2. http://www.shmoop.com/common-core-standards/ccss-hs-n-q-2.html Math.N-Q.2: At this website there are examples and multiple choice questions concerning descriptive modeling. S-ID 6a 1. http://illuminations.nctm.org/LessonDetail.aspx?ID=U180 Determining Functions Using Regression: In this lesson, students collect data and use technology to find functions that best describe the data. Students should be able to determine what family of function best describes the trend. Activities 1, 2 and 3 apply to Math. 2. http://illuminations.nctm.org/LessonDetail.aspx?ID=L738 Egg Launch Contest: In this lesson students will represent quadratic functions as a table, with CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 263 of 399 Columbus City Schools 6/28/13 a graph, and with an equation. They will compare data and move between representations. 3. http://education.ti.com/en/us/activity/detail?id=8199F5A7AD60470082E865BD93FBC3EE Modeling with a Quadratic Function: In this handheld lesson, students use a quadratic function to model the flight path of a basketball. They will interpret the parameters of the quadratic model to answer questions related to the path of the basketball. 4. http://education.ti.com/en/us/activity/detail?id=0D9CB1B9A7ED43739A6AF9EBC59A0C49 Quadratic Functions and Stopping Distance: In this calculator activity students will analyze data in real-life applications of the quadratic function. 5. http://www.dlt.ncssm.edu/algebra/HTML/10.htm Football and Braking Distance: Model Data with Quadratic Functions: In this lesson, students are given data to fit to a quadratic function using least squares regression. 6. http://education.ti.com/en/us/activity/detail?id=06825EC64C1B42ED8091F9CFED3C0977 Transforming Relationships: In this activity, students assess the strength of a linear relationship using a residual plot. They will also calculate the correlation coefficient and coefficient of determination to assess the data set. Students will then learn to transform one or two variables in the relationship to create a linear relationship. This is a Nspire activity. 7. http://education.ti.com/xchange/US/Math/Statistics/11524/Stat_Transform_TI84.pdf Transforming Relationships: In this lesson, students asses the strength of a linear relationship using a residual plot. This is a TI-84 activity. 8. http://calculator.maconstate.edu/quad_regression/index.html Quadratic Regression: At this site there are directions on using graphing technology to determine quadratic regressions. 9. Students should complete the “Ball Bounce Activity” (included in this Curriculum Guide) in order to see a real life situation that creates a parabolic graph. Students will collect data with a CBR to create the distance vs. time graph. 10. Students should complete the “Water Fountain Activity” (included in this Curriculum Guide) in order to fit a quadratic model to a real life situation. Please note that there is no answer key because students’ work and answers will vary for the whole activity. S-ID 6a 1. https://commoncorealgebra1.wikispaces.hcpss.org/Unit+2 Calculating Residuals: This website offers a lesson plan and student practice calculating residuals 2. https://commoncorealgebra1.wikispaces.hcpss.org/Unit+2 Analyzing Residuals: This website offers a lesson plan and student practice analyzing residuals. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 264 of 399 Columbus City Schools 6/28/13 A-REI 7 1. http://www.regentsprep.org/regents/math/algtrig/ate5/quadlinearsys.htm This website offers an explanation of the standard and several examples. 2. http://eucc2011.wikispaces.com/file/view/See_You_Later_Alligator_Student.pdf/317649886/S ee_You_Later_Alligator_Student.pdf This website offers practice problems for solving linear – quadratic systems. 3. http://secondaryiiinutah.wikispaces.com/file/view/Secondary%20II%20%20Non%20linear%20Systems.pdf/353991416/Secondary%20II%20%20Non%20linear%20Systems.pdf This website offers problem solving situations using linear-quadratic systems. 4. http://www.mathwarehouse.com/system-of-equations/how-to-solve-linear-quadraticsystem.php This site has notes and interactive practice on solving systems of linear and quadratic equations. This website offers an explanation of the standard and practice problems. 5. http://www.phschool.com/atschool/new_york/phmath07_intalg/IANYSENY06.pdf Systems of Linear and Quadratic Equations: This site has textbook notes and practice problems for solving systems of linear and quadratic equations. 6. http://mathbits.com/MathBits/TISection/Algebra1/LinQuad.htm Solving a Linear Quadratic System: At this site there are directions on the use of a graphing calculator to determine the solution of a linear quadratic system. 7. http://learnzillion.com/lessonsets/263-solve-simple-systems-of-equations-with-linear-andquadratic-equations Solve Simple Systems of Equations with Linear and Quadratic Equations: This website contains a series of tutorials for solving systems consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Reteach: http://www.nsa.gov/academia/_files/collected_learning/high_school/algebra/catapult_trajectori es.pdf Catapult Trajectories: Don’t Let Parabolas Throw You: In this activity students will find the equation of a parabola from given points. They will CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 265 of 399 Columbus City Schools 6/28/13 determine the effects of a, b, and c using graphing calculators. In the lessons provided at the website below, students can review graphing parabolas. Introduction to Parabolas (p. 6) Graphing Parabolas (pp. 8–10) Behavior of Parabolas (pp. 14–15) http://digitalcommons.brockport.edu/cgi/viewcontent.cgi?article=1140&context=ehd_theses Lessons can be found at this site for review of quadratic equations for this topic. Cannonball Trajectory Motion Applet (p. 84) Calculator Tables and Graphs (p. 87) Guiding Questions (p. 88) Investigate Average Rate of Change: Falling Book Problem (pp. 89 – 91) Applets to Compare Meaning of Coefficients to Linear and Quadratic Functions (p. 92) Applets to Compare Standard and Vertex Form (p. 93) Translations Applet: Will It Make the Hoop? (p. 94) http://www.algebralab.org/Word/Word.aspx?file=Algebra_QuadraticRegression.xml Word Problems: Quadratic Regression A re-teach lesson and practice on quadratic regression. Extensions: http://www.nsa.gov/academia/_files/collected_learning/high_school/algebra/catapult_trajectori es.pdf Catapult Lab Investigation At the website below there is a lab to investigate parabolic paths (pp. 18–21). Post Lab Activity (p. 26) Finding the Exact Quadratics (p. 28) Using TI Transform with Parabolas (pp. 31-34) Target Practice (p. 38) Determine the Equation: Students should find a picture that includes a parabola, preferably from architecture. They should trace the picture and transfer it to graph paper. They should then determine the equation of the parabola as well as an appropriate scale for their graph. What is a Catenary?: Instruct students to research catenaries. Explain what they are and determine the similarities and differences between them and parabolas. Give examples of catenaries. McDonald’s Arches: Have students find a picture of the McDonald’s arches. They should trace the picture and transfer to graph paper then determine the equation of the curve as well as an appropriate scale for their graph. What’s the Story? Given a quadratic graph, students should create a real-life situation that CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 266 of 399 Columbus City Schools 6/28/13 models the graph. 48 44 40 36 32 28 24 20 16 12 8 4 -2 -1 1 2 3 4 5 6 7 8 St. Louis Arch: Give students a graph of the St. Louis Arch, and have them determine its equation. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 267 of 399 Columbus City Schools 6/28/13 Textbook References: Textbook: Algebra I, Glencoe (2005): pp. 524-532, 533-538, 545, 553 Supplemental: Algebra I, Glencoe (2005): Chapter 10 Resource Masters Reading to Learn Mathematics, pp. 583, 589 Study Guide and Intervention, pp. 579-580, 585-586 Skills Practice, pp. 581, 587 Practice, pp. 582, 588 Enrichment, pp. 584, 590 Textbook: 841 Algebra 2,Glencoe (2003): pp. 286-293, 294-299, 320-321, 322-328, 329-335, 839- Supplemental: Algebra 2,Glencoe (2003): Chapter 6 Resource Masters Reading to Learn Mathematics, p. 317, 323, 347, 353 Study Guide and Intervention, pp. 313-314, 319-320, 343-344, 349-350 Skills Practice, p. 315, 321, 345, 352 Practice, p. 316, 322, 346, 352 Enrichment, p. 318, 334, 348, 354 Textbook: Integrated Mathematics: Course 3, McDougal Littell (2002): pp. 45-52, 72-73, 9596, 359-363, 651 Textbook: Advanced Mathematical Concepts, Glencoe (2004): pp. 159-168, 169-170, 171-179, 213-221 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 268 of 399 Columbus City Schools 6/28/13 F-BF 3 Name ___________________________________ Date __________________ Period ________ Quadratics Inquiry Project Teacher Notes: Objective: After completion of the quadratics project with a group of peers, the students will be able to successfully answer 8 out of 10 questions regarding quadratic equations in vertex form, and the shifts of quadratic graphs in vertex form. Day One: Students will be placed in groups of three to four, and be given a set of graphs of nine different parabolas (see next three pages). The students will figure out a way to place the graphs into two distinct categories, and will cut and paste them onto a large sheet of paper. They will continue this process until they have completed it a total of four times, using different categories each time. Once they have completed this they will begin a list of questions. These should be questions that they believe they need to know the answers to in order to be able to change one graph to another graph. (At the top of this paper should be their “guiding question”, “How does manipulating the equation affect the graph?”) Day Two: Show the class a graph of y = (x)2. Their task, with a graphing calculator, is to get an equation for each of the nine graphs. They should continue trying different equations until they believe they are as close as possible to the picture they started with. Day Three: Ask each group to give you their equation for a particular graph. Then graph the original equation with all of their equations, and see which group got the closest. Each group will note the best equation. (Students still will not see or be told what the original equations were. If the equation is entered into y10, it will not be visible to the students.) Day Four: Without a calculator ask students to determine which of the nine equations, which will now be given to them (see the following pages), goes with each of the graphs they have. They will label their graphs with these equations. When everyone is finished, check the answers again with the graphing calculator as a class. Day Five: Students will be given ten questions regarding how to transform one quadratic equation in vertex form into another quadratic equation in vertex form(See following pages). They will also be given self and peer evaluations to fill out (see the following pages). When all evaluations have been handed in discuss if they have answered their original questions themselves. Day Six/ Follow Up: Discuss what a parent function is, what variables are used in the equation, and how this relates to the vertex. At this point the students will be asserting what they know, and will now be given the correct terminology to use. (Students can post their graphs under categories on a large chart so that the class can establish "rules" on how changing the equation changes the graph). CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 269 of 399 Columbus City Schools 6/28/13 Assessment: Students can fill out a self evaluation as well as a peer evaluation. The teacher can do two, one relating to the group aspect, while the other relates to the mathematical concepts used. Finally there is a ten question follow up at the end (as noted on days five and six). The teacher may choose to grade/evaluate based on different methods for your particular class. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 270 of 399 Columbus City Schools 6/28/13 F-BF 3 Name ___________________________________ Date __________________ Period ________ 1. 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 2 4 6 8 10 -2 -4 -6 -8 -10 2. 10 8 6 4 2 -10 -8 -6 -4 -2 -2 -4 -6 -8 -10 3. 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 271 of 399 Columbus City Schools 6/28/13 4. 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 5. 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 2 4 6 8 10 -2 -4 -6 -8 -10 6. 10 8 6 4 2 -10 -8 -6 -4 -2 -2 -4 -6 -8 -10 7. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 272 of 399 Columbus City Schools 6/28/13 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 2 4 6 8 10 -2 -4 -6 -8 -10 8. 10 8 6 4 2 -10 -8 -6 -4 -2 -2 -4 -6 -8 -10 9. 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 273 of 399 Columbus City Schools 6/28/13 F-BF 3 Name ___________________________________ Date __________________ Period ________ 1. y = .5(x – 5)2 + 3 2. y = 2(x + 2)2 – 1 3. y = .2(x – 2)2 – 4 4. y = 8(x + 7)2 + 6 5. y = - 2(x + 4)2 + 2 2 6. y = - (x – 2)2 + 8 7 7. y = - 2(x + 5)2 – 3 8. y = .5(x + 6)2 + 1 9. y = - .2(x – 2)2 – 2 F-BF 3 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 274 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Quadratics Inquiry Based on your “research” during the last week, answer the following questions to the best of your ability. 1. How would you make a quadratic graph shift to the right? 2. How would you make a quadratic graph flip from opening up to opening down? 3. How would you make a quadratic graph shift up? 4. How would you make a quadratic graph wider? 5. How would you make a quadratic graph shift down? CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 275 of 399 Columbus City Schools 6/28/13 6. How would you make a quadratic graph become narrower? 7. How would you make a quadratic graph shift to the left? 8. How would you make a quadratic function that is opening down open up? 9. What is the number one thing that you learned from doing this project? 10. What questions do you still have about quadratics? CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 276 of 399 Columbus City Schools 6/28/13 Evaluator's Name: Student Name: CATEGORY 4 Contributions Routinely provides useful ideas when participating in the group and in classroom discussion. A definite leader who contributes a lot of effort. 3 Usually provides useful ideas when participating in the group and in classroom discussion. A strong group member who tries hard! Attitude Never is publicly critical of the project or the work of others. Always has a positive attitude about the task(s). Rarely is publicly critical of the project or the work of others. Often has a positive attitude about the task(s). Timemanagement Routinely uses time well throughout the project to ensure things get done on time. Group does not have to adjust deadlines or work responsibilities because of this person's procrastination. Monitors Group Effectiveness Routinely monitors the effectiveness of the group, and makes suggestions to make it more effective. Usually uses time well throughout the project, but may have procrastinated on one thing. Group does not have to adjust deadlines or work responsibilities because of this person's procrastination. Routinely monitors the effectiveness of the group and works to make the group more effective. 2 Sometimes provides useful ideas when participating in the group and in classroom discussion. A satisfactory group member who does what is required. Occasionally is publicly critical of the project or the work of other members of the group. Usually has a positive attitude about the task(s). Tends to procrastinate, but always gets things done by the deadlines. Group does not have to adjust deadlines or work responsibilities because of this person's procrastination. Occasionally monitors the effectiveness of the group and works to make the group more effective. Working with Others Almost always listens to, shares with, and supports the efforts of others. Tries to keep people working well together. Usually listens to, shares, with, and supports the efforts of others. Does not cause "waves" in the group. Often listens to, shares with, and supports the efforts of others, but sometimes is not a good team member. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 277 of 399 1 Rarely provides useful ideas when participating in the group and in classroom discussion. May refuse to participate. Rarely listens to, shares with, and supports the efforts of others. Often is not a good team player. Often is publicly critical of the project or the work of other members of the group. Often has a positive attitude about the task(s). Rarely gets things done by the deadlines AND group has to adjust deadlines or work responsibilities because of this person's inadequate time management. Rarely monitors the effectiveness of the group and does not work to make it more effective. Columbus City Schools 6/28/13 Student Name: CATEGORY Mathematical Concepts Math - Problem Solving : Quadratics 4 Explanation shows complete understanding of the mathematical concepts used to solve the problem(s). 3 Explanation shows substantial understanding of the mathematical concepts used to solve the problem(s). 2 Explanation shows some understanding of the mathematical concepts needed to solve the problem(s). Student was an engaged partner, listening to suggestions of others and working cooperatively throughout lesson. Explanation is detailed and clear. Student was an engaged partner but had trouble listening to others and/or working cooperatively. Student cooperated with others, but needed prompting to stay on-task. Explanation is clear. Explanation is a little difficult to understand, but includes critical components. Neatness and Organization The work is presented in a neat, clear, organized fashion that is easy to read. The work is presented in a neat and organized fashion that is usually easy to read. Completion All problems are completed. Strategy/ Procedures Typically, uses an efficient and effective strategy to solve the problem(s). All but 1 of the problems are completed. Typically, uses an effective strategy to solve the problem(s). Mathematical Reasoning Uses complex and refined mathematical reasoning. The work is presented in an organized fashion but may be hard to read at times. All but 2 of the problems are completed. Sometimes uses an effective strategy to solve problems, but does not do it consistently. Some evidence of mathematical reasoning. Working with Others Explanation Uses effective mathematical reasoning CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 278 of 399 1 Explanation shows very limited understanding of the underlying concepts needed to solve the problem(s) OR is not written. Student did not work effectively with others. Explanation is difficult to understand and is missing several components OR was not included. The work appears sloppy and unorganized. It is hard to know what information goes together. Several of the problems are not completed. Rarely uses an effective strategy to solve problems. Little evidence of mathematical reasoning. Columbus City Schools 6/28/13 F-BF 3 Name ___________________________________ Date __________________ Period ________ Graphs Graph each group of functions on the same screen. Compare and contrast the graphs. Group 1 Group 2 Group 3 Group 4 Group 5 f(x) = - x f(x) = x f(x) = x f(x) = x f(x) = x f(x) = 4 x f(x) = 1 x 2 f(x) = - 2x f(x) = (x – 2) f(x) = x -1 f(x) = 8 x f(x )= 6 x 5 f(x) = - 5x f(x) = (x + 3) f(x) = x + 3 Group 1 Group 2 Group 3 Group 4 Group 4 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 279 of 399 Group 5 Columbus City Schools 6/28/13 Group Group 1 Compare and contrast Group 2 Group 3 Group 4 Group 5 Write an equation of a parabola that moves the parent graph y = x : A. 3 units to the right and 2 units up B. 1 unit to the left and 3 units up C. 2 units to the right and 3 units down D. reflects and moves 2.5 units down E. reflects and moves to the left 1 units 2 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 280 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Graphs Answer Key Graph each group of functions on the same screen. Compare and contrast the graphs. Group 1 Group 2 Group 3 Group 4 Group 5 f(x) = x f(x) = x f(x) = 4 x f(x) = f(x) = 8 x f(x) = Group 1 f(x) = x f(x) = x 1 x 2 f(x) = - 2x f(x) = (x – 2) f(x) = x -1 6 x 5 f(x) = - 5x f(x) = (x + 3) f(x) = x + 3 Group 2 Group 4 Group Group 1 f(x) = x f(x) = 4 x f(x) = 8 x Group 2 f(x) = x 1 f(x) = x 2 6 f(x) = x 5 Group 3 f(x) = -x Group 3 Group 5 Compare and contrast 4 x² is narrower than x² 8 x² is narrower than both of the other graphs. 1 x² is wider than x² 2 6 x² is narrower than the other 2 graphs 5 - 2x² is narrower than x² CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 281 of 399 Columbus City Schools 6/28/13 f(x) =- x f(x) = - 2x f(x) = - 5x Group 4 f(x) = x f(x) = (x – 2) f(x) = (x + 3) Group 5 f(x) = x f(x) = x -1 f(x) = x + 3 - 5x² is narrower than the other 2 graphs (x – 2)² moves 2 units to the right (x + 3)² moves 3 units to the left x² -1 moves down 1 unit x² + 3 moves up 3 units Write an equation of a parabola that moves the parent graph y = x : A. 3 units to the right and 2 units up f(x) = (x – 3)² + 2 B. 1 unit to the left and 3 units up f(x) = (x + 1)² + 3 C. 2 units to the right and 3 units down f(x) = (x - 2)² - 3 D. reflects and moves 2.5 units down f(x) = -x² - 2.5 E. reflects and moves to the left ½ units f(x) = -(x + ½ )² CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 282 of 399 Columbus City Schools 6/28/13 F-BF 3 Name ___________________________________ Date __________________ Period ________ Families of Graphs Graph each equation with a graphing calculator. Sketch the graphs on the grids provided. How did the graphs move as compared to y = x²? Determine the vertex of each graph. Graph of y = x² Equation y=x +2 Graph Comparison Vertex y=x -2 y = (x - 2) CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 283 of 399 Columbus City Schools 6/28/13 y = (x - 2) + 2 y = (x + 2) y = (x + 2) - 2 y=-x y=-x +2 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 284 of 399 Columbus City Schools 6/28/13 y = - (x – 2) y = -x - 4x + 4 Compare the equations below to y = x². Equation Graph y = 2x Changes in graph Vertex y = 6x CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 285 of 399 Columbus City Schools 6/28/13 y= 1 x 2 y= 1 x 6 y = -2 x y= 1 x 8 y = 2 x - 4x + 2 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 286 of 399 Columbus City Schools 6/28/13 y = (2x + 4) y = (2x - 4) y = (2x + 4) + 1 1. What changes (moves, transformations) are you noticing about the graphs? 2. Make a list from your examinations of the graphs above. Ex. Graphs stretch (get wider) Graphs compress (get narrower) CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 287 of 399 Columbus City Schools 6/28/13 Families of Graphs Answer Key Graph each equation with a graphing calculator. Sketch the graphs on the grids provided. How did the graphs move as compared to y = x²? Determine the vertex of each graph. Graph of y = x² Equation y=x +2 Graph Comparison Vertex Moves up 2 units (0,2) Moves down 2 units (0, -2) Moves right 2 units (2, 0) y=x -2 y = (x - 2) CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 288 of 399 Columbus City Schools 6/28/13 y = (x - 2) + 2 Moves right 2 units (2, 2) and up 2 units. y = (x + 2) Moves left 2 units. (-2, 0) Moves left 2 units and down 2 units. (-2, -2) Reflection (0,0) y = (x + 2) - 2 y=-x y=-x +2 Reflection and up 2 (0, 2) units. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 289 of 399 Columbus City Schools 6/28/13 y = - (x – 2) Reflection and right 2 units. (2,0) Reflection and left 2 units and up 8 units. (-2, 8) y = -x - 4x + 4 Compare the equations below to y = x². Equation Graph y = 2x Changes in graph Vertex Compressed (0,0) Compressed (0,0) y = 6x CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 290 of 399 Columbus City Schools 6/28/13 y= 1 x 2 Stretched (0,0) y= 1 2 x 6 Stretched (0,0) Reflection and compressed (0,0) Reflection and stretched (0,0) Compressed and moves right 1 unit. (1, 0) y = -2 x y= 1 2 x 8 y = 2 x - 4x + 2 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 291 of 399 Columbus City Schools 6/28/13 y = (2x + 4) Left 2 units and compressed. (0, -2) Right 2 units and compressed. (2, 0) Left 2 units and up 1 unit and compressed. (-2,1) y = (2x - 4) y = (2x + 4) + 1 1. What changes (moves, transformations) are you noticing about the graphs? Wider left right reflections Narrower down up 2. Make a list from your examinations of the graphs above. Ex. Graphs stretch (get wider) Graphs compress (get narrower) Wider: coefficient of x² 1 Left: x² in parenthesis, + c Right: x² in parenthesis, - c Reflections: negative sign before x² Narrower: coefficient of x² < 1 Down: (- c) not in parenthesis Up: (+c) not in parenthesis CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 292 of 399 Columbus City Schools 6/28/13 F-BF 3 Name ___________________________________ Date __________________ Period ________ Tables Quadratics Graphs Equations 1. Graph each point in the table on the given set of axes. x 2 3 4 5 6 7 y 9 6 5 6 9 14 2. Describe the resulting graph in the space provided. Make at least 3 specific observations. 3. Graph and label at least 3 more points that lie on this graph. 4. Write an equation either in vertex form or in standard form that fits this data. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 293 of 399 Columbus City Schools 6/28/13 5. Graph each point in the table on the given set of axes. x 0 1 2 3 4 5 y 1 7 9 7 1 -9 6. Describe the resulting graph in the space provided. Make at least 3 specific observations. 7. Graph and label at least 3 more points that lie on this graph. 8. Write an equation either in vertex form or in standard form that fits this data. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 294 of 399 Columbus City Schools 6/28/13 9. Graph each point in the table on the given set of axes. x -2 -1 0 1 2 y -5 -2 -1 -2 -5 10. Describe the resulting graph in the space provided. Make at least 3 specific observations. 11. Graph and label at least 3 more points that lie on this graph. 12. Write an equation either in vertex form or in standard form that fits this data. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 295 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Quadratics Tables Graphs Equations Answer Key 1. Graph each point in the table on the given set of axes. x 2 3 4 5 6 7 y 9 6 5 6 9 14 2. Describe the resulting graph in the space provided. Make at least 3 specific observations. Answers may vary. The graph is parabolic; the second difference is constant; the graph opens up; the vertex is at (4, 5); and there are mirror image points at (3, 6) and (5, 6), and (2, 9) and (6, 9) 3. Graph and label at least 3 more points that lie on this graph. Answers may vary. Examples would be (1, 14) (8, 21) (9, 30). 4. Write an equation either in vertex form or in standard form that fits this data. vertex form: y = (x – 4)2 + 5 standard form: y = x2 – 8x + 21 5. Graph each point in the table on the given set of axes. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 296 of 399 Columbus City Schools 6/28/13 x 0 1 2 3 4 5 y 1 7 9 7 1 -9 6. Describe the resulting graph in the space provided. Make at least 3 specific observations. Answers may vary. The graph is parabolic;; the second difference is constant and it’s a negative number; the graph opens down; the vertex is at (2, 9); and there are mirror image points at (1, 7) and (3, 7), and (0, 1) and (4, 1) 7. Graph and label at least 3 more points that lie on this graph. Answers may vary. Examples would be (-2, -23) (-1, -9) (6, -23). 8. Write an equation either in vertex form or in standard form that fits this data. vertex form: y = -2(x – 2)2 + 9 standard form: y = -2x2 + 8x + 1 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 297 of 399 Columbus City Schools 6/28/13 9. Graph each point in the table on the given set of axes. x -2 -1 0 1 2 y -5 -2 -1 -2 -5 10. Describe the resulting graph in the space provided. Make at least 3 specific observations. Answers may vary. The graph is parabolic;; the second difference is constant and it’s a negative number; the graph opens down; the vertex is at (0, -1); and there are mirror image points at (-2, -5) and (2, -5), and (-1, -2) and (1, -2) 11. Graph and label at least 3 more points that lie on this graph. Answers may vary. Examples would be (3, -10), (-3, -10), (4, -17) 12. Write an equation either in vertex form or in standard form that fits this data. vertex form: y = -(x – 0)2 – 1 standard form: y = -x2 – 1 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 298 of 399 Columbus City Schools 6/28/13 F-BF 3 Name ___________________________________ Date __________________ Period ________ Vertex Form and Transformations Work with your partner to complete this assignment. Do not use a graphing calculator and show all your work. Examine the table, the graph, and the equation. What do you notice about each set? 1. Equation 2 f(x) = x Table x Graph -3 -2 -1 0 1 2 10 3 8 f(x) 9 4 1 0 1 2 9 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 2 4 6 8 10 -2 -4 -6 -8 -10 2. Equation f(x) = x2 + 2 Table Graph x -3 f(x) 11 -2 -1 0 1 2 3 10 6 3 2 3 6 11 8 6 4 2 -10 -8 -6 -4 -2 -2 -4 -6 -8 -10 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 299 of 399 Columbus City Schools 6/28/13 3. Equation f(x)= (x + 2)2 Table x Graph -3 -2 -1 0 1 2 3 10 f(x) 1 0 1 4 9 16 25 8 6 4 2 -10 -8 -6 -4 -2 2 4 2 4 6 8 10 -2 -4 -6 -8 -10 4. Equation f(x)= (x - 2)2+1 Table x f(x) Graph -1 10 0 5 1 2 2 1 3 2 4 5 5 10 10 8 6 4 2 -10 -8 -6 -4 -2 6 8 10 -2 -4 -6 -8 -10 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 300 of 399 Columbus City Schools 6/28/13 5. Given: f(x) = a(x – h)2 + k Determine the axis of symmetry for each graph. Determine the vertex. Compare this information to the general equation above. 6. Predict the axis of symmetry and vertex for y = (x – 4)2. 7. Predict the axis of symmetry and vertex for y = x2 + 5. 8. Predict the axis of symmetry and vertex for y = (x + 2)2 – 7. 9. Describe the effects of a, h and k on the axis of symmetry and the vertex. 10. What is the vertex and y-intercept for y = -3x2 + 1 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 11. Compare the following graphs with their equations. y = x2 y = 5x2 y = 1/2x2 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 301 of 399 Columbus City Schools 6/28/13 -10 -8 -6 -4 10 10 8 8 6 6 6 4 4 4 2 2 2 -2 2 4 6 8 10 -10 -8 -6 -4 -2 10 8 2 4 6 8 10 -10 -8 -6 -4 -2 2 -2 -2 -4 -4 -4 -6 -6 -6 -8 -8 -8 -10 -10 -10 4 6 8 10 -2 How does the coefficient affect the graphs? CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 302 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Vertex Form and Transformations Answer Key Work with your partner to complete this assignment. Do not use a graphing calculator and show all your work. Examine the table, the graph, and the equation. What do you notice about each set? 1. Equation Table Graph 2 f(x) = x x -3 -2 -1 0 1 2 10 3 8 f(x) 9 4 1 0 1 2 9 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 6 8 10 -2 -4 -6 -8 -10 Answers may vary: U-shaped graph that is symmetric; Line of symmetry is x = 0; y-intercept is (0, 0); Vertex (0, 0) 2. Equation Table Graph 2 f(x) = x + 2 x -3 -2 -1 0 1 2 10 3 8 f(x) 11 6 3 2 3 6 11 6 4 2 -10 -8 -6 -4 -2 2 4 -2 -4 -6 -8 -10 Answers may vary: U-shaped graph that is symmetric; Line of symmetry is x = 0; y-intercept is (0, 2) Vertex is (0, 2) CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 303 of 399 Columbus City Schools 6/28/13 3. Equation f(x)= (x + 2)2 Table x Graph -3 -2 -1 0 1 2 3 10 y= x2+4x+4 f(x) 1 0 1 4 9 16 25 8 6 4 2 -10 -8 -6 -4 -2 2 4 2 4 6 8 10 -2 -4 -6 -8 -10 Answers may vary: U-shaped graph that is symmetric Line of symmetry is x = -2 y-intercept is (0, 4) and the symmetric point is (-4, 4) Vertex is (-2, 0) 4. Equation f(x)= (x - 2)2+1 Table x f(x) Graph -1 10 0 5 1 2 2 1 3 2 4 5 10 5 10 8 6 4 2 -10 -8 -6 -4 -2 6 8 10 -2 -4 -6 -8 -10 Answers may vary: U-shaped graph that is symmetric Line of symmetry is x = 2 y-intercept is (0, 5) and the symmetric point is (4, 5) Vertex is (2, 1) CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 304 of 399 Columbus City Schools 6/28/13 5. Given: f(x) = a(x – h)2 + k Determine the axis of symmetry for each graph. Determine the vertex. x=h (h, k) Compare this information to the general equation above. 6. Predict the axis of symmetry and vertex for y = (x – 4)2. x=4 (4, 0) 7. Predict the axis of symmetry and vertex for y = x2 + 5. x=0 (0, 5) 8. Predict the axis of symmetry and vertex for y = (x + 2)2 – 7. x = -2 (-2, -7) 9. Describe the effects of a, h and k on the axis of symmetry and the vertex. “a” affects the shape of the parabola;; “h” affects the movement left or right of each x coordinate;; “k” effects the movement up or down of each y-coordinate. 10. What is the vertex and y-intercept for y = -3x2 + 1 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 The vertex is (0, 1); the y-intercept is (0, 1). 11. Compare the following graphs with their equations. y = x2 y = 5x2 -10 -8 -6 -4 y = 1/2x2 10 10 10 8 8 8 6 6 6 4 4 4 2 2 2 -2 2 4 6 8 10 -10 -8 -6 -4 -2 2 4 6 8 10 -10 -8 -6 -4 -2 2 -2 -2 -4 -4 -4 -6 -6 -6 -8 -8 -8 -10 -10 -10 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 305 of 399 4 6 8 10 -2 Columbus City Schools 6/28/13 How does the coefficient affect the graphs? y = x2, parent graph “a” is 1 y = 5x2 “a” is 5 Graph is more narrow (stretched vertically) 2 y = 1/2x “a” is ½ Graph is wider (compressed vertically) CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 306 of 399 Columbus City Schools 6/28/13 F-BF 3 Name ___________________________________ Date __________________ Period ________ Graphing Quadratic Functions Graph the following quadratic functions using a graphing calculator. Record how many times the graph intercepts the x-axis. 1. f(x) = x2 + 2x – 15 Intercepts ___________________ time(s) 2. f(x) = x2 – 6x + 9 Intercepts ___________________ time(s) 3. f(x) = x2 – 4x + 7 Intercepts ___________________ time(s) 4. f(x) = 4x2 + 12x + 9 Intercepts ___________________ time(s) 5. f(x) = 3x2 – 4x + 3 Intercepts ___________________ time(s) 6. f(x) = ( 1 2 ) x2 + ( 32 ) x + 9 Intercepts ___________________ time(s) 7. f(x) = -x2 + 6x – 14 Intercepts ___________________ time(s) 8. f(x) = -2x2 + 3x + 8 Intercepts ___________________ time(s) 9. f(x) = ( 14 ) x2 – x + 1 Intercepts ___________________ time(s) CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 307 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Graphing Quadratic Functions Answer Key Graph the following quadratic functions using a graphing calculator. Record how many times the graph intercepts the x-axis. 1. f(x) = x2 + 2x – 15 Intercepts two time(s) 2. f(x) = x2 – 6x + 9 Intercepts one time(s) 3. f(x) = x2 – 4x + 7 Intercepts zero time(s) 4. f(x) = 4x2 + 12x + 9 Intercepts one time(s) 5. f(x) = 3x2 – 4x + 3 Intercepts zero time(s) ) x2 + ( 32 ) x + 9 Intercepts two time(s) 6. f(x) = ( 1 2 7. f(x) = -x2 + 6x – 14 Intercepts zero time(s) 8. f(x) = -2x2 + 3x + 8 Intercepts two time(s) 9. f(x) = ( 14 ) x2 – x + 1 Intercepts one time(s) CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 308 of 399 Columbus City Schools 6/28/13 F-BF 3 Name ___________________________________ Date __________________ Period ________ Graph in a Flash 1. Enter the equation into your graphing calculator. 2. Decide on a viewing window by first determining the coordinates of the vertex, and if the graph opens upward or downward. Then choose your window to include the vertex and the x-axis. Record the coordinates of the vertex on this recording sheet. 3. Press GRAPH. 4. Bring up the CALC menu (by pressing 2nd TRACE). 5. Select the second option, zero. 6. Move the cursor to a point that is to the left of the first x-intercept, press ENTER. Then move the cursor to a point that is to the right of the first x-intercept, press ENTER. Then press ENTER to have the calculator make a best guess of the intercept. The calculator will now display the x and y values for the x-intercept you selected. 7. Repeat for any other x-intercepts. 8. Record the x-intercepts (zeros) on this recording sheet. Equation Vertex Zeros 1. y 2 x2 19 x 90 2. y 4 x2 32 x 20 3. y 3x2 55x 429 4. y 1 2 x 4 3 x 5 8 5. y 1 x2 25 6. y 3 2 x 4 7. y 8. y 2 x 15 45 7 745x 1962 7 x2 19 x 45 63 x2 100 7 x 80 29 4 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 309 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Graph in a Flash (Answer Key) 1. Enter the equation into your graphing calculator. 2. Decide on a viewing window by first determining the coordinates of the vertex, and if the graph opens upward or downward. Then choose your window to include the vertex and the x-axis. Record the coordinates of the vertex on this recording sheet. 3. Press GRAPH. 4. Bring up the CALC menu (by pressing 2nd TRACE). 5. Select the second option, zero. 6. Move the cursor to a point that is to the left of the first x-intercept, press ENTER. Then move the cursor to a point that is to the right of the first x-intercept, press ENTER. Then press ENTER to have the calculator make a best guess of the intercept. The calculator will now display the x and y values for the x-intercept you selected. 7. Repeat for any other x-intercepts. 8. Record the x-intercepts (zeros) on this recording sheet. Equation Vertex Zeros x = -3.46964111 1. y 2 x 2 19 x 90 (4.75, -135.125) or x = 12.96964111 2. y 4x 3. y 3x2 55x 429 4. y 32 x 20 2 1 2 x 4 1 x2 25 6. y 3 2 x 4 7. y 8. y (4, 44) 2 x 15 45 7 745x 1962 7 x2 19 x 45 63 x2 100 7 x 80 ( 9.16 , 681.083 ) x = 24.23410765 or x = -5.900774316 (-0.75, -5.140625) x = -5.284589287 or x = 3.784589287 _ 3 x 5 8 5. y x = 7.31662479 or x = 0.6833752096 29 4 _ (- 1.6 , -6.539683) x = -14.45307 or x = 11.119735 2 1 ( 496 , 186,970 ) 3 3 x = -2.626612 or x = 995.95994 (-1.357143, -57.89286) x = -4.232975 or x = 1.5186889 (-0.069444, -7.253038194) x = -3.46249 or x = 3.3236012 _ CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 310 of 399 Columbus City Schools 6/28/13 F-BF 3 Name ___________________________________ Date __________________ Period ________ Properties of a Parabola With the assistance of a graphing calculator, sketch the graph of each quadratic equation in pencil. Label the vertex of each parabola. 1. y = x2 – 4x + 9 Is there a line of symmetry? If so, use a pen to draw in the line of symmetry. Write the equation for the line of symmetry. What are the coordinates of the y-intercept? Does the parabola have a minimum or a maximum? What is the minimum or maximum value of the function? Identify the domain and the range of the parabola. Domain: Range: 2. y = - 0.5x2 + 2x – 5 Is there a line of symmetry? If so, use a pen to draw in the line of symmetry. Write the equation for the line of symmetry. What are the coordinates of the y-intercept? Does the parabola have a minimum or a maximum? What is the minimum or maximum value of the function? Identify the domain and the range of the parabola. Domain: Range: CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 311 of 399 Columbus City Schools 6/28/13 3. y = 3x2 – 6x + 7 Is there a line of symmetry? If so, use a pen to draw in the line of symmetry. Write the equation for the line of symmetry. What are the coordinates of the y-intercept? Does the parabola have a minimum or a maximum? What is the minimum or maximum value of the function? Identify the domain and the range of the parabola. Domain: Range: ______ 4. y = - x2 – 10x – 27 Is there a line of symmetry? If so, use a pen to draw in the line of symmetry. Write the equation for the line of symmetry. What are the coordinates of the y-intercept? Does the parabola have a minimum or a maximum point? What is the minimum or maximum value of the function? Identify the domain and the range of the parabola. Domain: Range: What connections can you make between the vertex, the line of symmetry, and the minimum or maximum of the graph? Are there any other connections that you notice? CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 312 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Properties of a Parabola Answer Key With the assistance of a graphing calculator, sketch the graph of each of the following quadratic equations in pencil. Label the vertex of each parabola. 1. y = x2 – 4x + 9 Is there a line of symmetry? Yes If so, use a pen to draw in the line of symmetry. Write the equation for the line of symmetry. x=2 What are the coordinates of the y-intercept? (0,9) Does the parabola have a minimum or a maximum? minimum What is the minimum or maximum value of the function? 5 Identify the domain and the range of the parabola. , Domain: Range: [5, ) 2. y = - 0.5x2 + 2x – 5 Is there a line of symmetry? yes If so, use a pen to draw in the line of symmetry. Write the equation for the line of symmetry. x=2 What are the coordinates of the y-intercept? (0,- 5) Does the parabola have a minimum or a maximum? maximum What is the minimum or maximum value of the function? - 3 Identify the domain and the range of the parabola. , Domain: Range: ( , 3] CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 313 of 399 Columbus City Schools 6/28/13 3. y = 3x2 – 6x + 7 Is there a line of symmetry? yes If so, use a pen to draw in the line of symmetry. Write the equation for the line of symmetry. x=1 What are the coordinates of the y-intercept? (0,7) Does the parabola have a minimum or a maximum? minimum What is the minimum or maximum value of the function? 4 Identify the domain and the range of the parabola. , Domain: Range: 4, 4. y = - x2 – 10x – 27 Is there a line of symmetry? yes If so, use a pen to draw in the line of symmetry. Write the equation for the line of symmetry. x=-5 What are the coordinates of the y-intercept? (0,- 27) Does the parabola have a minimum or a maximum? maximum What is the minimum or maximum value of the function? - 2 Identify the domain and the range of the parabola. , , 2 Domain: Range: What connections can you make between the vertex, the line of symmetry, and the minimum or maximum of the graph? Are there any other connections that you notice? The x-coordinate of the vertex is the same number used in the equation of the line of symmetry. The y-coordinate of the vertex is the minimum or maximum value of the function. The range will include the y-coordinate of the vertex as either the lowest or highest number. When the leading coefficient is positive, the parabola will open upward and will have a minimum value and when the leading coefficient is negative, the parabola CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 314 of 399 Columbus City Schools 6/28/13 will open downward and will have a maximum value. The y-intercept is equivalent to the constant in the equation. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 315 of 399 Columbus City Schools 6/28/13 F-IF 7 Name ___________________________________ Date __________________ Period ________ What Will My Parabola Look Like? The standard form of a quadratic function is f(x) = ax2 + bx + c. In this exploration, you will see how your graphs will change by altering one of the coefficients a, b, or c one at a time while fixing the other 2 coefficients. 1. Variations in c: a) Given the following equations, identify c and then graph each equation on your graphing calculator on the same screen. Sketch the graphs below. y1 = x2 + 2x c = ________ y2 = x2 + 2x + 2 c = _________ y3 = x2 + 2x + 4 c = _________ y4 = x2 + 2x – 3 c = _________ b) What are the similarities and differences between each of the above graphs? How does the c affect each graph? c) Find the vertex for each of the equations. What do you notice about the ordered pairs of the vertex from each graph? How are they related to each other? y1 = x2 + 2x vertex = ___________ y2 = x2 + 2x + 2 vertex = ___________ y3 = x2 + 2x + 4 vertex = __________ y4 = x2 + 2x – 3 vertex = ___________ d) Without graphing the equation y1 = x2 + 2x – 1 on your calculator, what do you think this graph will look like in comparison to the above graphs? What would the vertex be? Now, graph this and check if your predictions were correct. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 316 of 399 Columbus City Schools 6/28/13 e) Why do you think the c affected the graph the way it did? Be specific. 2. Variations in b: a) Given the following equations, identify b and then graph each equation on your graphing calculator on the same screen. Sketch the graphs below. y1 = x2 + 1 b = ________ y2 = x2 + 2x + 1 b = _________ y3 = x2 + 4x + 1 b = _________ y4 = x2 + 6x + 1 b = _________ y5 = x2 + 1 b = ________ y6 = x2 – 2x + 1 b = _________ y7 = x2 – 4x + 1 b = _________ y8 = x2 – 6x + 1 b = _________ b) What are the similarities and differences between each of the above graphs? How does the b affect each graph? c) Find the vertex for each of the equations. What do you notice about the ordered pairs of the vertex from each graph? How are they related to each other? y1 = x2 + 1 vertex = ___________ y5 = x2 + 1 y2 = x2 + 2x + 1 vertex = ___________ y6 = x2 – 2x + 1 vertex = ___________ vertex = ___________ CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 317 of 399 Columbus City Schools 6/28/13 y3 = x2 + 4x + 1 vertex = ___________ y7 = x2 – 4x + 1 vertex = ___________ y4 = x2 + 6x + 1 vertex = ___________ y8 = x2 – 6x + 1 vertex = ___________ d) Without graphing the equation y = x2 + 8x + 1 on your calculator, what do you think this graph will look like in comparison to the above graphs? Now, graph this and check if your predictions were correct. e) Why do you think the b affected the graph the way it did? Be specific. 3. Variations in a: a) Given the following equations, identify a and then graph each equation on your graphing calculator on the same screen. Sketch the graphs below. y1 = x2 + x + 1 a = ________ y2 = 0.5x2 + x + 1 a = _________ y3 = 0.3x2 + x + 1 a = _________ y4 = 0.1x2 + x + 1 a = _________ y5 = x2 + x + 1 a = ________ y6 = 2x2 + x + 1 a = _________ y7 = 3x2 + x + 1 a = _________ y8 = 10x2 + x + 1 a = _________ CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 318 of 399 Columbus City Schools 6/28/13 b) What are the similarities and differences between each of the above graphs? How does the a affect each graph? c) Without graphing the equation y = 100x2 + x + 1 on your calculator, what do you think this graph will look like in comparison to the above graphs? Now, graph this and check if your predictions were correct. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 319 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ What Will My Parabola Look Like? Answer Key The standard form of a quadratic function is f(x) = ax2 + bx + c. In this exploration, you will see how your graphs will change by altering one of the coefficients a, b or c one at a time while fixing the other 2 coefficients. 1. Variations in c: a) Given the following equations, identify c and then graph each equation on your graphing calculator on the same screen. Sketch the graphs below. y1 = x2 + 2x c= 0 y2 = x2 + 2x + 2 c= 2 y3 = x2 + 2x + 4 c= 4 y4 = x2 + 2x – 3 c= -3 b) What are the similarities and differences between each of the above graphs? How does the c affect each graph? The graph of y2 is shifted up 2 units from y1. The graph of y3 is shifted up 4 units from y1. The graph of y4 is shifted down 3 units from y1. All four graphs have the same width. The c is the number that tells how many units to shift from y1. c) Find the vertex for each of the equations. What do you notice about the ordered pairs of the vertex from each graph? How are they related to each other? y1 = x2 + 2x vertex = (-1, -1) y2 = x2 + 2x + 2 vertex = (-1, 1) y3 = x2 + 2x + 4 vertex = (-1, 3) y4 = x2 + 2x – 3 vertex = (-1, -4) CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 320 of 399 Columbus City Schools 6/28/13 All of the vertices have an x-coordinate of -1 because we have fixed the a and b -b coefficients and the formula to find the vertex is x . The y-value of the vertex 2a is different because the c is not zero in y2, y3, and y4. For example, in y2 the y-value is 1, which is two more units than the y-value in y1, which is -1. d) Without graphing the equation y = x2 + 2x – 1 on your calculator, what do you think this graph will look like in comparison to the above graphs? What would the vertex be? Now, graph this and check if your predictions were correct. This graph should be shifted down 1 unit from y1 because the c value is -1. The vertex should be (-1, -2). e) Why do you think the c affected the graph the way it did? Be specific. Since we fixed coefficients a and b, the c is just a constant, hence it’s just an addition to all of the y-values of all points in y1. For example, y2 = y1 + 2, y3 = y1 + 4, y4 = y1 – 3. 2. Variations in b: a) Given the following equations, identify b and then graph each equation on your graphing calculator on the same screen. Sketch the graphs below. y1 = x2 + 1 b= 0 y2 = x2 + 2x + 1 b= 2 y3 = x2 + 4x + 1 b= 4 y4 = x2 + 6x + 1 b= 6 y5 = x2 + 1 b= 0 y6 = x2 – 2x + 1 b= -2 y7 = x2 – 4x + 1 b= -4 y8 = x2 – 6x + 1 b= -6 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 321 of 399 Columbus City Schools 6/28/13 b) What are the similarities and differences between each of the above graphs? How does the b affect each graph? The y-intercepts are all the same, which is (0, 1). The width of all the parabolas are all the same. As the b values increase from 0 to 2, 4, and 6, the parabolas shifted left and down. But as the b values decreased from 0 to -2, -4 and -6, the parabolas shifted right and down. y2, y3, y4 are symmetric to y6, y7, y8 about the y-axis. c) Find the vertex for each of the equations. What do you notice about the ordered pairs of the vertex from each graph? How are they related to each other? y1 = x2 + 1 vertex = (0, 1) y5 = x2 + 1 vertex = (0, 1) y2 = x2 + 2x + 1 vertex = (-1, 0) y6 = x2 – 2x + 1 vertex = (1, 0) y3 = x2 + 4x + 1 vertex = (-2, -3) y7 = x2 – 4x + 1 vertex = (-2, -3) y4 = x2 + 6x + 1 vertex = (-3, -8) y8 = x2 – 6x + 1 vertex = (3, -8) The x-coordinate of the vertex in y2 is the opposite of the x-coordinate in y6 because the b value in y2 is the opposite of the b value in y6. The same idea is applied for y3 and y7 and y4 and y8. d) Without graphing the equation y = x2 + 8x + 1 on your calculator, what do you think this graph will look like in comparison to the above graphs? Now, graph this and check if your predictions were correct. This graph will have the same width as all of the above graphs but the vertex will be to the left and down from y1 because the b value is 8. e) Why do you think the b affected the graph the way it did? Be specific. -b Since the x-coordinate of the vertex is x , changing the b value would affect the 2a x-coordinate of the vertex. If b is a positive value, the x-coordinate of the vertex will shift left because of the -b in the formula. If b is a negative value, the x-coordinate of the vertex will shift right because of the -b in the formula. This is assuming that the a value is positive. 3. Variations in a: a) Given the following equations, identify a and then graph each equation on your graphing calculator on the same screen. Sketch the graphs below. y1 = x2 + x + 1 a= 1 y2 = 0.5x2 + x + 1 a= 0.5 y3 = 0.3x2 + x + 1 a= 0.3 y4 = 0.1x2 + x + 1 a= 0.1 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 322 of 399 Columbus City Schools 6/28/13 y5 = x2 + x + 1 a= 1 y6 = 2x2 + x + 1 a= 2 y7 = 3x2 + x + 1 a= 3 y8 = 10x2 + x + 1 a= 10 b) What are the similarities and differences between each of the above graphs? How does the a affect each graph? All of the y-intercepts are the same, which is (0, 1). As the absolute value of the a value decreases, the parabola gets wider. As the absolute value of the a value increases, the parabola becomes more narrow. c) Without graphing the equation y = 100x2 + x + 1 on your calculator, what do you think this graph will look like in comparison to the above graphs? Now, graph this and check if your predictions were correct. This graph will have the same y-intercept and will be very narrow compared to the above graphs. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 323 of 399 Columbus City Schools 6/28/13 F-IF 7 Name ___________________________________ Date __________________ Period ________ What Do You Need for the Graph? Work with a partner to solve the quadratic equation by graphing. Determine the vertex, b b ,f 2a 2a , y-intercept, the point symmetrical to the y-intercept, and the x-intercept(s). 1. What is the vertex of 2x2 + 6x + 4 = 0? 2. What is the y-intercept? 3. What is the point symmetrical to the y-intercept? 4. How many x-intercepts will this function have? Explain your reasoning. 5. Select another x value, and determine the ordered pair for that x value. Plot this ordered pair, and its reflection point. 6. Use the information above to plot your graph. 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 7. Explain why these points were needed for an accurate graph? CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 324 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ What Do You Need for the Graph? Answer Key Work with a partner to solve the quadratic equation by graphing. Determine the vertex b b ,f 2a 2a , y-intercept, the point symmetrical to the y-intercept, and the x-intercept(s). 1. What is the vertex of 2x2 + 6x + 4 = 0? (-3/2, -1/2) 2. What is the y-intercept? (0, 4) 3. What is the point symmetrical to the y-intercept? (-3, 4) 4. How many x-intercepts will this function have? Explain your reasoning. Two intercepts since the coordinates of the vertex are (-x, -y) and the function opens upward. 5. Select another x value, and determine the ordered pair for that x value. Plot this ordered pair, and its reflection point. (-2, 0) and (-1, 0) 6. Use the information above to plot your graph. 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 7. Explain why these points were needed for an accurate graph? The quadratic graph is represented by a parabola. The vertex is the lowest point (minimum) of the graph. A parabola is U-shaped and symmetrical so any point to the left of the vertex has a symmetrical point to the right. The y-intercept and x-intercepts have symmetrical points. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 325 of 399 Columbus City Schools 6/28/13 F-IF 7 Name ___________________________________ Date __________________ Period ________ Linear, Exponential and Quadratic Functions Compare the key features of the following graphs. A. y = 3x B. y = 3x2 -10 -8 -6 -4 C. y = 3x 10 10 10 8 8 8 6 6 6 4 4 4 2 2 2 -2 2 4 6 8 10 -10 -8 -6 -4 -2 2 4 6 8 10 -10 -8 -6 -4 -2 2 -2 -2 -2 -4 -4 -4 -6 -6 -6 -8 -8 -8 -10 -10 -10 4 6 8 10 Describe each of the graphs below, taking into consideration, key features. Make a table of values from the graphs. A. y = a(b)x x f(x) B. x f(x) y = ax2 + bx + c C. x f(x) y = ax + b y = a(x – h)2 +k 1. How would you tell from the table that a graph is linear? 2. How would you tell from the table that a graph is exponential? 3. How would you tell from the table that a graph is quadratic? 4. Describe the shapes of each graph. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 326 of 399 Columbus City Schools 6/28/13 5. Give an equation and graph a linear equation that is flatter than the parent graph, y = x, and has a y-intercept of +6 and a negative slope. 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 6. Give an equation and graph a quadratic function that is “narrower” than y = x2 with a yintercept of 4. 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 7. Give an equation and graph an exponential function that has a y-intercept of -6. 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 327 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Linear, Exponential and Quadratic Functions Answer Key Compare the key features of the following graphs. A. y = 3x B. y = 3x2 -10 -8 -6 -4 C. y = 3x 10 10 10 8 8 8 6 6 6 4 4 4 2 2 2 -2 2 4 6 8 10 -10 -8 -6 -4 -2 2 4 6 8 10 -10 -8 -6 -4 -2 2 -2 -2 -2 -4 -4 -4 -6 -6 -6 -8 -8 -8 -10 -10 -10 4 6 8 10 Describe each of the graphs below, taking into consideration, key features. Make a table of values from the graphs. A. y = a(b)x x -1 0 2 3 4 5 f(x) 1/3 1 9 27 81 243 B. x f(x) y = ax2 + bx + c -2 12 y = a(x – h)2 +k -1 3 C. x f(x) y = ax + b -2 -6 -1 -3 0 0 0 0 1 3 1 3 2 12 2 6 3 27 3 9 1. How would you tell from the table that a graph is linear? Rate of change is constant. Change is +3, 1st difference is constant. 2. How would you tell from the table that a graph is exponential? Rate of change is constant. Change is x 3. 3. How would you tell from the table that a graph is quadratic? Second difference is constant. (+6) 4. Describe the shapes of each graph. The shape of a linear graph is a line. The shape of the exponential graph is a curve. The shape of a quadratic graph is a parabola. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 328 of 399 Columbus City Schools 6/28/13 5. Give an equation and graph a linear equation that is flatter than the parent graph, y = x, and has a y-intercept of +6 and a negative slope. 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 A linear equation flatter than y = x, with a negative slope and has a y-intercept of 6 is y = -1/3x + 6. 6. Give an equation and graph a quadratic function that is “narrower” than y = x2 with a yintercept of 4. 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 An example of a quadratic function is y = 6x2 + 4 7. Give an equation and graph an exponential function that has a y-intercept of -6. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 329 of 399 Columbus City Schools 6/28/13 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 An example of an exponential function that has a y-intercept of -6 is y = 3x – 6. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 330 of 399 Columbus City Schools 6/28/13 F-IF 7 Name ___________________________________ Date __________________ Period ________ Zeros of Quadratic Functions Calculator Discovery Materials: graphing calculator Directions: Graph the following equations on your graphing calculator. Sketch the graph and then identify the x-intercepts. Next, identify that these x-intercepts are the zeros of what function? Write the function in standard from (f(x) = ax2 + bx + c). 1. y = (x + 1)(x – 3) 2. y = (x + 4)(x + 2) 10 10 8 8 6 6 4 4 2 2 -10 -8 -6 -4 -2 2 4 6 8 10 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -2 -4 -4 -6 -6 -8 -8 -10 -10 x-intercept(s): x-intercept(s): These x-intercept(s) are zeros of the function: These x-intercept(s) are zeros of the function: 3. y = x(x – 4) 4. y = (x – 3)(x + 2) 10 10 8 8 6 6 4 4 2 2 -10 -8 -6 -4 -2 2 -2 -4 -6 -8 -10 x-intercept(s): 4 6 8 10 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 x-intercept(s): CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 331 of 399 Columbus City Schools 6/28/13 These x-intercept(s) are zeros of the function: These x-intercept(s) are zeros of the function: 5. y = -2(x – 2)(x + 1) 6. y = (2x + 1)(2x – 5) -10 -8 -6 -4 10 10 8 8 6 6 4 4 2 2 -2 2 4 6 8 10 -10 -8 -6 -4 -2 2 -2 -2 -4 -4 -6 -6 -8 -8 -10 -10 4 6 8 10 x-intercept(s): x-intercept(s): These x-intercept(s) are zeros of the function: These x-intercept(s) are zeros of the function: Analysis: 1. What conjectures can you make about the xintercepts and the factors of each quadratic equation? Answer in complete sentences. 7. y = (2x – 4)2 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 x-intercept(s): These x-intercept(s) are zeros of the 2. Can you come up with an algebraic rule that allows you to find the zeros of a quadratic function without graphing? Use specific examples to support your rule. Check with a graphing calculator. function: CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 332 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Zeros of Quadratic Functions Calculator Discovery Answer Key Materials: graphing calculator Directions: Graph the following equations on your graphing calculator. Sketch the graph and then identify the x-intercepts. Next, identify that these x-intercepts are the zeros of what function? Write the function in standard from (f(x) = ax2 + bx + c). 1. y = (x + 1)(x – 3) 2. y = (x + 4)(x + 2) 10 10 8 8 6 6 4 4 2 2 -10 -8 -6 -4 -2 2 4 6 8 10 -10 -8 -6 -4 -2 2 -2 -2 -4 -4 -6 -6 -8 -8 -10 -10 x-intercept(s): x = (-1, 3) f(x) = x2 – 2x – 3 8 10 These x-intercept(s) are zeros of the function: f(x) = x2 + 6x + 8 4. y = (x – 3)(x +102) 10 8 8 6 6 4 4 2 2 -10 -8 -6 -4 -2 6 x-intercept(s): x = (-4, -2) These x-intercept(s) are zeros of the function: 3. y = x(x – 4) 4 2 4 6 8 10 -10 -8 -6 -4 -2 2 -2 -2 -4 -4 -6 -6 -8 -8 -10 -10 x-intercept(s): x = (0, 4) 4 6 8 10 x-intercept(s): x = (-2, 3) CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 333 of 399 Columbus City Schools 6/28/13 These x-intercept(s) are zeros of the function: These x-intercept(s) are zeros of the function: f(x) = x2 – 4x f(x) = x2 – x – 6 5. y = -2(x –102)(x + 1) 6. y = (2x + 1)(2x – 5) 8 10 6 8 4 6 2 4 -10 -8 -6 -4 -2 2 4 6 8 10 2 -2 -10 -8 -6 -4 -2 -4 2 4 6 8 10 -2 -6 -4 -8 -6 -10 -8 x-intercept(s): x = (-1, 2) x-intercept(s):-10x = (-½, 2.5) These x-intercept(s) are zeros of the function: These x-intercept(s) are zeros of the function: f(x) = 4x2 – 8x – 5 f(x) = -2x2 + 2x + 4 7. y = (2x – 4)2 Analysis: 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 1. What conjectures can you make about the xintercepts and the factors of each quadratic equation? Answer in complete sentences. Possible answer: The x-intercepts are found by finding the value of x when each factor is set equal to 0. -4 -6 -8 -10 x-intercept(s): x = (2, 0) 2. Can you come up with an algebraic rule that allows you to find the zeros of a quadratic function without graphing? Use specific examples to support your rule. Check with a graphing calculator. If ab = 0, then a = 0 or b = 0 These x-intercept(s) are zeros of the Possible example: y = (3x – 6)(x – 5) function: To find the x-intercepts: f(x) = 4x2 – 16x + 16 0 = (3x – 6)(x – 5) 3x – 6 = 0 or x – 5 = 0 x=2 or CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 334 of 399 x=5 Columbus City Schools 6/28/13 F-IF 7 Name ___________________________________ Date __________________ Period ________ Quadratic Qualities Complete the table for each function, then make a graph of the function using the points. After you have made all the graphs, look at them and discuss the common characteristics with your partner. 5 1. x f(x) f(x) = x2 – 6x + 7 -5 5 -5 5 2. x f(x) f(x) = 2x2 – 12x + 13 -5 5 -5 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 335 of 399 Columbus City Schools 6/28/13 5 3. x f(x) f(x) = -x2 – 4x – 1 -5 5 -5 5 4. x f(x) f(x) = ½x2 – 2x – 3 -5 5 -5 5 5. x f(x) f(x) = -2x2 – 4x + 3 -5 5 -5 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 336 of 399 Columbus City Schools 6/28/13 6. Determine the domain and range for the function, axis of symmetry, minimum or maximum value of: f(x) = x2 – 6x + 7 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 7. Graph a function with a domain of (- , ), range of (- , 4], and axis of symmetry x = -2. 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 Compare your graph with your partner. Are they the same graphs? If not, how do they differ? What characteristics would you need provided to have the same graphs? 8. Write a verbal description of the graphs for: a. f(x) = x2 – 6x + 7 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 337 of 399 Columbus City Schools 6/28/13 b. g(x) = 2x2 – 12x + 13 c. h(x) = -x2 – 4x – 1 d. f(x) = ½x2 – 2x – 3 e, g(x) = -2x2 – 4x + 3 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 338 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Quadratic Qualities Answer Key Complete the table for each function, then make a graph of the function using the points. After you have made all the graphs, look at them and discuss the common characteristics with your partner. 5 Answers will vary. Sample answers are provided below. 1. f(x) = x2 – 6x + 7 x f(x) 1 2 2 -1 3 -2 4 -1 5 2 -5 5 -5 5 2. f(x) = 2x2 – 12x + 13 x f(x) 1 3 2 -3 3 -5 4 -3 5 3 -5 5 -5 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 339 of 399 Columbus City Schools 6/28/13 5 3. f(x) = -x2 – 4x – 1 x f(x) -4 -1 -3 2 -2 3 -1 2 0 -1 -5 5 -5 5 4. f(x) = ½x2 – 2x – 3 x 0 f(x) -3 1 9 2 2 3 9 2 -5 4 -3 -5 5 -5 5 5. f(x) = -2x2 – 4x + 3 x f(x) -3 -3 -2 3 -1 5 0 3 1 -3 -5 5 -5 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 340 of 399 Columbus City Schools 6/28/13 6. Determine the domain and range for the function, axis of symmetry, minimum or maximum value of: f(x) = x2 – 6x + 7 The domain is (- , ). The range is [-2, ). The axis of symmetry is x = 3 and the minimum point is (3, -2). 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 7. Graph a function with a domain of (- , ), range of (- , 4], and axis of symmetry x = -2. Answers may vary; one possible graph is given below. 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 Compare your graph with your partner. Are they the same graphs? Answers will vary If not, how do they differ? Answers will vary What characteristics would you need provided to have the same graphs? Answers will vary -2 -4 -6 -8 -10 8. Write a verbal description of the graphs for: Answers will vary; possible answers are given below. a. f(x) = x2 – 6x + 7 The y-intercept is 7 The graph opens up The vertex is (3, -2) Average width Axis of symmetry x=3 Domain all real numbers CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 341 of 399 Columbus City Schools 6/28/13 Range 2, b. g(x) = 2x2 – 12x + 13 The y-intercept is 13 The graph opens up The vertex is (3, -5) More narrow Axis of symmetry x=3 Domain all real numbers Range 5, c. h(x) = -x2 – 4x – 1 The y-intercept is -1 The graph opens down The vertex is (-2, 3) Average width Axis of symmetry x= -2 Domain all real numbers ,3 Range d. f(x) = ½ x2 – 2x – 3 The y-intercept is -3 The graph opens up The vertex is (2, -5) Wider Axis of symmetry x=2 Domain all real numbers Range 5, e, g(x) = -2x2 – 4x + 3 The y-intercept is 3 The graph opens down The vertex is (-1, 5) More narrow Axis of symmetry x= -1 Domain all real numbers ,5 Range CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 342 of 399 Columbus City Schools 6/28/13 F-IF 7a Name ___________________________________ Date __________________ Period ________ Quadratic Qualities II 1.) Determine the domain and range for the function, axis of symmetry, minimum or maximum value of: f(x) = x2 – 6x + 7 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 2.) Graph a function with a domain of (- , ), range of (- , 4], and axis of symmetry x = -2. 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 Compare your graph with your partner. Are they the same graphs? If not, how do they differ? CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 343 of 399 Columbus City Schools 6/28/13 What characteristics would you need provided to have the same graphs? 3.) Write a verbal description of the graphs for: a. f(x) = x2 – 6x + 7 b. g(x) = 2x2 – 12x + 13 c. h(x) = -x2 – 4x – 1 d. f(x) = 1/2x2 – 2x – 3 e, g(x) = -2x2 – 4x + 3 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 344 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Quadratic Qualities II Answer Key 1.) Determine the domain and range for the function, axis of symmetry, minimum or maximum value of: f(x) = x2 – 6x + 7 The domain is (- , ). The range is [-2, ). The axis of symmetry is x = 3 and the minimum point is (3, -2). 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 2. Graph a function with a domain of (- , ), range of (- , 4], and axis of symmetry x = -2. 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 Compare your graph with your partner. Are they the same graphs? Probably not. If not, how do they differ? x-intercepts, width of the parabola CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 345 of 399 Columbus City Schools 6/28/13 What characteristics would you need provided to have the same graphs? x-intercepts 3. Write a verbal description of the graphs for: a. f(x) = x2 – 6x + 7 The y-intercept is 7 and the axis of symmetry is x = 3. The vertex is (3, -2). The domain is (- , ) and the range is [-2, ). b. g(x) = 2x2 – 12x + 13 The y-intercept is 13 and the axis of symmetry is x = 3. The vertex is (3, -5). The domain is (- , ) and the range is [-5, ). c. h(x) = -x2 – 4x – 1 The y-intercept is -1 and the axis of symmetry is x = -2. The vertex is (-2, 3). The domain is (- , ) and the range is (- , 3]. d. f(x) = 1/2x2 – 2x – 3 The y-intercept is -3 and the axis of symmetry is x = 2. The vertex is (2, -5). The domain is (- , ) and the range is [-5, ). e. g(x) = -2x2 – 4x + 3 The y-intercept is 3 and the axis of symmetry is x = -1. The vertex is (-1, 5). The domain is (- , ) and the range is (- , 5]. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 346 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Match the Graphs –Equations y = (x + 3)2 – 4 y = - 2(x – 1)2 – 2 y = - 2(x – 2)2 – 1 y = (x – 4)2 + 3 y = - (x + 5)2 + 6 y = - (x – 5)2 + 6 y = ½(x + 8)2 – 7 y = ½(x – 7)2 + 8 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 347 of 399 Columbus City Schools 6/28/13 Match the Graph-Graphs CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 348 of 399 Columbus City Schools 6/28/13 Match the Graphs Answer Key CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 349 of 399 Columbus City Schools 6/28/13 F-IF 4 Name ___________________________________ Date __________________ Period ________ Graph It! Work with a partner to plot the quadratic graphs. 1. Graph the vertex (2, 4) and x-intercepts (0, 0) and (4, 0). 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 Plot the point (1, 3) and the point symmetric to it. Plot a point (5, -5) and the point symmetric to it. Make a table with the points you plotted. 2. Make a graph of a parabola with a range of [0, ). 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 Make a table with the vertex and two symmetric points. Plot the points on the graph above. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 350 of 399 Columbus City Schools 6/28/13 3. Plot the coordinate points in the table. x -6 -5 -4 f(x) 8 3 0 -3 -1 -2 0 -1 3 0 8 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 3. Graph the axis of symmetry, x = -4. Create a graph of a parabola where the range is [-3, ) and the y-intercept is (0, 13). 18 15 12 9 6 3 -20 -16 -12 -8 -4 -3 -6 -9 -12 -15 -18 4 8 12 16 20 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 351 of 399 Columbus City Schools 6/28/13 5. Plot the coordinates in the table below. x -2 -1 f(x) 13 7 0 5 1 7 2 13 18 15 12 9 6 3 -20 -16 -12 -8 -4 4 -3 -6 -9 -12 -15 -18 8 12 16 20 6. Plot the x-intercepts, (-2, 0) and (2, 0). Create a graph of a parabola where the range is [-6, ). 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 Determine where the function is increasing. Determine where the function is decreasing. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 352 of 399 Columbus City Schools 6/28/13 7. Write a description of a quadratic graph, including the intercepts, the vertex, the end behavior and the intervals where the function is decreasing and increasing. Use these attributes to graph the quadratic function. 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 8. Determine which graph goes with the appropriate equations listed below. Explain your reasoning. A. y = 4x2 B. f(x) = 1.5(x - 2)(x + 2) C. y = (x + 4)2 – 3 D. y = -(x – 2)2 + 4 E. f(x) = (x + 3)2 – 1 F. y = 2x2 + 5 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 353 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Graph It! Answer Key Work with a partner to plot the quadratic graphs. 1. Graph the vertex (2, 4) and x-intercepts (0, 0) and (4, 0). 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 Plot the point (1, 3) and the point symmetric to it. (3, 3) Plot a point (5, -5) and the point symmetric to it. (-1, -5) Make a table with the points you plotted. x -1 0 1 2 f(x) -5 0 3 4 3 3 4 0 5 -5 2. Make a graph of a parabola with a range of [0, ). 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 354 of 399 Columbus City Schools 6/28/13 Make a table with the vertex and two symmetric points. Plot the points on the graph above. x -2 f(x) 3 3. Plot the coordinate points in the table. x -6 -5 -4 f(x) 8 3 0 0 0 -3 -1 2 3 -2 0 -1 3 0 8 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 4. Graph the axis of symmetry, x = -4. Create a graph of a parabola where the range is [-3, ) and the y-intercept is (0, 13). 18 15 12 9 6 3 -20 -16 -12 -8 -4 -3 -6 -9 -12 -15 -18 4 8 12 16 20 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 355 of 399 Columbus City Schools 6/28/13 5. Plot the coordinates in the table below. x -2 -1 f(x) 13 7 0 5 1 7 2 13 18 15 12 9 6 3 -20 -16 -12 -8 -4 4 -3 -6 -9 -12 -15 -18 8 12 16 20 6. Plot the x-intercepts, (-2, 0) and (2, 0). Create a graph of a parabola where the range is [-6, ). 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 Determine where the function is increasing. Domain: [0, ) R: [-6, ) Determine where the function is decreasing. Domain: (- ,0] R: (- , -6] CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 356 of 399 Columbus City Schools 6/28/13 7. Write a description of a quadratic graph, including the intercepts, the vertex, the end behavior and the intervals where the function is decreasing and increasing. Use these attributes to graph the quadratic function. Answers will vary. 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 8. Determine which graph goes with the appropriate equations listed below. Explain your reasoning. Reasons may vary; possible solutions are listed below. A. y = 4x2 #2 The y-value of the vertex is zero. B. f(x) = 1.5(x - 2)(x + 2) The x-intercepts are 2 and -2. #6 C. y = (x + 4)2 – 3 #4 The axis of symmetry and the range tell me that the vertex is (-4, -3). D. y = -(x – 2)2 + 4 #1 This is the only equation that the x-intercepts are solutions for; the vertex is (2, 4). E. f(x) = (x + 3)2 – 1 #3 All of the ordered pairs from the table work in this equation. F. y = 2x2 + 5 #5 All of the ordered pairs from the table work in this equation. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 357 of 399 Columbus City Schools 6/28/13 F-IF 4 Name ___________________________________ Date __________________ Period ________ Quadratic Functions and Rates of Change What do you know about linear functions and rates of change? Follow the directions below and complete the given table. a) Enter the numbers: - 4, -3, - 2, - 1, 0, 1, 2, 3, and 4 into List1. (Press the STAT key, then the ENTER key – if numbers are in L1, clear the entries by using your arrow keys to highlight L1, then press the CLEAR key followed by the ENTER key. Enter each number followed by ENTER.) b) Clear the entries in L2, if necessary using the method described in part a). c) Once L2 is empty, highlight L2 (L2= should be at the bottom of your screen) and enter the following: “L12”. (To obtain the quotation marks, press the ALPHA + keys, L1 can be obtained by pressing the 2nd STAT.) Press ENTER. Place the calculator entries into the table below. So the L2 column is the square of L1. d) The third list is going to be used to find the first difference between the squared values. Clear L3, if necessary. Highlight L3 and press the following keys: ALPHA +, 2nd STAT, move the cursor to OPS, 7: List(, 2nd STAT, 2:L2, ), ALPHA +, ENTER. The bottom of the screen should display List(L2). Then press enter. Place the entries into your table below. e) The fourth list is going to be used to find the second difference between the squared values. Clear L4, if necessary, then highlight L4. Use the following key strokes: ALPHA +, 2nd STAT, OPS, 7: List(, 2nd STAT, 3:L3, ), ALPHA +, ENTER. Place the entries into the table below. If the values in L1 represent values for x, what does L12 represent? L1 L2 L3 L4 -4 -3 -2 -1 0 1 2 3 4 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 358 of 399 Columbus City Schools 6/28/13 Try a different equation for L2. To do this, highlight L2, press ENTER, CLEAR. This will unlock the formula in L2. Enter the new expression: “2L12 + 4”. If the values in L1 represent values for x, what does 2L12 + 4 represent? Complete the table below. L1 L2 L3 L4 -4 -3 -2 -1 0 1 2 3 4 Try another equation for L2. Enter: “- 3L12 + 2L1 – 5”. If the values in L1 represent values for x, what does - 3L12 + 2L1 – 5 represent? Complete the table below. L1 L2 L3 L4 -4 -3 -2 -1 0 1 2 3 4 What observations have you made? What prediction can be made about quadratic functions? CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 359 of 399 Columbus City Schools 6/28/13 Create your own quadratic and enter it into L2. Is your prediction still true? What do you think will happen with a cubic function? (A cubic function is a function in which the largest exponent is three.) CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 360 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Quadratic Functions and Rates of Change Answer Key What do you know about linear functions and rates of change? The change in the y-values with respect to the x-values is constant. The rate of change between any two points is the same. Follow the directions below and complete the given table. a) Enter the numbers: - 4, -3, - 2, - 1, 0, 1, 2, 3, and 4 into List1. (Press the STAT key, then the ENTER key – if numbers are in L1, clear the entries by using your arrow keys to highlight L1, then press the CLEAR key followed by the ENTER key. Enter each number followed by ENTER.) b) Clear the entries in L2, if necessary using the method described in part a). c) Once L2 is empty, highlight L2 (L2= should be at the bottom of your screen) and enter the following: “L12”. (To obtain the quotation marks, press the ALPHA + keys, L1 can be obtained by pressing the 2nd STAT.) Press ENTER. Place the calculator entries into the table below. So the L2 column is the square of L1. d) The third list is going to be used to find the first difference between the squared values. Clear L3, if necessary. Highlight L3 and press the following keys: ALPHA +, 2nd STAT, move the cursor to OPS, 7: List(, 2nd STAT, 2:L2, ), ALPHA +, ENTER. The bottom of the screen should display List(L2). Then press enter. Place the entries into your table below. e) The fourth list is going to be used to find the second difference between the squared values. Clear L4, if necessary, then highlight L4. Use the following key strokes: ALPHA +, 2nd STAT, OPS, 7: List(, 2nd STAT, 3:L3, ), ALPHA +, ENTER. Place the entries into the table below. If the values in L1 represent values for x, what does L12 represent? L12 represents x2. L1 L2 L3 L4 -4 16 -7 2 -3 9 -5 2 -2 4 -3 2 -1 1 -1 2 0 0 1 2 1 1 3 2 2 4 5 2 3 4 9 16 7 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 361 of 399 Columbus City Schools 6/28/13 Try a different equation for L2. To do this, highlight L2, press ENTER, CLEAR. This will unlock the formula in L2. Enter the new expression: “2L12 + 4”. If the values in L1 represent values for x, what does 2L12 + 4 represent? 2L12+4 represents 2x2+4. Complete the table below. L1 L2 L3 L4 -4 36 - 14 4 -3 22 - 10 4 -2 12 -6 4 -1 6 -2 4 0 4 2 4 1 6 6 4 2 12 10 4 3 22 14 4 36 Try another equation for L2. Enter: “- 3L12 + 2L1 – 5”. If the values in L1 represent values for x, what does - 3L12 + 2L1 – 5 represent? – 3L12 + 2L1 – 5 represents - 3x2 + 2x – 5. Complete the table below. L1 L2 L3 L4 -4 - 61 23 -6 -3 - 38 17 -6 -2 - 21 11 -6 -1 - 10 5 -6 0 -5 -1 -6 1 -6 -7 -6 2 3 4 - 13 - 26 - 45 - 13 - 19 -6 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 362 of 399 Columbus City Schools 6/28/13 What observations have you made? What prediction can be made about quadratic functions? The second differences are a constant value. If you take half of the second difference, it is the same as the leading coefficient. If the second differences are the same, then the function is quadratic. Create your own quadratic and enter it into L2. Is your prediction still true? The quadratics will vary but the prediction will hold true. What do you think will happen with a cubic function? (A cubic function is a function in which the largest exponent is three.) If you have a cubic function, then the third differences will be the same. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 363 of 399 Columbus City Schools 6/28/13 F-IF 4 Name ___________________________________ Date __________________ Period ________ Linear or Quadratic? Determine if each table represents a linear function or a quadratic function by using finite differences. Underline or circle your choice between Linear or Quadratic below each table. Bonus: Develop an equation for each function. I. x y −3 −9 −2 −6 −1 −3 0 0 1 3 2 6 3 9 4 12 5 15 Linear or Quadratic II. x y −3 12 −2 8 −1 4 0 0 1 −4 2 −8 3 −12 4 −16 5 −20 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 364 of 399 Columbus City Schools 6/28/13 Linear or Quadratic? III. x y 0 5 1 6 2 9 3 14 4 21 5 30 6 41 7 54 8 69 Linear or Quadratic? IV. x y 0 −6 1 −5 2 −2 3 3 4 10 5 19 6 30 7 43 8 58 Linear or Quadratic? CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 365 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Linear or Quadratic? Answer Key Determine if the table represents a linear function or a quadratic function by using finite differences. Underline or circle your choice between Linear or Quadratic below each table. Bonus: See if you can come up with the equation for each one. I. x y −3 −9 −2 −6 −1 −3 0 0 1 3 2 6 3 3 9 3 4 12 3 5 15 3 3 3 y = 3x 3 3 Linear or Quadratic II. x y −3 12 −2 8 −1 4 0 0 1 −4 2 −8 3 −12 4 −16 5 −20 -4 -4 -4 y = -4x -4 -4 -4 -4 -4 Linear or Quadratic CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 366 of 399 Columbus City Schools 6/28/13 III. x y 0 5 1 6 2 9 3 14 4 21 5 30 6 41 7 54 8 69 1 3 5 7 9 11 13 15 2 2 2 2 y = x2 + 5 2 2 2 Linear or Quadratic IV. x y 0 −6 1 −5 2 −2 3 3 1 3 5 7 4 10 5 19 9 6 30 11 7 43 13 58 15 8 2 2 2 2 y = x2 – 6 2 2 2 Linear or Quadratic CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 367 of 399 Columbus City Schools 6/28/13 F-IF 6 Name ___________________________________ Date __________________ Period ________ Don’t Change That Perimeter! Close-It-In Fencing Company is selling a new type of flexible fencing material that Angelica wants to purchase to house her new puppy. She decides to purchase a 48-ft section of the fence to explore various sizes (areas) for the dog house. 1. On a separate sheet of paper, draw as many rectangles as you can with a perimeter of 48 ft using only whole numbers as widths. Label the length of each side and area of each rectangle. 2. Complete the following table of width, length, and area for all of the rectangles from #1. Width Length Area x 24 – x CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 368 of 399 Columbus City Schools 6/28/13 3. What do you notice about the sum of the length and width? Explain your findings. 4. Choose several points from the table to make a graph of the relationship between width and area of the rectangles. Plot the width on the x-axis and area on the y-axis. The graph can also be done on a graphing calculator by using lists. Enter the values for width in L1 and values for area in L2. Then do a STATPLOT to graph the points entered in the table. Sketch the graph on the paper provided. 5. What is the shape of the graph? Is this the graph of a linear or non-linear function? If it is non-linear, is it quadratic or exponential? (Hint: Use finite differences.) 6. Observe the values you entered in your table and the graph to reflect on the following cases: a. width < length b. width > length 7. What does the lowest (minimum) or highest (maximum) point on the graph represent? What do you notice about the width and length at this point on the graph? 8. Which of the following equations gives the correct area for a rectangle with a perimeter of 48 ft? Let A(x) stand for area and x stand for width. a. A(x) = x (48 – x) b. A(x) = x (24 – x) c. A(x) = x (48 – 2x) CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 369 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Don’t Change That Perimeter! Answer Key Close-It-In Fencing Company is selling a new type of flexible fencing material that Angelica wants to purchase to house her new puppy. She decides to purchase a 48-ft section of the fence to explore various sizes (areas) for the dog house. 1. On a separate sheet of paper, draw as many rectangles as you can with a perimeter of 48 ft using only whole numbers as widths. Label the length of each side and area of each rectangle. 2. Complete the following table of width, length, and area for all of the rectangles from #1. Width Length Area 1 23 23 2 22 44 3 21 63 4 20 80 5 19 95 6 18 108 7 17 119 8 16 128 9 15 135 10 14 140 11 13 143 12 12 144 13 11 143 14 10 140 15 9 135 16 8 128 17 7 119 18 6 108 19 5 95 20 4 80 21 3 63 22 2 44 23 1 24 – x 23 x x(24 – x) 3. What do you notice about the sum of the length and width? Explain your findings. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 370 of 399 Columbus City Schools 6/28/13 The sum of the length and width is always 24. Since perimeter = 2w + 2l = 48, w + l = 24. 4. Choose several points from the table to make a graph of the relationship between width and area of the rectangles. Plot the width on the x-axis and area on the y-axis. The graph can also be done on a graphing calculator by using lists. Enter the values for width in L1 and values for area in L2. Then do a STATPLOT to graph the points entered in the table. Sketch the graph on the paper provided. 144 132 120 108 96 84 72 60 48 36 24 12 -8 -4 4 8 12 16 20 24 28 5. What is the shape of the graph? Is this the graph of a linear or non-linear function? If it is non-linear, is it quadratic or exponential? (Hint: Use finite differences.) The graph’s shape is a parabola. This is a non-linear function. It is quadratic since the differences on the 2nd level are the same. 6. Observe the values you entered in your table and the graph to reflect on the following cases: a. width < length – The graph is increasing (area values are increasing). b. width > length – The graph is decreasing (area values are decreasing). 7. What does the lowest (minimum) or highest (maximum) point on the graph represent? What do you notice about the width and length at this point on the graph? Since the parabola opens downward, it has a highest (maximum) point, which represents the greatest area for 48 ft of rectangular fencing. The width = length. 8. Which of the following equations gives the correct area for a rectangle with a perimeter of 48 ft? Let A(x) stand for area and x stand for width. a. A(x) = x (48 – x) b. A(x) = x (24 – x) c. A(x) = x (48 – 2x) Answer: b The area, A = width ● length. In this scenario, the total (sum) of the width and length is always 24 ft, so if the width is x, the length would be the difference of x and 24. A common error would be to pick “a” as the answer, but the rectangle’s perimeter is 2w + 2l and simply w + l CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 371 of 399 Columbus City Schools 6/28/13 F-IF 6 Name ___________________________________ Date __________________ Period ________ Toothpicks and Models The goal is to determine a model that will permit your group to predict the number of toothpicks required to construct a square of any size that is subdivided into 1 x 1 squares. Below you will find examples of the 1 x 1 and 2 x 2 squares. 1. Complete the table using the toothpicks to make larger squares. Number of toothpicks Total number of per side of the square toothpicks 0 1 2 3 4 5 6 7 8 9 2. Do you see a pattern in the table of values? Explain the pattern that you have discovered. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 372 of 399 Columbus City Schools 6/28/13 3. What type of a function could you use to model this pattern? 4. Set up a system of linear equations using the data in the table above. Solve the system to create your function. Show all of your work! CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 373 of 399 Columbus City Schools 6/28/13 F-IF 6 Name ___________________________________ Date __________________ Period ________ Toothpicks and Models Answer Key The goal is to determine a model that will permit your group to predict the number of toothpicks required to construct a square of any size that is subdivided into 1 x 1 squares. Below you will find examples of the 1 x 1 and 2 x 2 squares. 1. Complete the table using the toothpicks to make larger squares. Number of toothpicks Total number of per side of the square toothpicks 0 0 1 4 2 12 3 24 4 40 5 60 6 84 7 112 8 144 9 170 2. Do you see a pattern in the table of values? Explain the pattern that you have discovered. Some students may see that consecutive multiples of 4 are being added to the previous term. Some students may use the first and second differences to determine that the pattern is quadratic (see the table below). Number of toothpicks per side of the square 0 1 2 3 4 5 6 7 8 9 Total number of toothpicks 0 4 12 24 40 60 84 112 144 170 First difference Second difference 4 8 12 16 20 24 28 32 36 4 4 4 4 4 4 4 4 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 374 of 399 Columbus City Schools 6/28/13 3. What type of a function could you use to model this pattern? Using the first and second differences, the student will discover that the model should be quadratic (see the table in #2). 4. Set up a system of linear equations using the data in the table above. Solve the system to create your function. Show all of your work! Students will select three ordered pairs, use the general form of the quadratic equation and create a system of linear equations in three variables. For example if the student uses the ordered pairs: (0, 0), (2, 12), and (4, 40), the student will generate the system: 0a 0b c 0 4a 2b c 12 16a 4b c 40 Students can solve by using linear combinations or they may choose to use matrices. 0a 0b c 0 0a 0b c 0 4a 2b c 12 16a 4b c 40 multiply row 2 by - 2 and add to row three 8a 0b c 16 16a 4b c 40 Therefore, c = 0, a = 2, and b = 2, giving students the function f x 2x2 2x . If matrices are used, the following work would be required. 0 0 1 a 0 4 2 1 b 16 4 1 c 12 40 a b c 0.125 - 0.25 0.125 - 0.75 1 - 0.25 1 0 0 a b c 2 2 0 0 12 40 Therefore, the student will obtain the function f x 2x2 2x . CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 375 of 399 Columbus City Schools 6/28/13 F-IF 6 Name ___________________________________ Date __________________ Period ________ Patterns with Triangles Figure 1 Figure 2 Figure 3 The number of equal triangles of each figure depends on the figure number (it is a function of the figure number). Complete the table based on the pattern you observe. If necessary, draw more figures based on the pattern you see. Figure Number 1 Number of Triangles 2 3 4 5 6 a) Describe any relationship (pattern) you see between the number of equal triangles to the figure number. Include a description of the first and second differences. b) Write an equation to describe the relationship between the number of equal triangles, t, to the figure number, n. c) According to your function in part (b), how many equal triangles are there in Figure 15? d) If a certain figure number had 100 equal triangles, what figure number is this? CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 376 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Patterns with Triangles Answer Key Figure 1 Figure 2 Figure 3 The number of equal triangles of each figure depends on the figure number (it is a function of the figure number). Complete the table based on the pattern you observe. If necessary, draw more figures based on the pattern you see. Figure Number 1 2 3 4 5 6 Number of Triangles 1 4 9 16 25 36 a) Describe any relationship (pattern) you see between the number of equal triangles to the figure number. Include a description of the first and second differences. The number of triangles is acquired from squaring the figure number. The first difference in the number of triangles is increasing as the figure number increases by 1 (i.e., 1st difference is 3, 5, 7, 9, etc.). The second difference is constant – it is 2. b) Write an equation to describe the relationship between the number of equal triangles, t, to the figure number, n. t = n2 c) According to your function in part (b), how many equal triangles are there in Figure 15? t = (15)2 = 225 triangles d) If a certain figure number had 100 equal triangles, what figure number is this? n2 = 100, n2 = 100 , n = 10 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 377 of 399 Columbus City Schools 6/28/13 F-IF 6 Name ___________________________________ Date __________________ Period ________ Patterns with Stacking Pennies Figure 1 Figure 2 Figure 3 The number of pennies for each figure depends on the figure number (it is a function of the figure number). Complete the table based on the pattern you observe. If necessary, construct more stacks of pennies based on the pattern you see. Figure Number 1 Number of Pennies 2 3 4 5 6 a) Describe any relationship (pattern) you see between the number of pennies to the figure number. Include a description of the first and second differences. b) Write an equation to describe the relationship between the number of pennies, p, to the figure number, n. c) According to your function in part (b), how many pennies are there in Figure 12? d) If a certain figure number had 36 pennies, what figure number is this? CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 378 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Patterns with Stacking Pennies Answer Key Figure 1 Figure 2 Figure 3 The number of pennies for each figure depends on the figure number (it is a function of the figure number). Complete the table based on the pattern you observe. If necessary, construct more stacks of pennies based on the pattern you see. Figure Number Number of Pennies 1 1 2 3 3 6 4 10 5 15 6 21 1. Describe any relationship (pattern) you see between the number of pennies to the figure number. Include a description of the first and second differences. The first difference in the number of pennies is increasing as the figure number increases by 1 (i.e., 1st difference is 2, 3, 4, 5, 6 etc.). The second difference is constant – it is 1. 2. Write an equation to describe the relationship between the number of pennies, p, to the figure number, n. p = 0.5n2 + 0.5n 3. According to your function in part (b), how many pennies are there in Figure 12? p = 0.5(12)2 + 0.5(12) = 78 pennies d) If a certain figure number had 36 pennies, what figure number is this? 0.5n2 + 0.5n = 36; n2 + n = 72; n2 + n – 72 = 0; (n – 8)(n + 9) = 0; n = 8, n = -9; Figure 8 has 36 pennies CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 379 of 399 Columbus City Schools 6/28/13 F-BF 1 Name ___________________________________ Date __________________ Period ________ Leap Frog Investigation These two frogs need your help. Each of them would like to continue on their journey to the other side of the pond but they have a problem. They both cannot be on the same lily pad at the same time and they can only jump over one lily pad at a time. How can they pass each other so they can continue on their way? How many hops does it take for these two frogs to pass? 1. Following the restrictions above, what is the smallest number of moves required for the two frogs to exchange places? 2. Count the minimum number of moves required for these two sets of frogs to exchange places. 3. Continue this process by adding another frog to each side. Number of frogs on each side 1 Minimum number of moves required 2 3 4 5 6 7 Using the appropriate number of lily pads below and your ‘frogs’, determine the number of moves for each situation in the chart. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 380 of 399 Columbus City Schools 6/28/13 4. Look at the table; describe any relationship or pattern that you observe. Include a description of the first and second differences. 5. Write the equation to describe the relationship between the number of moves, m, to the number of frogs, f. 6. Use your calculator’s STAT menu to make a scatterplot then enter your equation in the Y= menu. Check the graph to see if your equation fits the data. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 381 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Leap Frog Investigation Answer Key These two frogs need your help. Each of them would like to continue on their journey to the other side of the pond but they have a problem. They both cannot be on the same lily pad at the same time and they can only jump over one lily pad at a time. How can they pass each other so they can continue on their way? How many hops does it take for these two frogs to pass? 1. Following the restrictions above, what is the smallest number of moves required for the two frogs to exchange places? 3 2. Count the minimum number of moves required for these two sets of frogs to exchange places. 8 3. Continue this process by adding another frog to each side. Number of frogs on Minimum number of each side moves required 1 3 2 3 4 5 6 7 8 15 24 35 48 63 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 382 of 399 Columbus City Schools 6/28/13 Using the appropriate number of lily pads below and your ‘frogs’, determine the number of moves for each situation in the chart. 4. Look at the table; describe any relationship or pattern that you observe. Include a description of the first and second differences. The first differences are 5, 7, 9, 11, . . . etc. The second differences are a constant of 2. This relationship between the number of frogs and the moves required is quadratic. 5. Write the equation to describe the relationship between the number of moves, m, to the number of frogs, f. y = x2 + 2x 6. Use your calculator’s STAT menu to make a scatterplot then enter your equation in the Y= menu. Check the graph to see if your equation fits the data. 100 90 80 70 60 50 40 30 20 10 1 2 3 4 5 6 7 8 9 10 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 383 of 399 Columbus City Schools 6/28/13 F-BF 1 Name ___________________________________ Date __________________ Period ________ Area Application 1. Using two pipe cleaners, bend them to form rectangles of various widths (whole numbers only). Record the corresponding lengths in the table and calculate the area. Width (inches) Length (inches) Area (square inches) 1 2. Make a scatterplot of the data from the above table, comparing area (y-values) to width (xvalues). 36 24 AREA 12 1 2 3 4 5 6 WIDTH 7 8 9 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 384 of 399 10 11 12 Columbus City Schools 6/28/13 3. What type of relationship exists between the area of a rectangle and its width? Explain your reasoning. Your explanation should include a description of first and second differences and how they relate to the graph. 4. What are the dimensions of the rectangle with no area? Label the point(s) on the graph. 5. What are the dimensions of the rectangle with maximum area? Label the point(s) on the graph. 6. Identify a relationship between the width and length of each rectangle from the table. Write an expression that relates the length, l, to the width, w. 7. Using the expression from #6, write an equation for the area, A, of a rectangle in terms of the width, w, made from your pipe cleaners. 8. If you could make a circle from your pipe cleaners, what would the radius of the circle be? (C = 2 r) 9. Calculate the area of this circle. (A = r2) 10. Compare the area of this circle with the areas of the rectangles. What conclusions can you make about creating maximum area with a given perimeter? CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 385 of 399 Columbus City Schools 6/28/13 11. Think of a rectangle made from 60 inches of pipe cleaners. a) What are the dimensions of the rectangle that has maximum area? b) Write an equation for the area of a rectangle made from these pipe cleaners given the width, w. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 386 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Area Application Answer Key 1. Using two pipe cleaners, bend them to form rectangles of various widths (whole numbers only). Record the corresponding lengths in the table and calculate the area. Width (inches) 1 2 3 4 5 6 7 8 9 10 11 12 Length (inches) 11 10 9 8 7 6 5 4 3 2 1 0 Area (square inches) 11 20 27 32 35 36 35 32 27 20 11 0 2. Make a scatterplot of the data from the above table, comparing area (y-values) to width (xvalues). 36 24 AREA 12 1 2 3 4 5 6 WIDTH 7 8 9 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 387 of 399 10 11 12 Columbus City Schools 6/28/13 3. What type of relationship exists between the area of these rectangles and their widths? Explain your reasoning. Your explanation should include a description of first and second differences and how they relate to the graph. There is a quadratic relationship between the area of a rectangle and its width, given a fixed perimeter. If you look at the pattern that occurs in the area column, the second differences are a constant of 2. The numbers increase at a decreasing rate until you reach 36, then they decrease at an increasing rate. This relates to the graph because it looks like a parabola, which increases until a maximum height then is decreasing. 4. What are the dimensions of the rectangle with no area? Label the point(s) on the graph. There are two rectangles that could have no area. If the length is 0 and the width is 12 or if the length is 12 and the width is 0. These points would be the x-intercepts of the graph. 5. What are the dimensions of the rectangle with maximum area? Label the point(s) on the graph. The rectangle with maximum area is the 6 in. 6 in. rectangle, or square. This point is at the maximum on the graph. 6. Identify a relationship between the width and length of each rectangle from the table. Write an expression that relates the length, l, to the width, w. w + l = 12 or l = 12 – w 7. Using the expression from #6, write an equation for the area, A, of a rectangle in terms of the width, w, made from your pipe cleaners. A = l • w A = (12 – w)w A = 12w – w2 8. If you could make a circle from your pipe cleaners, what would the radius of the circle be? (C = 2 r) 24 in = 2 r r = 24 in / 2 r 3.82 in 9. Calculate the area of this circle. (A = r2) A = (3.82 in)2 A = 45.85 in2 10. Compare the area of this circle with the areas of the rectangles. What conclusions can you make about creating maximum area with a given perimeter? The area of this circle is more than the area of any of the rectangles. If the shape must be rectangular, then a square maximizes area. 11. Think of a rectangle made from 60 inches of pipe cleaners. a) What are the dimensions of the rectangle that has maximum area? The square that is 15 in 15 in b) Write an equation for the area of a rectangle made from these pipe cleaners given the width, w. A = (30 – w)w or A = 30w – w2 or A = (60 – 2w) 12 w CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 388 of 399 Columbus City Schools 6/28/13 F-BF 1 Name ___________________________________ Date __________________ Period ________ Toothpick Patterns Figure Figure 2 Figure 3 1. Complete the table based on the pattern you observe. Create more figures if necessary. Figure Number of Number Toothpicks 1 4 Figure Perimeter Number 1 4 Figure Area Number 1 1 2 2 2 3 3 3 4 4 4 5 5 5 Area Perimeter Number of Toothpicks 2. Graph each of the above sets of data below. Figure Number Figure Number Figure Number 3. Which graph is linear? Which is quadratic? Explain why they are linear or quadratic. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 389 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Toothpick Patterns Answer Key Figure 1 Figure 2 Figure 3 1. Complete the table based on the pattern you observe. Create more figures if necessary. Figure Number 1 2 3 4 5 Number of Toothpicks 4 Figure Perimeter Number 1 4 2 10 8 3 18 12 4 28 16 5 40 20 Figure Area Number 1 1 2 3 4 5 3 6 10 15 Figure Number Area Perimeter Number of Toothpicks 1. Graph each of the above sets of data below. Figure Number Figure Number 2. Which graph is linear? Which is quadratic? Explain why they are linear or quadratic. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 390 of 399 Columbus City Schools 6/28/13 The Number of Toothpicks vs. Figure Number and the Area vs. Figure Number are both quadratic functions because the 2nd difference is constant. The Perimeter vs. Figure Number function is linear because the 1st difference is constant. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 391 of 399 Columbus City Schools 6/28/13 S-ID 6a Name ___________________________________ Date __________________ Period ________ Ball Bounce Activity Materials: CBR, TI-82 or TI-83 Calculator, Link Cable, Ball INSTRUCTIONS: A. Setting up the calculator and ranger. 1. If you are using a TI-83 Plus, go to step 3. If you are using a TI-82 or TI-83: Select PRGM on the keypad. If the program RANGER is on the list select it and go to step 4. 2. If RANGER is not on the list, connect your calculator to the Ranger. On the calculator, select 2nd Link. (It’s on the X key). Use the right arrow to highlight RECEIVE and hit ENTER. The calculator will display Waiting… Open the RANGER and push the button 82/83. The calculator should display Receiving then RANGER PRGM and then DONE. Go to step 4. 3. On the TI-83 Plus, choose APPS and choose CBL/CBR. (If it is not on the list, follow the instructions for the TI-82 or TI-83.) Press any key. On the next screen select RANGER. Go the step 4. 4. Hit ENTER. Select #3 Applications. When prompted for UNITS, select #2 FEET. Choose #3 Ball Bounce. B. Ball Bounce. 1. Be sure that the ball is bounced on a smooth, level surface. Do not allow anything to obstruct the path between the Ranger and the ball while the data is being collected. 2. Follow the instructions on the calculator. 3. Your data should look like a series of parabolas, decreasing in height. Decide if you want to try again or not. 4. Hit ENTER. If you did not like your graph, select #5 REPEAT SAMPLE and go back to step #B2. If you like your graph, go to step 5. 5. Choose #4 PLOT TOOLS. On PLOT TOOLS choose #1 SELECT DOMAIN. Pick out your best parabola. For LEFT BOUND, use the right or left arrow to move the cursor to the lowest point on the left side of the parabola you chose. Hit ENTER. For RIGHT BOUND, use the right arrow to move to the lowest point on the right side of your parabola. Hit ENTER. Choose #7 QUIT. After the calculator displays: L1=TIME L2=DIST L3=VEL L4=ACCEL Done 6. Select GRAPH on the keypad. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 392 of 399 Columbus City Schools 6/28/13 ANALYSIS: 1. On your graph, what is measured on the x-axis? the y-axis? What is measured on 2. The ball was bouncing straight up and down. Why is the graph a series of parabolas? What orce makes the ball fall after each bounce? Why do the heights of the bounces decrease for each bounce? 3. Use TRACE to locate the approximate position of the vertex. 4. Remember that the vertex form of the equation of a parabola is y = a(x – h)2 + k. What is h for your parabola? What is k for your parabola? 5. Is a positive or negative? How do you know? 6. Guess a number for a and enter y = a(x – h)2 + k into the y= menu of your calculator, using the vertex for h and k and your guess for a. Check your guess by graphing your equation with the stat plot. If your parabola does not match your stat plot, make another guess for a. How does the steepness of your graph compared to the steepness of the stat plot help you make your next guess? Keep guessing until the graphs are nearly identical. Give your equation here. Simplify your previous answer so that it is in form y = ax2 + bx + c. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 393 of 399 Columbus City Schools 6/28/13 To check your work, graph this equation to see if it coincides with the vertex form of the equation. 7. The data from your parabola are stored in L1 and L2. Your calculator can find an equation that models your data. Such an equation is called a regression. To calculate a quadratic regression, push STAT and arrow to the right to highlight CALC. Choose QuadReg. DO NOT PUSH ENTER. With QuadReg on the calculator, on the same line, enter L1, L2. (L1 is 2nd 1 and L2 is 2nd 2, so that the command looks like QuadReg L1, L2. Press enter. The comma is the key above the 7.) Write the equation here. Enter the equation into Y2 and graph. How well does it match your data? 8. The acceleration of gravity, g, is 32 ft/sec2. The formula for a falling object is y = 12 gt2+ v0t + s0, where g is the acceleration of gravity, v0 is the initial velocity, and s0 is the initial height. Can you make any connections to the equation of your parabola? CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 394 of 399 Columbus City Schools 6/28/13 Name ___________________________________ Date __________________ Period ________ Ball Bounce Activity Answer Key Materials: CBR, TI-82 or TI-83 Calculator, Link Cable, Ball The activity works much better with large balls. Basketballs, volleyballs, and soccer balls work well. Also, the play balls available in the CPS warehouse work very well. INSTRUCTIONS: This is obviously much easier if everyone has the same kind of calculator, but once you get to step 4 everything works the same way. A. Setting up the calculator and ranger 1. If you are using a TI-83 Plus, go to step 3. If you are using a TI-82 or TI-83: Select PRGM on the keypad. If the program RANGER is on the list select it and go to step 4. 2. If RANGER is not on the list, connect your calculator to the Ranger. On the calculator, select 2nd Link. (It’s on the X key). Use the right arrow to highlight RECEIVE and hit ENTER. The calculator will display Waiting… Open the RANGER and push the button 82/83. The calculator should display Receiving then RANGER PRGM and then DONE. Go to step 4. 3. On the TI-83 Plus, choose APPS and choose CBL/CBR. (If it is not on the list, follow the instructions for the TI-82 or TI-83.) Press any key. On the next screen select RANGER. Go the step 4. 4. Hit ENTER. Select #3 Applications. When prompted for UNITS, select #2 FEET. Choose #3 Ball Bounce. B. Ball Bounce. The students should hold the ball by placing hands on the side of the ball. To release it, they just spread their hands. This keeps the hands out of the way of the motion detector and eliminates the problem of having an initial velocity from throwing the ball down. 1. Be sure that the ball is bounced on a smooth, level surface. Do not allow anything to obstruct the path between the Ranger and the ball while the data is being collected. 2. Follow the instructions on the calculator. 3. Your data should look like a series of parabolas, decreasing in height. Decide if you want to try again or not. 4. Hit ENTER. If you did not like your graph, select #5 REPEAT SAMPLE and go back to step #B2. If you like your graph, go to step 5. A good graph will definitely look parabolic. If the data looks jagged or linear, the students should repeat the data collection. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 395 of 399 Columbus City Schools 6/28/13 5. Choose #4 PLOT TOOLS. On PLOT TOOLS choose #1 SELECT DOMAIN. Pick out your best parabola. For LEFT BOUND, use the right or left arrow to move the cursor to the lowest point on the left side of the parabola you chose. Hit ENTER. For RIGHT BOUND, use the right arrow to move to the lowest point on the right side of your parabola. Hit ENTER. Choose #7 QUIT. After the calculator displays: L1=TIME L2=DIST L3=VEL L4=ACCEL Done 6. Select GRAPH on the keypad. If you want all students to have the data on their calculators, you must link the calculators. ANALYSIS: The ranger program automatically switches the data so that it displays the distance from the floor rather than the distance from the motion detector. Most students never question this, but once in a while someone notices. 1. On your graph, what is measured on the x-axis? TIME IN SECONDS What is measured on the y-axis? DISTANCE FROM THE GROUND IN FEET 2. The ball was bouncing straight up and down. Why is the graph a series of parabolas? What force makes the ball fall after each bounce? Why do the heights of the bounces decrease for each bounce? As time passes, the ball hits the ground and bounces up. It is slowing down as it reaches its maximum height, stops at the top, and falls back down at an increasing speed. Gravity makes the ball fall after each bounce. Friction makes the bounces decrease. 3. Use TRACE to locate the approximate position of the vertex. Answers vary _ 4. Remember that the vertex form of the equation of a parabola is y = a(x – h)2 + k. What is h for your parabola? Answers vary_ What is k for your parabola? Answers vary 5. Is a positive or negative? How do you know? a is negative because the parabolas open down. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 396 of 399 Columbus City Schools 6/28/13 6. Guess a number for a and enter y = a(x – h)2 + k into the y= menu of your calculator, using the vertex for h and k and your guess for a. Check your guess by graphing your equation with the stat plot. If your parabola does not match your stat plot, make another guess for a. How does the steepness of your graph compared to the steepness of the statplot help you make your next guess? If the graph is not as steep as the statplot, the absolute value of a should be increased. If the graph is steeper than the statplot, the absolute value of a should be decreased. Keep guessing until the graphs are nearly identical. Give your equation here. Answers vary Simplify your previous answer so that it is in form y = ax2 + bx + c. Answers vary To check your work, graph this equation to see if it coincides with the vertex form of the equation. 7. The data from your parabola are stored in L1 and L2. Your calculator can find an equation that models your data. Such an equation is called a regression. To calculate a quadratic regression, push STAT and arrow to the right to highlight CALC. Choose QuadReg. DO NOT PUSH ENTER. With QuadReg on the calculator, on the same line, enter L1, L2. (L1 is 2nd 1 and L2 is 2nd 2, so that the command looks like QuadReg L1, L2. Press enter. The comma is the key above the 7.) Write the equation here. Answers vary Enter the equation into Y2 and graph. How well does it match your data? If you wish to do this in one step, on the QuadReg command add Y1 at the end, so that the command is QuadReg L1, L2, Y2. You will find Y1 in VARS. On the TI-83, choose VARS, arrow to YVARS, select #1 function and select #2 Y2. On the TI-82, choose 2nd YVARS, #1 function, #2 Y2. 8. The acceleration of gravity, g, is -32ft/sec2. The formula for a falling object is y = 12 gt2 + v0t + s0, where g is the acceleration of gravity, v0 is the initial velocity, and s0 is the initial height. Can you make any connections to the equation of your parabola? The value for a should be about -16, b should be close to 0, and c should be the height of the selected bounce. CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 397 of 399 Columbus City Schools 6/28/13 S-ID 6a Name ___________________________________ Date __________________ Period ________ Water Fountain Activity Materials: Tape Measure Scotch Tape Wire or Pipe Cleaner Ruler Graph Paper TI-82 or TI-83 Graphing Calculator The path of the water from a water fountain approximates a parabola. In this activity, you will find an equation to model the path of the water. INSTRUCTIONS 1. You will need to go to the water fountain to gather your data. One person will turn on the fountain while another person bends the wire or pipe cleaner to approximate the path of the water. The model should resemble a parabola. 2. Draw a pair of axes on a piece of graph paper and then tape your model of the water flow to the graph paper. Make sure that the vertex of your parabola is on the y-axis and that the parabola opens down. 3. Pick ten points on your parabola and estimate the order pairs, being careful to choose points from both sides of the parabola and the vertex. Write them on the graph and record them below. Put them into List 1 and List 2 on your calculator. x y L1 L2 CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 398 of 399 Columbus City Schools 6/28/13 4. Make a scatterplot of your data, using the data to decide upon a good window. Give your window below. Xmin Xmax Xscl Ymin Ymax Yscl 5. What is the ordered pair for the vertex of your parabola? Remember 2 that the vertex form for the equation of a parabola is y = a(x – h) + k. What are h and k in the equation of your parabola? 6. Is the a in your parabola positive or negative? 7. Pick a value for a and use the h and k from the vertex to write a possible equation for your parabola. Write your equation here. 8. Put your equation from #7 in Y1 on the calculator and graph it with your statplot. Sketch your result here. How well did your equation fit your data? 9. Change your value for a in the equation and try again. What is the new value? Did it fit better or worse? How can you change it to make the fit better? 10. Keep changing the value for a until you have a good fit. What is your final equation? 11. How did you know the correct way to change a? How did changing a make the parabola wider or narrower? CCSSM II Comparing Functions and Different Representations of Quadratic Functions F – IF 4, 5, 6, 7, 7a, 9, F – BF 1, 1a, 1b, 3, A – CED 1, 2, F – LE 3, N – Q 2, S – ID 6a, 6b, A – REI 7 Quarter 2 Page 399 of 399 Columbus City Schools 6/28/13 High School CCSS Mathematics II Curriculum Guide -Quarter 3- Columbus City Schools Page 1 of 144 Table of Contents Math Practices Rationale .............................................................................................................................................. 3 RUBRIC – IMPLEMENTING STANDARDS FOR MATHEMATICAL PRACTICE .................................................................. 11 Mathematical Practices: A Walk-Through Protocol .................................................................................................... 16 Curriculum Timeline .................................................................................................................................................... 19 Scope and Sequence.................................................................................................................................................... 20 Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 ....................... 30 Teacher Notes .......................................................................................................................................................... 32 Families of Graphs #2 .......................................................................................................................................... 58 Solving By Factoring ............................................................................................................................................ 64 Polynomial Cards ................................................................................................................................................. 66 Finding the Greatest Common Monomial Using Algebra Tiles ........................................................................... 70 Polynomial Cards ................................................................................................................................................. 77 Drawkcab Problems............................................................................................................................................. 80 Discovering the Difference of Two Squares ........................................................................................................ 88 Factoring Using the Greatest Common Factor .................................................................................................... 92 Factoring By Grouping ......................................................................................................................................... 94 Factoring Worksheet ........................................................................................................................................... 96 Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 ...................................................................................................................................... 100 Teacher Notes:....................................................................................................................................................... 102 Sorting Activity .................................................................................................................................................. 118 Learning How to Complete the Square “Completely” ...................................................................................... 122 Transformations and Completing the Square Notes ......................................................................................... 126 Completing the Square and Transformations Practice ..................................................................................... 134 Discovery of Completing the Square ................................................................................................................. 138 Page 2 of 144 Math Practices Rationale CCSSM Practice 1: Make sense of problems and persevere in solving them. Why is this practice important? What does this practice look like when students are doing it? What can a teacher do to model this practice? What questions could a teacher ask to encourage the use of this practice? What does proficiency look like in this practice? What actions might the teacher make that inhibit the students’ use of this practice? Helps students to develop critical thinking skills. Teaches students to “think for themselves”. Helps students to see there are multiple approaches to solving a problem. Students immediately begin looking for methods to solve a problem based on previous knowledge instead of waiting for teacher to show them the process/algorithm. Students can explain what problem is asking as well as explain, using correct mathematical terms, the process used to solve the problem. Frame mathematical questions/challenges so they are clear and explicit. Check with students repeatedly to help them clarify their thinking and processes. “How would you go about solving this problem?” “What do you need to know in order to solve this problem?” What methods have we studied that you can use to find the information you need? Students can explain the relationships between equations, verbal descriptions, tables, and graphs. Students check their answer using a different method and continually ask themselves, “Does this make sense?” They understand others approaches to solving complex problems and can see the similarities between different approaches. Showing the students shortcuts/tricks to solve problems (without making sure the students understand why they work). Not giving students an adequate amount of think time to come up with solutions or processes to solve a problem. Giving students the answer to their questions instead of asking guiding questions to lead them to the discovery of their own question. Page 3 of 144 CCSSM Practice 2: Reason abstractly and quantitatively. Why is this practice important? What does this practice look like when students are doing it? What can a teacher do to model this practice? What questions could a teacher ask to encourage the use of this practice? What does proficiency look like in this practice? What actions might the teacher make that inhibit the students’ use of this practice? Students develop reasoning skills that help them to understand if their answers make sense and if they need to adjust the answer to a different format (i.e. rounding) Students develop different ways of seeing a problem and methods of solving it. Students are able to translate a problem situation into a number sentence or algebraic expression. Students can use symbols to represent problems. Students can visualize what a problem is asking. Ask students questions about the types of answers they should get. Use appropriate terminology when discussing types of numbers/answers. Provide story problems and real world problems for students to solve. Monitor the thinking of students. “What is your unknown in this problem? “What patterns do you see in this problem and how might that help you to solve it?” Students can recognize the connections between the elements in their mathematical sentence/expression and the original problem. Students can explain what their answer means, as well as how they arrived at it. Giving students the equation for a word or visual problem instead of letting them “figure it out” on their own. Page 4 of 144 CCSSM Practice 3: Construct viable arguments and critique the reasoning of others Why is this practice important? What does this practice look like when students are doing it? What can a teacher do to model this practice? What questions could a teacher ask to encourage the use of this practice? What does proficiency look like in this practice? What actions might the teacher make that inhibit the students’ use of this practice? Students better understand and remember concepts when they can defend and explain it to others. Students are better able to apply the concept to other situations when they understand how it works. Communicate and justify their solutions Listen to the reasoning of others and ask clarifying questions. Compare two arguments or solutions Question the reasoning of other students Explain flaws in arguments Provide an environment that encourages discussion and risk taking. Listen to students and question the clarity of arguments. Model effective questioning and appropriate ways to discuss and critique a mathematical statement. How could you prove this is always true? What parts of “Johnny’s “ solution confuses you? Can you think of an example to disprove your classmates theory? Students are able to make a mathematical statement and justify it. Students can listen, critique and compare the mathematical arguments of others. Students can analyze answers to problems by determining what answers make sense. Explain flaws in arguments of others. Not listening to students justify their solutions or giving adequate time to critique flaws in their thinking or reasoning. Page 5 of 144 CCSSM Practice 4: Model with mathematics Why is this practice important? Helps students to see the connections between math symbols and real world problems. What does this practice look like when students are doing it? Write equations to go with a story problem. Apply math concepts to real world problems. What can a teacher do to model this practice? Use problems that occur in everyday life and have students apply mathematics to create solutions. Connect the equation that matches the real world problem. Have students explain what different numbers and variables represent in the problem situation. Require students to make sense of the problems and determine if the solution is reasonable. How could you represent what the problem was asking? How does your equation relate to the problems? How does your strategy help you to solve the problem? Students can write an equation to represent a problem. Students can analyze their solutions and determine if their answer makes sense. Students can use assumptions and approximations to simplify complex situations. Not give students any problem with real world applications. What questions could a teacher ask to encourage the use of this practice? What does proficiency look like in this practice? What actions might the teacher make that inhibit the students’ use of this practice? Page 6 of 144 CCSSM Practice 5: Use appropriate tools strategically Why is this practice important? What does this practice look like when students are doing it? What can a teacher do to model this practice? What questions could a teacher ask to encourage the use of this practice? Helps students to understand the uses and limitations of different mathematical and technological tools as well as which ones can be applied to different problem situations. Students select from a variety of tools that are available without being told which to use. Students know which tools are helpful and which are not. Students understand the effects and limitations of chosen tools. Provide students with a variety of tools Facilitate discussion regarding the appropriateness of different tools. Allow students to decide which tools they will use. How is this tool helping you to understand and solve the problem? What tools have we used that might help you organize the information given in this problem? Is there a different tool that could be used to help you solve the problem? What does proficiency look like in this practice? Students are sufficiently familiar with tools appropriate for their grade or course and make sound decisions about when each of these tools might be helpful. Students recognize both the insight to be gained from the use of the selected tool and their limitations. What actions might the teacher make that inhibit the students’ use of this practice? Only allowing students to solve the problem using one method. Telling students that the solution is incorrect because it was not solved “the way I showed you”. Page 7 of 144 CCSSM Practice 6: Attend to precision. Why is this practice important? What does this practice look like when students are doing it? What can a teacher do to model this practice? What questions could a teacher ask to encourage the use of this practice? What does proficiency look like in this practice? What actions might the teacher make that inhibit the students’ use of this practice? Students are better able to understand new math concepts when they are familiar with the terminology that is being used. Students can understand how to solve real world problems. Students can express themselves to the teacher and to each other using the correct math vocabulary. Students use correct labels with word problems. Make sure to use correct vocabulary terms when speaking with students. Ask students to provide a label when describing word problems. Encourage discussions and explanations and use probing questions. How could you describe this problem in your own words? What are some non-examples of this word? What mathematical term could be used to describe this process. Students are precise in their descriptions. They use mathematical definitions in their reasoning and in discussions. They state the meaning of symbols consistently and appropriately. Teaching students “trick names” for symbols (i.e. the alligator eats the big number) Not using proper terminology in the classroom. Allowing students to use the word “it” to describe symbols or other concepts. Page 8 of 144 CCSSM Practice 7: Look for and make use of structure. Why is this practice important? When students can see patterns or connections, they are more easily able to solve problems What does this practice look like when students are doing it? Students look for connections between properties. Students look for patterns in numbers, operations, attributes of figures, etc. Students apply a variety of strategies to solve the same problem. Ask students to explain or show how they solved a problem. Ask students to describe how one repeated operation relates to another (addition vs. multiplication). How could you solve the problem using a different operation? What pattern do you notice? What can a teacher do to model this practice? What questions could a teacher ask to encourage the use of this practice? What does proficiency look like in this practice? Students look closely to discern a pattern or structure. What actions might the teacher make that inhibit the students’ use of this practice? Provide students with pattern before allowing them to discern it for themselves. Page 9 of 144 CCSSM Practice 8: Look for and express regularity in repeated reasoning Why is this practice important? What does this practice look like when students are doing it? What can a teacher do to model this practice? What questions could a teacher ask to encourage the use of this practice? When students discover connections or algorithms on their own, they better understand why they work and are more likely to remember and be able to apply them. Students discover connections between procedures and concepts Students discover rules on their own through repeated exposures of a concept. Provide real world problems for students to discover rules and procedures through repeated exposure. Design lessons for students to make connections. Allow time for students to discover the concepts behind rules and procedures. Pose a variety of similar type problems. How would you describe your method? Why does it work? Does this method work all the time? What do you notice when…? What does proficiency look like in this practice? Students notice repeated calculations. Students look for general methods and shortcuts. What actions might the teacher make that inhibit the students’ use of this practice? Providing students with formulas or algorithms instead of allowing them to discover it on their own. Not allowing students enough time to discover patterns. Page 10 of 144 RUBRIC – IMPLEMENTING STANDARDS FOR MATHEMATICAL PRACTICE Using the Rubric: Task: Teacher: Is strictly procedural. Does not require students to check solutions for errors. NEEDS IMPROVEMENT (students take ownership) EXEMPLARY Task: PROFICIENT Task: Teacher: Differentiates to keep advanced students challenged during work time. Integrates time for explicit meta-cognition. Expects students to make sense of the task and the proposed solution. Allows for multiple entry points and solution paths. Requires students to defend and justify their solution by comparing multiple solution paths. (teacher mostly models) Is overly scaffolded or procedurally “obvious”. Requires students to check answers by plugging in numbers. EMERGING Task: Teacher: Teacher: Allows ample time for all students to struggle with task. Expects students to evaluate processes implicitly. Models making sense of the task (given situation) and the proposed solution. Is cognitively demanding. Has more than one entry point. Requires a balance of procedural fluency and conceptual understanding. Requires students to check solutions for errors using one other solution path. (teacher does thinking) Review each row corresponding to a mathematical practice. Use the boxes to mark the appropriate description for your task or teacher action. The task descriptors can be used primarily as you develop your lesson to make sure your classroom tasks help cultivate the mathematical practices. The teacher descriptors, however, can be used during or after the lesson to evaluate how the task was carried out. The column titled “proficient” describes the expected norm for task and teacher action while the column titled “exemplary” includes all features of the proficient column and more. A teacher who is exemplary is meeting criteria in both the proficient and exemplary columns. PRACTICE Make sense of problems and persevere in solving them. Does not allow for wait time; asks leading questions to rush through task. Does not encourage students to individually process the tasks. Is focused solely on answers rather than processes and reasoning. Allots too much or too little time to complete task. Encourages students to individually complete tasks, but does not ask them to evaluate the processes used. Explains the reasons behind procedural steps. Does not check errors publicly. Page 11 of 144 PRACTICE Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Is either ambiguously stated. Does not expect students to interpret representations. Expects students to memorize procedures withno connection to meaning. Lacks context. Does not make use of multiple representations or solution paths. NEEDS IMPROVEMENT Task: Teacher: Task: Teacher: Does not ask students to present arguments or solutions. Expects students to follow a given solution path without opportunities to make conjectures. Task: EMERGING Does not help students differentiate between assumptions and logical conjectures. Asks students to present arguments but not to evaluate them. Allows students to make conjectures without justification. Is not at the appropriate level. Expects students to model and interpret tasks using a single representation. Explains connections between procedures and meaning. Is embedded in a contrived context. (teacher does thinking) Teacher: Task: Teacher: Institute for Advanced Study/Park City Mathematics Institute Secondary School Teachers Program/Visualizing Functions Page 12 of 144 PROFICIENT Expects students to interpret and model using multiple representations. Provides structure for students to connect algebraic procedures to contextual meaning. Links mathematical solution with a question’s answer. Avoids single steps or routine algorithms. Teacher: EXEMPLARY Helps students differentiate between assumptions and logical conjectures. Prompts students to evaluate peer arguments. Expects students to formally justify the validity of their conjectures. Expects students to interpret, model, and connect multiple representations. Prompts students to articulate connections between algebraic procedures and contextual meaning. (teacher mostly models) (students take ownership) Task: Task: Has realistic context. Has relevant realistic Requires students to context. frame solutions in a context. Teacher: Has solutions that can be expressed with multiple representations. Teacher: Task: Teacher: Identifies students’ assumptions. Models evaluation of student arguments. Asks students to explain their conjectures. Summer 2011 PRACTICE Model with mathematics. Use appropriate tools strategically. NEEDS IMPROVEMENT Requires students to Task: identify variables and to perform necessary computations. Teacher: Identifies appropriate variables and procedures for students. Does not discuss appropriateness of model. Does not incorporate Task: additional learning tools. Teacher: additional learning tools. Does not incorporate EMERGING (teacher does thinking) Requires students to Task: identify variables and to compute and interpret results. Teacher: Verifies that students have identified appropriate variables and procedures. Explains the appropriateness of model. Lends itself to one learning Task: tool. Does not involve mental computations or estimation. Teacher: Demonstrates use of appropriate learning tool. Page 13 of 144 PROFICIENT Requires students to (teacher mostly models) Task: identify variables, compute and interpret results, and report findings using a mixture of representations. the mathematics involved. Illustrates the relevance of Requires students to identify extraneous or missing information. Teacher: Asks questions to help students identify appropriate variables and procedures. Facilitates discussions in evaluating the appropriateness of model. Lends itself to multiple Task: learning tools. Gives students opportunity to develop fluency in mental computations. Teacher: Chooses appropriate learning tools for student use. estimation. Models error checking by EXEMPLARY Requires students to (students take ownership) Task: identify variables, compute and interpret results, report findings, and justify the reasonableness of their results and procedures within context of the task. Teacher: Expects students to justify their choice of variables and procedures. Gives students opportunity to evaluate the appropriateness of model. Requires multiple learning Task: tools (i.e., graph paper, calculator, manipulative). demonstrate fluency in Requires students to mental computations. Teacher: appropriate learning tools. Allows students to choose appropriate alternatives Creatively finds where tools are not available. PRACTICE Attend to precision. Look for and make use of structure. Requires students to automatically apply an algorithm to a task without evaluating its appropriateness. Does not intervene when students are being imprecise. Does not point out instances when students fail to address the question completely or directly. Gives imprecise instructions. NEEDS IMPROVEMENT Task: Teacher: Task: Teacher: Does not recognize students for developing efficient approaches to the task. Requires students to apply the same algorithm to a task although there may be other approaches. Task: EMERGING Identifies individual students’ efficient approaches, but does not expand understanding to the rest of the class. Demonstrates the same algorithm to all related tasks although there may be other more effective approaches. Requires students to analyze a task before automatically applying an algorithm. Inconsistently intervenes when students are imprecise. Identifies incomplete responses but does not require student to formulate further response. Has overly detailed or wordy instructions. (teacher does thinking) Teacher: Task: Teacher: Institute for Advanced Study/Park City Mathematics Institute Secondary School Teachers Program/Visualizing Functions Page 14 of 144 PROFICIENT Requires students to analyze a task and identify more than one approach to the problem. Consistently demands precision in communication and in mathematical solutions. Identifies incomplete responses and asks student to revise their response. Teacher: Task: Teacher: EXEMPLARY Prompts students to identify mathematical structure of the task in order to identify the most effective solution path. Encourages students to justify their choice of algorithm or solution path. Requires students to identify the most efficient solution to the task. Demands and models precision in communication and in mathematical solutions. Encourages students to identify when others are not addressing the question completely. Includes assessment criteria for communication of ideas. (teacher mostly models) (students take ownership) Task: Task: Has precise instructions. Teacher: Task: Teacher: Facilitates all students in developing reasonable and efficient ways to accurately perform basic operations. Continuously questions students about the reasonableness of their intermediate results. Summer 2011 PRACTICE Look for and express regularity in repeated reasoning. Is disconnected from prior and future concepts. Has no logical progression that leads to pattern recognition. NEEDS IMPROVEMENT Task: Teacher: Does not show evidence of understanding the hierarchy within concepts. Presents or examines task in isolation. Task: EMERGING Hides or does not draw connections to prior or future concepts. Is overly repetitive or has gaps that do not allow for development of a pattern. (teacher does thinking) Teacher: Institute for Advanced Study/Park City Mathematics Institute Secondary School Teachers Program/Visualizing Functions Page 15 of 144 PROFICIENT Reviews prior knowledge and requires cumulative understanding. Lends itself to developing a pattern or structure. (teacher mostly models) Task: Teacher: Connects concept to prior and future concepts to help students develop an understanding of procedural shortcuts. Demonstrates connections between tasks. EXEMPLARY Addresses and connects to prior knowledge in a nonroutine way. Requires recognition of pattern or structure to be completed. (students take ownership) Task: Teacher: Encourages students to connect task to prior concepts and tasks. Prompts students to generate exploratory questions based on the current task. Encourages students to monitor each other’s intermediate results. Summer 2011 Mathematical Practices: A Walk-Through Protocol *Note: This document should also be used by the teacher for planning and self-evaluation. Mathematical Practices MP.1. Make sense of problems and persevere in solving them MP.2. Reason abstractly and quantitatively. MP.3. Construct viable arguments and critique the reasoning of others. Observations Students are expected to______________: Engage in solving problems. Explain the meaning of a problem and restate in it their own words. Analyze given information to develop possible strategies for solving the problem. Identify and execute appropriate strategies to solve the problem. Check their answers using a different method, and continually ask “Does this make sense?” Teachers are expected to______________: Provide time for students to discuss problem solving. Students are expected to______________: Connect quantity to numbers and symbols (decontextualize the problem) and create a logical representation of the problem at hand. Recognize that a number represents a specific quantity (contextualize the problem). Contextualize and decontextualize within the process of solving a problem. Teachers are expected to______________: Provide appropriate representations of problems. Students are expected to____________________________: Explain their thinking to others and respond to others’ thinking. Participate in mathematical discussions involving questions like “How did you get that?” and “Why is that true?” Construct arguments that utilize prior learning. Question and problem pose. Practice questioning strategies used to generate information. Analyze alternative approaches suggested by others and select better approaches. Justify conclusions, communicate them to others, and respond to the arguments of others. Compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument, explain what it is. Teachers are expected to______________: Provide opportunities for students to listen to or read the conclusions and arguments of others. CCSSM National Professional Development Page 16 of 144 Mathematical Practices MP.4. Model with mathematics. MP 5. Use appropriate tools strategically Observations Students are expected to______________: Apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. Make assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. Experiment with representing problem situations in multiple ways, including numbers, words (mathematical language), drawing pictures, using objects, acting out, making a chart or list, creating equations, etc. Identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts, and formulas. Evaluate their results in the context of the situation and reflect on whether their results make sense. Analyze mathematical relationships to draw conclusions. Teachers are expected to______________: Provide contexts for students to apply the mathematics learned. Students are expected to______________: Use tools when solving a mathematical problem and to deepen their understanding of concepts (e.g., pencil and paper, physical models, geometric construction and measurement devices, graph paper, calculators, computer-based algebra or geometry systems.) Consider available tools when solving a mathematical problem and decide when certain tools might be helpful, recognizing both the insight to be gained and their limitations. Detect possible errors by strategically using estimation and other mathematical knowledge. Teachers are expected to______________: CCSSM National Professional Development Page 17 of 144 Mathematical Practices MP.6. Attend to precision. MP.7. Look for and make use of structure. MP.8. Look for and express regularity in repeated reasoning. Observations Students are expected to______________: Use clear and precise language in their discussions with others and in their own reasoning. Use clear definitions and state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. Specify units of measure and label parts of graphs and charts. Calculate with accuracy and efficiency based on a problem’s expectation. Teachers are expected to______________: Emphasize the importance of precise communication. Students are expected to______________: Describe a pattern or structure. Look for, develop, generalize, and describe a pattern orally, symbolically, graphically and in written form. Relate numerical patterns to a rule or graphical representation Apply and discuss properties. Teachers are expected to______________: Provide time for applying and discussing properties. Students are expected to______________: Describe repetitive actions in computation Look for mathematically sound shortcuts. Use repeated applications to generalize properties. Use models to explain calculations and describe how algorithms work. Use models to examine patterns and generate their own algorithms. Check the reasonableness of their results. Teachers are expected to______________: CCSSM National Professional Development Page 18 of 144 High School Common Core Math II Curriculum Timeline Topic Intro Unit Similarity Trigonometric Ratios Other Types of Functions Comparing Functions and Different Representations of Quadratic Functions Modeling Unit and Project Quadratic Functions: Solving by Factoring Quadratic Functions: Completing the Square and the Quadratic Formula Probability Geometric Measurement Geometric Modeling Unit and Project Standards Covered G – SRT 1 G – SRT 1a G – SRT 1b G – SRT 6 G – SRT 2 G – SRT 3 G – SRT 4 G – SRT 7 G – SRT 5 Grading Period 1 1 No. of Days 5 20 G – SRT 8 1 20 A – CED 1 A – CED 4 A – REI 1 N – RN 1 N – RN 2 N – RN 3 F – IF 4 F – IF 5 F – IF 6 F – IF 7 F – IF 7a F– IF 9 F – IF 4 F – IF 7b F – IF 7e F – IF 8 F – IF 8b F– BF1 A– CED 1 A– CED 2 F– BF 1 F– BF 1a F – BF 1b F– BF 3 F – BF 1a F – BF 1b F – BF 3 A – SSE 1b N–Q2 2 15 F – LE 3 N– Q 2 S – ID 6a S – ID 6b A – REI 7 2 20 2 10 A – APR 1 A – REI 1 A – REI 4b F – IF 8a A – CED 1 A – SSE 1b A – SSE 3a 3 20 A – REI 1 A – REI 4 A – REI 4a A – REI 4b A – SSE 3b F – IF 8 F – IF 8a A – CED 1 N – CN 1 N – CN 2 N – CN 7 3 20 S – CP 1 S – CP 2 S – CP 3 G – GMD 1 S – CP 4 S – CP 5 S – CP 6 G – GMD 3 S – CP 7 4 20 4 10 G – MG 1 G – MG 2 G – MG 3 4 15 Page 19 of 144 High School Common Core Math II 1st Nine Weeks Scope and Sequence Intro Unit – IO (5 days) Topic 1 – Similarity (20 days) Geometry (G – SRT): 1) Similarity, Right Triangles, and Trigonometry: Understand similarity in terms of similarity transformations. G – SRT 1: Verify experimentally the properties of dilations given by a center and a scale factor. G – SRT 1a: A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. G – SRT 1b: The dilation of a line segment is longer or shorter in the ratio given by the scale factor. G – SRT 2: Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. G – SRT 3: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Geometry (G – SRT): 2) Similarity, Right Triangles, and Trigonometry: Prove theorems involving similarity. G – SRT 4: Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. G – SRT 5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Topic 2 – Trigonometric Ratios (20 days) Geometry (G – SRT): 3) Similarity, Right Triangles, and Trigonometry: Define trigonometric ratios and solve problems involving .right triangles G – SRT 6: Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. G – SRT 7: Explain and use the relationship between the sine and cosine of complementary angles. G – SRT 8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Page 20 of 144 High School Common Core Math II 2nd Nine Weeks Scope and Sequence Topic 3 – Other Types of Functions (15 days) Creating Equations (A – CED): 4) Create equations that describe numbers or relationships A – CED 1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. A – CED 4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R. Reasoning with Equations and Inequalities (A – REI): 5) Understand solving equations as a process of reasoning and explain the reasoning. A – REI 1: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. The Real Number System (N – RN): 6) Extend the properties of exponents to rational exponents. N – RN 1: Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5. N – RN 2: Rewrite expressions involving radicals and rational exponents using the properties of exponents. The Real Number System (N – RN): 7) Use properties of rational and irrational numbers. N – RN 3: Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Interpreting Functions (F – IF): 8) Interpret functions that arise in applications in terms of the context. F – IF 4*: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.* Interpreting Functions (F – IF): 9) Analyze functions using different representations. F – IF 7b: Graph square root, cube root, and absolute value functions. F – IF 7e: Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Page 21 of 144 F – IF 8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. F – IF 8b: Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay. Building Functions (F – BF): 10) Build a function that models a relationship between two quantities. F – BF 1: Write a function that describes a relationship between two quantities. F – BF 1a: Determine an explicit expression, a recursive process, or steps for calculation from a context. F – BF 1b: Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Building Functions (F – BF): 11) Build new functions from existing functions. F – BF 3: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Seeing Structure in Expressions (A – SSE): 12) Interpret the structure of expressions. A – SSE 1b: Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)n as the product of P and a factor not depending on P. Quantities (NQ): 13) Reason quantitatively and use units to solve problems. N – Q 2: Define appropriate quantities for the purpose of descriptive modeling. Topic 4 – Comparing Functions and Different Representations of Quadratic Functions (20 days) Interpreting Functions (F – IF): 14) Interpret functions that arise in applications in terms of the context. F – IF 4*: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.* F – IF 5*: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number Page 22 of 144 of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.* F – IF 6: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Interpreting Functions (F – IF): 15) Analyze functions using different representations. F – IF 7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. F – IF 7a*: Graph linear and quadratic functions and show intercepts, maxima, and minima.* F – IF 9: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Creating Equations (A – CED): 16) Create equations that describe numbers or relationships. A – CED 1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. A – CED 2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Building Functions (F – BF): 17) Build a function that models a relationship between two quantities. F – BF 1: Write a function that describes a relationship between two quantities. F – BF 1a: Determine an explicit expression, a recursive process, or steps for calculation from a context. F – BF 1b: Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Building Functions (F – BF): 18) Build new functions from existing functions. F – BF 3: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Linear and Exponential Models (F – LE): 19) Construct and compare linear and exponential models and solve problems. Page 23 of 144 F- LE 3: Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. Quantities (N-Q): 20) Reason quantitatively and use units to solve problems. N – Q 2: Define appropriate quantities for the purpose of descriptive modeling. Interpreting Categorical and Quantitative Data (S – ID): 21) Summarize, represent, and interpret data on two categorical and quantitative variables. S – ID 6a: Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear and exponential models. S – ID 6b: Informally assess the fit of a function by plotting and analyzing residuals. Reasoning with Equations and Inequalities (A – REI): 22) Solve systems of equations. A – REI 7: Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = -3x and the circle x2 + y2 = 3. Modeling Unit and Project –(10 days) Page 24 of 144 High School Common Core Math II 3rd Nine Weeks Scope and Sequence Topic 5–Quadratic Functions – Solving by factoring (20 days) Arithmetic with Polynomials and Rational Expressions (A – APR): 23) Perform arithmetic operations on polynomials. A – APR 1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Reasoning with Equations and Inequalities (A – REI): 24) Understand solving equations as a process of reasoning and explain the reasoning. A – REI 1: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Reasoning with Equations and Inequalities (A – REI): 25) Solve equations and inequalities in one variable. A – REI 4b: Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Interpreting Functions (F – IF): 26) Analyze functions using different representations. F – IF 8a: Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Creating Equations (A – CED): 27) Create equations that describe numbers or relationships. A – CED 1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Seeing Structure in Expressions (A – SSE): 28) Interpret the structure of expressions. A – SSE 1b: Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)n as the product of P and factor not depending on P. Seeing Structure in Expressions (A – SSE): 29) Write expressions in equivalent forms to solve problems. A – SSE 3a: Factor a quadratic expression to reveal the zeros of the function it defines. Topic 6– Quadratic Functions – Completing the Square/Quadratic Formula (20 days) Page 25 of 144 Reasoning with Equations and Inequalities (A – REI): 30) Understand solving equations as a process of reasoning and explain the reasoning. A – REI 1: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Reasoning with Equations and Inequalities (A – REI): 31) Solve equations and inequalities in one variable. A – REI 4: Solve quadratic equations in one variable. A – REI 4a: Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. A – REI 4b: Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Seeing Structure in Expressions (A – SSE): 32) Write expressions in equivalent forms to solve problems. A – SSE 3b: Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. Interpreting Functions (F – IF): 33) Analyze functions using different representations. F – IF 8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. F – IF 8a: Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Creating Equations (A – CED): 34) Create equations that describe numbers or relationships. A – CED 1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. The Complex Number System (N – CN): 35) Perform arithmetic operations with complex numbers. N – CN 1: Know there is a complex number i such that i 2 number has the form a+bi with a and b real. 1 , and every complex N – CN 2: Use the relation i 2 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. The Complex Number System (N – CN): 36) Use complex numbers in polynomial identities and equations. Page 26 of 144 N – CN 7: Solve quadratic equations with real coefficients that have complex solutions. Page 27 of 144 High School Common Core Math II 4th Nine Weeks Scope and Sequence Topic 7 –Probability (20 days) Conditional Probability and the Rules of Probability (S – CP): 37) Understand independence and conditional probability and use them to interpret data. S – CP 1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”). S – CP 2: Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. S – CP 3: Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. S – CP 4: Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from you school will favor science given that the student is in the tenth grade. Do the same for other subjects and compare the results. S – CP 5: Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. Conditional Probability and the Rules of Probability (S – CP): 38) Use the rules of probability to compute probabilities of compound events in a uniform probability model. S – CP 6: Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. S – CP 7: Apply the Addition Rule, P(A or B) = P(B) – P(A and B), and interpret the answer in terms of the model. Topic 8 – Geometric Measurement (10 days) Geometric Measurement and Dimension (G – GMD): 39) Explain volume formulas and use them to solve problems. G – GMD 1: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. Page 28 of 144 G – GMD 3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Geometric and Modeling Project-(15 days) *Modeling with Geometry (G – MG): 40) Apply geometric concepts in modeling situations. G – MG 1*: Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).* G – MG 2*: Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).* G – MG 3*: Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).* Page 29 of 144 COLUMBUS CITY SCHOOLS HIGH SCHOOL CCSS MATHEMATICS II CURRICULUM GUIDE Topic 5 CONCEPTUAL CATEGORY TIME Quadratic Functions: Solving by Algebra and Functions RANGE 20 days Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Domain: Arithmetic with Polynomials and Rational Expressions (A – APR): Cluster 23) Perform arithmetic operations on polynomials. GRADING PERIOD 3 Domain: Reasoning with Equations and Inequalities (A – REI): Cluster 24) Understand solving equations as a process of reasoning and explain the reasoning. 25) Solve equations and inequalities in one variable. Domain: Interpreting Functions (F – IF): Cluster 26) Analyze functions using different representations. Domain: Creating Equations (A – CED): Cluster 27) Create equations that describe numbers or relationships. Domain: Seeing Structure in Expressions (A – SSE): Cluster 28) Interpret the structure of expressions. 29) Write expressions in equivalent forms to solve problems. Standards 23) Perform arithmetic operations on polynomials. A - APR 1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. 24) Understand solving equations as a process of reasoning and explain the reasoning. A – REI 1: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. 25) Solve equations and Inequalities in one variable. CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 30 of 144 Columbus City Schools 12/1/13 A – REI 4b: Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. 26) Analyze functions using different representations. F – IF 8a: Use the process of factoring and completing the square in a quadratic function to show zero, extreme values, and symmetry of the graph, and interpret these in terms of a context. 27) Create equations that describe numbers or relationships. A – CED 1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. 28) Interpret the structure of expressions. A – SSE 1b: Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)n as the product of P and factor not depending on P. 29) Write expressions in equivalent forms to solve problems. A – SSE 3a: Factor a quadratic expression to reveal the zeros of the function it defines. CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 31 of 144 Columbus City Schools 12/1/13 TEACHING TOOLS Vocabulary: binomial, degree, factored form of a quadratic function, factoring, factors, FOIL method, function, leading coefficient, like terms, monomial, parabola, polynomial, quadratic, quadratic equation, quadratic function, quadratic inequality, quadratic term, roots, solutions, Square Root Property, trinomial, Zero Product Property, zeros Teacher Notes: Factoring Polynomials Factoring, a method of breaking down polynomials into their parts, can be used to solve Quadratic equations. Follow the steps listed below to solve the quadratic ax2 + bx + c = 0; a 0. Step 1 - Factor out any common factors. Step 2 - Factor the remaining expression by determining which two integers when added = b, and when multiplied = c. Step 3 - Use the Principle of Zero Products, if ab = 0 then a = 0 or b = 0, to find the roots. Example: Find the roots of 2x2+ 10x + 12 = 0 by factoring. Step 1 - Factor out the common factor of 2. 2( x2 + 5 x + 6) = 0 Step 2 - Determine which integers when added = 5 and when multiplied = 6 by examining the factor pairs of 6: 1, 6; -1, -6; 2, 3; -2 , -3. The sum of the factor pair 2, 3 is 5. The quadratic equation factors as: 2( x + 2)( x + 3) = 0 Step 3 - Using the Principle of Zero Products we conclude that either 2 = 0, which is not true, or x + 2 = 0 or x + 3 = 0. Solving these equations we find that the solutions to the quadratic equation 2x2 + 10x + 12 = 0 are x = - 2 and x = -3. When simplifying the quotient of two trinomials, factor two primes and cancel (numerator paired to denominator) common factors. x2 2 x 3 x 2 7 x 12 2 x 2x 3 x 2 7 x 12 x 3 x 1 ( x 3)( x 4) Factor the numerator and denominator. x 1 ; x 3, 4 Since the denominator’s factors are (x – 3) and (x – 4), x 3 and x 4 x 4. Otherwise the denominator would equal zero, making the fraction undefined. The restrictions will need to be stated. The Algebra 2 textbook covers solving quadratic equations by graphing, factoring, completing the square, and the quadratic formula. You may wish to review multiplying binomials and factoring quadratics. The factoring worksheets can be used as additional review; however, the order is set up so that it lends itself to the “ac” method. The “ac” method utilizes factoring by grouping. CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 32 of 144 Columbus City Schools 12/1/13 Roots – Zeros The roots or zeros of a polynomial function are those values of x that make the equation true when set equal to zero, i.e. f(x) = 0 as shown below. Zero is the term used to describe the solution of a polynomial function and root is the term used to describe the solution of a polynomial equation. f(x) = x2 + 5x + 6, set f(x) = 0 giving the equation: x2 + 5x + 6 = 0, factor: (x + 3)(x + 2) = 0 Solve: x = -3, x = -2 -3 and -2 are the roots of the equation. Notice that the polynomial above has a degree of 2 and two roots. Assume you want to factor 3x2 + 10x – 8. 1. Multiply the quadratic term and the constant term. (3x2 -8 = -24x2) 2. Find the factors of product -24x2 that provide a sum of the linear term10x. (12x + (-2x) = 10x) 3. Replace the linear term in the original expression with the factors of -24x2 that provide a sum of 10x. (3x2 + 12x – 2x – 8) 4. Factor the expression by grouping: 3x( x 4) 2( x 4) ( x 4)(3x 2) This process will work with any quadratic expression! Factoring polynomials Always attempt to factor out what is common first. Here are some general guidelines of factoring based on the number of terms. Number of Terms Any number Technique Greatest Monomial Factor Two terms Difference of squares Difference of cubes Sum of cubes Three terms Perfect square trinomial Factoring a trinomial Four terms or more Factoring by grouping CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 33 of 144 Columbus City Schools 12/1/13 Suggestion: make a chart of this for your wall! x2 – 49 x2 –7x + 7x – 49 x(x – 7) + 7(x – 7) (x – 7)(x + 7) Factors of - 49x2 (- 7x)(7x) = - 49x2 Sum of 0x - 7x + 7x = 0x 4x2 + 16x + 15 4x2 + 6x + 10x + 15 2x(2x + 3) + 5(2x + 3) (2x + 3)(2x + 5) Factors of 60x2 (6x)(10x) = 60x2 Sum of 16x 6x + 10x = 16x 2x2 – 7x – 3 none of the combinations work not factorable Factors of – 6x2 (1x)(- 6x) = - 6x2 (- 1x)(6x) = - 6x2 (2x)(- 3x) = - 6x2 (- 2x)(3x) = - 6x2 Sum of – 7x 1x + (- 6x) = - 5x - 1x + 6x = 5x 2x + (- 3x) = - 1x - 2x + 3x = 1x Teacher Notes for A-CED 1 http://www.purplemath.com/modules/ineqquad.htm Written notes on solving quadratic inequalities can be found on this website. Misconceptions/Challenges: Students make mistakes when factoring quadratic expressions, because they fail to recognize the difference between when “a” is equal to one and when “a” is not equal to one. Students make mistakes with arithmetic when factoring. Instructional Strategies: A – CED 1: 1) Provide students with a copy of “More Area Applications” (included in this Curriculum Guide). Students will solve each problem by drawing a picture, writing an equation, and finding the solution both algebraically and graphically. A – REI 4b 1) Have students practice factoring to solve equations using the “Equation Cards” (included in this Curriculum Guide). Solving Factorable Quadratic Equations, http://www.regentsprep.org/Regents/math/ALGEBRA/AE5/indexAE5.htm CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 34 of 144 Columbus City Schools 12/1/13 2) This website provides instruction for solving quadratic equations by factoring. 3) Practice: http://www.regentsprep.org/Regents/math/ALGEBRA/AE5/PFacEq.htm This additional website has practice problems. 4) Quadratics: https://www.khanacademy.org/math/algebra/quadratics A series of links on solving quadratics through factoring, completing the square, graphing, and the quadratic equation are provided. 5) Solving a Quadratic Equation by Factoring: https://www.khanacademy.org/math/algebra/quadratics/factoring_quadratics/v/Example%20 1:%20Solving%20a%20quadratic%20equation%20by%20factoring A video tutorial is provided demonstrating how to solve a quadratic equation by use of factoring. 6) Solving a Quadratic Equation by Factoring: https://www.khanacademy.org/math/algebra/quadratics/factoring_quadratics/v/Example%20 2:%20Solving%20a%20quadratic%20equation%20by%20factoring A video tutorial is provided demonstrating how to solve a trinomial in the form ax2 + bx + c by use of factoring. 7) Solving a Quadratic Equation by Factoring: https://www.khanacademy.org/math/algebra/quadratics/factoring_quadratics/v/Example%20 3:%20Solving%20a%20quadratic%20equation%20by%20factoring A video tutorial is provided demonstrating how to problem solve using factoring. 8) Solving Quadratic Equations by Factoring: http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Solving%20Quadratic%20F actoring.pdf Students practice solving quadratic equations, written in different forms, by factoring. 9) Solving Quadratic Equations by Factoring: http://www.kutasoftware.com/FreeWorksheets/Alg2Worksheets/Quadratic%20Equations%2 0By%20Factoring.pdf Students solve quadratic equations using factoring with the practice problems found at this site. 10) Solving Quadratic Equations by Factoring: http://www.montereyinstitute.org/courses/Algebra1/U09L2T2_RESOURCE/index.html A warm up, video presentation, practice and review are provided as lessons on solving quadratic equations by factoring. 11) Solve an Equation using the Zero Product Property: http://www.ixl.com/math/algebra1/solve-an-equation-using-the-zero-product-property Students are provided problems to determine the solution of a quadratic equation by using the Zero Product Property. A tutorial is provided is the solutions offered are incorrect. CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 35 of 144 Columbus City Schools 12/1/13 12) Solve a Quadratic Equation by Factoringhttp://www.ixl.com/math/algebra-1/solve-aquadratic-equation-by-factoring: Students are provided problems to determine the solution of a quadratic equation by factoring and using the zero product property. A tutorial is provided is the solutions offered are incorrect. F –IF 8a 1) Students will complete the activity “Factor Me If You Can” (included in this Curriculum Guide) to connect solving by factoring and graphing. 2) Have students use the “Connecting Zeros, Roots, x-intercepts, and Solutions” worksheet (included int his Curriculum Guide), to see the relationship between solutions obtained by factoring and the x-intercepts or zeros of the quadratic function. The students should be able to solve any quadratic function with real solutions graphically. Students are to determine the minimum point, maximum point, roots and number of solutions of various functions with and without technology in “Families of Graphs # 2” (included in this Curriculum Guide). 3) Have the students use the “Solving Quadratics Graphically” activity (included in this Curriculum Guide) to reinforce the connection between zeros and solutions. 4) Have students use the “Solving by Factoring” worksheet (included in this Curriculum Guide) to practice solving quadratics by factoring. 5) Solve a Quadratic by Factoring: http://www.ixl.com/math/algebra-1/solve-a-quadraticequation-by-factoring This site offers a set of interactive practice problems and an explanation for an incorrect solution. 6) Factoring Trinomials Part 1: http://education.ti.com/en/us/activity/detail?id=E581F8E30F8A4C689F2A226A183FDC75 Students use technology to factor trinomials of the form x2 + bx + c, where b and c are positive integers and relate factoring a quadratic trinomial to an area model. 7) Factoring Trinomials Part 2: http://education.ti.com/en/us/activity/detail?id=1BEE8F88204147B6B8CD213556E97915 Students use technology to explore trinomials of the form x2 + bx + c, where b is negative and c is positive using an area model to factor trinomials in this form. 8) Exploring Polynomials: Factors, Roots, and Zeros: http://education.ti.com/en/us/activity/detail?id=384FB053735B4C86BBF76AA6E018891C Students use graphing technology to discover the zeros of the linear factors are the zeros of the polynomial function; connect the algebraic representation to the geometric representation; and see the effects of a double and/or triple root on the graph of a cubic function of the leading coefficient on a cubic function. CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 36 of 144 Columbus City Schools 12/1/13 9) Zeros of a Quadratic Function: http://education.ti.com/en/us/activity/detail?id=E9C63B78A29F47DFAA53DE57B74E212C Students merge graphical and algebraic representations of a quadratic function and its linear factors. 10) Factoring Trinomials (a = 1) (Easy): http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Factoring%201.pdf Students practice factoring trinomials in the form ax2 + bx + c and ax2 + bx – c. 11) Factoring Trinomials (a 1) (Hard): http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Factoring%202.pdf Students practice factoring trinomials in the form ax2 + bx + c and ax2 + bx – c. 12) Factoring Special Cases: http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Factoring%20Special%20C ases.pdf Students factor perfect square and difference of squares trinomials. 13) Factor by Grouping: http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Factoring%20By%20Group ing.pdf Students factor trinomials by grouping. 14) Factoring Trinomials: http://www.algebrahelp.com/lessons/factoring/trinomial/ This site has written explanations for factoring quadratics. 15) Factoring Quadratics: The Simple Case: http://www.purplemath.com/modules/factquad.htm Students factor quadratics that looks like ax2 + bx + c where a is 1. 16) Factoring Quadratics: The Hard Case: The Modified "a-b-c" Method, or "Box": http://www.purplemath.com/modules/factquad2.htm Students factor trinomials that looks like ax2 + bx + c where a is not 1. 17) Factoring Perfect Square Trinomials – Ex 1: http://patrickjmt.com/factoring-perfect-square-trinomials-ex1/ This site offers a video tutorial of a perfect square trinomial. 18) Factoring Perfect Square Trinomials – Ex 2: http://patrickjmt.com/factoring-perfect-square-trinomials-ex-2/ This site offers a second video tutorial of a perfect square trinomial. 19) Factoring Perfect Square Trinomials – Ex 3: http://patrickjmt.com/factoring-perfect-square-trinomials-ex3/ This site offers another video tutorial of a perfect square trinomial. CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 37 of 144 Columbus City Schools 12/1/13 20) Factoring Trinomials: Factor by Grouping – Ex 1: http://patrickjmt.com/factoring-trinomials-factor-by-grouping-ex-1/ This site offers a video tutorial for factoring a tutorial by grouping. 21) Factoring Trinomials: Factor by Grouping – Ex 2: http://patrickjmt.com/factoring-trinomials-factor-by-grouping-ex-2/ This site offers a second video tutorial for factoring a tutorial by grouping. 22) Factoring Trinomials: Factor by Grouping – Ex 3: http://patrickjmt.com/factoring-trinomials-factor-by-grouping-ex-3/ This site offers another video tutorial for factoring a tutorial by grouping. 23) Factoring Trinomials (A quadratic Trinomial) by Trial and Error: http://patrickjmt.com/factoring-trinomials-a-quadratic-trinomial-by-trial-and-error/ This site offers a video tutorial for factoring using the technique of trial and error. 24) Factoring Trinomials by Trial and Error – Ex 2: http://patrickjmt.com/factoring-trinomials-by-trial-and-error-ex-2/ This site offers another video tutorial for factoring using the technique of trial and error. 25) Solving Quadratic Equations by Factoring – Basic Examples: http://patrickjmt.com/solving-quadratic-equations-by-factoring-basic-examples/ This site offers a video tutorial for solving quadratic equations by factoring. 26) Solving Quadratic Equations by Factoring – Another Example: http://patrickjmt.com/solving-quadratic-equations-by-factoring-another-example/ This site offers another video tutorial for solving quadratic equations by factoring. 27) Factoring the Difference of Two Squares – Ex 1: http://patrickjmt.com/factoring-the-difference-of-two-squares-ex-1/ This site offers a video tutorial for factoring the difference of two squares. 28) Factoring the Difference of Two Squares – Ex 2: http://patrickjmt.com/factoring-the-difference-of-two-squares-ex-2-2/ This site offers a second video tutorial for factoring the difference of two squares. 29) Factoring the Difference of Two Squares – Ex 3: http://patrickjmt.com/factoring-the-difference-of-two-squares-ex-3-2/ This site offers a third video tutorial for factoring the difference of two squares. A –SSE 1b 1) Exploring Polynomials: Factors, Roots, and Zeros: http://education.ti.com/en/us/activity/detail?id=384FB053735B4C86BBF76AA6E018891 C Students will investigate graphical and algebraic representations of a polynomial function and its linear factors. They will determine the zeros of the polynomial function. CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 38 of 144 Columbus City Schools 12/1/13 2) Interpreting Algebraic Expressions: http://map.mathshell.org/materials/download.php?fileid=694 In this lesson students will translate between words, symbols, tables and area representations of algebraic expressions. 3) I Rule: http://www.uen.org/core/math/downloads/sec2_i_rule.pdf In this lesson students examine quadratic patterns in multiple representations. 4) I Rule: http://www.uen.org/core/math/downloads/sec2_i_rule_tn.pdf This site offers teacher notes for the lesson. 5) Look Out Below: http://www.uen.org/core/math/downloads/sec2_look_out_below.pdf In this lesson students examine quadratic functions on various sized intervals to determine average rates of change. 6) Look Out Below: http://www.uen.org/core/math/downloads/sec2_look_out_below_tn.pdf This site offers teacher notes for the lesson. 7) Something to Talk About: http://www.uen.org/core/math/downloads/sec2_something_to_talk_about.pdf In this lesson student are introduced to quadratic functions, designed to elicit representations and surface a new type of pattern and change. 8) Something to Talk About: http://www.uen.org/core/math/downloads/sec2_something_to_talk_about_tn.pdf Teacher notes are provided for this lesson. A – SSE 3a 1) Factoring Fanatic: http://alex.state.al.us/lesson_view.php?id=4152 In this lesson, students are provided practice for finding the correct factors for trinomial. They are provided with a Tic-Tac sheet to help them determine the pattern between the two numbers. 2) Math.A-SSE.3a: http://www.shmoop.com/common-core-standards/ccss-hs-a-sse-3a.html Written instructions for solving quadratic equations by factoring can be found at this site. 3) Learning Progression for CCSSM A-SSE 3a: http://www.google.com/url?sa=t&rct=j&q=&esrc=s&frm=1&source=web&cd=8&ved=0CF QQFjAH&url=http%3A%2F%2Foursland.edublogs.org%2Ffiles%2F2013%2F06%2FLearn CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 39 of 144 Columbus City Schools 12/1/13 ing-Progression-for-CCSSM-A-SSE.3a-HW11cs90f0.doc&ei=BnQCUrG9JpKCyAHe2YDYBw&usg=AFQjCNFpNq4CRjVE8YttcE0zih jOJfWwEQ&sig2=X168Hq2ME_eUazensjhcPw&bvm=bv.50310824,d.aWc This document provides instruction on solving quadratic equations by factoring and a problem concerning suspension bridges. 4) Challenging Factoring Quadratics: https://app.activateinstruction.org/playlist/resourcesview/id/5036aaa7efea65014c000022/rid/5021ad53efea65235f000a27/bc0/explore/bc1/playl ist A student practice sheet for solving by quadratics is provided at this site. A – APR 1 1) Polynomial Puzzler: http://illuminations.nctm.org/LessonDetail.aspx?id=L798 In this activity Students solve polynomials by solving a puzzle. Students will factor polynomials and multiply monomials and binomials. 2) Factoring Trinomials when (a = 1) http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Factoring%201.pdf At this website students practice factoring trinomials. 3) Factoring Trinomials (a ≠ 1) http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Factoring%202.pdf At this website students practice factoring trinomials. 4) Factoring Special Cases http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Factoring%20Special%20C ases.pdf At this website students practice factoring special cases. 5) Factoring by Grouping http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Factoring%20By%20Group ing.pdf At this website students practice factoring by grouping. 6) Factoring Quadratic Functions: https://www.khanacademy.org/math/algebra/quadratics/factoring_quadratics/v/factoringquadratic-expressions A video tutorial on factoring quadratic expressions can be found at the site below. 7) Factoring Simple Quadratics: https://www.khanacademy.org/math/algebra/quadratics/factoring_quadratics/v/factoringpolynomials-1 A video tutorial provides an example of factoring simple quadratic equations. CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 40 of 144 Columbus City Schools 12/1/13 8) Factoring Quadratic Expressions: https://www.khanacademy.org/math/algebra/quadratics/factoring_quadratics/v/factoringtrinomials-with-a-leading-1-coefficient A video tutorial provides an explanation on factoring a trinomial expression. 9) Factoring Polynomials 1: https://www.khanacademy.org/math/algebra/quadratics/factoring_quadratics/e/factoring_pol ynomials_1 This site provides interactive practice on factoring trinomials. If students need help, a tutorial is provided. 10) A Geometric Investigation of (a + b)2 http://illuminations.nctm.org/Activity.aspx?id=4089 This geometric demonstration show the value of the square of the binomial (a + b). 11) An easy way to find the common monomial factor of a polynomial is to write the prime factorization of each monomial and then identify the factors that are common to every monomial and factor it out. Arrange students into groups of three and give each group a polynomial that has three monomials. Each student takes one monomial and writes the prime factorization for it. The group then compares the monomials and selects any prime factors that are common to all three. The common factor will be the product of the selected factors. Give each group one card. When the group has finished working with a polynomial they can trade cards with another group. Have the groups continue trading cards until all groups have found the common factor for all of the polynomials. Use the “Polynomial Cards for Use with Prime Factorization” (included in this Curriculum Guide). 12) Use Algebra Tiles to model how to “Find the Greatest Common Monomial Factor Using Algebra Tiles” (included in this Curriculum Guide). Work through several problems on the overhead, while students work the same problems at their desks using the tiles (a recording sheet with problems is in this Curriculum Guide). Students should write, in algebraic form, what they are doing with the manipulatives to encourage making the connection between the concrete and abstract models. Have students count out Algebra Tiles to represent the polynomial to be factored. Students then arrange the tiles into a rectangle. Students should be led, if necessary, to arrange the tiles into the most compact rectangle possible, this will ensure that one of the factors is the greatest common factor. See the following example: To factor 2x2 + 6x, first count out 2 x2-tiles and 6 x-tiles. Then arrange them into as compact a rectangle as possible. Then, look at the width and length of the rectangle to find the factors. The width is 2x and the length is x + 3, therefore 2x2 + 6x can be rewritten as 2x(x + 3). This polynomial could also have been arranged into a rectangle with length and width of x and 2x + 6, however that would not have given the greatest common factor of 2x as one of the dimensions. CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 41 of 144 Columbus City Schools 12/1/13 13) Another method for factoring polynomials is factoring by grouping. Sometimes, polynomials can be factored by grouping terms. A polynomial may have a common factor that is a binomial. For example, 6x2 + 3x – 4x – 2 can be rewritten as 3x (2x + 1) – 2(2x + 1) where 2x + 1 is a common factor that is a binomial. We can use the Distributive Property to write 3x (2x + 1) – 2(2x + 1) as (2x + 1) (3x – 2). You can visualize this factoring procedure with a geometric model. The model shows the same total area using the two different arrangements. To make a model of 6x2 + 3x – 4x – 2, select Algebra Tiles and arrange them into a rectangle. Then look at the width and length of the rectangle to find the factors. These are the same factors found when factoring by grouping. 14) Use “Polynomial Cards for Factoring by Grouping” (included in this Curriculum Guide), so that students can work with a partner to factor and model the problems. Students should also find the simplified product for each polynomial (e.g., 6x2 + 3x – 4x – 2 = 6x2 – x – 2). This will enable students to take polynomials in a simplified form and rewrite them so that they can be factored using the grouping method. 15) Use “Drawkcab Problems” (included in this Curriculum Guide): Another method for factoring trinomials of the form ax2 + bx + c is to work the multiplication process backwards. This method incorporates the factoring by grouping method. To help students begin developing an understanding of the process involved for this method, give students a trinomial that can be factored. When using Algebra Tiles, students are required to add zero pairs to make the product rectangle. 16) Students will complete the activity “Discovering the Difference of Two Squares” (included in this Curriculum Guide) to discover the pattern for factoring a difference of two squares. 17) Have the students use “Factoring Using the Greatest Common Factor” activity (included in this Curriculum Guide). 18) Have the students use “Factoring by Grouping”, and the “Factoring Worksheet” activities (included in this Curriculum Guide). 19) Have students practice working backwards using Algebra Tiles to make zero pairs to make the product rectangle. Factor the following trinomial: x2 – 2x – 15. Step One – Place tiles that represent the trinomial on an Algebra Tile Mat. Place the unit tiles so that they form a rectangle. This will allow you to finish the larger rectangle using zero pairs of tiles Step Two – add zero pairs of Algebra Tiles (those tiles are outlined) so that they complete the rectangle. The sides of the rectangle are the factors for the trinomial. CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 42 of 144 Columbus City Schools 12/1/13 Step One Step Two The factors would be x + 3 and x – 5. Once students feel comfortable adding in the zero pairs, use the same trinomials to develop the steps for factoring by working backwards. Show students an example (see below) of multiplying a pair of factors that were found using the Algebra Tiles. (x + 3)(x – 5) x – 5x + 3x – 15 x2 – 2x – 15 2 Encourage students to notice the relationship between the coefficients of the x terms in the second two steps. Once students have realized that the coefficient of the x term in the product is found by adding or subtracting numbers that are a factor pair for the product of the coefficient of x2 and the constant term, they can be taught the following general steps for factoring a trinomial of the form ax2 + bx + c. To factor 2x2 – 5x – 12 (a trinomial of the form ax2 + bx + c) follow these steps: a. Find the product of (ax2) and (c) (ax2)(c) = (2x2) (-12) = -24x2 b. Find a pair of factors of (a) (c) (x2) that have the sum of bx Factor pairs Sum (-8x)(3x) = -24x2 -8x + 3x = -5x c. Rewrite the polynomial, expressing bx as the sum of a factor pair. 2x2 – 8x + 3x – 12 30) d. Use factoring by grouping to remove the GCF from the first two terms, and the GCF from the last two terms. Then use the distributive property to write as a product of two binomial factors. 2x2 – 8x + 3x – 12 = 2x(x – 4) + 3(x – 4) = (2x + 3) (x – 4) Reteach/Extension Reteach: 1) Solving Quadratic Equations: http://advancedCCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 43 of 144 Columbus City Schools 12/1/13 algebra.gcs.hs.schoolfusion.us/modules/locker/files/get_group_file.phtml?gid=4 45175&fid=1732539&sessionid= This is a re-teach practice sheet with an answer key provided on solving quadratic equations by graphing and factoring. 2) Solving Quadratic Equations: http://advancedalgebra.gcs.hs.schoolfusion.us/modules/locker/files/get_group_file.phtml?gid=445175&fid=1 732539&sessionid= This is a re-teach practice sheet with an answer key provided on solving quadratic equations by graphing and factoring. Extensions: 1) Performance Task: http://insidemathematics.org/common-core-math-tasks/high-school/HS-A2006%20Quadratic2006.pdf Students will find graphical properties of a quadratic function given by its formula and will have to factor for some problems. 2) Performance Task: http://insidemathematics.org/common-core-math-tasks/high-school/HS-F2007%20Graphs2007.pdf This problem involves working with linear and quadratic functions and their graphs and equations. Students will solve by factoring to justify their answer. Textbook References Textbook: Algebra I, Glencoe (2005): pp. 481-486, 487-488, 489-494, 495-500, 501506, 509-514, 840, 841 Supplemental: Algebra I, Glencoe (2005): Chapter 9 Resource Masters Reading to Learn Mathematics, pp. 533, 539, 545, 551, 557 Study Guide and Intervention, pp. 529-530, 535-536, 541-542, 547-548, 553-554 Skills Practice, pp. 531, 537, 543, 549, 555 Practice, pp. 532, 538, 544, 550, 556 Enrichment, pp. 534, 546, 558 Textbook: Algebra 2,Glencoe (2003): pp. 239-244, 270-275, 301-305, 837 Supplemental: Algebra 2,Glencoe (2003): Chapter 5 Resource Masters Reading to Learn Mathematics, pp. 261, 291 Study Guide and Intervention, pp. 257-258, 28-288 Skills Practice, pp. 259, 289 Practice, pp. 260, 290 Enrichment, pp. 262, 292 CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 44 of 144 Columbus City Schools 12/1/13 Chapter 6 Resource Masters Reading to Learn Mathematics, pp. 329 Study Guide and Intervention, pp. 325-326 Skills Practice, pp. 327 Practice, pp. 328 Textbook: Integrated Mathematics: Course 3, McDougal Littell (2002): pp. 45-52, 72-73, 645 Textbook: Advanced Mathematical Concepts, Glencoe (2004): pp. 141, 159-16, 169-170 Textbook: Mathematics II Common Core, Pearson, pp. 665-671, 672 – 678, 679 – 687, 688 – 694, 695 – 704, 738-740. CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 45 of 144 Columbus City Schools 12/1/13 A – CED 1 Name_______________________________________________ Date___________ Period_____ More Area Application For each problem below, draw a picture, write an equation, solve the problem algebraically, and support your work graphically. 1. Given a rectangle with an area of 45 cm2, find the dimensions of the rectangle if the length is 4 cm more than the width. 2. Given a triangle with an area of 16 in2, find the height of the triangle if it is twice the length of the base. 3. Given a circle with an area of 30 m2, find the radius and circumference of the circle. 4. Rectangle #1 has a length that is 5 less than twice a number and a width of 4 more than that number. Rectangle #2 has a length of 1 less than the number and the width is the number. Find the value of the number if the area of Rectangle #1 is equal to the area of Rectangle #2. CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 46 of 144 Columbus City Schools 12/1/13 A – CED 1 Name_______________________________________________ Date___________ Period_____ More Area Application Answer Key 1. Given a rectangle with an area of 45 cm2, find the dimensions of the rectangle if the length is 4 cm more than the width. w w+4 45 = w(w + 4) 45=w2 + 4w w2 + 4w – 45 = 0 (w + 9)(w – 5) = 0 w = (-9, 5) The width cannot be negative so w = 5. The length is w + 4 = 5 + 4 = 9 2. Given a right triangle with an area of 16 in2, find the height of the triangle if it is twice the length of the base. 16 = 12 b • 2b 16 = b2 4=b The base cannot be negative, so the height is 8. 2b b 3. Given a circle with an area of 30 m2, find the radius and circumference of the circle. 30 = r2 C=2 r 2 30/ = r C = 2 (3.09) r = 3.09 C = 19.4 4. Rectangle #1 has a length that is 5 less than twice a number and a width of 4 more than that number. Rectangle #2 has a length of 1 less than the number and the width is the number. Find the value of the number if the area of Rectangle #1 is equal to the area of Rectangle #2. Rectangle #2 Rectangle #1 x+4 x–1 2x – 5 (2x – 5)(x + 4) = x(x – 1) x 2x2 + 3x – 20 = x2 – x x2 + 4x – 20 = 0 x = 2.9, -6.9 The dimensions of a rectangle cannot be negative so the only reasonable answer is 2.9 CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 47 of 144 Columbus City Schools 12/1/13 A – REI 4b Name_______________________________________________ Date___________ Period_____ Equation Cards (to be solved by factoring) x2 – x – 20 = 0 x2 + 9x + 18 = 0 2x2 + 9x – 5 = 0 6x2 + 7x = 20 2x2 – 15x = 27 x2 = 7x – 12 12x2 – 2x – 4 = 0 x(4x + 1) = 5 x(15x + 1) – 2 = 0 x(125 – x) = 2500 CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 48 of 144 Columbus City Schools 12/1/13 A – REI 4b Name_______________________________________________ Date___________ Period_____ Equation Cards to be solved by factoring Answer Key x2 – x – 20 = 0 (x – 5)(x + 4) = 0 x = 5 and x = -4 x2 + 9x + 18 = 0 (x + 6)(x + 3) = 0 x = -6 and x = -3 2x2 + 9x – 5 = 0 6x2 + 7x = 20 2x2 – 15x = 27 x2 = 7x – 12 12x2 – 2x – 4 = 0 x(4x + 1) = 5 x(15x + 1) – 2 = 0 x(125 – x) = 2500 (x + 5)(2x – 1) = 0 x = -5 and x = 1/2 (2x + 3)(x – 9) = 0 x = -3/2 and x = 9 (4x + 2)(3x – 2) = 0 x = -1/2 and x = 2/3 (3x – 1)(5x + 2) = 0 x = 1/3 and x = -2/5 (3x – 4)(2x + 5) = 0 x = 4/3 and x = -5/2 (x – 4)(x – 3) = 0 x = 4 and x = 3 (x – 1)(4x + 5) = 0 x = 1 and x = -5/4 (x – 25)(x – 100) = 0 x = 25 and x = 100 CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 49 of 144 Columbus City Schools 12/1/13 F – IF 8a Name_______________________________________________ Date___________ Period_____ Factor Me If You Can Find the zeros of the following quadratic functions by factoring. 1. y = x² - 8x + 7 2. y = x² + 2x – 8 3. y = x² + 6x + 9 4. y = x² + 6x + 8 5. y = x² - 2x + 1 6. y = x² + 5x + 4 Use the graphing calculator to verify your answer. Sketch each equation on the grids provided below. Use trace to find the x-intercepts graphically. Find the zeros of the following functions by factoring if possible. CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 50 of 144 Columbus City Schools 12/1/13 7. y = x² - 7x – 8 8. y = x² + 3x + 5 9. y = x² + 6x – 7 10. y = x² + 3x + 6 11. y = x² + 5 12. y = x² + 4x Use the graphing calculator to verify your answer. Sketch the graphs of each of the functions. CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 51 of 144 Columbus City Schools 12/1/13 F – IF 8a Name_______________________________________________ Date___________ Period_____ Factor Me If You Can Answer Key Find the zeros of the following quadratic functions by factoring. 1. y = x² - 8x + 7 (x – 1)(x – 7) 2. y = x² + 2x – 8 (x + 4)(x – 2) 3. y = x² + 6x + 9 (x + 3)(x + 3) 4 y = x² + 6x + 8 (x + 4)(x + 2) 5. y = x² - 2x + 1 (x – 1)(x – 1) 6. y = x² + 5x + 4 (x + 4)(x + 1) Use the graphing calculator to verify your answer. Sketch each equation on the grids provided below. Use trace to find the x-intercepts graphically. Find the zeros of the following functions by factoring if possible. CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 52 of 144 Columbus City Schools 12/1/13 7. y = x² - 7x – 8 (x – 8)(x + 1) 8. y = x² + 3x + 5 Not factorable 9. y = x² + 6x – 7 (x + 7)(x – 1) 10. y = x² + 3x + 6 Not factorable 11. y = x² + 5 Not factorable 12. y = x² + 4x x (x + 4) Use the graphing calculator to verify your answer. Sketch the graphs of each of the functions. 7. 8. 9. 10 24 21 18 15 12 9 6 3 -25 -20 -15 -10 -5 -3 -6 -9 -12 -15 -18 -21 -24 6 4 2 5 10 15 20 25 -10 -8 -6 -4 -2 2 4 6 8 10 -20 -16 -12 -8 -4 -3 -6 -9 -12 -15 -18 -2 -4 -6 -8 -10 10. -10 -8 -6 18 15 12 9 6 3 8 11. 10 10 8 8 8 6 6 6 4 4 4 2 2 2 2 4 6 8 10 -10 -8 -6 8 12 16 20 2 4 6 12. 10 -4 -2 4 -4 -2 2 4 6 8 10 -10 -8 -6 -4 -2 -2 -2 -2 -4 -4 -4 -6 -6 -6 -8 -8 -8 -10 -10 -10 CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 53 of 144 8 10 Columbus City Schools 12/1/13 F – IF 8a Name_______________________________________________ Date___________ Period_____ Connecting Zeros, Roots, x-intercepts and Solutions A) Graph the following quadratic equations, identify the x-intercepts. B) Solve the quadratic equation by factoring. 1. A) y x2 3x 10 2. A) y - x2 7 x 6 x-intercept(s): x-intercept(s): B) x2 3x 10 0 B) - x2 7 x 6 0 CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 54 of 144 Columbus City Schools 12/1/13 3. A) y x2 4x 4 4. A) y 5x2 15x x-intercept(s): B) x2 4 x 4 0 x-intercept(s): B) 5x2 15x 0 What do you notice about the x-intercepts and the solutions you obtained by factoring? The terms zeros, roots, and solutions are used interchangeably when solving equations. The of the equation f x of the graph of a function are the 0 . These numbers are called the function. Solutions are also called of the . CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 55 of 144 Columbus City Schools 12/1/13 F – IF 8a Name_______________________________________________ Date___________ Period_____ Connecting Zeros, Roots, x-intercepts and Solutions Answer Key A) Graph the following quadratic equations, identify the x-intercepts. B) Solve the quadratic equation by factoring. 1. A) y x2 3x 10 2. A) y - x2 7 x 6 x y x y -2 -1 0 1 1.5 2 3 4 5 0 -6 - 10 - 12 - 12.25 - 12 - 10 -6 0 -7 -6 -5 -4 - 3.5 -3 -2 -1 0 -6 0 4 6 6.25 6 4 0 -6 x-intercept(s): (- 2, 0) and (5, 0) x-intercept(s): (- 6, 0) and (- 1, 0) B) x2 3x 10 0 (x – 5)(x + 2) = 0 x – 5 = 0 or x + 2 = 0 x = 5 or x = - 2 B) - x2 7 x 6 0 - (x2 + 7x + 6) = 0 - (x + 6)(x + 1) = 0 (x + 6) = 0 or x + 1 = 0 x = - 6 or x = - 1 CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 56 of 144 Columbus City Schools 12/1/13 3. A) y x2 4x 4 x y 0 1 2 3 4 4 1 0 1 4 x-intercept(s): B) x2 4 x 4 0 (x – 2)2 = 0 x–2=0 x=2 4. A) y 5x2 15x (2,0) x y 0 1 1.5 2 3 0 - 10 - 11.25 - 10 0 x-intercept(s): (0,0) and (3,0) B) 5x2 15x 0 5x(x – 3) = 0 5x = 0 or x – 3 = 0 x = 0 or x = 3 What do you notice about the x-intercepts and the solutions you obtained by factoring? The x-coordinates of the x-intercepts are the same as the solutions obtained when solving for x. When solving the equations, you are trying to determine which x-values will give you a yvalue of zero. All x-intercepts will have a y-coordinate of zero. Therefore, when solving an equation, the solutions correspond to the x-intercepts. The terms zeros, roots, and solutions are used interchangeably when solving equations. The x-intercepts of the graph of a function are the solutions of the equation f x 0 . These numbers are called the zeros of the function. Solutions are also called roots CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 57 of 144 Columbus City Schools 12/1/13 F – IF 8a Name_______________________________________________ Date___________ Period_____ Families of Graphs #2 1. Using the Families of Graphs activity that you previously completed, fill in the table below. Function Minimum Point Maximum Point Roots Number of Solutions 2 y=x +2 y = x2 – 2 y = (x - 2)2 y = (x - 2)2 + 2 y = (x + 2)2 y = (x + 2)2 – 2 y = -x2 y = -x2 + 2 y = -(x – 2)2 y = -x2 – 4x + 4 Without graphing, determine the vertex, roots, and number of solutions for the following functions. Show all work. 1. f(x) = x2 – 2x – 8 2. f(x) = 2x2 + 8x – 10 3. f(x) = -x2 + 6x – 6 CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 58 of 144 Columbus City Schools 12/1/13 F – IF 8a Name_______________________________________________ Date___________ Period_____ Families of Graphs #2 Answer Key 1. Using the Families of Graphs activity that you previously completed, fill in the table below. Function Minimum Point Maximum Point Roots Number of Solutions 2 y=x +2 (0, 2) None None No real Solutions 2 y=x -2 (0, -2) None 2 real solutions ±2 y = (x - 2)2 (2, 0) None 2 1 real solution y = (x - 2)2 + 2 (2, 2) None None y = (x + 2)2 (-2, 0) None -2 No real solutions 1 real solution y = (x + 2)2 - 2 (-2, -2) None -2 ± 2 2 real solutions y = -x2 None (0, 0) 0 1 real solution y = -x2 + 2 None (0, 2) y = -(x – 2)2 None (2, 0) 2 1 real solution y = -x2 – 4x + 4 None (2, -8) -2 ± 2 2 2 real solutions ±2 2 real solutions Without graphing, determine the vertex, roots, and number of solutions for the following functions. Show all work. 1. f(x) = x2 – 2x – 8 Vertex: (1, -9); Roots: x = 4 and x = -2; 2 solutions 2. f(x) = 2x2 + 8x – 10 Vertex: (-2, -18); Roots: x = 1 and x = -5; 2 solutions 3. f(x) = -x2 + 6x – 6 Vertex (3, 3); Roots: x = 3 3 ; 2 solutions CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 59 of 144 Columbus City Schools 12/1/13 F – IF 8a Name_______________________________________________ Date___________ Period_____ Solving Quadratics Graphically Sketch a graph of each quadratic equation; state the vertex, domain and range, x-intercepts (if they exist), and the y-intercept for each of the graphs. Solve each quadratic equation by factoring or the square root method. 1. y x2 2 x 8 Vertex: Range: y-intercept: 2. y -2x2 4x 2 Vertex: Range: y-intercept: Solve: x2 2 x 8 0 Domain: x-intercept(s): Solve: -2 x2 4 x 2 0 Domain: x-intercept(s): CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 60 of 144 Columbus City Schools 12/1/13 3. y 6 x 2 5x 4 Vertex: Range: y-intercept: 4. y - x2 4 Vertex: Range: y-intercept Solve: 6x2 5x 4 0 Domain: x-intercept(s): Solve: - x2 4 0 Domain: x-intercept(s): CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 61 of 144 Columbus City Schools 12/1/13 F – IF 8a Name_______________________________________________ Date___________ Period_____ Solving Quadratics Graphically Answer Key Sketch a graph of each quadratic equation; state the vertex, domain and range, x-intercepts (if they exist), and the y-intercept for each of the graphs. Solve each quadratic equation by factoring or the square root method. 1. y x -4 -3 -2 -1 0 1 2 x2 2 x 8 y 0 -5 -8 -9 -8 -5 0 Solve: x2 2 x 8 0 (x + 4)(x – 2) = 0 x + 4 = 0 or x – 2 = 0 x = - 4 or x = 2 - , Vertex: (- 1, - 9) Domain: Range: - 9, x-intercept(s): (- 4,0) and (2,0) y-intercept: (0,- 8) 2. y -2x2 4x 2 x y -3 -2 -1 0 1 -8 -2 0 -2 -8 Solve: -2 x2 4 x 2 0 - 2(x2 + 2x + 1) = 0 - 2(x + 1) 2 = 0 (x + 1) 2 = 0 x+1=0 x=-1 - , Vertex: (- 1,0) Domain: Range: - , 0 y-intercept: (0,- 2) x-intercept(s): (- 1,0) CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 62 of 144 Columbus City Schools 12/1/13 3. y 6 x 2 5x 4 x y -1 0 5 12 1 2 7 -4 1 -5 24 -3 10 Solve: 6x2 5x 4 0 (3x - 4)(2x + 1) = 0 3x – 4 = 0 or 2x + 1 = 0 4 1 x or x 3 2 Domain: - , Range: - 5.0417, 4 1 , 0 and - , 0 3 2 y-intercept: (0,- 4) x-intercept(s): Vertex: 4. y - x2 4 x y -2 -1 0 1 2 -8 -5 -4 -5 -8 Solve: - x2 4 0 - x2 = 4 x2 = - 4 x -4 no real solution Vertex: (0,- 4) Range: x 5 1 ,-5 or 0.4167, - 5.0417 12 24 Domain: -4 - , x-intercept(s): no x-intercepts y-intercept: (0,- 4) CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 63 of 144 Columbus City Schools 12/1/13 F – IF 8a Name_______________________________________________ Date___________ Period_____ Solving By Factoring Solve each of the following equations by factoring. 1. x2 13x 36 0 2. x2 2x 63 0 3. x2 2 x 8 0 4. x2 5x 24 5. x2 9 10 x 6. x 2 16 0 7. 4x2 25 0 8. 4x2 9. x3 4 x 0 10. 4x2 7 29 x 11. x3 12x2 32x 0 12. 12 x2 10 7 x 2 13. 4x 4 x 3 0 14. x 6 x 1 x 12 CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 64 of 144 Columbus City Schools 12/1/13 F – IF 8a Name_______________________________________________ Date___________ Period_____ Solving by Factoring Answer Key Solve each of the following equations by factoring. 1. x2 13x 36 0 2. x2 2x 63 0 (x – 9)(x – 4) = 0 (x + 9)(x – 7) = 0 x = 9 or x = 4 x = - 9 or x = 7 3. x2 2 x 8 0 (x + 4)(x – 2) = 0 x = - 4 or x = 2 4. x2 5x 24 x2 – 5x – 24 = 0 x = 8 or x = - 3 5. x2 9 10 x x2 – 10x + 9 = 0 (x – 9)(x – 1) = 0 x = 9 or x = 1 6. x 2 16 0 (x + 4)(x – 4) = 0 x = - 4 or x = 4 7. 4x2 25 0 (2x + 5)(2x – 5) = 0 8. 4x2 x 4x2 – x = 0 x(4x – 1) = 0 5 5 or x 2 2 3 9. x 4 x 0 x(x2 – 4) = 0 x(x + 2)(x – 2) = 0 x = 0 or x + 2 = 0 or x – 2 = 0 1 4 2 10. 4x 7 29 x 4x2 – 29x + 7=0 (4x – 1)(x – 7) = 0 4x – 1 = 0 or x – 7 = 0 1 or x = 7 x 4 12. 12 x2 10 7 x 12x2 – 7x – 10 = 0 (3x + 2)(4x – 5) = 0 3x + 2 = 0 or 4x – 5 = 0 2 5 or x x 3 4 x 6 x 1 12 14. x x = 0 or x = - 2 or x = 2 11. x3 12x2 32x 0 x(x2 – 12x + 32) = 0 x(x – 8)(x – 4) = 0 x = 0 or x – 8 = 0 or x – 4 = 0 x = 0 or x = 8 or x = 4 2 13. 4x 4 x 3 0 (2x – 1)(2x + 3) = 0 2x – 1 = 0 or 2x + 3 = 0 1 3 or x x 2 2 x = 0 or x 6x2 – x – 12 = 0 (3x + 4)(2x – 3) = 0 3x + 4 = 0 or 2x – 3 = 0 x 4 or x 3 3 2 CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 65 of 144 Columbus City Schools 12/1/13 A– APR 1 Name_______________________________________________ Date___________ Period_____ Polynomial Cards Use for Prime Factorization Teacher note: Label the back of each card with a number or letter to make switching cards between groups easier. 16x2y + 42xy2 – 20x2y2 24x3 + 32x2 – 48x 6y4 – 15y2 + 24y 4x2y + 12x2y2 + 20xy3 6x3 – 14x2 – 20x CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 66 of 144 Columbus City Schools 12/1/13 -3x5 + 15x3 + 6x2 3 2 8x – 6x – 18x 3x4 + 12x2 – 9x 3 2 12x + 8x + 20x 5 3 2 30y – 18y + 54y CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 67 of 144 Columbus City Schools 12/1/13 A – APR 1 Name_______________________________________________ Date___________ Period_____ Polynomial Cards Use for Prime Factorization Answer Key Polynomial: 16x2y + 42xy2 – 20x2y2 Prime Factorization: 16x2y factors to: 2 • 2 • 2 • 2 • x • x • y 42xy2 factors to: 2 • 3 • 7 • x • y • y -20x2y2 factors to: -1 • 2 • 2 • 5 • x • x • y • y GCF: 2xy Polynomial written as two factors: (2xy)(8x + 21y – 10xy) Polynomial: 24x3 + 32x2 – 48x Prime Factorization: 24x3 factors to: 2 • 2 • 2 • 3 • x • x • x 32x2 factors to: 2 • 2 • 2 • 2 • 2 • x • x -48x factors to: -1 • 2 • 2 • 2 • 2 • 3 • x GCF: 8x Polynomial written as two factors: (8x)(3x2 + 4x – 6) Polynomial: 6y4 – 15y2 + 24y Prime Factorization: 6y4 factors to: 2 • 3 • y • y • y • y -15y2 factors to: -1 • 3 • 5 • y • y 24y factors to: 2 • 2 • 2 • 3 • y GCF: 3y Polynomial written as two factors: (3y)(2y3 – 5y + 8) Polynomial: 4x2y + 12x2y2 + 20xy3 Prime Factorization: 4x2y factors to: 2 • 2 • x • x • y 12x2y2 factors to: 2 • 2 • 3 • x • x • y • y 20xy3 factors to: 2 • 2 • 5 • x • y • y • y GCF: 4xy Polynomial written as two factors: (4xy)(x + 3xy + 5y2) Polynomial: 6x3 – 14x2 – 20x Prime Factorization: 6x3 factors to: 2 • 3 • x • x • x -14x2 factors to: -1 • 2 • 7 • x • x -20x factors to: -1 • 2 • 2 • 5 • x GCF: 2x Polynomial written as two factors: (2x)(3x2 – 7x – 10) Polynomial: -3x5 + 15x3 + 6x2 Prime Factorization: -3x5 factors to: -1 • 3 • x • x • x • x • x 15x3 factors to: 3 • 5 • x • x • x CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 68 of 144 Columbus City Schools 12/1/13 6x2 factors to: 2 • 3 • x • x GCF: 3x2 Polynomial written as two factors: (3x2)(-x3 + 5x + 2) Polynomial: -8x3 – 6x2 – 18x Prime Factorization: -8x3 factors to: -1 • 2 • 2 • 2 • x • x • x -6x2 factors to: -1 • 2 • 3 • x • x -18x factors to: -1 • 2 • 3 • 3 • x GCF: -2x Polynomial written as two factors: (-2x)(4x2 + 3x + 9) Polynomial: 3x4 + 12x2 – 9x Prime Factorization: 3x4 factors to: 3 • x • x • x • x 12x2 factors to: 2 • 2 • 3 • x • x -9x factors to: -1 • 3 • 3 • x GCF: 3x Polynomial written as two factors: (3x)(x3 + 4x – 3) Polynomial: 12x3 + 8x2 + 20x Prime Factorization: 12x3 factors to: 2 • 2 • 3 • x • x • x 8x2 factors to: 2 • 2 • 2 • x • x 20x factors to: 2 • 2 • 5 • x GCF: 4x Polynomial written as two factors: (4x)(3x2 + 2x + 5) Polynomial: 30y5 – 18y3 + 54y2 Prime Factorization: 30y5 factors to: 2 • 3 • 5 • y • y • y • y • y -18y3 factors to: -1 • 2 • 3 • 3 • y • y • y 54y2 factors to: 2 • 3 • 3 • 3 • y • y 2 GCF: 6y Polynomial written as two factors: (6y2)(5y3 – 3y + 9) CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 69 of 144 Columbus City Schools 12/1/13 A – APR 1 Name_______________________________________________ Date___________ Period_____ Finding the Greatest Common Monomial Using Algebra Tiles First, count out Algebra Tiles to represent the polynomial. Second, arrange the tiles into a rectangle. Sketch the rectangle on this sheet. Third, look at the width and length of the rectangle. This represents the factors of the polynomial. 1. 3x2 + 6x = 2. 2x2 – 3x = CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 70 of 144 Columbus City Schools 12/1/13 3. 3x2 – 15x = 4. 4x2 + 6x = CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 71 of 144 Columbus City Schools 12/1/13 5. 3x + 6 = 6. 2x2 – x = CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 72 of 144 Columbus City Schools 12/1/13 7. 4x2 + 12x = 8. –2x2 + 4x = CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 73 of 144 Columbus City Schools 12/1/13 A – APR 1 Name_______________________________________________ Date___________ Period_____ Finding the Greatest Common Monomial Using Algebra Tiles Answer Key First, count out Algebra Tiles to represent the polynomial. Second, arrange the tiles into a rectangle. Sketch the rectangle on this sheet. Third, look at the width and length of the rectangle. This represents the factors of the polynomial. 1. 3x2 + 6x = 3x(x + 2) 2. 2x2 – 3x = x(2x – 3) CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 74 of 144 Columbus City Schools 12/1/13 3. 3x2 – 15x = 3x(x – 5) 4. 4x2 + 6x = 2x(2x + 3) 5. 3x + 6 = 3(x + 2) CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 75 of 144 Columbus City Schools 12/1/13 6. 2x2 – x = x(2x – 1) 7. 4x2 + 12x = 4x(x + 3) 8. –2x2 + 4x = -2x(x – 2) CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 76 of 144 Columbus City Schools 12/1/13 A – APR 1 Name_______________________________________________ Date___________ Period_____ Polynomial Cards Use for Factoring by Grouping Teacher note: Label the back of each card with a number or letter to make it easier for students to record their work. 2 x + 2x + 7x + 14 2 x – 9x + 4x – 36 x2 + 2x + 3x + 6 x2 – 3x + 6x – 18 CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 77 of 144 Columbus City Schools 12/1/13 6x2 – 3x + 4x – 2 2 3x – 6x – 4x + 8 2x2 – 6x + 4x – 12 x2 + 4x – 3x – 12 CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 78 of 144 Columbus City Schools 12/1/13 A – APR 1 Name_______________________________________________ Date___________ Period_____ Polynomial Cards Use for Factoring by Grouping Answer Key x2 + 2x + 7x + 14 (x2 + 2x) + (7x + 14) x(x + 2) + 7(x + 2) (x + 7)(x + 2) Simplified product: x2 + 9x + 14 Polynomial: Factors: x2 – 9x + 4x – 36 (x2 – 9x) + (4x – 36) x(x – 9) + 4(x – 9) (x + 4)(x – 9) Simplified product: x2 – 5x – 36 Polynomial: Factors: x2 + 2x + 3x + 6 (x2 + 2x) + (3x + 6) x(x + 2) + 3(x + 2) (x + 3)(x + 2) Simplified product: x2 + 5x + 6 Polynomial: Factors: x2 – 3x + 6x – 18 (x2 – 3x) + (6x – 18) x(x – 3) + 6(x – 3) (x + 6)(x – 3) Simplified product: x2 + 3x – 18 Polynomial: Factors: 6x2 – 3x + 4x – 2 (6x2 – 3x) + (4x – 2) 3x(2x – 1) + 2(2x – 1) (3x + 2)(2x – 1) Simplified product: 6x2 + 1x – 2 Polynomial: Factors: 3x2 – 6x – 4x + 8 (3x2 – 6x) + (-4x + 8) or could be written as (3x2 – 6x) – (4x – 8) 3x(x – 2) – 4(x – 2) (3x – 4)(x – 2) Simplified product: 3x2 – 10x + 8 Polynomial: 2x2 – 6x + 4x – 12 Factors: (2x2 – 6x) + (4x – 12) 2x(x – 3) + 4(x – 3) (2x + 4)(x – 3) Simplified product: 2x2 – 2x – 12 Polynomial: Factors: x2 + 4x – 3x – 12 x(x + 4) – 3(x + 4) (x – 3)(x + 4) Simplified product: x2 + x – 12 Polynomial: Factors: CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 79 of 144 Columbus City Schools 12/1/13 A – APR 1 Name_______________________________________________ Date___________ Period_____ Drawkcab Problems (Backward Problems) Factor each polynomial by working backwards. Use Algebra Tiles to make a model of the polynomial. 1. x2 + 2x – 8 = 2. x2 + x – 6 = CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 80 of 144 Columbus City Schools 12/1/13 3. x2 – x – 12 = 4. x2 + 2x – 15 = CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 81 of 144 Columbus City Schools 12/1/13 5. x2 – 3x – 10 = 6. 2x2 – 9x + 4 = CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 82 of 144 Columbus City Schools 12/1/13 7. 6x2 + 17x + 5 = 8. 3x2 + 10x – 8 = CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 83 of 144 Columbus City Schools 12/1/13 A – APR 1 Name_______________________________________________ Date___________ Period_____ Drawkcab Problems (Backward Problems) Answer Key Factor each polynomial by working backwards. Use Algebra Tiles to make a model of the polynomial. 1. x2 + 2x – 8 =(x + 4)(x – 2) Teacher note: Zero pairs of Algebra Tiles that were added to make a complete rectangle are outlined to make them more obvious. Students may need to be guided to add these zero pairs. 2. x2 + x – 6 = (x + 3)(x – 2) CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 84 of 144 Columbus City Schools 12/1/13 3. x2 – x – 12 = (x + 3)(x – 4) 4. x2 + 2x – 15 = (x + 5)(x – 3) CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 85 of 144 Columbus City Schools 12/1/13 5. x2 – 3x – 10 = (x – 5)(x + 2) 6. 2x2 – 9x + 4 = (2x – 1)(x – 4) CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 86 of 144 Columbus City Schools 12/1/13 7. 6x2 + 17x + 5 = (3x + 1)(2x + 5) 8. 3x2 + 10x – 8 = (3x – 2)(x + 4) CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 87 of 144 Columbus City Schools 12/1/13 A – APR 1 Name_______________________________________________ Date___________ Period_____ Discovering the Difference of Two Squares 1. Draw a square, using a ruler to measure each side, and label each side as “a”. a a 2. Draw a smaller square inside the upper left corner of your current square, and label each side as “b”. b a b a 3. Shade the original square, leaving out the new square. b a b a 4. Find an expression for the area of the shaded region, in terms of “a” and “b”. CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 88 of 144 Columbus City Schools 12/1/13 5. Cut the non-shaded area off, and determine an expression for each side of the remaining figure. b b a a a a 6. Cut the shaded area along the dotted line to make two separate rectangles, and then place them together to form one rectangle, labeling each side of the new rectangle in terms of “a” and “b”. a a 7. Determine another expression for the area of the shaded region, using the new rectangle. The shaded region has not been changed, just re-arranged, therefore the two expressions must be equal. CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 89 of 144 Columbus City Schools 12/1/13 A- – APR 1 Name_______________________________________________ Date___________ Period_____ Discovering the Difference of Two Squares Answer Key 1. Draw a square, using a ruler to measure each side, and label each side as “a”. a a 2. Draw a smaller square inside the upper left corner of your current square, and label each side as “b”. b a b a 3. Shade the original square, leaving out the new square. b a b a 4. Find an expression for the area of the shaded region, in terms of “a” and “b”. A = a2 – b2 CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 90 of 144 Columbus City Schools 12/1/13 5. Cut the non-shaded area off, and determine an expression for each side of the remaining figure, in terms of a and b. a-b b b a b a a-b a 6. b a Cut the shaded area along the dotted line to make two separate rectangles, and then place them together to form one rectangle, labeling each side of the new rectangle in terms of “a” and “b”. a-b b b a-b a a-b a b a 7. Determine another expression for the area of the shaded region, using the new rectangle. A = (a – b)(a + b) The shaded region has not been changed, just re-arranged, therefore the two expressions must be equal. CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 91 of 144 Columbus City Schools 12/1/13 A – APR 1 Name_______________________________________________ Date___________ Period_____ Factoring Using the Greatest Common Factor Factor each polynomial as the product of its greatest common factor and another polynomial. 1. 6 x 12 2. 14 x 12 4. 4 x 8 y 12 5. 14s 2 21st 7. 8x3 16 x2 8. 15x2 9 x 3. 9 x2 6 x 12 6. 10x3 5x2 15x 9. r 2h 2 r 2 CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 92 of 144 Columbus City Schools 12/1/13 A – APR 1 Name_______________________________________________ Date___________ Period_____ Factoring Using the Greatest Common Factor Answer Key Factor each polynomial as the product of its greatest common factor and another polynomial. 1. 6 x 12 2. 14 x 12 6(x + 2) 2(7x – 6) 4. 4 x 8 y 12 5. 14s 2 21st 4(x + 2y – 3) 7. 8x3 16 x2 8x2(x + 2) 3. 9 x2 6 x 12 3(3x2 + 2x – 4) 6. 10x3 5x2 15x 5x(2x2 – x + 3) 7s(2s + 3t) 8. 15x2 9 x 3x(5x – 3) 9. r 2h 2 r 2 r 2 (h 2) CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 93 of 144 Columbus City Schools 12/1/13 A – APR 1 Name_______________________________________________ Date___________ Period_____ Factoring By Grouping Factor. Check by multiplying the factors. 1. 3 x y x x y 2. 2 x x 4 3. 5 x x 3 x 3 4. 5 x 4 5. 6 x 2 x 1 5 2x 1 6. 4 x 2 3 x pr 8. 6 x 3 y 2 xz yz 7. pq 2qr 2r 2 7 x 4 x 5x 4 7 3x 2 9. ab 2b ac 2c 10. x3 2 x2 3x 6 11. 2 x3 12. 2x3 6x2 5x 15 x2 8x 4 CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 94 of 144 Columbus City Schools 12/1/13 A – APR 1 Name_______________________________________________ Date___________ Period_____ Factoring By Grouping Answer Key Factor. Check by multiplying the factors. 1. 3 x y x x y (x + y)(3 + x) 2. 2 x x 4 7 x 4 (x – 4)(2x + 7) x 3 3. 5 x x 3 (x + 3)(5x – 1) 4. 5 x 4 x 5 x 4 (5x + 4)(1 – x) 5. 6 x 2 x 1 5 2 x 1 6x(2x – 1) – 5(2x – 1) (2x – 1)(6x – 5) or (- 2x + 1)(- 6x + 5) 6. 4 x 2 3 x 7 3 x 2 4x(2 – 3x) + 7(- 3x + 2) (2 – 3x)(4x + 7) or (3x – 2)(- 4x – 7) 7. pq 2qr 2r 2 pr q(p + 2r) + r(2r + p) (p + 2r)(q + r) 8. 6 x 3 y 2 xz yz 3(2x – y) + z(2x – y) (2x – y)(3 + z) 9. ab 2b ac 2c b(a – 2) + c(a – 2) (a – 2)(b + c) 10. x3 2 x2 3x 6 x2(x – 2) + 3(x – 2) (x – 2)(x2 + 3) 11. 2 x3 x2 8x 4 x2 (2x + 1) + 4(2x + 1) (2x + 1)(x2 + 4) 12. 2x3 6x2 5x 15 2x2 (x – 3) – 5(x – 3) (x – 3)(2x2 – 5) CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 95 of 144 Columbus City Schools 12/1/13 A – APR 1 Name_______________________________________________ Date___________ Period_____ Factoring Worksheet Completely factor the following polynomials. Rewrite the problem, show all of your work and the answer on a separate piece of paper. 1. 2 x2 5x 3 2. 7 x2 8x 1 3. x 2 4x 5 4. x2 2 x 35 5. x2 12 x 24 6. 49 x2 81 7. 49 14x x2 8. 81x4 16 9. 2x2 4x 2 10. 4 x 2 11. x2 3x 54 12. x2 15x 44 13. 64x2 16xy y 2 14. 64 121c4 x 3 CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 96 of 144 Columbus City Schools 12/1/13 15. x2 13x 42 16. x2 14 x 51 17. x2 20 x 51 18. x2 3x 40 19. 7 x2 18x 8 20. 10 x 2 x2 21. 6 23x 4 x2 22. 9x2 25x 6 23. 4 x2 12 x 9 24. x4 25. 7 x2 19 x 6 26. 36 x2 5x 24 27. 144 x2 169 28. 20 x2 27 x 8 29. 12 x2 7 x 10 30. 3x2 7 x 6 x2 56 CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 97 of 144 Columbus City Schools 12/1/13 A – APR 1 Name_______________________________________________ Date___________ Period_____ Factoring Worksheet Answer Key Completely factor the following polynomials. Rewrite the problem; show all of your work and the answer on a separate piece of paper. 1. 2 x2 5x 3 2. 7 x2 8x 1 (2x + 3)(x + 1) (7x – 1)(x – 1) 3. x 2 4 x 5 (x + 5)(x – 1) 4. x2 2 x 35 (x + 7)(x – 5) 5. x2 12 x 24 prime 6. 49 x2 81 (7x + 9)(7x – 9) 7. 49 14x x2 (7 – x)2 8. 81x4 16 (9x2 + 4)(9x2 – 4) (9x2 + 4)(3x + 2)(3x – 2) 9. 2x2 4x 2 2(x2 – 2x + 1) 2(x – 1) 2 10. 4 x 2 x 3 (4x + 3)(x – 1) 11. x2 3x 54 (x + 9)(x – 6) 12. x2 15x 44 (x + 11)(x + 4) 13. 64x2 16xy y 2 (8x – y) 2 14. 64 121c4 (8 + 11c2)(8 – 11c2) 15. x2 13x 42 (x – 6)(x – 7) 16. x2 14 x 51 (x + 17)(x – 3) 17. x2 20 x 51 (x + 17)(x + 3) 18. x2 3x 40 (x – 8)(x + 5) CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 98 of 144 Columbus City Schools 12/1/13 19. 7 x2 18x 8 (7x – 4)(x – 2) 20. 10 x 2 x2 (5 – 2x)(2 + x) 21. 6 23x 4 x2 (6 + x)(1 – 4x) 22. 9x2 25x 6 (9x + 2)(x – 3) 23. 4 x2 12 x 9 (2x + 3) 2 24. x4 x2 56 (x2 – 8)(x2 + 7) 25. 7 x2 19 x 6 (7x – 2)(x + 3) 26. 36 x2 5x 24 (9x – 8)(4x + 3) 27. 144 x2 169 (12x + 13)(12x – 13) 28. 20 x2 27 x 8 (5x + 8)(4x – 1) 29. 12 x2 7 x 10 (4x – 5)(3x + 2) 30. 3x2 7 x 6 (3x + 2)(x – 3) CCSSM II Quadratic Functions: Solving by Factoring A –CED 1, A – REI 4b, F – IF 8a, A – SSE 1b, 3a, A – APR 1 Quarter 3 Page 99 of 144 Columbus City Schools 12/1/13 COLUMBUS CITY SCHOOLS HIGH SCHOOL CCSS MATHEMATICS II CURRICULUM GUIDE Topic 6 CONCEPTUAL CATEGORY TIME GRADING Quadratic Functions: Completing Algebra, Functions, Number RANGE PERIOD 20 days the Square and the Quadratic and Quantity 3 Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Domain: Reasoning with Equations and Inequalities (A – REI): Cluster 30) Understand solving equations as a process of reasoning and explain the reasoning. 31) Solve equations and inequalities in one variable. Domain: Seeing Structure in Expressions (A – SSE): Cluster 32) Write expressions in equivalent forma to solve problems. Domain: Interpreting Functions (F – IF): Cluster 33) Analyze functions using different representations. Domain: Creating Equations (A – CED): Cluster 34) Create equations that describe numbers or relationships. Domain: The Complex Number System (N – CN): Cluster 35) Perform arithmetic operations with complex numbers. 36) Use complex numbers in polynomial identities and equations. Standards 30) Understand solving equations as a process of reasoning and explain the reasoning. A – REI 1: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. 31) Solve equations and Inequalities in one variable. A – REI 4: Solve quadratic equations in one variable. A – REI 4a: Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 100 of 144 Columbus City Schools 12/1/13 A – REI 4b: Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. 32) Write expressions in equivalent forms to solve problems. A – SSE 3b: Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. 33) Analyze functions using different representations. F – IF 8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. F – IF 8a: Use the process of factoring and completing the square in a quadratic function to show zero, extreme values, and symmetry of the graph, and interpret these in terms of a context. 34) Create equations that describe numbers or relationships A – CED 1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. 35) Perform arithmetic operations with complex numbers. N – CN 1: Know there is a complex number i such that i 2 has the form a+bi with a and b real. 1 , and every complex number 1 and the commutative, associative, and distributive N – CN 2: Use the relation i 2 properties to add, subtract, and multiply complex numbers. 36) Use complex numbers in polynomial identities and equations. N – CN 7: Solve quadratic equations with real coefficients that have complex solutions. CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 101 of 144 Columbus City Schools 12/1/13 TEACHING TOOLS Vocabulary: binomial, coefficient, completing the square, complex conjugates, complex number, complex roots, constant term, degree, discriminant, extraneous solution, function, imaginary number, imaginary part, imaginary unit, leading coefficient, polynomial, principal root, pure imaginary number, quadratic equation, quadratic, quadratic equation, Quadratic Formula, quadratic function, quadratic inequality, quadratic term, real number, real part, roots, solutions, Square Root Property, square roots, standard form, trinomial, zeros Teacher Notes: Note: In the previous topic students solved by factoring, and taking square roots. In this topic, students will solve by graphing and the use of the Quadratic Formula. The Algebra 2 textbook covers solving quadratic equations by graphing, factoring, completing the square, and the quadratic formula. You may wish to review multiplying binomials and factoring quadratics. Quadratic Formula: Another algebraic method of finding the roots of a quadratic equation is the quadratic formula: ±√ 𝑥= and 𝑎 ≠ 0. , where a, b and c represent the same values as the a, b and c in y = ax2 + bx + c Example: Solve 2x2 6 x 3 0 by using the quadratic formula. x x 6 62 4 2 3 2 2 6 2 3 4 3 6 3 2 or x 36 24 4 6 2 3 4 6 a = 2, b = 6, c = 3: 12 4 3 3 2 The nature of the solutions of a quadratic function can be determined by examining the value of the discriminant; b2 – 4ac. Discriminant b2 – 4ac > 0 b2 – 4ac = 0 b2 – 4ac < 0 Number of Roots 2 real roots 1 real root with multiplicity 0 real roots. The roots are imaginary numbers. The section on solving quadratics by using the quadratic formula includes both real and complex solutions. You may wish to select problems with real solutions first. The complex number system is discussed in chapter 5, section 9 of the Algebra 2 text. The mode on the TI-84 can be changed to work with imaginary numbers. To change the mode from Real to Complex, press the MODE key, move your cursor down to Real, press the right arrow key one time, then press ENTER. Upon completion of the Complex Number section, you will need to resume work using the quadratic formula. CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 102 of 144 Columbus City Schools 12/1/13 Complex Numbers Complex numbers take the standard form a + bi, where a and b are real numbers and i, the 1 . In a complex number, a is the real part and bi is the complex imaginary part is equal to part, thus real numbers are actually complex numbers with b = 0. Powers of i The values of the powers of i repeat in a regular pattern every 4th power. i 1= i 1 i2 = i i = 1 i3 = i2 i = 1 i = i i4 = i3 i = i i= 1 1 2 1 1 1 2 ( 1) 1 To find the value of a power of i greater than 4, divide the exponent by 4 and examine the remainder. If the remainder is: 1234- the power of i has the same value as i1 which is equal to i. the power of i has the same value as i2 which is equal to 1. the power of i has the same value as i3 which is equal to i. the power of i has the same value as i4 which is equal to 1. Example: Find the value of i22. 22 4 5 , remainder 2, therefore i22 = i2 = 1. Example with the quadratic formula: To find the nonreal zeros of the function f(x) = x2 + x + 1, set x2 + x + 1 = 0, and apply the quadratic formula. x -1 1 4(1)(1) -1 -3 2 2 -1 i 3 -1 3 i 2 2 2 Operations on Complex Numbers When adding complex numbers, add the real parts and add the imaginary parts as shown in example 1 and example 2 below. Example 1: (a + bi) + (c + di) = (a + c) + (b + d)i Example 2: (3 + 4i) + (6 + 2i) = 9 + 6i To multiply complex numbers, use the same method as you would use when multiplying two binomials. CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 103 of 144 Columbus City Schools 12/1/13 Example 1: (a + bi)(c + di) = ac + adi + bci + bdi2 = ac + (ad + bc)i + -(bd), since i = -1 Example 2: (2 + 3i)(4 + 5i) = 8 + 10i +12i + 15i2 = 8 + 22i + -15 = -7 + 22i Complex Conjugates The conjugate of a complex number is that number by which you can multiply a complex number to obtain a real number. The conjugate of a + bi is a – bi. Just as in the case of a difference of squares, multiplying a complex number by its conjugate causes the middle term containing the complex part to drop out leaving the real parts. (a + bi)(a – bi) = a2 – abi + abi – (bi)2 = a2 + b2 When the denominator of a fraction is a complex number, multiplying the numerator and denominator by the complex conjugate, will rationalize the denominator as shown below. 3 2i 5 6i (3 2i)(5 6i) (5 6i)(5 6i) 15 18i 10i 12i 2 25 36 27 8i 61 Completing the Square When a quadratic with real roots doesn’t appear to be factorable, it can be forced to factoring by completing the square. Completing the square converts the left hand side of the equation into a perfect square trinomial. After factoring, the solution to the quadratic can be found by taking the square root of each side of the equation and solving for the variable as shown in the general case and the example below. To complete the square of the quadratic subtract c from each side x2 + bx + c = 0, x2 + bx = -c, 2 add b to each side of the equation 2 factor the left hand side take the square root of each side of the equation solving for x x 2 b 2 bx b 2 x x b x 2 2 c 2 c b 2 b 2 b 2 , 2 , 2 b 2 c b c+ 2 2 2 ’ 2 , CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 104 of 144 Columbus City Schools 12/1/13 b 2 x b c+ 2 2 . Example: Find the roots of the quadratic x2 + 4x – 9 = 0 by completing the square. subtract - 9 from each side of the equation x2 + 4x = 9, 2 add 4 to each side of the equation 2 x2 4 x factor the left hand side and simplify the right hand side x 2 x 2 x 2 take the square root of each side solve for x x 2 13 , and , x 2 2 4 2 2 9 4 2 2 , 13 , 13 , 13 , 13 Please note that if a is not equal to one, it must be factored out of the equation prior to making the left hand side into a perfect square trinomial. This value must be taken into consideration when adding it to both sides of the equation. (See the example below.) Find the roots of the quadratic 2x2 + 4x – 9 = 0 by completing the square. subtract - 9 from each side of the equation 2x2 + 4x = 9, factor the 2 out of the left hand side of the equation 2(x2 + 2x) = 9 2 add 2 2 to each side of the equation 2 2(x + 2x) + 2 2 2 2 =9+ 2 factor the left hand side and simplify the right hand side divide both sides by the factor take the square root of each side 2(x + 1)2 = 10, solve for x x=-1 5, x = - 1 + 5 , and x = - 1 - 2 , (x + 1)2 = 5 x+1= 5, 5 Misconceptions/Challenges: Students make mistakes when evaluating the quadratic formula, because they do not understand the difference between ( b) 2 and b 2 . CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 105 of 144 Columbus City Schools 12/1/13 Students use the incorrect values for a, b, and c in the quadratic formula, because they do not put the equations in standard form first. Students do not understand how to complete the square for a perfect square trinomial. Students make mistakes with rational numbers. Students incorrectly multiply polynomials; they believe they can just distribute the exponent through the binomial, or when multiplying two different polynomials they forget to multiply the inside terms. Students make mistakes when finding the conjugate of a complex number; they often multiply by the same binomial instead, but still cancel out the middle terms, therefore they get the wrong sign on the last term. Students do not recognize the pattern with imaginary numbers. Instructional Strategies: A – REI 1 1) Solving quadratic functions http://www.shmoop.com/common-core-standards/ccss-hs-a-rei-1.html This site provides written explanations and a practice sheet for solving quadratic functions. A – REI 4 1) Throwing an Interception: http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod3_quadequ_tn _062213.pdf In this lesson students develop the quadratic formula to determine the x-intercepts of the function. A – REI 4a 1) Completing the Square http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Factoring%20By%20Groupi ng.pdf This website provides practice for completing the square. 2) Students will complete the “Sorting Activity” (included in this Curriculum Guide). In this activity students will look at the different ways a quadratic equation can be represented (e.g. vertex form, trinomial, factored, vertex form, as a parabola). This activity can be used as a precursor to the “Completing the Square” activity. It is important at the conclusion of the activity to emphasize that completing the square makes it easier to: (i) determine the minimum value of a quadratic with a positive leading term; (ii) find the roots; and (iii) draw the graph. 3) Have students complete the activity “Learning How to Complete the Square “Completely”” (included in this Curriculum Guide). The activity is designed to emphasize the two primary benefits for using the technique of “Completing the Square” to simplify/solve a quadratic equation. In particular, it will allow students to (i) locate the CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 106 of 144 Columbus City Schools 12/1/13 minimum point of a quadratic curve y = x2 + bx + c, and (ii) put a quadratic polynomial into vertex form which will aid students in finding the roots. 4) Have the students use the “Transformations and Completing the Square Notes” and “Completing the Square and Transformations Practice” worksheets (included in this Curriculum Guide). The students will be able to graph any quadratic using transformations. The students will also understand how the vertex of a parabola relates to the vertex form of a quadratic equation. 5) Students should complete “Discovery of Completing the Square” (included in this Curriculum Guide) to be able to convert standard form quadratic equations into vertex form. Some teacher instruction may be required in addition to this activity. 6) Give students manipulatives (i. e., Algebra Tiles or Algeblocks) to multiply, and factor quadratic equations. Instruct students to solve quadratic equations by completing the square using manipulatives. Use the resource: Virtual Manipulatives (NLVM: Algebra Tiles: http://nlvm.usu.edu/en/nav/grade_g_4.html 7) Proof Without Words: http://illuminations.nctm.org/ActivityDetail.aspx?ID=132 At this site you can find an interactive geometric proof x2 + ax = (x + a/2)2 – (a/2)2. 8) Transform a Quadratic Equation by Completing the Square: http://learnzillion.com/lessons/1240-transform-a-quadratic-equation-by-completing-thesquare In this lesson students will learn how to transform a quadratic equation by completing the square. 9) Transform a Quadratic Equation by Completing the Square, a=1: http://learnzillion.com/lessons/1239-transform-a-quadratic-equation-by-completing-thesquare-a1 In this lesson students will learn how to transform a quadratic equation by completing the square. 10) Derive the Quadratic Formula: Completing the Square: http://learnzillion.com/lessons/268-derive-the-quadratic-formula-completing-the-square In this lesson students will learn how to derive the quadratic formula by completing the square. 11) Solve a Quadratic Equation: Completing the Square (1): http://learnzillion.com/lessons/265-solve-a-quadratic-equation-completing-the-square-1 This is 1 of 2 lessons in which students will learn how to solve a quadratic equation by completing the square. This lesson teaches you how to complete the square with a leading coefficient of 1. CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 107 of 144 Columbus City Schools 12/1/13 12) Solve a Quadratic Equation: Completing the Square (2): http://learnzillion.com/lessons/266-solve-a-quadratic-equation-completing-the-square-2 This is 1 of 2 lessons in which students will learn how to solve a quadratic equation by completing the square. This lesson teaches you how to complete the square with a leading coefficient other than 1. 13) Completing the Square: http://www.ixl.com/math/algebra-1/complete-the-square Students are provided interactive problems to fill in the number that makes the polynomial a perfect-square quadratic. A tutorial is provided is the solutions offered are incorrect. 14) “Factoring by Mack”: http://alex.state.al.us/lesson_view.php?id=24082 In this lesson students will learn a strategy to factor trinomials. 15) Completing the Square: http://education.ti.com/en/us/activity/detail?id=0DB3F0D2FA0D4F028119DB20332F99CE In this activity students complete the square in an algebraic expression. Students will use algebra tiles to build a geometric model of a perfect square quadratic. 16) Completing the Square Algebraically: http://education.ti.com/en/us/activity/detail?id=F38582092FBD46FCB8F3DCEBBBA3D496 In this Nspire lesson students will complete the square algebraically to rewrite a quadratic expression. 17) Quadratic Formula: How to Derive: http://patrickjmt.com/deriving-the-quadratic-formula/ This site offers a tutorial on deriving the quadratic formula. A – REI 4b 1) Curbside Rivalry: http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod3_quadfun_0 62213.pdf In this lesson (pp. 39-44), students examine how different forms of a quadratic equation can facilitate the solving of the equations. 2) Perfecting My Quads: http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod3_quadfun_0 62213.pdf Students building fluency with solving quadratic equations in this lesson (pp. 45-52) 3) To Be Determined: http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod3_quadfun_0 62213.pdf Students focus on the discriminant and the roots that are complex in this lesson (pp. 53-59) CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 108 of 144 Columbus City Schools 12/1/13 4) My Irrational and Imaginary Friends: http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod3_quadfun_0 62213.pdf Students work with arithmetic with imaginary numbers and complex numbers in this lesson (pp. 60-66). 5) iNumbers: http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod3_quadfun_0 62213.pdf Students practice working with arithmetic of complex numbers and pure imaginary numbers (pp. 67 -74). 6) Quadratics: https://www.khanacademy.org/math/algebra/quadratics A series of links on solving quadratics through factoring, completing the square, graphing, and the quadratic equation are provided. 7) Quadratic Functions: http://www.mcclenahan.info/sfhs/Algebra2/LectureNotes/76_Quadratic_Functions.pdf At this site there is a lesson on determining the intercepts and minimum and maximum points. 8) Completing the Square (easy): http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Complete%20the%20Squar e.pdf Students complete the square to determine the value of “c” in a trinomial expression. 9) Completing the Square (harder): http://www.kutasoftware.com/FreeWorksheets/Alg2Worksheets/Completing%20the%20Squ are.pdf Students determine the value of “c” in a trinomial by completing the square. 10) Solving Quadratic Equations with Square Roots (Easy): http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Solving%20Quadratic%20R oots.pdf Practice problems, on solving equations that contain square roots, can be found at this website. 11) Quadratic Equations with Square Roots (Hard): http://www.kutasoftware.com/FreeWorksheets/Alg2Worksheets/Quadratic%20Equations%2 0Square%20Roots.pdf Students practice solving quadratic equations with square roots with real and complex solutions. CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 109 of 144 Columbus City Schools 12/1/13 12) Solving Equations by Completing the Square (Hard): http://www.kutasoftware.com/FreeWorksheets/Alg2Worksheets/Quadratic%20Equations%2 0By%20Completing%20the%20Square.pdf Students will solve equations using completing the square. 13) Using the Quadratic Formula (Easy): http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Quadratic%20Formula.pdf Students will solve equations using the quadratic formula for problems with real number solutions. 14) Using the Quadratic Formula (Harder): http://www.kutasoftware.com/FreeWorksheets/Alg2Worksheets/Quadratic%20Formula.pdf Students determine real and complex solutions to quadratic functions by using the quadratic formula. 15) Solving Equations by Completing the Square (Easy): http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Solving%20Completing%20 Square.pdf Students will solve equations using completing the square. 16) Discriminant: http://www.kutasoftware.com/FreeWorksheets/Alg2Worksheets/The%20Discriminant.pdf Students determine the number of real and imaginary solutions by determining the value of the discriminant. 17) Solving Quadratic Equations using the Quadratic Formula: http://www.montereyinstitute.org/courses/Algebra1/U10L1T3_RESOURCE/index.html A warm up, video presentation, practice and review are provided as a lesson on solving quadratic equations using the quadratic formula. 18) Solving Quadratic Equations by Completing the Square: http://www.montereyinstitute.org/courses/Algebra1/U10L1T2_RESOURCE/index.html A warm up, video presentation, practice and review are provided as a lesson on solving quadratic equations by completing the square. 19) Solving Quadratic Equations: Cutting Corners: http://map.mathshell.org.uk/materials/lessons.php?taskid=432 Students will solve quadratics in one variable by solving quadratic equations by taking square roots, completing the square, using the quadratic formula, and factoring. 20) Quadratic Formula: http://patrickjmt.com/using-the-quadratic-formula/ This site offers a video tutorial on use of the quadratic formula. CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 110 of 144 Columbus City Schools 12/1/13 21) Quadratic Equations – Factoring and Quadratic Formula: http://patrickjmt.com/quadratic-equations-factoring-and-quadratic-formula/ This tutorial offers examples for solving quadratic equations using either factoring or the quadratic formula. 22) Solving Quadratic Equations using the Quadratic Formula – Ex 1: http://patrickjmt.com/solving-quadratic-equations-using-the-quadratic-formula-ex-1/ The tutorial offers examples for solving equations using the quadratic formula. 23) Solving Quadratic Equations using the Quadratic Formula – Ex 2: http://patrickjmt.com/solving-quadratic-equations-using-the-quadratic-formula-ex-2/ The tutorial offers more examples for solving equations using the quadratic formula. 24) Solving Quadratic Equations using the Quadratic Formula – Ex 2: http://patrickjmt.com/solving-quadratic-equations-using-the-quadratic-formula-ex-3/ The tutorial offers more examples for solving equations using the quadratic formula. 25) Quadratic Equations, Discriminant, Quadratic Formula: http://www.regentsprep.org/Regents/math/algtrig/ATE3/indexATE3.htm Lessons, practice and teacher resources are provided for solving quadratic equations using the quadratic formula. 26) Solve a Quadratic Equation using Square Roots: http://www.ixl.com/math/algebra-1/solve-a-quadratic-equation-using-square-roots Students are provided problems to determine the solution to a quadratic equation by taking square roots. A tutorial is provided is the solutions offered are incorrect. 27) Solve a Quadratic Equation by Completing the Square: http://www.ixl.com/math/algebra-1/solve-a-quadratic-equation-by-completing-the-square Students are provided problem to determine the solution of a quadratic equation by completing the square. A tutorial is provided is the solutions offered are incorrect. 28) Solve a Quadratic Equation using the Quadratic Formula: http://www.ixl.com/math/algebra-1/solve-a-quadratic-equation-using-the-quadratic-formula Students are provided problems to determine the solution of a quadratic equation with the quadratic formula. A tutorial is provided is the solutions offered are incorrect. 29) Using the Discriminant: http://www.ixl.com/math/algebra-1/using-the-discriminant Students are provided problems to determine the number of solutions for a quadratic equation. A tutorial is provided is the solutions offered are incorrect. 30) Solve Quadratic Equations: http://www.ixl.com/math/geometry/solve-quadratic-equations Students are provided problems to determine the solution of a quadratic equation using CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 111 of 144 Columbus City Schools 12/1/13 different methods. A tutorial is provided is the solutions offered are incorrect. A – SSE 3b 1) Math.A-SSE.3b: http://www.shmoop.com/common-core-standards/ccss-hs-a-sse-3b.html Written instructions for solving quadratic equations by completing the square can be found at this site. 2) Completing the Square: http://www.mathworksheetsland.com/algebra/5squareinquad/ip.pdf Printable worksheets and lessons are provided for students to practice determining the “c” of a perfect square trinomial. F – IF 8 1) Use the task, “Quadratic (2009),” found at the Inside Mathematics website. Have students demonstrate their understanding of quadratic functions given different representations. Students will interpret rates of change given graphical and numerical data. Ask them to identify the minimum points and determine the solutions of these functions algebraically. In this activity students use a verbal description to create an equation in vertex form, and expand it to standard form. Students are then asked to examine the graph, along with a horizontal line and another linear graph. Students must identify the vertex, as well as the intersection points for the different lines with the parabola. Students must also complete the algebra to get the same results, and then go a step further and identify where the graph of the parabola equals zero. http://insidemathematics.org/common-core-math-tasks/highschool/HS-A-2009%20Quadratic2009.pdf 2) Give students instructions on creating a graphic organizer. Instruct them to use the organizer to compare quadratic functions using the process of factoring, completing the square and graphing. There are websites for with examples of graphic organizers. http://www.teachnology.com/worksheets/graphic/ 3) Building the Perfect Square: http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod2_structure_0 62213.pdf In this lesson (pp. `14-22), students use visual and algebraic approaches to completing the square. F – IF 8a 1) Lining Up Quadratics: http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod2_structure_0 62213.pdf In this lesson (pp. 23-28), students will focus on the vertex and intercepts for quadratics. CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 112 of 144 Columbus City Schools 12/1/13 2) Area "FOILed" Again! http://education.ti.com/en/us/activity/detail?id=E0A02061CC2B4007B4EC672574B28016 Students practice finding rectangular areas with algebraic expressions for the lengths of the sides. 3) Factor Fixin’: http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod2_structure_0 62213.pdf In this lesson (pp. 29-33), students focus on connecting the factored and expanded or standard forms of a quadratic. 4) I’ve Got a Fill-in: http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod2_structure_0 62213.pdf In this lesson (pp. 34-41), students build fluency in rewriting and connecting different forms of a quadratic. 5) Forming Quadratics: http://map.mathshell.org/materials/lessons.php?taskid=224 In this lesson students demonstrate their understanding of the factored form of the function and can identify the roots of the graphs; understand how the completed square form of the function can identify the maximum or minimum of a graph; and understand how the standard form can provide the graphs’ intercepts. 6) Proof without Words: Completing the Square: http://illuminations.nctm.org/ActivityDetail.aspx?ID=132 This site provides an interactive geometric proof for students to understand the concept of completing the square. 7) Practice: http://www.ixl.com/math/algebra-1/solve-a-quadratic-equation-by-completing-the-square Students complete the square and write their answers as integers, proper or improper fractions in simplest form, or decimals rounded to the hundredths place. 8) Completing the Square: http://ccssmath.org/?s=F-IF+8+quadratics Students use algebra tiles to build a geometric model of a perfect square trinomial. They will complete the square and recognize the characteristics of a perfect square. A – CED 1 1) Applications of Quadratic Functions: http://www.montereyinstitute.org/courses/Algebra1/U10L2T1_RESOURCE/index.html A warm up, video presentation, practice and review problems are provided for creating algebraic models for quadratic situations and solving them. 2) Math in Basketball Lesson Plan: CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 113 of 144 Columbus City Schools 12/1/13 http://www.thirteen.org/get-the-math/files/2012/08/Math-in-Basketball-Full-Lesson-FINAL8.16.12.pdf Using video segments and interactive on the web student explore quadratic functions. This site offers the lesson plan, student activity sheets and answer keys. N – CN 1 1) Determine whether a square root is real or imaginary: http://learnzillion.com/lessons/225-determine-whether-a-square-root-is-real-or-imaginary In this lesson students will learn how to determine whether a square root is real or imaginary. 2) Classifying Complex Numbers: http://alex.state.al.us/lesson_view.php?id=11364 This lesson has 4 activities. The first section is a teacher Power Point presentation of the relationship between the sets of complex, real and imaginary numbers. The class group or individual activity is a Power Point lesson where students select the appropriate set (strictly complex number, strictly real numbers and strictly imaginary numbers) by clicking on it. A second activity asks students to classify complex numbers into subsets of strictly complex, strictly real or strictly imaginary. The fourth section is a test. 3) Write the square root of negative number as imaginary: http://learnzillion.com/lessons/226-write-the-sq-root-of-neg-number-as-imaginary In this lesson you will learn how to write the square root of a negative number as imaginary. 4) Classify complex numbers as real or imaginary: http://learnzillion.com/lessons/227-classify-complex-numbers-as-real-or-imaginary In this lesson you will learn how to classify complex numbers as real or imaginary. N – CN 2 1) Complex Number Addition: http://education.ti.com/en/us/activity/detail?id=07EF321269B64BB398EABD1C0E0D9061 This lesson involves the addition of two complex numbers. Students compute the sum of two complex numbers and visually and geometrically describe the sum. 2) Complex Numbers: http://www.regentsprep.org/Regents/math/algtrig/ATO6/ImagineLes.htm This website provides lessons, practice and teacher resources for complex numbers. 3) Operations with Complex Numbers: http://www.kutasoftware.com/FreeWorksheets/Alg2Worksheets/Operations%20with%20Co mplex%20Numbers.pdf Students practice simplifying complex numbers with this assignment. CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 114 of 144 Columbus City Schools 12/1/13 4) Properties of Complex Numbers: http://www.kutasoftware.com/FreeWorksheets/Alg2Worksheets/Properties%20of%20Compl ex%20Numbers.pdf Students determine the absolute value of complex numbers and graph complex numbers with this assignment. 5) Rationalizing Imaginary Denominators: http://www.kutasoftware.com/FreeWorksheets/Alg2Worksheets/Rationalizing%20Imaginary %20Denominators.pdf Students simplify expressions by rationalizing the denominators. 6) Adding and Subtracting Complex Numbers: http://www.regentsprep.org/Regents/math/algtrig/ATO6/lessonadd.htm This lesson provides instructions and practice problems to add and subtract complex numbers. 7) Complex Number Multiplication: http://education.ti.com/en/us/activity/detail?id=95F04720818B470A9B87DB4E44A23E44 This lesson involves the product of complex numbers, powers of i and complex conjugates. 8) Complex Numbers: http://education.ti.com/en/us/activity/detail?id=6FD90593B6FF446CB9BE76C9AF380ECE Students calculate problems from the student worksheet to determine the rules for adding, subtracting, multiplying, and dividing complex numbers. 9) Multiplying and Dividing Complex Numbers: http://www.regentsprep.org/Regents/math/algtrig/ATO6/multlesson.htm This lesson provides instructions and practice problems to multiply and divide complex numbers. 10) Practice with Arithmetic of Complex Numbers: http://www.regentsprep.org/Regents/math/algtrig/ATO6/practicepageadd.htm Practice is provided on adding and subtracting complex numbers. 11) Practice with Multiplying and Dividing Complex Numbers: http://www.regentsprep.org/Regents/math/algtrig/ATO6/multprac.htm Practice is provided on multiplying and dividing complex numbers. N – CN 7 1) Complex Numbers and the Quadratic Formula: http://www.purplemath.com/modules/complex3.htm This site provides a written description of how to use the quadratic formula to determine imaginary solutions. CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 115 of 144 Columbus City Schools 12/1/13 2) Solving Quadratic Equations with Complex Roots: http://www.regentsprep.org/Regents/math/algtrig/ATE3/quadcomlesson.htm The lesson provides notes and examples on solving quadratic equations with complex roots. 3) Practice Solving Quadratic Equations with Complex Roots: http://www.regentsprep.org/Regents/math/algtrig/ATE3/quadcompractice.htm A set of practice problems are provided at this site. 4) Complex Numbers Introduction: http://www.purplemath.com/modules/complex.htm A video presentation and notes are provided on complex numbers. 5) Complex Roots from the Quadratic Formula: http://www.youtube.com/watch?feature=player_embedded&v=dnjK4DPqh0k A Khan Academy video presentation of determining complex roots is provided at this site. 6) Determine whether a number is real or imaginary: http://learnzillion.com/lessons/228determine-whether-a-number-is-real-or-imaginary-isolating-the-quadratic-term In this lesson students will learn how to determine whether a number is real or imaginary by isolating the quadratic term. 7) Solve quadratic equations with real coefficients: http://learnzillion.com/lessons/230solve-quadratic-equations-with-real-coefficients-using-the-quadratic-formula In this lesson you will learn how to solve quadratic equations with real coefficients by using the quadratic formula. 8) Determine whether a number is real or imaginary: http://learnzillion.com/lessons/229-determine-whether-a-number-is-real-or-imaginarycalculating-the-value-of-the-discriminant In this lesson you will learn how to determine whether a number is real or imaginary by calculating the value of the discriminant. 9) Solve equations: http://learnzillion.com/lessons/231-solve-equations-completing-the-square In this lesson you will learn how to solve equations by completing the square. Reteach: 1) Forming Quadratics: http://map.mathshell.org/materials/download.php?fileid=700 In this lesson, students will work with different algebraic forms of a quadratic function to understand the properties of different representations (graphical). Students will identify roots CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 116 of 144 Columbus City Schools 12/1/13 by factoring the quadratic equations, complete the square to determine minimum or maximum points and use the standard form of the equation to find the y-intercept. 2) Completing the Square: http://advancedalgebra.gcs.hs.schoolfusion.us/modules/locker/files/get_group_file.phtml?gid=445175&fid =1732523&sessionid= This is a re-teach practice sheet with an answer key provided for students to solve quadratic equations using completing the square. 3) Quadratic Formula: http://advancedalgebra.gcs.hs.schoolfusion.us/modules/locker/files/get_group_file.phtml?gid=445175&fid =1732543&sessionid= This is a re-teach practice sheet with an answer key provided for students to solve quadratic equations with the quadratic formula. Extensions: 1) Horseshoes in Flight: http://www.nctm.org/uploadedFiles/Journals_and_Books/Books/FHSM/RSMTask/Horseshoes.pdf The height of the thrown horseshoe depends on the time that has passed since it was released. Students will derive information about the flight of a horseshoe from the graph and the four given equivalent algebraic expressions that describe its flight and complete the activity sheet. 2) Bridging the Gap: http://www.oame.on.ca/main/files/OMCA%20MCF3M/Unit%204%20Midterm%20SP%2 0Task.pdf This midterm summative performace task has a series of lessons in which students will: solve a problem by creating a scale model, collect data and create an algebraic model; demonstrate their understanding of connections between numeric, graphical, and algebraic representations of quadratic functions; and solve real-world problems. 3) Performance Task: http://insidemathematics.org/commoncore-math-tasks/high-school/HS-A2009%20Quadratic2009.pdf Students will work with a quadratic function in various forms. CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 117 of 144 Columbus City Schools 12/1/13 A – REI 4a Name_______________________________________________ Date___________ Period_____ Sorting Activity Work with a partner to cut and sort the cards. Label a column in your notes so that it looks like the column on the left-hand margin of this page. Tape or glue two sets of six cards each in your notes. Quadratic Equation Factored Completing the Square/Vertex Form Minimum Point Solutions y = x2 + 5x + 6 y = x2 + 5x – 6 y = (x + 2)(x + 3) y = (x – 2)(x – 3) y 5 x 2 Minimum at - 2 49 4 y 5 49 ,2 4 y = 0 when x = - 5 2 1 2 5 x 2 Minimum at - 2 1 4 5 1 ,2 4 y = 0 when x = - 5 2 7 2 Graph CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 118 of 144 Columbus City Schools 12/1/13 Quadratic Equation Factored Completing the Square/Vertex Form y = x2 – 5x + 6 y = x2 – 5x – 6 y = (x + 1)(x – 6) y = (x – 1)(x + 6) y 5 x 2 Minimum at Minimum Point 2 49 4 y 5 1 ,2 4 y = 0 when x = 5 2 7 2 5 x 2 Minimum at 2 1 4 5 49 ,2 4 y = 0 when x = 5 2 1 2 Solutions Graphs CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 119 of 144 Columbus City Schools 12/1/13 A – REI 4a Name_______________________________________________ Date___________ Period_____ Sorting Activity Answer Key The answers students should have are aligned in the two columns. Quadratic Equation y = x2 - 5x + 6 y = x2 + 5x + 6 Factored y = (x - 2)(x - 3) y = (x + 2)(x + 3) Completing the Square/Vertex Form Minimum Point Solutions y 5 x 2 Minimum at 2 1 4 y 5 1 ,2 4 y = 0 when x = 5 2 1 2 5 x 2 Minimum at - 2 1 4 5 1 ,2 4 y = 0 when x = - 5 2 1 2 Graphs CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 120 of 144 Columbus City Schools 12/1/13 Quadratic Equation y = x2 + 5x – 6 y = x2 – 5x – 6 y = (x – 1)(x + 6) y = (x + 1)(x – 6) Factored Completing the Square/Vertex Form Minimum Point Solutions y 5 x 2 Minimum at - 2 49 4 y 5 49 ,2 4 y = 0 when x = - 5 2 7 2 5 x 2 Minimum at 2 49 4 5 49 ,2 4 y = 0 when x = 5 2 7 2 Graphs CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 121 of 144 Columbus City Schools 12/1/13 A – REI 4a Name_______________________________________________ Date___________ Period_____ Learning How to Complete the Square “Completely” 1. In words, explain what is the same and what is different about the equations (x – 2)2 = 25 and x2 – 4x + 4 = 25. 2. x2 – 4x + 4 is called what kind of trinomial? Answer: _______________________________ 3. (x – 2)2 is called what kind of binomial? Answer: _______________________________ 4. List a few things you can say about the graph of y = (x – 2)2 . 5. Will (x – 2)2 ever be negative? Explain. 6. Will (x + 2)2 ever be negative? Explain. 7. List a few things about the graph of y = (x – 2)2 + 3. 8. What is the minimum value of y = (x – 2)2 – 10? Answer: _______________ What is the minimum value of y = (x + 2)2 + 10? Answer: _______________ 9. If y = (x – 2)2 + 3, then y is a quadratic in _____________________ form. 10. The vertex of the parabola y = (x – 2)2 + 3 is ___________________ . 11. The vertex of the parabola y = (x + 3)2 – 2 is ___________________ . 12. What is the location of the minimum point for y = (x + 3)2 – 2 ? Answer: ______________ 13. Completing the square allows us to write a quadratic in ___________________ form by changing the given trinomial into a _____________________________ trinomial and then factoring it into a binomial-squared. CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 122 of 144 Columbus City Schools 12/1/13 For problems 14-17, write the quadratic in vertex form by completing the square and give the minimum value. 14. y = x2 + 6x – 11 15. y = x2 – 10x + 16 Vertex form:___________________ Vertex form:__________________ Minimum: ___________________ Minimum: __________________ 16. y = x2+2x – 8 17. y = x2 + 5x + 6 Vertex form:___________________ Vertex form:__________________ Minimum: ___________________ Minimum: __________________ CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 123 of 144 Columbus City Schools 12/1/13 A – REI 4a Name_______________________________________________ Date___________ Period_____ Learning How to Complete the Square “Completely” Answer Key 1. In words, explain what is the same and what is different about the equations (x – 2)2 = 25 and x2 – 4x + 4 = 25. The equations look different, but algebraically they are the same because (x – 2)2 = x2 – 4x + 4 when multiplied out. Also, the roots are the same. 2. x2 – 4x + 4 is called what kind of trinomial? Answer:__perfect square trinomial_________ 3. (x – 2)2 is called what kind of binomial? Answer:__binomial squared_______________ 4. List a few things you can say about the graph of y = (x – 2)2 . (i) it is a parabola (iii) minimum value at y = 0 (v) when x = 2, then y = 0 (e.g. x-intercept = 2) (ii) it has the shape of y = x2 (iv) it is symmetric about the line x = 2 5. Will (x – 2)2 ever be negative? Explain. No, because the whole binomial is being squared and a ‘squared’ number will always be positive. 6. Will (x + 2)2 ever be negative? Explain. No, because the whole binomial is being squared and a ‘squared’ number will always be positive. 7. List a few things about the graph of y = (x – 2)2 + 3. The minimum value occurs at y = 3. It is a horizontal shift 2 to the right and a vertical shift 3-up of the graph of y = x2. 8. What is the minimum value of y = (x – 2)2 – 10? What is the minimum value of y = (x + 2)2 + 10? Answer: -10 Answer: -10 9. If y = (x – 2)2 + 3, then y is a quadratic in ____vertex______ form. 10. The vertex of the parabola y = (x – 2)2 + 3 is (2,3) . 11. The vertex of the parabola y = (x + 3)2 – 2 is (-3, -2) . 12. What is the location of the minimum point for y = (x + 3)2 – 2 ? Answer: (-3, -2) CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 124 of 144 Columbus City Schools 12/1/13 13. Completing the square allows us to write a quadratic in __vertex ______ form by changing the given trinomial into a ____perfect-square __________ trinomial and then factoring it into a binomial-squared. For problems 14-17, write the quadratic in vertex form by completing the square and give the minimum value. 14. y = x2 + 6x – 11 15. y = x2 – 10x + 16 Vertex form: y = (x + 3)2 – 20 Vertex form: y = (x – 5)2 – 9 Minimum: -20 Minimum: -9 16. y = x2+2x – 8 17. y = x2 + 5x + 6 Vertex form: y = (x + 1)2 – 9 Vertex form: y = (x + 5/2)2 – ¼__ Minimum: 9 Minimum: -1/4 CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 125 of 144 Columbus City Schools 12/1/13 A – REI 4a Name_______________________________________________ Date___________ Period_____ Transformations and Completing the Square Notes The function f x x 2 is the parent function of all quadratics. Every quadratic can be transformed from this graph. By using completing the square, all quadratics can be rewritten, in what some people call, the vertex form for a quadratic equation. 2 d . Think of this as a template. The vertex form is: y a x c Transformations: a If a 0 , then the graph is reflected about the x-axis. If a 1 , then a vertical stretch by a factor of a occurs. If 0 a 1 , then a vertical shrink by a factor of a occurs. c If c 0 , then the graph will shift c units to the right. If c 0 , then the graph will shift c units to the left. d If d If d 0 , then the graph will shift down d units. 0 , then the graph will shift up d units. So, if y -2 x 3 2 1 , the following transformations would occur to the graph of f x 1) Reflection about the x-axis, since a x2 . - 2 . (The y-coordinates will become opposites.) 2) Vertical stretch by a factor of 2, because - 2 2 . (Multiply the y-coordinates by 2.) 3) Horizontal shift 3 units to the right, because c = 3. (Add 3 to the x-coordinates.) 4) Vertical shift up 1 unit, since d 1 . (Add 1 to the y-coordinates.) CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 126 of 144 Columbus City Schools 12/1/13 Below you will see the transformations that are applied to f(x) = x2. The graph of f(x) = x2. A reflection about the x-axis. A vertical stretch by a factor of 2. Horizontal shift to the right 3 units. A vertical shift up 1 unit. What is the value of c? What is the value of d? What are the coordinates of the vertex for the parabola on the left? Is there a connection between the vertex and the value of c and d? CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 127 of 144 Columbus City Schools 12/1/13 Practice: State the transformations that would occur to f(x) = x2 and the coordinates of the vertex for each of the new graphs. Using the transformations and the points (-2,4), (-1,1), (0,0), (1,1), and (2,4) from the graph of f(x) = x2, find the coordinates of the transformed points and graph the new function. Show the mapping of the points. 1. y = 3(x – 1) 2 – 5 2. y 2 x 5 3 2 7 3. a) State the transformations that would occur to f(x) = x2, if y = - 3(x + 4) 2 + 2. b) Using the transformations stated in part a and the point (2,4) from the graph of y = x2, give the coordinates for the transformed point. CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 128 of 144 Columbus City Schools 12/1/13 How to complete the square Let Group the quadratic and linear term together. If the leading coefficient is not 1, factor it out. y x2 4 x 5 y x2 4x 5 y x2 4x -2 y x 2 Thus y 2 2 5 5 4 x 2 -2 2 x2 4 x 5 = x 2 You will need to create a perfect trinomial square (take half of the linear term and square it). But because you do not want to change the equivalence of the equation; you will need to add and subtract the same number to the one side of the equation. 2 Rewrite the perfect trinomial square in its factored form as a square of a binomial. 1 2 Simplify the constants. 1 State the transformations. What are the coordinates of the vertex? Completing the Square, when the leading coefficient is not 1. Group the quadratic and linear term together. 2 y -2x 6x 7 y - 2x2 6x 7 y - 2 x 2 3x 7 y - 2 x 2 3x - -2 x 3 2 2 y -2 x 3 2 2 y Factor the leading coefficient out of the linear and quadratic terms. 3 2 2 7 7 7 - -2 - -2 9 4 9 2 -2 x 3 2 Create a perfect trinomial square but remember to subtract the exact same number from the constant. Remember you have a multiplier in front that will need to be included when subtracting. 2 Rewrite the perfect trinomial square in it’s factored form: square of a binomial. 3 2 2 5 2 Simplify the constant. State the transformations. What are the coordinates for the vertex? CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 129 of 144 Columbus City Schools 12/1/13 A – REI 4a Name_______________________________________________ Date___________ Period_____ Transformations and Completing the Square Notes Answer Key The function f x x is the parent function of all quadratics. Every quadratic can be transformed from this graph. By using completing the square, all quadratics can be rewritten, in what some people call, the vertex form for a quadratic equation. 2 d . Think of this as a template. The vertex form is: y a x c 2 Transformations: a If a 0 , then the graph is reflected about the x-axis. If a 1 , then a vertical stretch by a factor of a occurs. If 0 a 1 , then a vertical shrink by a factor of a occurs. c If c 0 , then the graph will shift c units to the right. If c 0 , then the graph will shift c units to the left. d If d If d 0 , then the graph will shift down d units. 0 , then the graph will shift up d units. So, if y -2 x 3 2 1 , the following transformations would occur to the graph of f x 1) Reflection about the x-axis, since a x2 . - 2 . (The y-coordinates will become opposites.) 2) Vertical stretch by a factor of 2, because - 2 2 . (Multiply the y-coordinates by 2.) 3) Horizontal shift 3 units to the right, because c = 3. (Add 3 to the x-coordinates.) 4) Vertical shift up 1 unit, since d 1 . (Add 1 to the y-coordinates.) CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 130 of 144 Columbus City Schools 12/1/13 Below you will see the transformations that are applied to f(x) = x2. The graph of f(x) = x2. A reflection about the x-axis. A vertical stretch by a factor of 2. Horizontal shift to the right 3 units. A vertical shift up 1 unit. What is the value of c? 3 What is the value of d? 1 What are the coordinates of the vertex for the parabola on the left? (3,1) Is there a connection between the vertex and the value of c and d? Yes there is a connection. The coordinates for the vertex correspond to the c and d values. The vertex can be written as (c,d). CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 131 of 144 Columbus City Schools 12/1/13 Practice: State the transformations that would occur to f(x) = x2 and the coordinates of the vertex for each of the new graphs. Using the transformations and the points (-2,4), (-1,1), (0,0), (1,1), and (2,4) from the graph of f(x) = x2, find the coordinates of the transformed points and graph the new function. Show the mapping of the points. 1. y = 3(x – 1) 2 – 5 A vertical stretch by a factor of 3. A horizontal shift to the right 1 unit. A vertical shift down 5 units. The coordinates of the vertex are (1, - 5). - 2, 4 - 2, 12 -1, 12 -1, 1 -1, 3 0, 0 0, 0 1, 0 1, - 5 1, 1 1, 3 2, 3 2, - 2 2, 4 2, 12 0, 3 -1, 7 0, - 2 3, 12 3, 7 2 2 x 5 7 3 A reflection about the x-axis. 2. y - A vertical shrink by a factor of 2 . 3 A horizontal shift left 5 units. A vertical shift up 7 units. The coordinates of the vertex are (- 5, 7) - 2, 4 - 2, - 4 - 2, - 2.67 - 7, - 2.67 -1, 1 -1, -1 0, 0 0, 0 0, 0 1, 1 1, -1 1, - 0.67 2, 4 2, - 4 3. -1, - 0.67 - 6, - 0.67 - 5, 0 2, - 2.67 - 7, 4.33 - 6, 6.33 - 5, 7 - 4, - 0.67 - 3, - 2.67 - 4, 6.33 - 3, 4.33 a) State the transformations that would occur to f(x) = x2, if y = - 3(x + 4)2 + 2. A reflection about the x-axis. A vertical stretch by a factor of 3. A horizontal shift to the left 4. A vertical shift up 2 units. b) Using the transformations stated in part a and the point (2,4) from the graph of y = x2, give the coordinates for the transformed point. (2, 4) (2, - 4) (2, - 12) (- 2, - 12) (- 2, - 10) CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 132 of 144 Columbus City Schools 12/1/13 How to complete the square y x2 4 x 5 y x2 4x 5 y x2 4x -2 2 5 y x 2 Let Thus y 2 5 4 Group the quadratic and linear term together. x 2 If the leading coefficient is not 1, factor it out. -2 2 You will need to create a perfect trinomial square (take half of the linear term and square it). But because you do not want to change the equivalence of the equation; you will need to add and subtract the same number to the one side of the equation. 1 2 x2 4 x 5 = x 2 2 Rewrite the perfect trinomial square in its factored form as a square of a binomial. 1 Simplify the constants. Shift left 2 and up 1. Vertex (2, 1) Completing the Square, when the leading coefficient is different from 1. y -2x2 6x 7 State the transformations, and find the vertex. Group the quadratic and linear term y - 2x2 6x 7 together. y - 2 x 2 3x 7 Factor the leading coefficient out of the linear and quadratic terms. y - 2 x 2 3x - 2 y 3 -2 x 2 3 -2 x 2 2 y 7 7 3 2 2 7 -2 9 2 -2 - 3 2 Create a perfect trinomial square but remember to subtract the exact same number from the constant. Remember you have a multiplier in front that will need to be included when subtracting. 2 9 4 3 -2 x 2 2 5 2 Rewrite the perfect trinomial square in it’s factored form: square of a binomial. Simplify the constant. State are and the Reflect about the x-axis, a vertical stretch by a factor ofthe 2, atransformations. horizontal shiftWhat left one one-half units, and a vertical shift down 2 and one-half units. for Thethe coordinates coordinates vertex? for the 3 5 ,vertex are . 2 2 CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 133 of 144 Columbus City Schools 12/1/13 A – REI 4a Name_______________________________________________ Date___________ Period_____ Completing the Square and Transformations Practice Complete the square for each quadratic equation, state the transformations, show the transformation of the points (-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4), from the graph of y = x2, give the new coordinates of the vertex, and sketch the new graph. 1. y = x2 + 4x – 3 2. y = 3x2 – 6x + 7 CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 134 of 144 Columbus City Schools 12/1/13 3. y 1 2 x 2 x 4 4. y = - 2x2 + 4x + 1 CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 135 of 144 Columbus City Schools 12/1/13 A – REI 4a Name_______________________________________________ Date___________ Period_____ Completing the Square and Transformations Practice Answer Key Complete the square for each quadratic equation, state the transformations, show the transformation of the points (-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4), from the graph of y = x2,give the new coordinates of the vertex, and sketch the new graph using the transformed points. 1. y = x2 + 4x – 3 y = (x2 + 4x) – 3 y = (x2 + 4x + 4) – 3 – 4 y = (x + 2)2 – 7 (-2, 4) (- 1, 1) (0, 0) (1, 1) (2, 4) (- 4, 4) (- 3, 1) (- 2, 0) (- 1, 1) (0, 4) (- 4, - 3) (- 3, - 6) (- 2, - 7) (- 1,- 6) (0, - 3) The transformations are: A horizontal shift to the left 2 units. A vertical shift down 7 units. The coordinates of the vertex are (- 2, - 7). 2. y = 3x2 – 6x + 7 y = 3(x2 – 2x) + 7 y = 3(x2 – 2x + 1) + 7 – 3 y = 3(x – 1)2 + 4 (- 2, 4) (- 1, 1) (0, 0) (1, 1) (2, 4) (- 2, 12) (- 1, 3) (0, 0) (1, 3) (2, 12) (- 1, 12) (0, 3) (1, 0) (2, 3) (3, 12) (- 1, 16) (0, 7) (1, 4) (2, 7) (3, 16) The transformations are: A vertical stretch by a factor of 3. A horizontal shift right 1 unit. A vertical shift up 4 units. The coordinates of the vertex are (1, 4). CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 136 of 144 Columbus City Schools 12/1/13 3. y 1 2 x 2 x 4 1 2 x 2x 4 2 1 2 1 y x 2x 1 4 2 2 1 1 2 y x 1 4 2 2 The transformations are: y 1 . 2 A horizontal shift to the left 1 unit. 1 A vertical shift down 4 units. 2 A vertical shrink by a factor of (- 2, 4) (- 1, 1) (- 2, 2) - 1, (0, 0) (0, 0) (1, 1) 1, (2, 4) (2, 2) 1 2 - 3, - 2 (- 3, 2) 1 2 - 2, 1 2 (- 2, - 4) (- 1, 0) - 1, - 4 1 2 (0, - 4) 0, (1, 2) 1 2 1, - 2 1 2 1 2 The coordinates of the vertex are - 1, - 4 1 . 2 4. y = - 2x2 + 4x + 1 y = - 2(x2 - 2x) + 1 y = - 2(x2 - 2x + 1) + 1 + 2 y = - 2(x - 1)2 + 3 The transformations are: A reflection about the x-axis. A vertical stretch by a factor of 2. A horizontal shift to the right 1 unit. A vertical shift up 3 units. (- 2, 4) (- 2, - 4) (- 2, - 8) (- 1, - 8) (- 1, - 5) (- 1, 1) (- 1, - 1) (- 1, - 2) (0, - 2) (0, 1) (0, 0) (0, 0) (0, 0) (1, 0) (1, 3) (1, 1) (1, - 1) (1, - 2) (2, - 2) (2, 1) (2, 4) (2, - 4) (2, - 8) (3, - 8) (3, - 5) The coordinates of the vertex are (1, 3) CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 137 of 144 Columbus City Schools 12/1/13 A – REI 4a Name_______________________________________________ Date___________ Period_____ Discovery of Completing the Square Multiply the following binomial expressions and simplify. 1. (x + 4)(x + 4) 2. (a – 3)(a – 3) SHOW WORK ______ + ______ + ______ + ______ SIMPLIFY SHOW WORK ______ + ______ + ______ + ______ SIMPLIFY ______ + ______ + ______ ______ + ______ + ______ 3. (y – 1)(y – 1) 4. (w + 5)(w + 5) SHOW WORK SHOW WORK ______ + ______ + ______ + ______ SIMPLIFY ______ + ______ + ______ + ______ SIMPLIFY ______ + ______ + ______ ______ + ______ + ______ 5. What do you notice about all of the above problems? 6. Describe how the final result compares to the original problem? Be specific. CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 138 of 144 Columbus City Schools 12/1/13 Factor the following trinomials. In other words, do the reverse of #1-4. 7. x2 + 4x + 4 8. r2 – 12r + 36 9. p2 + 20p + 100 10. q2 – 6q + 9 Fill in the missing number to make the following problems perfect square trinomials. 11. x2 + 14x + _______ 12. x2 – 18x + _______ 13. y2 + _______ + 16 14. t2 + _______ + 25 15. m2 – 3m + ______ 16. k2 + 9k + _______ Each of the following problems is not a perfect square trinomial. Your job is to turn them into perfect squares by “completing the square.” Answer i) – iv) to help “complete the square” for each problem. 17. b2 +16b +30 = 0 i) In order for this to be a perfect square, the constant term should be __________. ii) I need to add __________ to the left side of the equation to make it a perfect square trinomial, but I also must add __________ to the right side of the equation to keep it balanced. Show this work. iii) Factor the left side into a perfect square. Show this work. iv) Bring the constant back over to the left side to set the equation equal to zero again. Show this work. v) The vertex of this equation is ________________. CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 139 of 144 Columbus City Schools 12/1/13 18. x2 + 4x + 7=0 i) In order for this to be a perfect square, the constant term should be __________. ii) I need to add __________ to the left side of the equation to make it a perfect square trinomial, but I also must add __________ to the right side of the equation to keep it balanced. Show this work. iii) Factor the left side into a perfect square. Show this work. iv) Bring the constant back over to the left side to set the equation equal to zero again. Show this work. v) The vertex of this equation is ________________. 19. f2 – 6f + 5 = 0 i) In order for this to be a perfect square, the constant term should be __________. ii) I need to add __________ to the left side of the equation to make it a perfect square trinomial, but I also must add __________ to the right side of the equation to keep it balanced. Show this work. iii) Factor the left side into a perfect square. Show this work. iv) Bring the constant back over to the left side to set the equation equal to zero again. Show this work. v) The vertex of this equation is ________________. 20. r2 – 10r – 4 = 0 CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 140 of 144 Columbus City Schools 12/1/13 i) In order for this to be a perfect square, the constant term should be __________. ii) I need to add __________ to the left side of the equation to make it a perfect square trinomial, but I also must add __________ to the right side of the equation to keep it balanced. Show this work. iii) Factor the left side into a perfect square. Show this work. iv) Bring the constant back over to the left side to set the equation equal to zero again. Show this work. v) The vertex of this equation is _______________. CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 141 of 144 Columbus City Schools 12/1/13 A – REI 4a Name_______________________________________________ Date___________ Period_____ Discovery of Completing the Square Answer Key Multiply the following binomial expressions and simplify. 1. (x + 4)(x + 4) 2. (a – 3)(a – 3) SHOW WORK x2 + 4x SHOW WORK + 4x + 16 a2 + -3a + -3a + 9 SIMPLIFY SIMPLIFY __x2__ + __8x__ + __16__ __a2__ + __-6a__ + __9__ 3. (y – 1)(y – 1) 4. (w + 5)(w + 5) SHOW WORK SHOW WORK __y2__ + __-y__ + __-y___ + __1___ __w2__ + __5w__ + __5w_ + __25__ SIMPLIFY SIMPLIFY __y2__ + __-2y__ + __1__ __w2__ + _10w__ + __25__ 5. What do you notice about all of the above problems? #1-#4 are all problems that have a binomial multiplied by the same binomial. 6. Describe how the final result compares to the original problem? Be specific. The middle term is two times the number in the original problem. The last term is the square of the number in the original problem. Factor the following trinomials. In other words, do the reverse of #1-4. 7. x2 + 4x + 4 8. r2 – 12r + 36 (x + 2)(x + 2) (r – 6)(r – 6) 9. p2 + 20p + 100 (p + 10)(p + 10) 10. q2 – 6q + 9 (q – 3)(q – 3) Fill in the missing number to make the following problems perfect square trinomials. 11. x2 + 14x + __49___ 12. x2 – 18x + __81___ CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 142 of 144 Columbus City Schools 12/1/13 13. y2 + __8y__ + 16 14. t2 + __10t__ + 25 15. m2 – 3m + _2.25_ 16. k2 + 9k + __20.25 _ Each of the following problems is not a perfect square trinomial. Your job is to turn them into perfect squares by “completing the square.” Answer i) – iv) to help “complete the square” for each problem. 17. b2 +16b +30 = 0 i) In order for this to be a perfect square, the constant term should be ____64____. ii) I need to add __ 34____ to the left side of the equation to make it a perfect square trinomial, but I also must add ___34____ to the right side of the equation to keep it balanced. Show this work. b2 + 16b +30 + 34 = 34 b2 + 16b + 64 = 34 iii) Factor the left side into a perfect square. Show this work. (b + 8)2 = 34 iv) Bring the constant back over to the left side to set the equation equal to zero again. Show this work. (b + 8)2 – 34 = 0 v) The vertex of this equation is __(-8, -34) _. 18. x2 + 4x + 7=0 i) In order for this to be a perfect square, the constant term should be ___ 4_____. ii) I need to add __ -3____ to the left side of the equation to make it a perfect square trinomial, but I also must add ___ -3____ to the right side of the equation to keep it balanced. Show this work. x2 + 4x + 7 – 3 = -3 x2 + 4x + 4 = -3 iii) Factor the left side into a perfect square. Show this work. (x + 2)2 = -3 iv) Bring the constant back over to the left side to set the equation equal to zero again. Show this work. (x + 2)2 + 3 = 0 v) The vertex of this equation is ___ (-2, 3)______. 19. f2 – 6f + 5 = 0 i) In order for this to be a perfect square, the constant term should be ___ 9_____. ii) I need to add ____4_____ to the left side of the equation to make it a perfect square trinomial, but I also must add ____4____ to the right side of the equation to keep it balanced. Show this work. f2 – 6f + 5 + 4 = 4 f2 – 6f + 9 = 4 iii) Factor the left side into a perfect square. Show this work. (f – 3)2 = 4 CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 143 of 144 Columbus City Schools 12/1/13 iv) Bring the constant back over to the left side to set the equation equal to zero again. Show this work. (f – 3)2 – 4 = 0 v) The vertex of this equation is ____(3, -4)____. 20. r2 – 10r – 4 = 0 i) In order for this to be a perfect square, the constant term should be ___ 25______. ii) I need to add ___ 29____ to the left side of the equation to make it a perfect square trinomial, but I also must add ___ 29____ to the right side of the equation to keep it balanced. Show this work. r2 – 10r – 4 + 29 = 29 r2 – 10r + 25 = 29 iii) Factor the left side into a perfect square. Show this work. (r – 5)2 = 29 iv) Bring the constant back over to the left side to set the equation equal to zero again. Show this work. (r – 5)2 – 29 = 0 v) The vertex of this equation is __ (5, -29)___. CCSSM II Quadratic Functions: Completing the Square and the Quadratic Formula: A – REI 1, 4, 4a, 4b; A – SSE 3b; F – IF 8, 8a; A – CED 1; N – CN 1, 2, 7 Quarter 3 Page 144 of 144 Columbus City Schools 12/1/13 High School CCSS Mathematics II Curriculum Guide -Quarter 4- Columbus City Schools Page 0 of 122 Contents RUBRIC – IMPLEMENTING STANDARDS FOR MATHEMATICAL PRACTICE ....................... 10 Mathematical Practices: A Walk-Through Protocol .............................................................................. 14 Curriculum Timeline .............................................................................................................................. 17 Scope and Sequence ............................................................................................................................... 18 Probability S-CP 1, 2, 3, 4, 5, 6, 7 ......................................................................................................... 29 Teacher Notes: ......................................................................................................................................... 30 The Titanic 1 ........................................................................................................................................ 49 The Titanic 2 ........................................................................................................................................ 53 Cards and Independence ...................................................................................................................... 56 Rain and Lightening ............................................................................................................................. 58 The Titanic 3 ........................................................................................................................................ 61 Breakfast Before School ...................................................................................................................... 66 How do you get to school? ................................................................................................................... 68 Coffee at Mom’s Diner ........................................................................................................................ 71 Geometric Measurement G-GMD 1, 3 .................................................................................................. 73 Teacher Notes: ......................................................................................................................................... 74 Discovering Pi ...................................................................................................................................... 82 Volume ................................................................................................................................................. 85 Centerpiece........................................................................................................................................... 94 Geometric Modeling and Project G-MG 1, 2, 3 .................................................................................... 97 Teacher Notes: ......................................................................................................................................... 98 Misconceptions/Challenges: .................................................................................................................... 98 G – MG 1 ................................................................................................................................................. 99 G – MG 2 ............................................................................................................................................... 100 G – MG 3 ............................................................................................................................................... 100 Tennis Ball in a Can ........................................................................................................................... 103 Toilet Roll .......................................................................................................................................... 108 Ice Cream Cone .................................................................................................................................. 111 Page 1 of 122 Math Practices Rationale CCSSM Practice 1: Make sense of problems and persevere in solving them. Why is this practice important? What does this practice look like when students are doing it? What can a teacher do to model this practice? What questions could a teacher ask to encourage the use of this practice? What does proficiency look like in this practice? What actions might the teacher make that inhibit the students’ use of this practice? Helps students to develop critical thinking skills. Teaches students to “think for themselves”. Helps students to see there are multiple approaches to solving a problem. Students immediately begin looking for methods to solve a problem based on previous knowledge instead of waiting for teacher to show them the process/algorithm. Students can explain what problem is asking as well as explain, using correct mathematical terms, the process used to solve the problem. Frame mathematical questions/challenges so they are clear and explicit. Check with students repeatedly to help them clarify their thinking and processes. “How would you go about solving this problem?” “What do you need to know in order to solve this problem?” What methods have we studied that you can use to find the information you need? Students can explain the relationships between equations, verbal descriptions, tables, and graphs. Students check their answer using a different method and continually ask themselves, “Does this make sense?” They understand others approaches to solving complex problems and can see the similarities between different approaches. Showing the students shortcuts/tricks to solve problems (without making sure the students understand why they work). Not giving students an adequate amount of think time to come up with solutions or processes to solve a problem. Giving students the answer to their questions instead of asking guiding questions to lead them to the discovery of their own question. Page 2 of 122 CCSSM Practice 2: Reason abstractly and quantitatively. Why is this practice important? What does this practice look like when students are doing it? What can a teacher do to model this practice? What questions could a teacher ask to encourage the use of this practice? What does proficiency look like in this practice? What actions might the teacher make that inhibit the students’ use of this practice? Students develop reasoning skills that help them to understand if their answers make sense and if they need to adjust the answer to a different format (i.e. rounding) Students develop different ways of seeing a problem and methods of solving it. Students are able to translate a problem situation into a number sentence or algebraic expression. Students can use symbols to represent problems. Students can visualize what a problem is asking. Ask students questions about the types of answers they should get. Use appropriate terminology when discussing types of numbers/answers. Provide story problems and real world problems for students to solve. Monitor the thinking of students. “What is your unknown in this problem? “What patterns do you see in this problem and how might that help you to solve it?” Students can recognize the connections between the elements in their mathematical sentence/expression and the original problem. Students can explain what their answer means, as well as how they arrived at it. Giving students the equation for a word or visual problem instead of letting them “figure it out” on their own. Page 3 of 122 CCSSM Practice 3: Construct viable arguments and critique the reasoning of others Why is this practice important? What does this practice look like when students are doing it? What can a teacher do to model this practice? What questions could a teacher ask to encourage the use of this practice? What does proficiency look like in this practice? What actions might the teacher make that inhibit the students’ use of this practice? Students better understand and remember concepts when they can defend and explain it to others. Students are better able to apply the concept to other situations when they understand how it works. Communicate and justify their solutions Listen to the reasoning of others and ask clarifying questions. Compare two arguments or solutions Question the reasoning of other students Explain flaws in arguments Provide an environment that encourages discussion and risk taking. Listen to students and question the clarity of arguments. Model effective questioning and appropriate ways to discuss and critique a mathematical statement. How could you prove this is always true? What parts of “Johnny’s “ solution confuses you? Can you think of an example to disprove your classmates theory? Students are able to make a mathematical statement and justify it. Students can listen, critique and compare the mathematical arguments of others. Students can analyze answers to problems by determining what answers make sense. Explain flaws in arguments of others. Not listening to students justify their solutions or giving adequate time to critique flaws in their thinking or reasoning. Page 4 of 122 CCSSM Practice 4: Model with mathematics Why is this practice important? Helps students to see the connections between math symbols and real world problems. What does this practice look like when students are doing it? Write equations to go with a story problem. Apply math concepts to real world problems. What can a teacher do to model this practice? Use problems that occur in everyday life and have students apply mathematics to create solutions. Connect the equation that matches the real world problem. Have students explain what different numbers and variables represent in the problem situation. Require students to make sense of the problems and determine if the solution is reasonable. How could you represent what the problem was asking? How does your equation relate to the problems? How does your strategy help you to solve the problem? Students can write an equation to represent a problem. Students can analyze their solutions and determine if their answer makes sense. Students can use assumptions and approximations to simplify complex situations. Not give students any problem with real world applications. What questions could a teacher ask to encourage the use of this practice? What does proficiency look like in this practice? What actions might the teacher make that inhibit the students’ use of this practice? Page 5 of 122 CCSSM Practice 5: Use appropriate tools strategically Why is this practice important? What does this practice look like when students are doing it? What can a teacher do to model this practice? What questions could a teacher ask to encourage the use of this practice? Helps students to understand the uses and limitations of different mathematical and technological tools as well as which ones can be applied to different problem situations. Students select from a variety of tools that are available without being told which to use. Students know which tools are helpful and which are not. Students understand the effects and limitations of chosen tools. Provide students with a variety of tools Facilitate discussion regarding the appropriateness of different tools. Allow students to decide which tools they will use. How is this tool helping you to understand and solve the problem? What tools have we used that might help you organize the information given in this problem? Is there a different tool that could be used to help you solve the problem? What does proficiency look like in this practice? Students are sufficiently familiar with tools appropriate for their grade or course and make sound decisions about when each of these tools might be helpful. Students recognize both the insight to be gained from the use of the selected tool and their limitations. What actions might the teacher make that inhibit the students’ use of this practice? Only allowing students to solve the problem using one method. Telling students that the solution is incorrect because it was not solved “the way I showed you”. Page 6 of 122 CCSSM Practice 6: Attend to precision. Why is this practice important? What does this practice look like when students are doing it? What can a teacher do to model this practice? What questions could a teacher ask to encourage the use of this practice? What does proficiency look like in this practice? What actions might the teacher make that inhibit the students’ use of this practice? Students are better able to understand new math concepts when they are familiar with the terminology that is being used. Students can understand how to solve real world problems. Students can express themselves to the teacher and to each other using the correct math vocabulary. Students use correct labels with word problems. Make sure to use correct vocabulary terms when speaking with students. Ask students to provide a label when describing word problems. Encourage discussions and explanations and use probing questions. How could you describe this problem in your own words? What are some non-examples of this word? What mathematical term could be used to describe this process. Students are precise in their descriptions. They use mathematical definitions in their reasoning and in discussions. They state the meaning of symbols consistently and appropriately. Teaching students “trick names” for symbols (i.e. the alligator eats the big number) Not using proper terminology in the classroom. Allowing students to use the word “it” to describe symbols or other concepts. Page 7 of 122 CCSSM Practice 7: Look for and make use of structure. Why is this practice important? When students can see patterns or connections, they are more easily able to solve problems What does this practice look like when students are doing it? Students look for connections between properties. Students look for patterns in numbers, operations, attributes of figures, etc. Students apply a variety of strategies to solve the same problem. Ask students to explain or show how they solved a problem. Ask students to describe how one repeated operation relates to another (addition vs. multiplication). How could you solve the problem using a different operation? What pattern do you notice? What can a teacher do to model this practice? What questions could a teacher ask to encourage the use of this practice? What does proficiency look like in this practice? Students look closely to discern a pattern or structure. What actions might the teacher make that inhibit the students’ use of this practice? Provide students with pattern before allowing them to discern it for themselves. Page 8 of 122 CCSSM Practice 8: Look for and express regularity in repeated reasoning Why is this practice important? What does this practice look like when students are doing it? What can a teacher do to model this practice? What questions could a teacher ask to encourage the use of this practice? When students discover connections or algorithms on their own, they better understand why they work and are more likely to remember and be able to apply them. Students discover connections between procedures and concepts Students discover rules on their own through repeated exposures of a concept. Provide real world problems for students to discover rules and procedures through repeated exposure. Design lessons for students to make connections. Allow time for students to discover the concepts behind rules and procedures. Pose a variety of similar type problems. How would you describe your method? Why does it work? Does this method work all the time? What do you notice when…? What does proficiency look like in this practice? Students notice repeated calculations. Students look for general methods and shortcuts. What actions might the teacher make that inhibit the students’ use of this practice? Providing students with formulas or algorithms instead of allowing them to discover it on their own. Not allowing students enough time to discover patterns. Page 9 of 122 RUBRIC – IMPLEMENTING STANDARDS FOR MATHEMATICAL PRACTICE Task: Lacks context. Does not make use of multiple representations or solution paths. NEEDS IMPROVEMENT Task: Teacher: Expects students to model and interpret tasks using a single representation. Explains connections between procedures and meaning. Is embedded in a contrived context. (teacher does thinking) EMERGING Task: Teacher: Expects students to interpret and model using multiple representations. Provides structure for students to connect algebraic procedures to contextual meaning. Links mathematical solution with a question’s answer. Has realistic context. Requires students to frame solutions in a context. Has solutions that can be expressed with multiple representations. (teacher mostly models) PROFICIENT Task: Teacher: Summer 2011 Expects students to interpret, model, and connect multiple representations. Prompts students to articulate connections between algebraic procedures and contextual meaning. Has relevant realistic context. (students take ownership) EXEMPLARY Using the Rubric: Review each row corresponding to a mathematical practice. Use the boxes to mark the appropriate description for your task or teacher action. The task descriptors can be used primarily as you develop your lesson to make sure your classroom tasks help cultivate the mathematical practices. The teacher descriptors, however, can be used during or after the lesson to evaluate how the task was carried out. The column titled “proficient” describes the expected norm for task and teacher action while the column titled “exemplary” includes all features of the proficient column and more. A teacher who is exemplary is meeting criteria in both the proficient and exemplary columns. RACTICE Reason abstractly and quantitatively. Teacher: Does not expect students to interpret representations. Expects students to memorize procedures with no connection to leaning. Institute for Advanced Study/Park City Mathematics Institute Secondary School Teachers Program/Visualizing Functions Page 10 of 122 PRACTICE Model with mathematics. Use appropriate tools strategically. NEEDS IMPROVEMENT Requires students to Task: identify variables and to perform necessary computations. Teacher: Identifies appropriate variables and procedures for students. Does not discuss appropriateness of model. Does not incorporate Task: additional learning tools. Teacher: additional learning tools. Does not incorporate EMERGING (teacher does thinking) Requires students to Task: identify variables and to compute and interpret results. Teacher: Verifies that students have identified appropriate variables and procedures. Explains the appropriateness of model. Lends itself to one learning Task: tool. Does not involve mental computations or estimation. Teacher: Demonstrates use of appropriate learning tool. Page 11 of 122 PROFICIENT Requires students to (teacher mostly models) Task: identify variables, compute and interpret results, and report findings using a mixture of representations. the mathematics involved. Illustrates the relevance of Requires students to identify extraneous or missing information. Teacher: Asks questions to help students identify appropriate variables and procedures. Facilitates discussions in evaluating the appropriateness of model. Lends itself to multiple Task: learning tools. Gives students opportunity to develop fluency in mental computations. Teacher: Chooses appropriate learning tools for student use. estimation. Models error checking by EXEMPLARY Requires students to (students take ownership) Task: identify variables, compute and interpret results, report findings, and justify the reasonableness of their results and procedures within context of the task. Teacher: Expects students to justify their choice of variables and procedures. Gives students opportunity to evaluate the appropriateness of model. Requires multiple learning Task: tools (i.e., graph paper, calculator, manipulative). demonstrate fluency in Requires students to mental computations. Teacher: appropriate learning tools. Allows students to choose appropriate alternatives Creatively finds where tools are not available. PRACTICE Attend to precision. Look for and make use of structure. Requires students to automatically apply an algorithm to a task without evaluating its appropriateness. Does not intervene when students are being imprecise. Does not point out instances when students fail to address the question completely or directly. Gives imprecise instructions. NEEDS IMPROVEMENT Task: Teacher: Task: Teacher: Does not recognize students for developing efficient approaches to the task. Requires students to apply the same algorithm to a task although there may be other approaches. (teacher does thinking) EMERGING (teacher mostly models) PROFICIENT (students take ownership) EXEMPLARY Summer 2011 Prompts students to identify mathematical structure of the task in order to identify the most effective solution path. Encourages students to justify their choice of algorithm or solution path. Task: Task: Task: Has overly detailed or Has precise instructions. Includes assessment wordy instructions. Teacher: criteria for Consistently demands communication of Teacher: precision in communication ideas. Inconsistently intervenes when and in mathematical Teacher: students are imprecise. solutions. Demands and models Identifies incomplete Identifies incomplete precision in responses but does not responses and asks student communication and in require student to to revise their response. mathematical formulate further solutions. response. Encourages students to identify when others are not addressing the question completely. Task: Task: Task: Requires students to Requires students to Requires students to analyze a task before analyze a task and identify identify the most automatically applying an algorithm. more than one approach efficient Teacher: to the problem. solution to the task. Identifies individual students’ efficient Teacher: Teacher: approaches, but does not expand Facilitates all students in understanding to developing reasonable and the rest of the class. efficient ways to accurately Demonstrates the same algorithm to perform basic operations. all related tasks although there may be Continuously questions other more effective students about the approaches. reasonableness of their intermediate results. Institute for Advanced Study/Park City Mathematics Institute Secondary School Teachers Program/Visualizing Functions Page 12 of 122 PRACTICE Look for and express regularity in repeated reasoning. Is disconnected from prior and future concepts. Has no logical progression that leads to pattern recognition. NEEDS IMPROVEMENT Task: Teacher: Does not show evidence of understanding the hierarchy within concepts. Presents or examines task in isolation. Task: EMERGING Hides or does not draw connections to prior or future concepts. Is overly repetitive or has gaps that do not allow for development of a pattern. (teacher does thinking) Teacher: Institute for Advanced Study/Park City Mathematics Institute Secondary School Teachers Program/Visualizing Functions Page 13 of 122 PROFICIENT Reviews prior knowledge and requires cumulative understanding. Lends itself to developing a pattern or structure. (teacher mostly models) Task: Teacher: Connects concept to prior and future concepts to help students develop an understanding of procedural shortcuts. Demonstrates connections between tasks. EXEMPLARY Addresses and connects to prior knowledge in a nonroutine way. Requires recognition of pattern or structure to be completed. (students take ownership) Task: Teacher: Encourages students to connect task to prior concepts and tasks. Prompts students to generate exploratory questions based on the current task. Encourages students to monitor each other’s intermediate results. Summer 2011 Mathematical Practices: A Walk-Through Protocol Mathematical Practices Observations *Note: This document should also be used by the teacher for planning and self-evaluation. MP.1. Make sense of problems and persevere in solving them Teachers are expected to______________: Provide appropriate representations of problems. Students are expected to______________: Connect quantity to numbers and symbols (decontextualize the problem) and create a logical representation of the problem at hand. Recognize that a number represents a specific quantity (contextualize the problem). Contextualize and decontextualize within the process of solving a problem. Teachers are expected to______________: Provide time for students to discuss problem solving. Students are expected to______________: Engage in solving problems. Explain the meaning of a problem and restate in it their own words. Analyze given information to develop possible strategies for solving the problem. Identify and execute appropriate strategies to solve the problem. Check their answers using a different method, and continually ask “Does this make sense?” MP.2. Reason abstractly and quantitatively. MP.3. Construct viable arguments and critique the reasoning of others. Students are expected to____________________________: Explain their thinking to others and respond to others’ thinking. Participate in mathematical discussions involving questions like “How did you get that?” and “Why is that true?” Construct arguments that utilize prior learning. Question and problem pose. Practice questioning strategies used to generate information. Analyze alternative approaches suggested by others and select better approaches. Justify conclusions, communicate them to others, and respond to the arguments of others. Compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument, explain what it is. CCSSM National Professional Development Page 14 of 122 Mathematical Practices MP.4. Model with mathematics. MP 5. Use appropriate tools strategically Observations Teachers are expected to______________: Provide opportunities for students to listen to or read the conclusions and arguments of others. Teachers are expected to______________: Detect possible errors by strategically using estimation and other mathematical knowledge. Consider available tools when solving a mathematical problem and decide when certain tools might be helpful, recognizing both the insight to be gained and their limitations. Students are expected to______________: Apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. Make assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. Experiment with representing problem situations in multiple ways, including numbers, words (mathematical language), drawing pictures, using objects, acting out, making a chart or list, creating equations, etc. Identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts, and formulas. Evaluate their results in the context of the situation and reflect on whether their results make sense. Analyze mathematical relationships to draw conclusions. Teachers are expected to______________: Provide contexts for students to apply the mathematics learned. Students are expected to______________: Use tools when solving a mathematical problem and to deepen their understanding of concepts (e.g., pencil and paper, physical models, geometric construction and measurement devices, graph paper, calculators, computer-based algebra or geometry systems.) CCSSM National Professional Development Page 15 of 122 Mathematical Practices MP.6. Attend to precision. MP.7. Look for and make use of structure. MP.8. Look for and express regularity in repeated reasoning. Observations Students are expected to______________: Use clear and precise language in their discussions with others and in their own reasoning. Use clear definitions and state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. Specify units of measure and label parts of graphs and charts. Calculate with accuracy and efficiency based on a problem’s expectation. Teachers are expected to______________: Emphasize the importance of precise communication. Students are expected to______________: Describe a pattern or structure. Look for, develop, generalize, and describe a pattern orally, symbolically, graphically and in written form. Relate numerical patterns to a rule or graphical representation Apply and discuss properties. Teachers are expected to______________: Provide time for applying and discussing properties. Students are expected to______________: Describe repetitive actions in computation Look for mathematically sound shortcuts. Use repeated applications to generalize properties. Use models to explain calculations and describe how algorithms work. Use models to examine patterns and generate their own algorithms. Check the reasonableness of their results. Teachers are expected to______________: CCSSM National Professional Development Page 16 of 122 Topic Intro Unit Similarity Trigonometric Ratios Other Types of Functions Comparing Functions and Different Representations of Quadratic Functions Modeling Unit and Project Quadratic Functions: Solving by Factoring Quadratic Functions: Completing the Square and the Quadratic Formula Probability Geometric Measurement Geometric Modeling Unit and Project High School Common Core Math II Curriculum Timeline Standards Covered G – SRT 1 G – SRT 1a G – SRT 1b G – SRT 6 G – SRT 2 G – SRT 3 G – SRT 4 G – SRT 7 G – SRT 5 Grading Period 1 1 No. of Days 5 20 G – SRT 8 1 20 A – CED 1 A – CED 4 A – REI 1 N – RN 1 N – RN 2 N – RN 3 F – IF 4 F – IF 5 F – IF 6 F – IF 7 F – IF 7a F– IF 9 F – IF 4 F – IF 7b F – IF 7e F – IF 8 F – IF 8b F– BF1 A– CED 1 A– CED 2 F– BF 1 F– BF 1a F – BF 1b F– BF 3 F – BF 1a F – BF 1b F – BF 3 A – SSE 1b N–Q2 2 15 F – LE 3 N– Q 2 S – ID 6a S – ID 6b A – REI 7 2 20 2 10 A – APR 1 A – REI 1 A – REI 4b F – IF 8a A – CED 1 A – SSE 1b A – SSE 3a 3 20 A – REI 1 A – REI 4 A – REI 4a A – REI 4b A – SSE 3b F – IF 8 F – IF 8a A – CED 1 N – CN 1 N – CN 2 N – CN 7 3 20 S – CP 1 S – CP 2 S – CP 3 G – GMD 1 S – CP 4 S – CP 5 S – CP 6 G – GMD 3 S – CP 7 4 20 4 10 G – MG 1 G – MG 2 G – MG 3 4 15 Page 17 of 122 High School Common Core Math II 1st Nine Weeks Scope and Sequence Intro Unit – IO (5 days) Topic 1 – Similarity (20 days) Geometry (G – SRT): 1) Similarity, Right Triangles, and Trigonometry: Understand similarity in terms of similarity transformations. G – SRT 1: Verify experimentally the properties of dilations given by a center and a scale factor. G – SRT 1a: A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. G – SRT 1b: The dilation of a line segment is longer or shorter in the ratio given by the scale factor. G – SRT 2: Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. G – SRT 3: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Geometry (G – SRT): 2) Similarity, Right Triangles, and Trigonometry: Prove theorems involving similarity. G – SRT 4: Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. G – SRT 5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Topic 2 – Trigonometric Ratios (20 days) Geometry (G – SRT): 3) Similarity, Right Triangles, and Trigonometry: Define trigonometric ratios and solve problems involving .right triangles G – SRT 6: Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Page 18 of 122 G – SRT 7: Explain and use the relationship between the sine and cosine of complementary angles. G – SRT 8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Page 19 of 122 High School Common Core Math II 2nd Nine Weeks Scope and Sequence Topic 3 – Other Types of Functions (15 days) Creating Equations (A – CED): 4) Create equations that describe numbers or relationships A – CED 1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. A – CED 4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R. Reasoning with Equations and Inequalities (A – REI): 5) Understand solving equations as a process of reasoning and explain the reasoning. A – REI 1: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. The Real Number System (N – RN): 6) Extend the properties of exponents to rational exponents. N – RN 1: Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5. N – RN 2: Rewrite expressions involving radicals and rational exponents using the properties of exponents. The Real Number System (N – RN): 7) Use properties of rational and irrational numbers. N – RN 3: Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Interpreting Functions (F – IF): 8) Interpret functions that arise in applications in terms of the context. F – IF 4*: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.* Interpreting Functions (F – IF): Page 20 of 122 9) Analyze functions using different representations. F – IF 7b: Graph square root, cube root, and absolute value functions. F – IF 7e: Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. F – IF 8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. F – IF 8b: Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay. Building Functions (F – BF): 10) Build a function that models a relationship between two quantities. F – BF 1: Write a function that describes a relationship between two quantities. F – BF 1a: Determine an explicit expression, a recursive process, or steps for calculation from a context. F – BF 1b: Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Building Functions (F – BF): 11) Build new functions from existing functions. F – BF 3: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Seeing Structure in Expressions (A – SSE): 12) Interpret the structure of expressions. A – SSE 1b: Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)n as the product of P and a factor not depending on P. Quantities (NQ): 13) Reason quantitatively and use units to solve problems. N – Q 2: Define appropriate quantities for the purpose of descriptive modeling. Topic 4 – Comparing Functions and Different Representations of Quadratic Functions (20 days) Page 21 of 122 Interpreting Functions (F – IF): 14) Interpret functions that arise in applications in terms of the context. F – IF 4*: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.* F – IF 5*: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.* F – IF 6: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Interpreting Functions (F – IF): 15) Analyze functions using different representations. F – IF 7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. F – IF 7a*: Graph linear and quadratic functions and show intercepts, maxima, and minima.* F – IF 9: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Creating Equations (A – CED): 16) Create equations that describe numbers or relationships. A – CED 1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. A – CED 2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Building Functions (F – BF): 17) Build a function that models a relationship between two quantities. F – BF 1: Write a function that describes a relationship between two quantities. F – BF 1a: Determine an explicit expression, a recursive process, or steps for calculation from a context. Page 22 of 122 F – BF 1b: Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Building Functions (F – BF): 18) Build new functions from existing functions. F – BF 3: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Linear and Exponential Models (F – LE): 19) Construct and compare linear and exponential models and solve problems. F- LE 3: Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. Quantities (N-Q): 20) Reason quantitatively and use units to solve problems. N – Q 2: Define appropriate quantities for the purpose of descriptive modeling. Interpreting Categorical and Quantitative Data (S – ID): 21) Summarize, represent, and interpret data on two categorical and quantitative variables. S – ID 6a: Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear and exponential models. S – ID 6b: Informally assess the fit of a function by plotting and analyzing residuals. Reasoning with Equations and Inequalities (A – REI): 22) Solve systems of equations. A – REI 7: Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = -3x and the circle x2 + y2 = 3. Modeling Unit and Project –(10 days) Page 23 of 122 High School Common Core Math II 3rd Nine Weeks Scope and Sequence Topic 5–Quadratic Functions – Solving by factoring (20 days) Arithmetic with Polynomials and Rational Expressions (A – APR): 23) Perform arithmetic operations on polynomials. A – APR 1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Reasoning with Equations and Inequalities (A – REI): 24) Understand solving equations as a process of reasoning and explain the reasoning. A – REI 1: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Reasoning with Equations and Inequalities (A – REI): 25) Solve equations and inequalities in one variable. A – REI 4b: Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Interpreting Functions (F – IF): 26) Analyze functions using different representations. F – IF 8a: Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Creating Equations (A – CED): 27) Create equations that describe numbers of relationships. A – CED 1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Seeing Structure in Expressions (A – SSE): 28) Interpret the structure of expressions. A – SSE 1b: Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)n as the product of P and factor not depending on P. Seeing Structure in Expressions (A – SSE): 29) Write expressions in equivalent forms to solve problems. Page 24 of 122 A – SSE 3a: Factor a quadratic expression to reveal the zeros of the function it defines. Topic 6– Quadratic Functions – Completing the Square/Quadratic Formula (20 days) Reasoning with Equations and Inequalities (A – REI): 30) Understand solving equations as a process of reasoning and explain the reasoning. A – REI 1: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Reasoning with Equations and Inequalities (A – REI): 31) Solve equations and inequalities in one variable. A – REI 4: Solve quadratic equations in one variable. A – REI 4a: Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. A – REI 4b: Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Seeing Structure in Expressions (A – SSE): 32) Write expressions in equivalent forms to solve problems. A – SSE 3b: Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. Interpreting Functions (F – IF): 33) Analyze functions using different representations. F – IF 8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. F – IF 8a: Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Creating Equations (A – CED): 34) Create equations that describe numbers or relationships. A – CED 1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. The Complex Number System (N – CN): Page 25 of 122 35) Perform arithmetic operations with complex numbers. N – CN 1: Know there is a complex number i such that i 2 number has the form a+bi with a and b real. 1 , and every complex N – CN 2: Use the relation i 2 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. The Complex Number System (N – CN): 36) Use complex numbers in polynomial identities and equations. N – CN 7: Solve quadratic equations with real coefficients that have complex solutions. Page 26 of 122 High School Common Core Math II 4th Nine Weeks Scope and Sequence Topic 7 –Probability (20 days) Conditional Probability and the Rules of Probability (S – CP): 37) Understand independence and conditional probability and use them to interpret data. S – CP 1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”). S – CP 2: Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. S – CP 3: Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. S – CP 4: Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from you school will favor science given that the student is in the tenth grade. Do the same for other subjects and compare the results. S – CP 5: Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. Conditional Probability and the Rules of Probability (S – CP): 38) Use the rules of probability to compute probabilities of compound events in a uniform probability model. S – CP 6: Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. S – CP 7: Apply the Addition Rule, P(A or B) = P(B) – P(A and B), and interpret the answer in terms of the model. Topic 8 – Geometric Measurement (10 days) Page 27 of 122 Geometric Measurement and Dimension (G – GMD): 39) Explain volume formulas and use them to solve problems. G – GMD 1: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. G – GMD 3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Geometric and Modeling Project-(15 days) *Modeling with Geometry (G – MG): 40) Apply geometric concepts in modeling situations. G – MG 1*: Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).* G – MG 2*: Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).* G – MG 3*: Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).* Page 28 of 122 COLUMBUS CITY SCHOOLS HIGH SCHOOL CCSS MATHEMATICS II CURRICULUM GUIDE Topic 7 Probability S-CP 1, 2, 3, 4, 5, 6, 7 CONCEPTUAL CATEGORY Statistics TIME GRADING RANGE PERIOD 20 4 Domain: Conditional Probability and the Rules of Probability (S – CP): Cluster 37) Understand independence and conditional probability and use them to interpret data. 38) Use the rules of probability to compute probabilities of compound events in a uniform probability model. Standards 37) Understand independence and conditional probability and use them to interpret data. S – CP 1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”). S – CP 2: Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. S – CP 3: Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. S – CP 4: Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from you school will favor science given that the student is in the tenth grade. Do the same for other subjects and compare the results. S – CP 5: Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. 38) Use the rules of probability to compute probabilities of compound events in a uniform probability model. S – CP 6: Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. S – CP 7: Apply the Addition Rule, P(A or B) = P(B) – P(A and B), and interpret the answer in terms of the model. CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 29 of 122 TEACHING TOOLS Vocabulary: A given B, association, categorical variable complement, complement, conditional, conditional probability, dependent, dependent events, element, empty set, event, experimental probability, independent, independent event, intersection, intersecting sets, joint probability, marginal probability, mutually exclusive, outcome, probability, quantitative, random sample, sample space, set, subset, theoretical probability, tree diagram, two-way frequency table, union, universal set, Venn diagram, , , P(A), P(A ∩ B), P(A ∪ B), P(A | B), { } Teacher Notes: I Can Statements S-CP1 I can describe subsets of a sample space in terms of outcomes, unions, intersections, and complements. I can find the theoretical probability of random phenomena. I can create the sample space for a random phenomenon. I can describe an event as a subset of a sample space using characteristics of the outcomes. I can describe the union of two or more events as a subset of a sample space using characteristics of the outcomes. I can describe the intersection of two or more events as a subset of a sample space using characteristics of the outcomes. I can construct a Venn diagram to find the union, intersection or complement of events. S-CP2 I can determine whether two events are independent based on their probability. I can use examples of random phenomena to show that two events are independent if the probability of their intersection is the product of their probabilities. I can use examples of random phenomena to show that if the product of two events is the probability of their intersection, the two events are independent. S-CP3 I can explain the conditional probability of A given B. I can explain independence of A and B using conditional probability. I can define independence and dependence between two events, A and B. I can define conditional probability. I can use a variety of methods to calculate probabilities (e.g., tree diagrams, Venn diagrams, two way tables and formulas). I can determine the probability of event A given event B by calculating the conditional probability. I can prove two events, A and B, are independent by applying the definition of conditional probability. I can describe the meaning of independence in terms of the formula P(A) = P(A|B). CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 30 of 122 S-CP4 I can construct and interpret two-way frequency tables of data when two categories are associated with each object. I can determine independence of events using a two-way table as a sample space. I can approximate conditional probabilities using a two-way table as a sample space. I can organize categorical data in two-way frequency tables. I can interpret joint probability in the context of the data. I can interpret marginal probability in the context of the data. I can interpret conditional probability in the context of the data. I can determine if two events are independent. S-CP5 I can distinguish between conditional probability and independence in everyday language and everyday situations. I can recognize the concepts of conditional probability based on everyday language and everyday situations. I can recognize the concepts of independence based on everyday language and everyday situations. I can use data to compare the values of A given B and B given A. I can use data to determine if A and B are independent. I can explain the concepts of conditional probability based on everyday language and everyday situations. I can explain the concepts of independence based on everyday language and everyday situations. S-CP6 I can determine the conditional probability of two events and interpret the solution within a given context. I can find the conditional probability between intersecting sets A and B (e.g., use a Venn Diagram or two-way table to find the conditional probability). I can interpret the conditional probability between sets A and B with intersecting sets. I can draw Venn Diagrams showing the relationship between sets A and B showing mutually exclusive, intersecting, and one a subset of the other. S-CP7 I can calculate the probability P(A or B) by using the Addition Rule. I can interpret the solution to P(A or B) in the given context. I know the Addition Rule. I can describe the conditions under which P(A or B) = P(A) + P(B) - P(A and B). I can describe the conditions under which P(A or B) = P(A) + P(B) I can apply the addition rule with intersecting sets A and B (e.g., a Venn Diagram or twoway table). I can interpret the Addition Rule with intersecting sets A and B. CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 31 of 122 Probability symbols: P(A) probability function probability of event A P(A ∩ B) probability of events intersection probability that of events A and B P(A ∪ B) probability of events Union probability that of events A or B P(A | B) conditional probability function probability of event A given event B occurred {} set notation a collection of elements A ∩ B intersection objects that belong to set A and set B A∪B union A⊆B subset subset has fewer elements or equal to the set A⊂B proper subset strict subset subset has fewer elements than the set A⊄B not subset left set not a subset of right set not complement Ac is not A c ` objects that belong to set A or set B. Outcome: an outcome is the result of an experiment. The set of all possible outcomes of an experiment is the sample space. Two events that have no common outcomes are mutually exclusive events. If two mutually exclusive events are the only ones that can possibly occur, we say these events are complementary. The complement of event A is Ā, where Ā represents the event that A does not occur. If A and Ā are complementary events, P(A) + P(Ā) = 1. The probability of the complement of A is P(A ) = 1 – P(A). Example: If P(A) = 0.65 then P(A ) is 0.35. At this website an example of both complementary and mutually exclusive events are given: http://www.shmoop.com/basic-statistics-probability/complementary-mutuallyexclusive-events.html At this site, there are worked solutions for examples and videos: http://www.onlinemathlearning.com/complementary-events.html CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 32 of 122 Notes, explaining complementary events, are given on this site: http://www.sunshinemaths.com/topics/probability/complementary-events/ Complement of an event – suppose A is an event in the universal set U, the complement of A (“not A”) consists of all the outcomes in U that are not in A. For example, if A is the event that two of three children are boys, then A c (complement of A) is the event that there are either zero, one, or three boys. For more info: http://www.mathgoodies.com/lessons/vo16/complement.html http://mathwordscom/c/complement_event.htm When knowledge of whether an event occurs affects the probability of a second event, the events are dependent. If knowledge of the first event does not affect the probability of the second, the events are independent. If A and B are independent events, then the probability that both A and B occur is P(A B) = P(A) P(B). Example: If P(A) = 1/3 and P(B) =1/4 then P(A B) = ¼ . A and B are independent events if and only if P(A and B) = P(A) P(B). Notes and examples are given for dependent and independent events: http://www.mathsisfun.com/data/probability-events-independent.html At the site, there are worked solutions for examples and videos: http://www.onlinemathlearning.com/independent-events.html Independent event – two events are independent if the outcome of one event has no effect on the outcome of the other. For more info: http://www.intermath-uga.gatech.edu/dictnary/descript.asp?termID=173 Conditional probability results when the probability of event B depends on event A. The conditional probability that B occurs and A has occurred, can be written as P(B A). For events A and B, P(A and B) = P(A) P(B A). P(A/B) = P(A and B)/P(B) where P(B) 0. Notes and an example for conditional probability can be found at this site: https://people.richland.edu/james/lecture/m170/ch05-cnd.html Notes on conditional probability can be found at the following sites: http://www.mathgoodies.com/lessons/vol6/conditional.html http://people.hofstra.edu/stefan_waner/realworld/tutorialsf3/frames6_5.html http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042jmathematics-for-computer-science-spring-2005/lecture-notes/l18_prob_cond.pdf http://www.regentsprep.org/regents/math/algebra/apr3/lconditional.htm Conditional Probability and the Rules of Probability: Different examples are given for conditional probability: http://www.shmoop.com/common-core-standards/ccss-hs-s-cp-2.html CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 33 of 122 Compound events – an event made of two or more simple events. For more info: http://www.harcourtschool.com/glossary/math2/define/gr6/compound_event6.html Conditional Probability – let A and B be two events. The probability that A will occur given that B has already occurred is the ‘conditional probability of A given B’ and is denoted by P A B For more info: http://www.mathwords.com/c/conditional_probability.htm http://www.cut-the-knot.org/fta/Buffon/ConditionalProbability.shtml The lesson contains rules for probability. http://stattrek.com/probability/probabilityrules.aspx Venn Diagrams and Set Notation explanations can be found at this site. http://www.purplemath.com/modules/venndiag2.htm Notes and Venn Diagrams used to explain probability. http://www.mathsisfun.com/data/probability-events-mutually-exclusive.html Subset - set A is the subset of B if all of the elements of set A are contend in set B. It is written as A B For more info: http://www.mathwords.com/s/subset.htm Http://mthworld.wolfram.com?Subset.html Union – combining the elements of two or more sets. Union is indicated by the For more info: http://ww.mathwords.com/u/union.htm (cup) symbol. Union of sets – the union of two sets A and B is the set obtained by combining the members of each set. If A = {1, 2, 3} and B= {2, 4, 6}, then A B = {1 2, 3, 4, 6}. For more info: http://www.intermath-uga.gatech.edu/dictnary/descript.asp?termID=376 Intersection of sets – the intersection of sets A and B, denoted by A B , is the set of elements that are in both A and B. For more info: http://www.mathwords.com/i/intersection.htm http://www.intermath-uga.gatech.edu/dictnary/descript.ap?termID=182 Vocabulary, suggested instructional strategies, sample formative assessment tasks, resources and problems tasks for S-CP 1 – 5 are provided at this site. http://www.schools.utah.gov/CURR/mathsec/Core/Secondary-II/II-4-S-CP-1-(1).aspx Probability The sample space is defined as the set of all possible outcomes of an event. Probability is the CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 34 of 122 overall likelihood of the occurrence of an event and can be determined in two ways. The first, experimental probability, is calculated by performing trials and is equal to the ratio of the number of favorable outcomes to the number of trials. Experimental probability favorable outcomes trials Theoretical probability assumes that all outcomes occur randomly and is equal to the ratio of the number of favorable outcomes in the event to the number of possible outcomes in the sample space. favorable outcomes Theoretical probability sample space or possible outcomes Probabilities can be calculated for simple single events such as figuring the chance of rolling a 3 on a fair number cube, or for a combination of events which are called compound events. There are two types of compound events. The first type, independent events are events where the outcome of one has no effect on the outcome of the other. The second type of compound events are ones in which the outcome of one event has an effect on the outcome of another event. These types are known as dependent events. Independent Events Box A contains: 3 pennies, 2 nickels, and 4 quarters Box B contains: 5 pennies, 3 nickels and 1 quarter If one coin is selected from each box, what is the probability a penny will be selected from Box A and a quarter from Box B? The probability of selecting a penny from Box A The probability of selecting a quarter from Box B # pennies # coins # quarters # coins 3 9 1. 3 1. 9 Notice that what is drawn from Box A has no impact on what is drawn from Box B. The probability that both will occur is: P A P B 1 1 3 9 1 . 27 In general, when events are independent, the probability of A and B, P(A and B) P( A) P( B) . Dependent Events If 2 coins are selected from Box A, which contains 3 pennies, 2 nickels and 4 quarters, what are the chances of selecting a penny followed by a quarter? CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 35 of 122 P( A) P( B) 3 4 9 8 12 72 1 6 or number of pennies number of coins 3 9 1 and 3 number of quarters number of coins 4 8 1 1 1 thus 3 2 2 1 6 By understanding the basics of how data is used to make claims, students will be able to determine for themselves the validity and sensibility of the statistical claims on which our society bases its decisions. This teacher introduction is not intended to cover all of the learning goals, but rather it is a supplement to the textbook. It is a tool to help with topics that may be unfamiliar, involve technology, or are not covered in depth in the textbook. Different Kinds of Probability Situations (Notes for teachers. This has been adapted from the book, Mathematics Is) As probability problems become more complex, it often helps to analyze them in terms of “kinds of events.” Mutually exclusive events, dependent events, and independent events will be considered. For each type of event, a rather informal description (not a precise mathematical definition) will be given and then some problems in which the particular type of event is involved will be analyzed. PROBLEM: A card is drawn at random from a deck of 52 playing cards. Find the probability that is a jack or a queen. SOLUTION: Use the obvious sample space where n(S) = 52. Let E1 be the event of “drawing a jack”;; therefore, E1 { js, jh, jd , jc} . Let E 2 be the event of “drawing a queen”;; thus, and it can be determined that: E 2 {qs, qh, qd , qc} . Therefore, E1 E 2 4 4 8 P( E1 E 2) P( E1) P( E 2) P( E1 E 2) 0 52 52 52 In the previous problem the events E1 and E 2 are called mutually exclusive events. In other words, if one card is drawn, then the events of “drawing a jack” and “drawing a queen” cannot both occur. Mathematically, this means that E1 E 2 . The following property is very helpful when working with mutually exclusive events. Property: If E1 and E 2 are mutually exclusive events ( E1 E 2 ), then P( E1 E 2) P( E1) P( E 2) . This property extends to any finite number of mutually exclusive events. CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 36 of 122 PROBLEM: Suppose a jar contains 5 white, 7 green, and 9 red marbles. If one marble is drawn at random from the jar, find the probability that it is white or green. SOLUTION: The events of “drawing a white marble” or “drawing a green marble” are mutually exclusive. Therefore, by the previously mentioned property, the probability of drawing a white or green marble is 5 7 12 4 P( W or G) 21 21 21 7 PROBLEM: A card is drawn at random from a deck of 52 playing cards. Find the probability that it is a jack, queen, or king. SOLUTION: The events of “drawing a jack”, “drawing a queen”, or “drawing a king” are mutually exclusive. Therefore, by the previously mentioned property: 4 4 4 12 3 P( j, q, or k ) 52 52 52 52 13 The concepts of dependent and independent events can be explained easily in terms of an example. Consider the “jar problem” again. Suppose there are 5 white, 7 green, and 9 red marbles in a jar. Now consider two sequences of events as follows: (1) Pull out a marble and then, without replacing it, pull out a second marble. (2) Pull out a marble and then replace it and pull out a second marble. (These two situations are often referred to as “drawing without replacement” and “drawing with replacement”.) In the first situation (without replacement), what happens on the second draw depends on what happens on the first draw. Thus, the events are dependent in (1). In the second situation, since the first marble drawn is being replaced, what occurs on the second draw is not dependent upon what happens on the first draw. Thus, the events are independent in (2). The following two properties indicate how probabilities dealing with dependent and independent events can be calculated. Property: If E1 and E 2 are dependent events, then the probability of E1 and E 2 occurring is given by P( E1) P( E1 / E 2) where P( E1 / E 2) represents the probability of E 2 occurring given that E1 has occurred. Property: If E1 and E 2 are independent events, then the probability of E1 and E 2 occurring is given by P( E1) P( E 2) . The following examples should help clarify the exact meaning of these properties. PROBLEM: A jar contains 5 white, 7 green, and 9 red marbles. If two marbles are drawn in succession, without replacement, find the probability that they are both white. SOLUTION: First it is necessary to recognize that the events are dependent. The probability CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 37 of 122 that the first marble drawn is white is 5 . The probability that the second marble drawn is white 21 4 . (After the first marble is drawn, there will be 20 marbles remaining, since the first 20 marble is not replaced. Of the 20 marbles remaining, 4 are white.) 5 4 20 1 Therefore: P(W and W ) . 21 20 420 21 is PROBLEM: A jar contains 5 white, 7 green, and 9 red marbles. If two marbles are drawn in succession, with replacement, find the probability that they are both white. SOLUTION: This time it is necessary to recognize that the events are independent. The 5 probability that the first marble drawn is white is . Now before the second marble is 21 drawn, the first marble is replaced. Therefore, the probability that the second marble drawn is 5 5 5 25 white is . Therefore: P(W and W ) 21 21 21 441 20 25 In the two previous problems, note that 0.048 and 0.057 . The probability of 420 441 drawing two successive white marbles is “a bit better” with replacement than without replacement. This should certainly seem reasonable. PROBLEM: Toss a pair of dice three times. Find the probability of getting a pair of 6’s all three times. SOLUTION: The three tosses of the pair of dice are independent events. Since the probability 1 of getting a pair of 6’s on one toss is , for three independent tosses 36 1 1 1 1 36 36 36 = 46,656 Now consider a few problems for which the concept of “mutually exclusive” is used, along with the ideas of dependent and independent events. As will be seen, some rather complex problems can be analyzed easily by combining these ideas. PROBLEM: A jar contains 5 white, 7 green, and 9 red marbles. If two marbles are drawn in succession, without replacement, find the probability that one of them is white and one of them is green. SOLUTION: The drawing of a white and green marble can occur in two different ways, namely, (1) by drawing a white marble first and then a green marble second, or (2) by drawing a green marble first and a white marble second. Thus, the events (1) and (2) are mutually exclusive, each of which is broken into dependent events of “first draw” and “second draw”. Therefore, the probability can be computed as follows: CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 38 of 122 5 21 7 20 7 21 5 20 35 420 35 420 70 420 1 6 P(white on the 1st) • P(green on the 2nd) + P(green on the 1st) • P(white on the 2nd) PROBLEM: Two cards are drawn in succession, with replacement, from a deck of 52 playing cards. Find the probability that a jack and a queen are drawn. SOLUTION: The drawing of a jack and a queen can occur two different ways, namely (1) a jack on the first draw and a queen on the second draw, or (2) a queen on the first draw and a jack on the second draw. These are mutually exclusive events and each is broken into independent events of “first draw” and “second draw” with replacement. Therefore the probability can be 4 4 4 4 16 16 32 2 computed as follows: . 52 52 52 52 2704 2704 2704 169 P(jack on first draw) • P(queen on second draw) + P(queen on first draw) • P(jack on second draw) 3 PROBLEM: The probability that Carol will win a certain game whenever she plays is . If 5 she plays twice, find the probability that she will win one and lose the other. (In this “game” you either win or lose; there are no ties.) SOLUTION: She can win one and lose the other game in two mutually exclusive ways. She can win the first game and lose the second game, or she can lose the first game and win the second game. Therefore, the probability of winning one game and losing one game is: 3 2 2 3 6 6 12 5 5 5 5 25 25 25 PROBLEM: Toss two coins. Find the probability of getting one head and one tail. SOLUTION: Suppose we toss a penny and a nickel are tossed. The event of “one head and one tail” can occur two mutually exclusive ways, namely, (1) a head on the penny and a tail on the nickel, or (2) a tail on the penny and a head on the nickel. Thus, the probability of getting one 1 1 1 1 1 1 2 1 head and one tail is: . 2 2 2 2 4 4 4 2 Misconceptions/Challenges: Students do not understand the difference between independent and dependent events. Students forget to subtract 1 from the number of possibilities each time they perform multiplication using the counting principal for dependent events. Students do not understand that the simple probability of two or more independent events can be found by adding the individual probabilities. CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 39 of 122 Students do not understand that the compound probability of two or more independent events is found by multiplying each individual probability by the other ones. Students make mistakes when using the addition rule. In the formula P(A U B) = P(A) + P(B) – P(A ∩ B), students leave off the last term. Instructional Strategies: The following instructional strategies cover several standards: 1) This website provides the core content, support for teachers, sample formative assessment tasks and problems tasks for S-CP 1-5. http://schools.utah.gov/CURR/mathsec/Core/Secondary-II/II-4-S-CP-1-(1).aspx 2) Modeling Conditional Probabilities 1, Lucky Dip, http://map.mathshell.org/materials/lessons.php?taskid=409&subpage=problem A lesson to help students to understand conditional probability. 3) Topic 7 in Analytical Geometry is a unit on probability. Students are provided with authentic tasks at this site to be used in math class using probability. Students will examine conditional probability and independence through this unit of study. This site provides vocabulary for the standards. The following lessons can be used from the site. Modeling Conditional Probabilities 2 has a lesson to help students understand conditional probability. How Odd? In this lesson students will determine the probability that one or both of the dice show odd values. They organize data in Venn diagrams and record their data in a two-way frequency table. This activity begins on page 8. This lesson covers standards S-CP 1 and SCP7. The Conditions are Right: Students learn about conditional probability in a series of activities in this unit. They practice recording their data in a two-way frequency table. This lesson begins on page 17. This lesson covers standards S-CP 2, 3, 4, 5, and 6. The Land of Independence: In this activity students show independence in probability by using the equation P(A B) = P(A) P(B). They are able to make statistical inferences with the data. This lesson begins on page 30. This lesson covers standards S-CP2, 3, 4, and 5. https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_9-12_AnalyticGeo_Unit7SE.pdf 4) CP 1-7 (A series of lessons they can choose from for the entire unit. Applications of Probability: Series of lessons on probability to download at this site. https://commoncoregeometry.wikispaces.hcpss.org/Unit+5 CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 40 of 122 The following instructional strategies contain printable worksheets: CP – 1: 1, 5, 6, 7, 9, 15 CP – 2: 1, 2, 3, 4, 5, 7, 8 CP – 3: 1, 2, 4, 5 CP – 4: 1, 2, 4, 5, 6, 7 CP – 5: 1, 2, 3, 4, 6 CP – 6: 1, 2, 3, 4, 5, 6, 7, 8 CP – 7: 1, 2, 3, 4 S – CP 1 1) Fried Freddy’s is a lesson on using samples to estimate probabilities on pages 17 – 24. http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod9_prob_tn_83 113.pdf a lesson on using samples to estimate probabilities on pages 17 – 24. 2) These websites have a series of videos to explain conditional probability. http://learnzillion.com/lessonsets/508-describe-events-as-subsets-of-a-sample-space-or-asunions-intersections-or-complements-of-other-events http://learnzillion.com/lessonsets/374-describe-events-as-subsets-of-a-sample-space-or-asunions-intersections-or-complements-of-other-events 3) The website below provides the lesson, practice and teacher resource for samples spaces and tree diagrams. http://regentsprep.org/Regents/math/ALGEBRA/APR4/indexAPR4.htm 4) lesson, practice and teacher resource for conditional probability. http://regentsprep.org/Regents/math/ALGEBRA/APR3/indexAPR3.htm 5) Conditional Probability Worksheet 1 provides a set of 10 practice problems http://www.shmoop.com/common-core-standards/handouts/s-cp_worksheet_1.pdf Answer sheet for Conditional Probability Worksheet 1. http://www.shmoop.com/common-core-standards/handouts/s-cp_worksheet_1_ans.pdf 6) Birthday Problem http://education.ti.com/en/us/activity/detail?id=723828BE0A8D47998C7F1A395B528327 An Nspire lesson for students to investigate the probability of two people having the same birthday in a crowd of a given size. 7) Conditional Probability http://education.ti.com/en/us/activity/detail?id=BC6AFC94F54A4AAB9C5284A64824A950 An Nspire lesson will investigate probability questions using tabular and graphical information. 8) On-line practice for compound events in finding the number of outcomes. http://www.ixl.com/math/algebra-1/compound-events-find-the-number-of-outcomes CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 41 of 122 9) Evaluating Statements About Probability: http://map.mathshell.org/materials/lessons.php?taskid=225 This lesson addresses common misconceptions about simple and compound events relating to probability. 10) Organize Sample Space Information: http://learnzillion.com/lessons/3957-organize-sample-space-information In this lesson students organize sample space information of independent, conditional, and disjoint events by using a Venn diagram and a table. This lesson may be used as teacher or class notes. 11) Identify the intersection of two events: http://learnzillion.com/lessons/3958-identify-the-intersection-of-two-events In this lesson students will learn how to identify the intersection of two subsets by organizing a sample space with a Venn diagram and two-way frequency table. This lesson may be used as teacher or class notes. 12) Identify the union of two events: http://learnzillion.com/lessons/3959-identify-the-union-of-two-events In this lesson students will learn how to identify the union of two subsets by organizing a sample space with a Venn diagram and a two-way frequency table. This lesson may be used as teacher or class notes. 13) Identify the complement of an event: http://learnzillion.com/lessons/3960-identify-the-complement-of-an-event In this lesson students will learn how to identify the complement of a subset in a sample space by organizing a sample space with a Venn diagram and a table. This lesson may be used as teacher or class notes. 14) Determine quantities belonging to subsets of a sample space: http://learnzillion.com/lessons/3961-determine-quantities-belonging-to-subsets-of-a-samplespace In this lesson studnts will learn how to determine quantities belonging to subsets of a sample space by identifying the union, intersection, and complement. This lesson may be used as teacher or class notes. 15) The Titanic 1: http://www.illustrativemathematics.org/illustrations/949 In this task students are asked a series of questions as they explore the concepts of probability as a fraction of outcomes. Students will develop an understanding of conditional probability. (activity sheet for this site included in this curriculum guide.) 16) On-line practice for theoretical probability is provided at this website. http://www.ixl.com/math/algebra-1/theoretical-probability CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 42 of 122 17) This website provides the lesson, practice and teacher resource for problems involving AND and OR. http://regentsprep.org/Regents/math/ALGEBRA/APR8/indexAPR8.htm 18) Calculate probabilities by using the complement and addition rule: http://learnzillion.com/lessons/2537-calculate-probabilities-by-using-the-complement-andaddition-rule In this lesson students will learn how to calculate probabilities by using the complement and addition rule. S – CP 2 1) Probability of Repeated Independent Events http://education.ti.com/en/us/activity/detail?id=278A996A76194FE3989499697FF4F9E0 This site provides a N-Spire lesson on investigating probability by simulating tossing a coin three times to calculate the probability of multiple independent events occurring. 2) Conditional Probability Worksheet 2 provides a set of 10 practice problems. http://www.shmoop.com/common-core-standards/handouts/s-cp_worksheet_2.pdf The answer key for Conditional Probability Worksheet 2 is provided at the website below. http://www.shmoop.com/common-core-standards/handouts/s-cp_worksheet_2_ans.pdf 3) Fried Freddy’s http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod9_prob_tn_83 113.pdf A lesson on using samples to estimate probabilities on pages 17 – 24. 4) Freddy Revisited http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod9_prob_tn_83 113.pdf A lesson on examining independence of events using two-way tables on pages 31 – 35. 5) Striving for Independence http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod9_prob_tn_83 113.pdf A lesson on using data in various representations to determine independence on pages 36 – 42. 6) Online Practice from IXL: http://www.ixl.com/math/algebra-1/identify-independent-and-dependent-events Practice is provided to identify independent and dependent events. 7) The Titanic 2: http://www.illustrativemathematics.org/illustrations/950 Students will develop their understanding of conditional probability and independence. (Activity sheet for this site included in this curriculum guide.). CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 43 of 122 8) Cards and Independence: http://www.illustrativemathematics.org/illustrations/943 Students explore the concept of independence of events in task. (An activity page for this site is included in this curriculum guide.) S – CP 3 1) * Freddy Revisited http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod9_prob_tn_83 113.pdf A lesson on examining independence of events using two-way tables on pages 31-35. 2) * Striving for Independence http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod9_prob_tn_83 113.pdf A lesson on using data in various representations to determine independence on pages 36 – 42. 3) This website provides a lesson on computing probability of basic problems and how to compute conditional probability http://alex.state.al.us/lesson_view.php?id=29364 4) Conditional Probability Worksheet 3 http://www.shmoop.com/common-core-standards/handouts/s-cp_worksheet_3.pdf A set of ten practice problems. Conditional Probability Worksheet 3: http://www.shmoop.com/common-core-standards/handouts/s-cp_worksheet_3_ans.pdf Answer key for the ten practice problems. 5) Rain and Lightning: http://www.illustrativemathematics.org/illustrations/1112 In this task students will explore different concepts of probability: (An activity page for this site is included in this curriculum guide.) S – CP 4 1) * Freddy Revisited http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod9_prob_tn_83 113.pdf A lesson on examining independence of events using two-way tables on pages 31 – 35. 2) * Striving for Independence http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod9_prob_tn_83 113.pdf A lesson on using data in various representations to determine independence on pages 36 – 42. CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 44 of 122 3) Pennies, Pennies and More Pennies 4) http://alex.state.al.us/lesson_view.php?id=23814 A cooperative group and interactive lesson using pennies to determine the geometric probability that the head of a pin will land on the penny and not on the floor between the pennies. 5) Conditional Probability Worksheet 4 http://www.shmoop.com/common-core-standards/handouts/s-cp_worksheet_4.pdf A set of ten practice problems. Conditional Probability Worksheet 4. http://www.shmoop.com/common-corestandards/handouts/s-cp_worksheet_4_ans.pdf The website below provides the answer key for Conditional Probability Worksheet 4. 6) Two-way Tables and Association http://education.ti.com/en/us/activity/detail?id=4DDB355E9373418EB24208275270E2F8 A lesson involving analyzing the results of a survey using a two-way frequency table. 7) The Titanic 3: http://www.illustrativemathematics.org/illustrations/951 In this last task about the Titanic, students have to formulate a plan to answer the question using a two-way frequency table. 8) False Positive: http://www.achieve.org/ccss-cte-classroom-tasks False positive and false negative results may occur during diagnostic tests. Students will investigate the accuracy of a medical test. 9) Conditional Probability Practice Problems: http://www.stat.illinois.edu/courses/stat100/Exams/Practice.pdf This site offers practice problems and an answer key. S – CP 5 1) Dartboard Probability http://alex.state.al.us/lesson_view.php?id=26387 A lesson in which students determine the probability of events presented in a geometric context. 2) Conditional Probability Worksheet 5 http://www.shmoop.com/common-core-standards/handouts/s-cp_worksheet_5.pdf A set of practice problems. Answer key http://www.shmoop.com/common-core-standards/handouts/s-cp_worksheet_5_ans.pdf Answer key for Conditional Probability Worksheet 5 CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 45 of 122 3) * Freddy Revisited http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod9_prob_tn_83 113.pdf A lesson on examining independence of events using two-way tables on pages 31 – 35. 4) * Striving for Independence http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod9_prob_tn_83 113.pdf A lesson on using data in various representations to determine independence on pages 36 – 42. 5) The following website provides the core content, support for teachers, sample formative assessment tasks and problems tasks for S-CP 6 -7 on pages 1 – 2. http://schools.utah.gov/CURR/mathsec/Core/Secondary-II/Unit-4---Use-the-Rules-ofProbability-to-Compute-P.aspx 6) Breakfast Before School: http://www.illustrativemathematics.org/illustrations/1019 In this task students learn to explain the meaning of independence in a simple context. (An activity page for this site included in this curriculum guide.) 7) Conditional Probability and Independence: http://www.montgomerycollege.edu/faculty/~jriseber/public_html/wquiz7-3a.htm This website offers interactive practice with probability with a quiz. 8) Conditional Probability and Probability of Simultaneous Events: http://www.shodor.org/interactivate/lessons/ConditionalProb/ This lesson contains several problems. Students use formulas connected with conditional probability and probability of simultaneous events. S – CP 6 1) TB or Not TB http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod9_prob_tn_83 113.pdf A lesson on estimating conditional probabilities and interpreting the meaning of a set of data on pages 3 – 8. 2) Chocolate versus Vanilla http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod9_prob_tn_83 113.pdf A lesson on examining conditional probability using multiple presentations on pages 9 – 16. 3) *Fried Freddy’s http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod9_prob_tn_83 113.pdf A lesson on using samples to estimate probabilities on pages 17 – 24. CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 46 of 122 4) Visualizing with Venn http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod9_prob_tn_83 113.pdf A lesson on creating Venn diagram’s using data while examining the addition rule for probability on pages 25 – 30. 5) Striving for Independence http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod9_prob_tn_83 113.pdf A lesson on using data in different representation to determine the independence of events on pages 36 – 42. 6) Functions Worksheet 6 http://www.shmoop.com/common-core-standards/handouts/s-cp_worksheet_6.pdf Sample problems. The answer key for Functions Worksheet 6 is provided at the link below. http://www.shmoop.com/common-core-standards/handouts/s-cp_worksheet_6_ans.pdf 7) How Do You Get to School? http://www.illustrativemathematics.org/illustrations/1025 In this task students use information in a two-way table to calculate the probability and conditional probability in a multiple choice question. (An activity sheet for this site is included in this curriculum guide.) 8) False Positive: http://www.achieve.org/ccss-cte-classroom-tasks False positive and false negative results may occur during diagnostic tests. Students will investigate the accuracy of a medical test. S – CP 7 1) *Fried Freddy’s http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod9_prob_tn_83 113.pdf A lesson on using samples to estimate probabilities on pages 17 – 24. 2) * Visualizing with Venn http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod9_prob_tn_83 113.pdf A lesson on creating Venn diagram’s using data while examining the addition rule for probability on pages 25 – 30. 3) Functions Worksheet 7 provides practice for students. http://www.shmoop.com/common-core-standards/handouts/s-cp_worksheet_7.pdf Functions Worksheet 7 Answer Key is provided at the website below. http://www.shmoop.com/common-core-standards/handouts/s-cp_worksheet_7_ans.pdf CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 47 of 122 4) Coffee at Mom’s Diner: http://www.illustrativemathematics.org/illustrations/1024 In this task students use the addition rule to compute a probability. (An activity sheet for this site is included in this curriculum guide.). Textbook References Textbook: Integrated Mathematics: Course 3, McDougal Littell (2002): pp. 386, 396 – 397, 397 – 398. Textbook: Algebra2, Glencoe (2005): pp.651-657, 658-663. Supplemental: Algebra 2, Glencoe (2005): Chapter 12 Resource Masters Reading to Learn Mathematics, pp. 721, 727 Study Guide and Intervention, pp.717-718, 723-724 Skills Practice, pp. 719, 725 Practice, pp. 720, 726 Enrichment: pp. 722, 728 On-line: www.pearsonsuccessnet.com Teacher resources: Modeling with Geometry 13-1 Experimental and Theoretical Probability 13-2 Probability Distributions and Frequency Tables 13-3 Permutations and Combinations 13-4 Compound Probability 13-5 Probability Models 13-6 Conditional Probability Formulas 13-7 Modeling Randomness Reteach/Extension Extension: 1) This lesson, Medical Testing, can be used to help students make sense of a real life situation, determine the math needed to apply to the problem, understand and calculate conditional probability. http://map.mathshell.org/materials/lessons.php?taskid=438&subpage=problem Reteach: 1) Conditional Probability, Independence, and Contingency Tables: These videos show students how to create two-way frequency tables and to approximate conditional probabilities. http://learni.st/users/S33572/boards/3419-conditional-probability-independence-and-contingencytables-common-core-standard-9-12-s-cp-4 2) Probability Theory: Sample lessons are given on topics such as the addition rules for probability, independent events, and conditional probability. http://www.mathgoodies.com/lessons/toc_vol6.html CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 48 of 122 S – CP 1 Name_______________________________________________ Date___________ Period_____ The Titanic 1 On April 15, 1912, the Titanic struck an iceberg and rapidly sank with only 710 of her 2,204 passengers and crew surviving. Data on survival of passengers are summarized in the table below. (Data source: http://www.encyclopedia-titanica.org/titanic-statistics.html) Survived Did not survive Total 201 123 324 Second class passengers 118 166 284 Third class passengers 181 528 709 500 817 1317 First class passengers Total passengers 1. Calculate the following probabilities. Round your answers to three decimal places. a. If one of the passengers is randomly selected, what is the probability that this passenger was in first class? b. If one of the passengers is randomly selected, what is the probability that this passenger survived? c. If one of the passengers is randomly selected, what is the probability that this passenger was in first class and survived? d. If one of the passengers is randomly selected from the first class passengers, what is the probability that this passenger survived? (That is, what is the probability that the passenger survived, given that this passenger was in first class?) e. If one of the passengers who survived is randomly selected, what is the probability that this passenger was in first class? CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 49 of 122 f. If one of the passengers who survived is randomly selected, what is the probability that this passenger was in third class? 2. Why is the answer to part (a.iv) larger than the answer to part (a.iii)? 3. Why is the answer to part (a.v) larger than the answer to part (a.vi)? 4. What other questions can you ask and answer using information in the given table? List at least three. CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 50 of 122 S – CP 1 Name_______________________________________________ Date___________ Period_____ The Titanic 1 Answer Key 1. Calculate the following probabilities: a. The probability of the passenger being in first class is the number of all first class passengers divided by total number of passengers , that is P(passenger being in first class)=3241317≈0.246 b. The probability that the passenger survived is the number of all passengers who survived divided by total number of passengers, that is P(passenger survived)=5001317≈0.380 c. This is the fraction of all passengers that are both in first class and survived, which is P[(passenger was in first class) and (passenger survived)]=2011317≈0.153 d. This is a conditional probability. To find the probability that the passenger survived, given this passenger was in first class, we calculate the fraction of first class passenger who survived, that is P(passenger survived|passenger was in first class)=201324≈0.620. e. This is a conditional probability: P(passenger was in first class|passenger survived). We can calculate it as the fraction of surviving passengers who were in first class, which is 201500≈0.402 f. This is a conditional probability: P(passenger was in third class|passenger survived). We can calculate it as the fraction of surviving passengers who were in third class, which is 181500≈0.362 2. Even though in both parts (a.iii) and (a.iv) we have the same numerator (201), in part (a.iii) the sample space consists of all the passengers, but in part (a.iv) the sample space is restricted to only the first class passengers. Since in part (a.iv) we divide by a smaller number, the answer in part (a.iv) is larger than in part (a.iii). 3. In both parts (a.v) and (a.vi) the sample space is restricted to all the passengers who survived. But since among that group there were more first class than third class passengers, the answer to part (a.v) is larger than the answer to part (a.vi). CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 51 of 122 4. There are many questions that can be answered using the given table. Possible answers may include, but are not limited, to the following: o If one of the passengers is randomly selected, what is the probability that this passenger was in second class? Answer: P(passenger was in second class)=2841317≈0.216. o If one of the passengers is randomly selected, what is the probability that this passenger was in second class and survived? Answer: P[(passenger was in second class) and (passenger survived)]=1181317≈0.090.. o If one of the passengers is randomly selected from among the second class passengers, what is the probability that this passenger survived? Answer: P(passenger survived | passenger was in second class)=118284≈0.415. o If one of the passengers who survived is randomly selected, what is the probability that this passenger was in second class? Answer: P(passenger was in second class | passenger survived)=118500≈0.236 CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 52 of 122 S – CP 2 Name_______________________________________________ Date___________ Period_____ The Titanic 2 On April 15, 1912, the Titanic struck an iceberg and rapidly sank with only 710 of her 2,204 passengers and crew surviving. Some believe that the rescue procedures favored the wealthier first class passengers. Data on survival of passengers are summarized in the table below. We will use this data to investigate the validity of such claims. (Data source: http://www.encyclopediatitanica.org/titanic-statistics.html) Survived Did not survive Total 201 123 324 Second class passengers 118 166 284 Third class passengers 181 528 709 500 817 1317 First class passengers Total passengers 1. Are the events “passenger survived” and “passenger was in first class” independent events? Support your answer using appropriate probability calculations. 2. Are the events “passenger survived” and “passenger was in third class” independent events? Support your answer using appropriate probability calculations. 3. Did all passengers aboard the Titanic have the same probability of surviving? Support your answer using appropriate probability calculations. CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 53 of 122 S – CP 2 Name_______________________________________________ Date___________ Period_____ The Titanic 2 Answer Key Answer sheet for Titanic 2 found at: http://www.illustrativemathematics.org/illustrations/950 1. We use the fact, that two events A and B are independent, if P(A|B)=P(A). In this case, we compare the conditional probability P(passenger survived|passenger was in first class) with the probability P(passenger survived). The probability of surviving, given that the passenger was in first class, is the fraction of first class passengers who survived. That is, we restrict the sample space to only first class passengers to obtain P(passenger survived|passenger was in first class)=201324≈0.620. The probability that the passenger survived is the number of all passengers who survived divided by the total number of passengers, that is P(passenger survived)=5001317≈0.380. Since 0.620≠0.380, the two given events are not independent. Moreover, we can say that being a passenger in first class increased the chances of surviving. Note, that we could also compare P(passenger was in first class/passenger survived)=201500≈0.402 and P(passenger was in first class)=3241317≈0.246. Again, since 0.402≠0.246, the two events are not independent. 2. Using similar reasoning as in part (a), we compare P(passenger survived|passenger was in third class)=181709≈0.255, and P(passenger survived)=5001317≈0.380. Since 0.255≠0.380, the two given events are not independent. Moreover, we can see that being a passenger in third class decreased the chances of being rescued. 3. One way to answer this question is to compare the probabilities of surviving for randomly chosen passengers in first, second, and third class, respectively. To do this, we calculate the following conditional probabilities: o In part (a) we calculated that P(passenger survived|passenger was in first class)=201324≈0.620. o The probability that the passenger survived, given that this passenger was in second class, is the fraction of passengers in second class who survived, that is P(passenger survived|passenger was in second class)=118284≈0.415. o In part (b) we calculated that P(passenger survived|passenger was in third class)=181709≈0.255. CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 54 of 122 Comparing these probabilities we can say that not all passengers aboard the Titanic had the same chance of surviving. More precisely, the chance of surviving depended on the class, with the first class passengers having the greatest, and the third class passengers having the smallest chance of being rescued. Note that there are different probabilities we could use to answer this question (for example we could compare probability that a randomly selected passenger survived P(passenger survived)=5001317≈0.380 with the conditional probability P(passenger survived|passenger was in first class)=201324≈0.620). However, the conclusion should always CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 55 of 122 S – CP 2 Name_______________________________________________ Date___________ Period_____ Cards and Independence One card is selected at random from the following set of 6 cards, each of which has a number and a black or white symbol: {2△,4□,8■,8⧫,5□,5■} 1. Let B be the event that the selected card has a black symbol, and F be the event that the selected card has a 5. Are the events B and F independent? Justify your answer with appropriate calculations. 2. Let B be the event that the selected card has a black symbol, and E be the event that the selected card has an 8. Are the events B and E independent? Justify your answer with appropriate calculations. CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 56 of 122 S – CP 2 Name_______________________________________________ Date___________ Period_____ Cards and Independence Answer Key The answers and commentary for this task is provided at the website: http://www.illustrativemathematics.org/illustrations/943 Solution: Solution 1 To test if the two events A and B are independent we check whether P(A and B)=P(A)⋅P(B). 1. Out of the six cards, there are three with a black symbol, two with a 5, and only one card has a 5 and a black symbol. Thus we have P(B)=36=12 P(F)=26=13 P(B and F)=16 Since P(B)⋅P(F)=12⋅13=16=P(B and F), the two events B and F are independent. 2. Out of the six cards, there are three with a black symbol, and two with an 8, both of them with a black symbol. Thus we have P(B)=36=12 P(E)=26=13 P(B and E)=26=13. Since P(B)⋅P(E)=12⋅13=16≠13=P(B and E), the two events B and E are not independent. Solution: Solution 2 To test if the two events A and B are independent we check whether P(A|B)=P(A). 1. Out of the six cards, there are two with a 5, so P(F)=26=13. Out of the three cards with black symbols, there is only one with a 5, so P(F|B)=13. Since P(F|B)=P(F), the two events B and F are independent. 2. Out of the six cards, there are two with an 8, so P(E)=26=13 . Out of the three cards with black symbols, there are two with an 8, so P(E|B)=23 . Since P(E|B)≠P(E) , the two events B and E are not independent. CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 57 of 122 S – CP 3 Name_______________________________________________ Date___________ Period_____ Rain and Lightening 1. Today there is a 55% chance of rain, a 20% chance of lightning, and a 15% chance of lightning and rain together. Are the two events “rain today” and ”lightning today” independent events? Justify your answer. 2. Now suppose that today there is a 60% chance of rain, a 15% chance of lightning, and a 20% chance of lightning if it’s raining. What is the chance of both rain and lightning today? 3. Now suppose that today there is a 55% chance of rain, a 20% chance of lightning, and a 15% chance of lightning and rain. What is the chance that we will have rain or lightning today? 4. Now suppose that today there is a 50% chance of rain, a 60% chance of rain or lightning, and a 15% chance of rain and lightning. What is the chance that we will have lightning today? CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 58 of 122 S – CP 3 Name_______________________________________________ Date___________ Period_____ Rain and Lightening Answer Key Answers and teacher notes can be found at the following website: http://www.illustrativemathematics.org/illustrations/1112 1. Given o P(rain)=.55, o P(lightning)=.2, and o P(lightning and rain)=.15. Two events are independent if P(lightning and rain)=P(lightning)⋅P(rain). Since P(lightning)⋅P(rain)=≠(.55)⋅(.2)=.11.15=P( lightning and rain) the two events are not independent. 2. Given o P(rain)=.6, o P(lightning)=.15, and o P(lightning | rain)=.2. We need to find P(rain and lightning). We use the formula P(lightning | rain)=P(lightning and rain)P(rain). Since we have two of the three pieces of information, we have to solve for the third one. Multiplying both sides of the equation by P(rain) we get P(lightning and rain)=P(lightning | rain)⋅P(rain)=(.2)⋅(.6)=.12 Answer: There is a 12% chance of both rain and lightning today. 3. Given o P(rain)=.55, o P(lightning)=.2, and o P(lightning and rain)=.15. CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 59 of 122 We need to find P(rain or lightning), which is the same as P(lightning or rain). Using the Addition Rule we obtain P(lightning or rain)=P(lightning)+P(rain)−P(lightning and rain)=.2+.55−.15=.6 Answer: There is a 60% chance of rain or lightning today. 4. Given o P(rain)=.5, o P(lightning)=.6, and o P(lightning and rain)=.15. We need to find P(lightning). We use the Addition Rule: P(rain or lightning)=P(rain)+P(lightning)−P(rain and lightning). Since we have three of the four pieces of information, we have to solve for the fourth one, the probability of lightning. Subtracting P(rain) and adding P(rain and lightning) to both sides of the equation, we obtain P(lightning)=P(rain or lightning)−P(rain)+P(rain and lightning)=.6−.5+15=.25 Answer: There is a 25% chance of lightning today. CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 60 of 122 S – CP 4 Name_______________________________________________ Date___________ Period_____ The Titanic 3 On April 15, 1912, the Titanic struck an iceberg and rapidly sank with only 710 of her 2,204 passengers and crew surviving. Some believe that the rescue procedures favored the wealthier first class passengers. Other believe that the survival rates can be explained by the ”women and children first” policy. Data on survival of passengers are summarized in the table below. Investigate what might and might not be concluded from the given data. (Data source: http://www.encyclopedia-titanica.org/titanic-statistics.html) Survived Did not survive Total Children in first class 4 1 5 Women in first class 139 4 143 Men in first class 58 118 176 Children in second class 22 0 22 Women in second class 83 12 95 154 167 Children in third class 30 50 80 Women in third class 91 88 179 Men in third class 60 390 450 Total passengers 500 817 1317 Men in second class 13 CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 61 of 122 S – CP 4 Name_______________________________________________ Date___________ Period_____ The Titanic 3 Answer Key The solution and teacher notes can be found at the following website: http://www.illustrativemathematics.org/illustrations/951 Commentary This is the last task in the series of three, which ask related questions, but use different levels of scaffolding. This task uses a more detailed version of the data table. This is a very open ended task. It poses the question, but the students have to formulate a plan to answer it, and use the two-way table of data to find all the necessary probabilities. The special emphasis is on developing their understanding of conditional probability and independence. This task could be used as a group activity where students cooperate to formulate a plan of how to answer the question and calculate the appropriate probabilities. The task could lead to extended class discussions about the different ways of using probability to justify general claims (i.e. Can we really say that first class passengers had a larger chance of being rescued? Why or why not? What was the role of gender in the rescue procedures?) The other tasks in this series are S-CP.1,4,6 The Titanic 1 and S-CP.3,4,5,6 The Titanic 2. Solution Note that there are different ways we could answer this question. First, we ignore the gender and compare the probability of surviving for a randomly chosen passenger in first class, to the probabilities of surviving for randomly selected second and third class passengers, respectively. To do this, we calculate the following conditional probabilities. The probability that the passenger survived, given that the this passenger was in first class, is the fraction of first class passengers who survived, that is P(passenger survived|passenger was in first class)=201324≈0.620. The probability that the passenger survived, given that the this passenger was in second class, is the fraction of second class passengers who survived, that is P(passenger survived|passenger was in second class)=118284≈0.415. The probability that the passenger survived, given that the this passenger was in third class, is the fraction of second class passengers who survived, that is P(passenger survived|passenger was in third class)=181709≈0.255. CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 62 of 122 These probabilities suggest that the chance of surviving depended on the class, with the first class passengers having the greatest, and the third class passengers having the smallest chance of surviving. Now we want to investigate if what appears to point to class discrimination could be explained in terms of gender of passengers. First, we ignore the class and take into consideration only the gender of the passengers. We can calculate the following conditional probabilities to compare the probabilities of surviving for a randomly selected child, woman, and man. The probability that the passenger survived, given that the this passenger was a child, is the fraction of children who survived, that is: P(passengersurvived|passengerwasachild)=56107≈0.523. The probability that the passenger survived, given that the she was a woman, is the fraction of women who survived, that is: P(passengersurvived|passengerwasawoman)=313417≈0.751. The probability that the passenger survived, given that the he was a man, is the fraction of men who survived, that is: P(passenger survived|passenger was a man)=131793≈0.165. These probabilities suggest that gender was an important factor with rescue procedures, with both women and children having a larger chance of surviving than men. Now we look at gender distribution between the three classes. Since women and children had large chance of surviving, we can consider them together and calculate the following conditional probabilities: The probability that the passenger was a child or a woman, given that the this passenger was in first class, is the fraction of first class passenger who were children or women, that is: P(passenger was child or woman|passenger was in first class)=148324≈0.457. The probability that the passenger was a child or a woman, given that the this passenger was in second class, is the fraction of second class passenger who were children or women, that is: P(passenger was child or woman|passenger was in second class)=117284≈0.412. The probability that the passenger was a child or a woman, given that the this passenger was in third class, is the fraction of third class passenger who were children or women, that is: P(passenger was child or woman|passenger was in third class)=259709≈0.365. Looking at these probabilities we can see that there were larger proportions of children and women in first and second class, than in third class. Now the question is if the difference in gender distribution together with different survival rates for different genders was the only CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 63 of 122 reason to explain the different survival rates for different classes. If that were the case, that is, if class was not a factor in rescue procedures, then any child, regardless of the class in which the child traveled, would have roughly the same chance of surviving (≈0.523). The same should hold for all women and all men. Thus we compare the survival rates for passengers of the same gender, but from different classes. First, consider children: The probability that a child survived, given that the child was in first class: P(child survived|child was in first class)=45≈0.800. The probability that a child survived, given that the child was in second class: P(child survived|child was in second class)=2222≈1.0. The probability that a child survived, given the child was in third class: P(child survived|child was in third class)=3080≈0.375. We can see that the children in first and second class had a larger chance of surviving than the children in the third class. We can do similar calculations for women and men. The probability that a woman survived, given that she was in first class: P(woman survived|woman was in first class)=139143≈0.972. The probability that a woman survived, given that she was in second class: P(woman survived|woman was in second class)=8395≈0.874. The probability that a woman survived, given she was in third class: $P(\text{woman survived} | \text{woman was in third class}) = \frac{91}{179} \approx 0.508. The probability that a man survived, given that he was in first class: P(man survived|man was in first class)=58176≈0.330. The probability that a man survived, given that he was in second class: P(man survived|man was in second class)=13167≈0.078. The probability that a man survived, given he was in third class: P(man survived|man was in third class)=60450≈0.133. The final conclusion: The survival rates for women (0.751) and children (0.523) were larger than for men (0.1651), which suggests that the rescue procedures favored women and children. However, a random passenger in first class of any gender had at least twice as large of a chance of surviving as a passenger of the same gender in third class. For example, 0.972 survival rate for women in first class compared to 0.508 survival rate for women in third class. Such discrepancy CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 64 of 122 cannot be justified with different gender distribution between the three classes. Therefore, the given data also suggests that the rescue procedures favored the first class passengers. CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 65 of 122 S – CP 5 Name_______________________________________________ Date___________ Period_____ Breakfast Before School Task: On school days, Janelle sometimes eats breakfast and sometimes does not. After studying probability for a few days, Janelle says, “The events ‘I eat breakfast’ and ‘I am late for school’ are independent.” Explain what this means in terms of the relationship between Janelle eating breakfast and her probability of being late for school in language that someone who hasn’t taken statistics would understand. CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 66 of 122 S – CP 5 Name_______________________________________________ Date___________ Period_____ Breakfast Before School Answer Key Teacher notes and the solution can be found at the following website: http://www.illustrativemathematics.org/illustrations/1019 Teacher Notes The purpose of this task is to assess a student's ability to explain the meaning of independence in a simple context. You might consider expanding this task by asking students to also explain what it would mean to say that the two events are not independent. You might also provide some probability values (such as "on days when Janelle eats breakfast, she is late to school about 20% of the time and on days when she does not eat breakfast, she is late to school about 10% of the time." Then ask if this indicates that the two events "eats breakfast" and "late to school" are independent or not independent. Solution: Possible Solution If the events “I eat breakfast” and “I am late for school” are independent, that means that the probability that one of the events happens is not influenced by whether or not the other event has happened (or is happening). In this case, what Janelle is saying is that the probability that she is late for school is the same whether or not she eats breakfast. CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 67 of 122 S – CP 6 Name_______________________________________________ Date___________ Period_____ How do you get to school? All of the upper-division students (juniors and seniors) at a high school were classified according to grade level and response to the question "How do you usually get to school?" The resulting data are summarized in the two-way table below. Car Bus Walk Totals Juniors 96 122 56 274 Seniors 184 58 30 272 Totals 280 180 86 546 1. If an upper-division student at this school is selected at random, what is the probability that this student usually takes a bus to school? 1. 58272 2. 180546 3. 122274 4. 58122 5. 272546 CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 68 of 122 2. If a randomly selected upper-division student says he or she is a junior, what is the probability that she usually walks to school? 1. 56546 2. 86546 3. 56274 4. 86274 5. 274546 CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 69 of 122 S – CP 6 Name_______________________________________________ Date___________ Period_____ How do you get to school? Answer Key Student solutions and teacher notes are provided at: http://www.illustrativemathematics.org/illustrations/1025 Teacher Notes: This task is designed as an assessment item. It requires students to use information in a two-way table to calculate a probability and a conditional probability. Although the item is written in multiple choice format, the answer choices could be omitted to create a short-answer task. Solution: 1. Answer is (ii). P(Bus)=180546=0.330 2. Answer is (iii). P(Walks|Junior)=56274=0.204 CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 70 of 122 S – CP 7 Name_______________________________________________ Date___________ Period_____ Coffee at Mom’s Diner At Mom’s diner, everyone drinks coffee. Let C= the event that a randomly-selected customer puts cream in their coffee. Let S= the event that a randomly-selected customer puts sugar in their coffee. Suppose that after years of collecting data, Mom has estimated the following probabilities: P(C)=0.6P(S)=0.5P(C or S)=0.7 Estimate P(C and S) and interpret this value in the context of the problem. CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 71 of 122 S – CP 7 Name_______________________________________________ Date___________ Period_____ Coffee at Mom’s Diner Answer Key Solution and teacher notes at the website below: http://www.illustrativemathematics.org/illustrations/1024 Teacher Notes: This task assesses a student's ability to use the addition rule to compute a probability and to interpret a probability in context. While the most obvious use of this task is as an assessment item, it could also be used in instruction as a practice problem, Solutions Using the addition rule, P(C or S)=P(C)+P(S)−P(C and S), it follows that: 0.7P(C and S)=0.6+0.5−P(C and S)=0.6+0.5−0.7=0.4 The probability that a randomly-selected customer at Mom’s has both cream and sugar in his or her coffee is 0.4. CCSSM II Probability S-CP 1, 2, 3, 4, 5, 6, 7 Quarter 4 Columbus City Schools 1/31/14 Page 72 of 122 COLUMBUS CITY SCHOOLS HIGH SCHOOL CCSS MATHEMATICS II CURRICULUM GUIDE Topic 8 CONCEPTUAL CATEGORY TIME Geometric Geometry RANGE Measurement G-GMD 10 days 1, 3 Domain: Geometric Measurement and Dimension (G-GMD): Cluster 39) Explain volume formulas and use them to solve problems. GRADING PERIOD 4 Standards 40) Explain volume formulas and use them to solve problems. G – GMD 1: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. G – GMD 3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. CCSSM II Geometric Measurement G-GMD 1, 3 Quarter 4 Columbus City Schools 1/31/14 Page 73 of 122 TEACHING TOOLS Vocabulary: Area, Argument (Informal Mathematical), Base, Cavalieri’s Principle, Circle, Circumference, Cone, Cube, Cylinder, Diameter, Dissection, Limit, Line, Parallel, Pi, Prism, Pyramid, Radius, Solid, Volume Teacher Notes: Circle Formulas: Circumference of a Circle C= 2 r r = radius of circle 2 Area of a Circle A= r r = radius of circle Volume Formulas: Cylinder V r 2 h r = radius of cylinder h = height of cylinder 1 Pyramid V Bh B = area of base of pyramid h = height of pyramid 3 1 2 Cone V= V r h r = radius of cone h = height of cone 3 4 3 Sphere V= V r = radius of sphere r 3 Cavalieri’s Principle (aka Method of Indivisibles): 2-dimensional case: Suppose that within a given plane, two regions are included between a pair of parallel lines. If every line parallel to these two lines intersects both regions in line segments of equal length, then the two regions have equal areas. 3-dimensional case: Suppose that within a given three-dimensional space, two regions (solids) are included between two parallel planes. If every plane parallel to these two planes intersects both regions in cross-sections of equal area, then the two regions have equal volumes. Dissection Arguments: In this context, “dissection argument” is a name given to geometric arguments for perimeter/circumference, area, and any other relevant geometric property that rely on breaking a shape or object into pieces. The crux of such arguments tends to rely on clever rearrangements of said pieces. Limit Arguments: In this context, “limit argument” is a name given to arguments that rely on something approaching infinity (or any given finite value). For example, a polygon with an arbitrarily large number of sides has a perimeter approaching that of the circle which it inscribes. CCSSM II Geometric Measurement G-GMD 1, 3 Quarter 4 Columbus City Schools 1/31/14 Page 74 of 122 Dissection arguments also often rely on limiting processes, as objects might be dissected an arbitrarily large number of times. Relationships Between Two-Dimensional and Three-Dimensional Objects Although many such relationships might be described, this cluster of standards is concerned with a few in particular. Notably, all 3-dimensional objects contain infinitely many 2-dimensional cross sections and 2-dimensional objects can be moved through 3-dimensional space (emphasis on rotations rather than translations in this case) in order to generate 3-dimensional objects. Note that some engineering software uses the latter process to create objects. Additionally, volume formulas are built from area formulas. For example, the formula for the volume of a cylinder contains the formula for the area of a circle. I can statements: G-GMD1 I can explain why the formula for the circumference of a circle works. I can explain why the formula for the area of a circle works. I can explain why the formulas for a cylinder, pyramid and cone work. I know the formula for the circumference of a circle. I know the formula for the area of a circle. I know the formula for the volume of a cylinder. I know the formula for the volume of a pyramid. I know the formula of the volume of a cone. I know Cavalieri's principle. I can use dissection, Cavalieri's principle, and/or limits to justify an informal argument for the circumference of a circle, the area of a circle, and the volume of a cylinder, pyramid, and cone. G-GMD3 I can apply formulas for cylinders, pyramids, cones and spheres to problems. I know the formula for the volume of a cone. I know the formula for the volume of a cylinder. I know the formula for the volume of a pyramid. I know the formula for the volume of a sphere. I know the formula for the surface area of a sphere. I can apply the formula for the volume of a cone, cylinder, pyramid, and sphere to solve problems. I can apply the formula for the surface area of a sphere to solve problems. CCSSM II Geometric Measurement G-GMD 1, 3 Quarter 4 Columbus City Schools 1/31/14 Page 75 of 122 Misconceptions/Challenges: G-GMD 1 This standard involves the derivation of specific area and volume formulae using informal mathematical argument. Student misconceptions will thus involve application and understanding of appropriate arguments. One of the primary challenges will simply be that students must have reached a sufficient level of abstract thought in order to understand these arguments. G-GMD 3 This standard involves raw application of volume formulae. As such, students will encounter the same challenges they would with the application of any other set of formulae. Notably, they may have difficulty picking the correct formula to use and they may have trouble figuring out where to plug in given values. If the question asks for a value that is not already isolated as the formula is written then they could also have trouble isolating said value. For example, a student given the circumference of a circle may have trouble solving for the radius. Instructional Strategies: The following instructional strategies cover several standards: 1) This website provides the core content, support for teachers, sample formative assessment tasks and problems tasks for G – GMD 4. http://schools.utah.gov/CURR/mathsec/Core/Secondary-Mathematics-III/Unit-4---VisualizeRelationships-Between-Two-Dimen.aspx The following instructional strategies contain printable worksheets: G – MD 1: 2, 3, 6, 7, 12, 13, 14, 15 G – MD 3: 1, 2, 3, 4, 5, 8, 11 G-GMD 1 1) Pi Line Lesson from NCTM: http://illuminations.nctm.org/LessonDetail.aspx?id=L575 Students collect data from several circular objects, plot Circumference vs. Diameter, and find the line of best fit to discover the slope is Pi. 2) “Discovering Pi” (included in this curriculum guide) activity develops the understanding of the relationship between circumference and diameter or radius. Students will use various items to “measure” the circumference and diameter of the circle, they then find their ratio. Students should start with the largest item of measuring such as a penny then using smaller and smaller items to improve the ratio of Circumference: Diameter. The more accurate their measurements, the closer their ratio should be to Pi. Possible items to use for measuring might be buttons, beads, sequins, or string. Another version of this strategy could be to use larger and larger circles and only one item as a measuring tool. Be sure students understand their ratio is approaching Pi because their measuring tool is more accurate as the circle gets larger, not because the ratio is different for each circle. 3) Apple Pi Lesson NCTM: http://illuminations.nctm.org/LessonDetail.aspx?ID=U159 This unit consists of two discovery lessons, one for Circumference and the other for Area. CCSSM II Geometric Measurement G-GMD 1, 3 Quarter 4 Columbus City Schools 1/31/14 Page 76 of 122 4) Circumference and Area of Circles, Volume of Cylinder, Cavalieri’s Principle: https://www.softchalkcloud.com/lesson/files/AG9dfynWh2i01b/Informal_Arguement_Form ulas_print.html This article describes how to prove (informally) the formulae for the circumference of a circle, the area of a circle, and the volume of a cylinder using limit arguments, dissection arguments, and Cavalieri’s Principle. 5) Volume of a Pyramid and a Cone from Enriching Mathematics: http://nrich.maths.org/1408 This article describes how to prove (informally) the volume formulae for pyramids and cones using relatively simple mathematics including dissection arguments 6) Exploring Cavalieri's Principle http://education.ti.com/en/us/activity/detail?id=DEA761964EB640B7A90CD198F42EF964 An Nspire activity where students explore the principle for cross sectional area and volume. 7) Unit 3: Extending to three dimensions: https://commoncoregeometry.wikispaces.hcpss.org/Pilot+Teacher+Resources This website contains several full length lessons, including instructional strategies, warm up activities, and homework activities, and will link you to several additional websites. 8) The following website provides explanations, examples and misconceptions for the geometric modeling and dimension standards on pages 225 – 234. http://katm.org/wp/wpcontent/uploads/flipbooks/High-School-CCSS-Flip-Book-USD-259-2012.pdf#page=225 9) This website provides the core content, support for teachers, sample formative assessment tasks and problems tasks for G – GMD 1 and 3. http://schools.utah.gov/CURR/mathsec/Core/Secondary-II/II-6-G-GMD-1.aspx 10) Cavalieri’s Principle https://commoncoregeometry.wikispaces.hcpss.org/Pilot+Teacher+Resources This lesson provides an activity where students stack congruent objects to from cylinders and prisms. They use provided tools to calculate the volume of the cylinders and prisms. 11) This website provides the core content, support for teachers, sample formative assessment tasks and problems tasks for G – GMD 1 on page 1. http://schools.utah.gov/CURR/mathsec/Core/Secondary-II/Unit-4---Use-Probability-toEvaluate-Outcomes-of-D.aspx 12) NCTM Geometry to Algebra 2 Pick a Peck of Packages Track & Field Pepperoni Packing Problem of the Day #27-29, 51-53, 61 CCSSM II Geometric Measurement G-GMD 1, 3 Quarter 4 Columbus City Schools 1/31/14 Page 77 of 122 13) A Day at the Beach is a performance task http://schools.nyc.gov/NR/rdonlyres/C03D80B2-9213-43A9-AAA3BB0032C62F4F/139657/NYCDOE_G10_ADayattheBeach_FINAL1.pdf 14) Planning the Gazebo http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod7_circgeo_se _83113.pdf This is a lesson in developing formulas for the perimeter and area of regular polygons on pages 22 - 25. 15) Sand Castles http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod7_circgeo_se _83113.pdf This is a lesson working with volume and scaling to see relationships on pages 52 – 57. 16) Water Tank Creations Part I, http://alex.state.al.us/lesson_view.php?id=8979 This is a lesson where students study the surface area and volume of three-dimensional shapes by creating a water tank using the shapes. 17) Car Caravan, http://threeacts.mrmeyer.com/carcaravan/ A picture of a circle made of cars is given and the question of how many cars it contains is asked. Students are asked a series of additional questions. G-GMD 3 1) Mathematical Assessment Project: Calculating Volumes of Compound Shapes-Glasses: http://map.mathshell.org/materials/download.php?fileid=684 This lesson looks at the volumes of drinking glasses with compound shapes. 2) Best Size Cans: http://map.mathshell.org/materials/tasks.php?taskid=284&subpage=expert This learning task asks students to optimize the amount of aluminum used in a can with a specified volume. 3) A Day at the Beach is a performance task. http://schools.nyc.gov/NR/rdonlyres/C03D80B29213-43A9-AAA3-BB0032C62F4F/139657/NYCDOE_G10_ADayattheBeach_FINAL1.pdf 4) Fish Tank https://commoncoregeometry.wikispaces.hcpss.org/Pilot+Teacher+Resources This lesson presents a situation and students are asked to describe what will happen and justify their answers using mathematical calculations. 5) Dan Meyer: You Pour, I Choose: http://mrmeyer.com/threeacts/youpourichoose/ Students watch soda being poured into two different glasses and have to choose which one they would want. If you could transform the glass to hold more soda, should you double the CCSSM II Geometric Measurement G-GMD 1, 3 Quarter 4 Columbus City Schools 1/31/14 Page 78 of 122 height or double the radius. Lastly, they decide what the height of the soda could be in each glass so that they have the same amount of soda. 6) Sand Castles is a lesson working with volume and scaling to see relationships on pages 52 – 57. (This is also found in G – MD 1.) http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod7_circgeo_se _83113.pdf 7) Water Tank https://commoncoregeometry.wikispaces.hcpss.org/Pilot+Teacher+Resources This is a problem situation about Earthoid, a water storage tank. Students are given several mathematical tasks to calculate for this sphere. This sphere was paint to resemble a globe. 8) Doctor’s Appointment http://www.illustrativemathematics.org/illustrations/527 In this task students are given a real world situation using a geometric model where students use geometric reasoning and their knowledge to volume formulas for cylinders and cones. In this task students will use the dimensions of a cone to discover different measurements related to the cone. The task uses a paper drinking cup as its model. (There is an activity sheet for this site included in this curriculum guide.) 9) Centerpiece http://www.illustrativemathematics.org/illustrations/514 This is a real world example where students determine the different volumes possible for the center piece given a cylinder and glass vase. (An activity sheet for his site is included in this curriculum guide.) CCSSM II Geometric Measurement G-GMD 1, 3 Quarter 4 Columbus City Schools 1/31/14 Page 79 of 122 Reteach/Extension Reteach: 1) Solid of revolutions: https://www.khanacademy.org/math/calculus/solid_revolution_topic/solid_of_revolution/v/solidof-revolution--part-1 Video series that is all about revolving shapes around axes. Heavy on calculus but that can be skipped over. 2) Glencoe Geometry Prerequisite Skills Workbook, pp. 21-22, 37-38, 43-44, 97-100 3) Using Cavalieri's Principle to Determine Volumes: This site offers a series of videos to explain using Cavalieri’s Principle. http://learni.st/users/S33572/boards/3155-using-cavalieri-s-principle-to-determine-volumescommon-core-standard-9-12-g-gmd-2 4) The Circumference and Area of Circles: This sheet provides practice for students in determining the circumference and area of circles. http://www.kutasoftware.com/FreeWorksheets/GeoWorksheets/11Circumference%20and%20Area%20of%20Circles.pdf 5) Volume of Prisms and Cylinders: This sheet provides practice for students in determining the volume of prisms and cylinders. http://www.kutasoftware.com/FreeWorksheets/GeoWorksheets/10Volume%20of%20Prisms%20and%20Cylinders.pdf 6) Volume of Pyramids and Cones: This sheet provides practice for students in determining the volume of pyramids and cones. http://www.kutasoftware.com/FreeWorksheets/GeoWorksheets/10Volume%20of%20Pyramids%20and%20Cones.pdf 7) Spheres: This sheet provides practice for students in determining the volume and surface area of spheres. http://www.kutasoftware.com/FreeWorksheets/GeoWorksheets/10-Spheres.pdf 8) Have students work on “Volume” (included in this Curriculum Guide) Extensions: 1) Tennis Balls in a Can http://www.illustrativemathematics.org/illustrations/512 A real life situation using a can of tennis balls and an x-ray machine at the airport to see the cross sections of the can, and to determine what the cross section would look like in different circumstances CCSSM II Geometric Measurement G-GMD 1, 3 Quarter 4 Columbus City Schools 1/31/14 Page 80 of 122 2) Glencoe Geometry Enrichment, pp. 628, 728, 734, 740 3) Glencoe Algebra 2 Textbook, section 8-6 (pp. 449-454) Resource Masters, pp. 485-490, 512 Teaching Algebra With Manipulatives Masters, pp. 268-269 4) Cutting Conics, http://illuminations.nctm.org/Lesson.aspx?id=2907 This is a lesson where students explore and discover conic sections by cutting a cone with a plane. Circles, ellipses, parabolas, and hyperbolas are examined using the Conic Section Explorer tool: 5) Using Conic Section Explorer, students explore different conic sections and their graphs. They use the Cone View to change the cone and the plane that creates the cross section and then observe how the graph changes: http://illuminations.nctm.org/Activity.aspx?id=3506 Textbook References Textbook: Geometry, Glencoe (2005): Textbook, Sections 11-3 (pp. 610-616), 13-1 (pp. 688-694), 13-2 (pp. 696-701), and 13-3 (pp. 702-706) Resource Masters, pp. 623-627, 655, 657, 723-727, 729-733, 735-739, 767, 769 School-to-Career Masters, pp. 25-26 Teaching Geometry with Manipulatives Masters, pp. 1, 9, 18, 184-186, 189, 206-207 Graphing Calculator and Computer Masters, pp. 41-42 On-line: Pearson: www.pearsonsuccessnet.com Teacher Resources 9-5 Surface areas and volumes of spheres Textbook: Mathematics II Common Core, Pearson (2014) pp. 755-758, 781-790, 791-793, 794802. CCSSM II Geometric Measurement G-GMD 1, 3 Quarter 4 Columbus City Schools 1/31/14 Page 81 of 122 G – GMD 1 Name_______________________________________________ Date___________ Period_____ Discovering Pi 1. Place the items you are using as a measuring tool around the circumference of the circle. Count the number of items. Record it in the table below. 2. Place the same items across a diameter of the circle. Count the number of items. Record it in the table below. 3. Calculate the ratio . Record it in the table below. 4. Repeat steps #1-3 for 2 other measuring tools. Name of measuring tool 5. The ratio of Circumference Diameter 𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 is getting closer to what irrational number? 6. Since it is difficult to measure the length of the circumference exactly, how could we calculate it using the length of the diameter? 7. How could we calculate the length of the circumference using the radius? CCSSM II Geometric Measurement G-GMD 1, 3 Quarter 4 Columbus City Schools 1/31/14 Page 82 of 122 CCSSM II Geometric Measurement G-GMD 1, 3 Quarter 4 Columbus City Schools 1/31/14 Page 83 of 122 G – GMD 1 Name_______________________________________________ Date___________ Period_____ Discovering Pi Answer Key 1. Place the items you are using as a measuring tool around the circumference of the circle. Count the number of items. Record it in the table below. 2. Place the same items across a diameter of the circle. Count the number of items. Record it in the table below. 3. Calculate the ratio . Record it in the table below. 4. Repeat steps #1-3 for 2 other measuring tools. Name of measuring tool Circumference Diameter 𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 Answers May Vary 5. The ratio of is getting closer to what irrational number? π 6. Since it is difficult to measure the length of the circumference exactly, how could we calculate it using the length of the diameter? C= πD 7. How could we calculate the length of the circumference using the radius? C=2πr CCSSM II Geometric Measurement G-GMD 1, 3 Quarter 4 Columbus City Schools 1/31/14 Page 84 of 122 G – GMD 3 Name_______________________________________________ Date___________ Period_____ Volume Draw a figure for each of the following problems. Label all moving parts with a variable and draw an arrow to show the direction of movement. Label all fixed quantities with their value. Show all work clearly and be prepared to explain your methods. 1. A tank in the shape of a square-based pyramid has the tip of the pyramid pointed downward. The area of the base is 81 ft2 and it is 12 ft. deep. Water is flowing into the tank at the rate of 30 . a) Express the volume of water in the tank as a function of its height. b) Find the volume of water in the tank when the water is 6 ft. deep. c) Two minutes after the water hit the 6 ft. deep mark, what is the volume of water in the tank? Find the depth of the water and the area of the water’s surface at this time. 2. A circle is inscribed in a square as shown. The perimeter of the square is increasing at a constant rate of 16 inches per second. As the circle expands, the square expands to maintain the condition of tangency. a) If the perimeter of the square is 40 inches, find the circumference of the circle. b) After 3 seconds have passed, find the new circumference of the circle. c) After 3 seconds have passed, find the area of the region enclosed between the circle and the square. CCSSM II Geometric Measurement G-GMD 1, 3 Quarter 4 Columbus City Schools 1/31/14 Page 85 of 122 3. At a sand and gravel plant, sand is falling off a conveyor and onto a conical pile at the rate of 5π . The diameter of the base of the cone is approximately three times the height. a) Express the volume of sand in the pile as a function of its radius. b) Find the volume of sand in the pile when the base is 15 ft. wide. c) Four minutes after the pile was 15 ft. wide, find the volume and corresponding radius of the sand pile. 4. As shown in the figure below, water is draining from a conical tank with a height of 12 ft. and diameter of 8 ft. into a cylindrical tank that has a base with area 400π ft2. The height of the water in the conical tank is dropping at a rate of 2 . a) Write an expression for the volume of water in the conical tank as a function of h. b) Assuming the conical tank was initially full, find the volume of water that it held. c) After 3 hours, find the volume of water in the conical tank. CCSSM II Geometric Measurement G-GMD 1, 3 Quarter 4 Columbus City Schools 1/31/14 Page 86 of 122 d) Assuming the cylindrical tank was initially empty, find the height of the water in that tank after 3 hours. 5. A cup for a frozen drink is a cylinder with a hemispherical lid. The total height of the cup with the lid is 9.5 in. while the height of the cup without the lid is 8 in. a) Find the capacity of the cup with the lid when it is completely full. b) After 2 minutes, ¼ of the drink is gone. What is the height of the drink left in the cup? c) After 2 more minutes, the same amount of the drink is gone. What is the height of the drink left in the cup? 6. A basketball has a circumference of 29.5 in. Air is being added at a rate of 2 a) What is the original volume? . b) After 10 sec., what is the circumference of the basketball? CCSSM II Geometric Measurement G-GMD 1, 3 Quarter 4 Columbus City Schools 1/31/14 Page 87 of 122 G – GMD 3 Name_______________________________________________ Date___________ Period_____ Volume Answer Key Draw a figure for each of the following problems. Label all moving parts with a variable and draw an arrow to show the direction of movement. Label all fixed quantities with their value. Show all work clearly and be prepared to explain your methods. 1. A tank in the shape of a square-based pyramid has the tip of the pyramid pointed downward. The area of the base is 81 ft2 and it is 12 ft. deep. Water is flowing into the tank at the rate of 30 . a) Express the volume of water in the tank as a function of its height. 𝒔 𝒉 𝟑 s = side length of base = 𝟏𝟐 𝒔 = 𝟒𝒉 𝟗 h = height of pyramid 𝟏 𝟏 𝟑 V = 𝟑 𝒔𝟐 𝒉 = 𝟑 𝟒 𝟑 𝒉𝟐 𝒉 = 𝟏𝟔 𝒉𝟑 b) Find the volume of water in the tank when the water is 6 ft. deep. 𝟑 V = 𝟏𝟔 𝟔𝟑 = 𝟒𝟎. 𝟓 𝒇𝒕𝟑 c) Two minutes after the water hit the 6 ft. deep mark, what is the volume of water in the tank? Find the depth of the water and the area of the water’s surface at this time. 𝟑 V = 40.5 𝒇𝒕𝟑 + 60 𝒇𝒕𝟑 = 100.5 𝒇𝒕𝟑 = 𝟏𝟔 𝒉𝟑 h = 8.1 ft s = 6.1 ft A = 37.21 𝒇𝒕𝟐 2. A circle is inscribed in a square as shown. The perimeter of the square is increasing at a constant rate of 16 inches per second. As the circle expands, the square expands to maintain the condition of tangency. a) If the perimeter of the square is 40 inches, find the circumference of the circle. s = side length of square P = 4s = 40 in s = 10 in = d r = 5 in r = radius of circle C = 2πr = 10π in ≈ 31.4 in d = diameter of circle b) After 3 seconds have passed, find the new circumference of the circle. P = 40 in + 48 in = 88 in 4s = 88 in s = 22 in = d r = 11 in C = 22 π in ≈ 69.1 in c) After 3 seconds have passed, find the area of the region enclosed between the circle and the square. 𝑨𝒓𝒆𝒂𝒔𝒒𝒖𝒂𝒓𝒆 − 𝑨𝒓𝒆𝒂𝒄𝒊𝒓𝒄𝒍𝒆 𝟐𝟐𝟐 − π𝟏𝟏𝟐 ≈ 103.9 𝒊𝒏𝟐 CCSSM II Geometric Measurement G-GMD 1, 3 Quarter 4 Columbus City Schools 1/31/14 Page 88 of 122 3. At a sand and gravel plant, sand is falling off a conveyor and onto a conical pile at the rate of 5π . The diameter of the base of the cone is approximately three times the height. a) Express the volume of sand in the pile as a function of its radius. 𝟑 𝟐 d = diameter of the base d = 3h r = 𝟐h h = 𝟑r 𝟏 𝟏 𝟐 𝟐 h = height of the cone V = 𝟑π𝒓𝟐 h = 𝟑π𝒓𝟐 𝟑 𝒓 = 𝟗 𝝅𝒓𝟑 r = radius of circle b) Find the volume of sand in the pile when the base is 15 ft. wide. 𝟐 r = 7.5 ft. V = 𝟗 𝝅(𝟕. 𝟓 𝒇𝒕)𝟑 = 93.75 𝝅 ft 3≈ 294.5 ft 3 c) Four minutes after the pile was 15 ft. wide, find the volume and corresponding radius of the sand pile. V = 93.75 𝝅 ft 3+ 20 𝝅 ft 3= 113.75 𝝅 𝒇𝒕3 ≈ 357.4 𝒇𝒕3 𝟐 V = 𝟗 𝝅𝒓𝟑 = 113.75 𝝅 𝒇𝒕3 r=8 4. As shown in the figure below, water is draining from a conical tank with a height of 12 ft. and diameter of 8 ft. into a cylindrical tank that has a base with area 400π ft2. The height of the water in the conical tank is dropping at a rate of 2 . a) Write an expression for the volume of water in the conical tank as a function of h. 𝒅 𝒉 𝟐 𝟏 d = diameter of the base of the cone 𝟖 = 𝟏𝟐 d = 𝟑𝒉 r = 𝟑𝒉 𝟏 𝟏 𝟏 𝟐 𝟏 r = radius of the base of the cone V = 𝟑π𝒓𝟐 h = 𝟑π 𝟑 𝒉 𝒉 = 𝟐𝟕 𝝅𝒉𝟑 h = height of the cone b) Assuming the conical tank was initially full, find the volume of water that it held. 𝟏 V = 𝟐𝟕 𝝅(𝟏𝟐 𝒇𝒕)𝟑 = 64 𝝅 ft3≈ 201.1 ft 3 c) After 3 hours, find the volume of water in the conical tank. 𝟏 h=6 V = 𝟐𝟕 𝝅(𝟔 𝒇𝒕)𝟑 = 8 𝝅 ft3≈ 25.1 ft 3 d) Assuming the cylindrical tank was initially empty, find the height of the water in that tank after 3 hours. 𝟕 𝑽𝒄𝒚𝒍𝒊𝒏𝒅𝒆𝒓 = 64 𝝅 ft3 - 8 𝝅 ft3 = 56 𝝅 ft3 = πr2h = 400πh h = 𝟓𝟎 𝒇𝒕 CCSSM II Geometric Measurement G-GMD 1, 3 Quarter 4 Columbus City Schools 1/31/14 Page 89 of 122 5. A cup for a frozen drink is a cylinder with a hemispherical lid. The total height of the cup with the lid is 9.5 in. while the height of the cup without the lid is 8 in. a) Find the capacity of the cup with the lid when it is completely full. 𝟐 r = radius of sphere/cylinder 𝑽𝒉𝒆𝒎𝒊𝒔𝒑𝒉𝒆𝒓𝒆 = 𝟑 𝝅𝒓𝟑 𝑽𝒄𝒚𝒍𝒊𝒏𝒅𝒆𝒓 = πr2h h = height of cylinder 𝟐 V = 𝟑 𝝅(𝟏. 𝟓 𝐢𝐧)𝟑 + π(𝟏. 𝟓 𝐢𝐧)𝟐 (𝟖 𝐢𝐧)= 𝟖𝟏 𝟒 π 𝐢𝐧𝟑 ≈ 63.6 𝐢𝐧𝟑 b) After 2 minutes, ¼ of the drink is gone. What is the height of the drink left in the cup? 𝟖𝟏 𝟖𝟏 𝟐𝟒𝟑 V = 𝟒 π 𝐢𝐧𝟑 - 𝟏𝟔π 𝐢𝐧𝟑 = 𝟏𝟔 π 𝐢𝐧𝟑 ≈ 47.7 𝐢𝐧𝟑 πr2h =π(𝟏. 𝟓 𝐢𝐧)2 h = 𝟐𝟒𝟑 𝟏𝟔 π 𝐢𝐧𝟑 h= 𝟐𝟕 𝟒 𝒊𝒏 = 6.75 in c) After 2 more minutes, the same amount of the drink is gone. What is the height of the drink left in the cup? 𝟐𝟒𝟑 𝟖𝟏 𝟖𝟏 V = 𝟏𝟔 π 𝐢𝐧𝟑 − 𝟏𝟔π 𝐢𝐧𝟑 = 𝟖 π 𝐢𝐧𝟑 πr2h =π(𝟏. 𝟓 𝐢𝐧)2 h = 𝟖𝟏 𝟖 𝟗 π 𝐢𝐧𝟑 h = 𝟐 𝒊𝒏 = 4.5 in 6. A basketball has a circumference of 29.5 in. Air is being added at a rate of 2 a) What is the original volume? 𝟓𝟗 C = 2 πr = 29.5 in r = 𝟒𝛑 𝐢𝐧 ≈ 4.7 in 𝟒 𝟒 V = 𝟑 𝛑𝒓𝟑 = 𝟑 𝛑 𝟓𝟗 𝐢𝐧 𝟒𝛑 𝟑 . ≈ 𝟒𝟑𝟑. 𝟓 𝐢𝐧𝟑 b) After 10 sec., what is the circumference of the basketball? 𝟒 V = 𝟑 𝛑𝒓𝟑 ≈ 453.5 in3 r ≈ 4.76 in C ≈ 29.9 in CCSSM II Geometric Measurement G-GMD 1, 3 Quarter 4 Columbus City Schools 1/31/14 Page 90 of 122 G – GMD 3 Name_______________________________________________ Date___________ Period_____ Doctor’s Appointment Jared is scheduled for some tests at his doctor’s office tomorrow. His doctor has instructed him to drink 3 liters of water today to clear out his system before the tests. Jared forgot to bring his water bottle to work and was left in the unfortunate position of having to use the annoying paper cone cups that are provided by the water dispenser at his workplace. He measures one of these cones and finds it to have a diameter of 7cm and a slant height (measured from the bottom vertex of the cup to any point on the opening) of 9.1cm. Note: 1 cm3=1 ml 1. How many of these cones of water must Jared drink if he typically fills the cone to within 1cm of the top and he wants to complete his drinking during the work day? 2. Suppose that Jared drinks 25 cones of water during the day. When he gets home he measures one of his cylindrical drinking glasses and finds it to have a diameter of 7cm and a height of 15cm. If he typically fills his glasses to 2cm from the top, about how many glasses of water must he drink before going to bed? CCSSM II Geometric Measurement G-GMD 1, 3 Quarter 4 Columbus City Schools 1/31/14 Page 91 of 122 G – GMD 3 Name_______________________________________________ Date___________ Period_____ Doctor’s Appointment Answer Key Solution and teacher notes are at the website below: http://www.illustrativemathematics.org/illustrations/527 Teacher Notes: The purpose of the task is to analyze a plausible real-life scenario using a geometric model. The task requires knowledge of volume formulas for cylinders and cones, some geometric reasoning involving similar triangles, and pays attention to reasonable approximations and maintaining reasonable levels of accuracy throughout. Submitted by Patrick Barringer to the Third illustrative Mathematics Task writing contest. Solutions 1. Since the slant height and radius are known (9.1 cm and 3.5 cm, respectively), we can use the Pythagorean Theorem to find the height of the cone as displayed in the equation. (Solving x2+3.52=9.12 gives x=8.4). Since we leave the top centimeter of the height of the cup empty, the height of the filled portion would then be 7.4cm. Using similar triangles we find that the radius of the filled area is in ratio 7.4:8.4 to the original radius of 3.5cm, i.e., the radius of the filled region is 7.48.4⋅3.5=3712≈3.08 cm. Using the formula for the volume of a cone we find that each cone of water contains approximately 13πr2h=13π(3.08)2(7.4)≈73.51 cubic centimeters (i.e., milliliters) of water. As Jared needs to drink 3000ml of water, dividing 300073.51≈40.81 tells us that he needs to drink

Top subcategories

Top subcategories

Top subcategories

Top subcategories

Top subcategories

Top subcategories

Top subcategories

Top subcategories

Download 1 - Instructional Information Services