Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
List of important publications in mathematics wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Fundamental theorem of algebra wikipedia , lookup
Elementary algebra wikipedia , lookup
System of polynomial equations wikipedia , lookup
Quadratic reciprocity wikipedia , lookup
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? 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?” What does proficiency look like in this practice? What actions might the teacher make that inhibit the students’ use of this practice? 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? 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 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? 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. 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: 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 NEEDS IMPROVEMENT Task: Make sense of problems and persevere in solving them. EMERGING PROFICIENT EXEMPLARY (teacher does thinking) (teacher mostly models) (students take ownership) Task: Is strictly procedural. Does not require students to check solutions for errors. Task: Is overly scaffolded or procedurally “obvious”. Requires students to check answers by plugging in numbers. 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. 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. 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. Allows for multiple entry points and solution paths. Requires students to defend and justify their solution by comparing multiple solution paths. Teacher: Teacher: Page 11 of 144 Task: 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. 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. PRACTICE Reason abstractly and quantitatively. EMERGING NEEDS IMPROVEMENT Lacks context. Does not make use of multiple representations or solution paths. Is embedded in a contrived context. Teacher: Expects students to Teacher: (teacher does thinking) Task: Task: Does not expect students to interpret representations. Expects students to memorize procedures withno connection to meaning. model and interpret tasks using a single representation. Explains connections between procedures and meaning. Task: Task: Construct viable arguments and critique the reasoning of others. Is either ambiguously stated. Teacher: EXEMPLARY (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 Expects students to expressed with multiple interpret, model, and representations. connect multiple representations. Prompts students to Teacher: Expects students to articulate connections interpret and model between algebraic using multiple procedures and contextual representations. meaning. Provides structure for students to connect algebraic procedures to contextual meaning. Links mathematical solution with a question’s answer. Task: Is not at the appropriate level. Teacher: Does not ask students to present arguments or solutions. Expects students to follow a given solution path without opportunities to make conjectures. PROFICIENT Teacher: Avoids single steps or routine algorithms. Identifies students’ assumptions. Models evaluation of student arguments. Asks students to explain their conjectures. Teacher: 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. Institute for Advanced Study/Park City Mathematics Institute Secondary School Teachers Program/Visualizing Functions Helps students differentiate between assumptions and logical conjectures. Prompts students to evaluate peer arguments. Expects students to formally justify the validity of their conjectures. Summer 2011 Page 12 of 144 PRACTICE Task: Model with mathematics. EMERGING NEEDS IMPROVEMENT (teacher does thinking) Task: Requires students to Requires students to identify variables and to perform necessary computations. Teacher: Identifies appropriate Does not discuss Requires students to identify variables and to compute and interpret report findings using a mixture of representations. Verifies that students have identified appropriate variables and procedures. appropriateness of model. Illustrates the relevance of the mathematics involved. Explains the EXEMPLARY (students take ownership) Task: Requires students to identify variables, compute and interpret results, and results. Teacher: variables and procedures for students. PROFICIENT (teacher mostly models) Task: Requires students to appropriateness of model. 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 identify extraneous or missing information. Teacher: their choice of variables and procedures. Gives students opportunity to evaluate the appropriateness of model. Asks questions to help students identify appropriate variables and procedures. Facilitates discussions in evaluating the appropriateness of model. Use appropriate tools strategically. Task: Task: Does not incorporate additional learning tools. Teacher: Task: Lends itself to one learning tool. additional learning tools. Does not involve mental Teacher: learning tools. computations or estimation. Does not incorporate Task: Lends itself to multiple Gives students opportunity to develop fluency in mental computations. Demonstrates use of appropriate learning tool. Chooses appropriate learning tools for student use. Models error checking by estimation. tools (i.e., graph paper, calculator, manipulative). Requires students to demonstrate fluency in Teacher: Requires multiple learning mental computations. Teacher: Allows students to choose appropriate learning tools. Creatively finds appropriate alternatives where tools are not available. Page 13 of 144 PRACTICE Attend to precision. EMERGING NEEDS IMPROVEMENT (teacher does thinking) Task: Task: Gives imprecise instructions. Has overly detailed or wordy instructions. Teacher: Look for and make use of structure. Teacher: Does not intervene when students are being imprecise. Does not point out instances when students fail to address the question completely or directly. Requires students to automatically apply an algorithm to a task without evaluating its appropriateness. Teacher: Teacher: Inconsistently intervenes when students are imprecise. Identifies incomplete responses but does not require student to formulate further response. Task: Task: 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. PROFICIENT (teacher mostly models) (students take ownership) Task: Task: Has precise instructions. Includes assessment Teacher: criteria for communication of ideas. Consistently demands precision in Teacher: communication and in Demands and models mathematical solutions. precision in Identifies incomplete communication and in responses and asks mathematical solutions. student to revise their Encourages students to response. identify when others are not addressing the question completely. Task: Requires students to analyze a task before automatically applying an algorithm. 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. Institute for Advanced Study/Park City Mathematics Institute Secondary School Teachers Program/Visualizing Functions EXEMPLARY Task: Requires students to analyze a task and identify more than one approach to the problem. Facilitates all students in developing reasonable and efficient ways to accurately perform basic operations. Continuously questions students about the reasonableness of their intermediate results. Requires students to identify the most efficient solution to the task. Teacher: Teacher: 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. Summer 2011 Page 14 of 144 PRACTICE Look for and express regularity in repeated reasoning. EMERGING NEEDS IMPROVEMENT Is disconnected from prior and future concepts. Has no logical progression that leads to pattern recognition. Does not show evidence of understanding the hierarchy within concepts. Presents or examines task in isolation. Is overly repetitive or has gaps that do not allow for development of a pattern. Teacher: Hides or does not draw connections to prior or Teacher: (teacher does thinking) Task: Task: future concepts. PROFICIENT EXEMPLARY (teacher mostly models) (students take ownership) Task: Task: Reviews prior knowledge Addresses and connects to and requires cumulative prior knowledge in a non understanding. routine way. Lends itself to Requires recognition of developing a pattern or structure to be pattern or structure. completed. Teacher: Teacher: Connects concept to prior and future concepts to help students develop an understanding of procedural shortcuts. Demonstrates connections between tasks. Institute for Advanced Study/Park City Mathematics Institute Secondary School Teachers Program/Visualizing Functions 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 Page 15 of 144 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 1 , and every complex number has the form a+bi with a and b real. 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. x 2 2 x 3 x 3 x 1 x 2 7 x 12 ( x 3)( x 4) Factor the numerator and denominator. x2 2 x 3 x 1 ; x 3, 4 Since the denominator’s factors are (x – 3) and (x – 4), x 3 and 2 x 7 x 12 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 = 2r 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 x2 + 9x + 18 = 0 (x – 5)(x + 4) = 0 x = 5 and x = -4 (x + 6)(x + 3) = 0 x = -6 and x = -3 2x2 + 9x – 5 = 0 6x2 + 7x = 20 (x + 5)(2x – 1) = 0 x = -5 and x = 1/2 (3x – 4)(2x + 5) = 0 x = 4/3 and x = -5/2 2x2 – 15x = 27 x2 = 7x – 12 (2x + 3)(x – 9) = 0 x = -3/2 and x = 9 (x – 4)(x – 3) = 0 x = 4 and x = 3 12x2 – 2x – 4 = 0 x(4x + 1) = 5 (4x + 2)(3x – 2) = 0 x = -1/2 and x = 2/3 (x – 1)(4x + 5) = 0 x = 1 and x = -5/4 x(15x + 1) – 2 = 0 x(125 – x) = 2500 (3x – 1)(5x + 2) = 0 x = 1/3 and x = -2/5 (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 11. 8 12 16 20 2 4 6 12. 10 10 10 8 8 8 6 6 6 4 4 4 2 2 2 -4 -2 4 -10 10. -10 -8 -6 18 15 12 9 6 3 8 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 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 graph of a function are the of the equation f x 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) ± 2 2 real solutions y = -(x – 2)2 None (2, 0) 2 1 real solution y = -x2 – 4x + 4 None (2, -8) -2 ± 2 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 2x 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 6x2 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 x2 2x 8 x y -4 -3 -2 -1 0 1 2 0 -5 -8 -9 -8 -5 0 Vertex: (- 1, - 9) Range: - 9, Solve: x2 2 x 8 0 (x + 4)(x – 2) = 0 x + 4 = 0 or x – 2 = 0 x = - 4 or x = 2 Domain: - , 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 y 6 x 2 5x 4 3. 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 x-intercept(s): , 0 and - , 0 3 2 y-intercept: (0,- 4) 1 5 Vertex: , - 5 or 0.4167, - 5.0417 24 12 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) Domain: Range: x - 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 x 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 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 x or x = 7 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 x or 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 3 or x 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 8x – 6x – 18x 3 2 3x4 + 12x2 – 9x 3 2 12x + 8x + 20x 30y – 18y + 54y 5 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 67 of 144 2 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 x – 9x + 4x – 36 2 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 3x – 6x – 4x + 8 2 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 b a b a a-b a 6. 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) 7s(2s + 3t) 8. 15x2 9 x 3x(5x – 3) 3. 9 x2 6 x 12 3(3x2 + 2x – 4) 6. 10x3 5x2 15x 5x(2x2 – x + 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 7 x 4 3. 5 x x 3 x 3 4. 5 x 4 x 5 x 4 5. 6 x 2 x 1 5 2 x 1 6. 4 x 2 3 x 7 3 x 2 7. pq 2qr 2r 2 pr 8. 6 x 3 y 2 xz yz 9. ab 2b ac 2c 10. x3 2 x2 3x 6 11. 2 x3 x2 8x 4 12. 2x3 6x2 5x 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 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) 3. 5 x x 3 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 4 x 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 x 3 11. x2 3x 54 12. x2 15x 44 13. 64x2 16xy y 2 14. 64 121c4 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 x2 56 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 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 1 , and every complex number has the form a+bi with a and b real. N – CN 2: Use the relation i 2 1 and the commutative, associative, and distributive 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: −𝑏±√𝑏 2 −4𝑎𝑐 𝑥= 2𝑎 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 6 62 4 2 3 6 36 24 6 12 22 4 4 x 6 2 3 3 3 6 2 3 3 3 or x 4 2 4 2 a = 2, b = 6, c = 3: 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 imaginary part is equal to 1 . In a complex number, a is the real part and bi is the complex 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 2 i2 = i i = 1 1 1 1 i3 = i2 i = 1 i = i 2 i4 = i3 i = i i = 1 1 1 (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. -1 1 4(1)(1) -1 -3 2 2 -1 i 3 -1 3 i 2 2 2 x 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 i² = -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 (3 2i)(5 6i) 15 18i 10i 12i 2 27 8i 5 6i (5 6i)(5 6i) 25 36 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 2 x2 + bx + c = 0, x2 + bx = -c, 2 2 b add to each side of the equation 2 b b x bx c , 2 2 factor the left hand side b b x c , 2 2 2 2 2 2 take the square root of each side of the equation 2 b b x c ’ 2 2 2 solving for x b b x c+ 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 2 b b x 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 4 add to each side of the equation 2 factor the left hand side and simplify the right hand side take the square root of each side solve for x 2 2 4 4 x2 4 x 9 , 2 2 2 x 2 13 , x 2 13 , x 2 13 , x 2 13 , and , x 2 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 2 2 2 add to each side of the equation 2 2 2 2(x + 2x) + = 9 + , 2 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 y = x2 + 5x + 6 y = x2 + 5x – 6 y = (x + 2)(x + 3) y = (x – 2)(x – 3) Factored 2 Completing the Square/Vertex Form Minimum Point 2 5 49 y x 2 4 5 49 Minimum at - , 4 2 Solutions y = 0 when x = - 5 1 2 2 5 1 y x 2 4 5 1 Minimum at - , - 2 4 y = 0 when x = - 5 7 2 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 y = x2 – 5x + 6 y = x2 – 5x – 6 y = (x + 1)(x – 6) y = (x – 1)(x + 6) Factored 2 Completing the Square/Vertex Form 2 5 49 y x 2 4 5 1 y x 2 4 5 1 Minimum at , - 2 4 5 49 Minimum at , 4 2 Minimum Point y = 0 when x = 5 7 2 2 y = 0 when x = 5 1 2 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) 2 Completing the Square/Vertex Form Minimum Point Solutions 2 5 1 y x 2 4 5 1 y x 2 4 5 1 Minimum at , - 2 4 5 1 Minimum at - , - 2 4 y = 0 when x = 5 1 2 2 y = 0 when x = - 5 1 2 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 2 2 5 49 y x 2 4 5 49 y x 2 4 5 49 Minimum at - , 4 2 5 49 Minimum at , 4 2 y = 0 when x = - 5 7 2 2 y = 0 when x = 5 7 2 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 The vertex form is: y a x c d . Think of this as a template. 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 0 , then the graph will shift down d units. If d 0 , then the graph will shift up d units. So, if y - 2 x 3 1 , the following transformations would occur to the graph of f x x 2 . 2 1) Reflection about the x-axis, since a - 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 2 x 5 7 3 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 5 - 2 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. y x 2 5 4 x 2 1 Rewrite the perfect trinomial square in its factored form as a square of a binomial. Thus y x2 4x 5 = x 2 1 Simplify the constants. 2 2 2 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 Factor the leading coefficient out of the linear and quadratic terms. y - 2 x 2 3x 7 2 2 3 3 y - 2 x 2 3x - 7 - 2 - 2 2 3 9 y - 2 x 7 - 2 2 4 2 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. Rewrite the perfect trinomial square in it’s factored form: square of a binomial. 2 3 3 5 9 y -2 x 7 - -2 x 2 2 2 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 The vertex form is: y a x c d . Think of this as a template. 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 0 , then the graph will shift down d units. If d 0 , then the graph will shift up d units. So, if y - 2 x 3 1 , the following transformations would occur to the graph of f x x 2 . 2 1) Reflection about the x-axis, since a - 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, 7 -1, 1 -1, 3 0, 3 0, - 2 0, 0 0, 0 1, 0 1, - 5 1, 1 1, 3 2, 3 2, - 2 2, 4 2, 12 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 - 7, 4.33 -1, 1 -1, -1 -1, - 0.67 - 6, - 0.67 - 6, 6.33 0, 0 0, 0 0, 0 - 5, 0 - 5, 7 1, 1 1, -1 1, - 0.67 - 4, - 0.67 - 4, 6.33 2, 4 2, - 4 2, - 2.67 - 3, - 2.67 - 3, 4.33 3. 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 Let Group the quadratic and linear term together. y x2 4x 5 If the leading coefficient is not 1, factor it out. y x2 4x - 2 5 - 2 2 2 y x 2 5 4 x 2 1 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. Thus y x2 4x 5 = x 2 1 Rewrite the perfect trinomial square in its factored form as a square of a binomial. 2 2 2 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. 2 2 3 3 y - 2 x 2 3x - 7 - 2 - 2 2 3 9 y - 2 x 7 - 2 2 4 2 2 2 3 3 5 9 y -2 x 7 - -2 x 2 2 2 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. 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 x4 2 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) (- 1, 16) (0, 3) (0, 7) (1, 0) (1, 4) (2, 3) (2, 7) (3, 12) (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 x4 2 1 2 x 2x 4 2 1 1 y x 2 2 x 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 1 (- 2, 4) (- 2, 2) (- 3, 2) - 3, - 2 2 1 1 (- 1, 1) - 1, - 2, (- 2, - 4) 2 2 1 (0, 0) (0, 0) (- 1, 0) - 1, - 4 2 1 1 (1, 1) 1, 0, (0, - 4) 2 2 1 (2, 4) (2, 2) (1, 2) 1, - 2 2 1 The coordinates of the vertex are - 1, - 4 . 2 2 4. y = - 2x + 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) A vertical shrink by a factor of 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