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High School CCSS Mathematics II Curriculum Guide -Quarter 1Columbus City Schools Page 1 of 162 Table of Contents RUBRIC – IMPLEMENTING STANDARDS FOR MATHEMATICAL PRACTICE ....................... 11 Mathematical Practices: A Walk-Through Protocol .............................................................................. 16 Curriculum Timeline .............................................................................................................................. 19 Scope and Sequence ............................................................................................................................... 20 Similarity 1, 1a, 1b, 2, 3, 4, 5 ................................................................................................................. 29 Teacher Notes .......................................................................................................................................... 30 Are You Golden? ................................................................................................................................. 43 The Gumps ........................................................................................................................................... 47 The Gumps and Similar Figures .......................................................................................................... 55 Draw Similar Triangles ........................................................................................................................ 61 Similar Quilt Blocks............................................................................................................................. 63 Quilt Calculations ................................................................................................................................ 64 Investigating Triangles with Two Pairs of Congruent Angles ............................................................. 67 Similar Triangles Application .............................................................................................................. 71 Find the Scale Factor............................................................................................................................ 72 Let’s Prove the Pythagorean Theorem ................................................................................................. 76 Proving the Pythagorean Theorem, Again!.......................................................................................... 80 Trigonometric Ratios G-SRT 6, 7, 8 ...................................................................................................... 82 Teacher Notes .......................................................................................................................................... 83 Exploring Special Right Triangles (45-45-90)..................................................................................... 97 Exploring Special Right Triangles (30-60-90)..................................................................................... 99 Discovering Trigonometric Ratios ..................................................................................................... 104 Make a Model: Trigonometric Ratios ................................................................................................ 108 Let’s Measure the Height of the Flagpole .......................................................................................... 112 Applications of Trigonometry Using Indirect Measurement ............................................................. 114 Find the Missing Side or Angle ......................................................................................................... 122 Between the Uprights ......................................................................................................................... 124 Solve the Triangle .............................................................................................................................. 129 Right Triangle Park ............................................................................................................................ 135 Find the Height................................................................................................................................... 136 Find the Height Data Sheet ................................................................................................................ 137 Applications of the Pythagorean Theorem ......................................................................................... 138 Memory Match – Up .......................................................................................................................... 140 Memory Match – Up Cards ............................................................................................................... 141 Similar Right Triangles and Trigonometric Ratios ............................................................................ 147 Similar Right Triangles and Trigonometric Ratios ............................................................................ 149 Hey, All These Formulas Look Alike!............................................................................................... 155 Problem Solving: Trigonometric Ratios ............................................................................................ 157 Grids and Graphics Addendum ............................................................................................................ 162 Page 2 of 162 Math Practices Rationale CCSSM Practice 1: Make sense of problems and persevere in solving them. Why is this practice important? What does this practice look like when students are doing it? What can a teacher do to model this practice? What questions could a teacher ask to encourage the use of this practice? What does proficiency look like in this practice? What actions might the teacher make that inhibit the students’ use of this practice? Helps students to develop critical thinking skills. Teaches students to “think for themselves”. Helps students to see there are multiple approaches to solving a problem. Students immediately begin looking for methods to solve a problem based on previous knowledge instead of waiting for teacher to show them the process/algorithm. Students can explain what problem is asking as well as explain, using correct mathematical terms, the process used to solve the problem. Frame mathematical questions/challenges so they are clear and explicit. Check with students repeatedly to help them clarify their thinking and processes. “How would you go about solving this problem?” “What do you need to know in order to solve this problem?” What methods have we studied that you can use to find the information you need? Students can explain the relationships between equations, verbal descriptions, tables, and graphs. Students check their answer using a different method and continually ask themselves, “Does this make sense?” They understand others approaches to solving complex problems and can see the similarities between different approaches. Showing the students shortcuts/tricks to solve problems (without making sure the students understand why they work). Not giving students an adequate amount of think time to come up with solutions or processes to solve a problem. Giving students the answer to their questions instead of asking guiding questions to lead them to the discovery of their own question. Page 3 of 162 CCSSM Practice 2: Reason abstractly and quantitatively. Why is this practice important? What does this practice look like when students are doing it? What can a teacher do to model this practice? What questions could a teacher ask to encourage the use of this practice? What does proficiency look like in this practice? What actions might the teacher make that inhibit the students’ use of this practice? Students develop reasoning skills that help them to understand if their answers make sense and if they need to adjust the answer to a different format (i.e. rounding) Students develop different ways of seeing a problem and methods of solving it. Students are able to translate a problem situation into a number sentence or algebraic expression. Students can use symbols to represent problems. Students can visualize what a problem is asking. Ask students questions about the types of answers they should get. Use appropriate terminology when discussing types of numbers/answers. Provide story problems and real world problems for students to solve. Monitor the thinking of students. “What is your unknown in this problem? “What patterns do you see in this problem and how might that help you to solve it?” Students can recognize the connections between the elements in their mathematical sentence/expression and the original problem. Students can explain what their answer means, as well as how they arrived at it. Giving students the equation for a word or visual problem instead of letting them “figure it out” on their own. Page 4 of 162 CCSSM Practice 3: Construct viable arguments and critique the reasoning of others Why is this practice important? What does this practice look like when students are doing it? What can a teacher do to model this practice? What questions could a teacher ask to encourage the use of this practice? What does proficiency look like in this practice? What actions might the teacher make that inhibit the students’ use of this practice? Students better understand and remember concepts when they can defend and explain it to others. Students are better able to apply the concept to other situations when they understand how it works. Communicate and justify their solutions Listen to the reasoning of others and ask clarifying questions. Compare two arguments or solutions Question the reasoning of other students Explain flaws in arguments Provide an environment that encourages discussion and risk taking. Listen to students and question the clarity of arguments. Model effective questioning and appropriate ways to discuss and critique a mathematical statement. How could you prove this is always true? What parts of “Johnny’s “ solution confuses you? Can you think of an example to disprove your classmates theory? Students are able to make a mathematical statement and justify it. Students can listen, critique and compare the mathematical arguments of others. Students can analyze answers to problems by determining what answers make sense. Explain flaws in arguments of others. Not listening to students justify their solutions or giving adequate time to critique flaws in their thinking or reasoning. Page 5 of 162 CCSSM Practice 4: Model with mathematics Why is this practice important? What does this practice look like when students are doing it? What can a teacher do to model this practice? What questions could a teacher ask to encourage the use of this practice? What does proficiency look like in this practice? What actions might the teacher make that inhibit the students’ use of this practice? Helps students to see the connections between math symbols and real world problems. Write equations to go with a story problem. Apply math concepts to real world problems. Use problems that occur in everyday life and have students apply mathematics to create solutions. Connect the equation that matches the real world problem. Have students explain what different numbers and variables represent in the problem situation. Require students to make sense of the problems and determine if the solution is reasonable. How could you represent what the problem was asking? How does your equation relate to the problems? How does your strategy help you to solve the problem? Students can write an equation to represent a problem. Students can analyze their solutions and determine if their answer makes sense. Students can use assumptions and approximations to simplify complex situations. Not give students any problem with real world applications. Page 6 of 162 CCSSM Practice 5: Use appropriate tools strategically Why is this practice important? What does this practice look like when students are doing it? What can a teacher do to model this practice? What questions could a teacher ask to encourage the use of this practice? 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 162 CCSSM Practice 6: Attend to precision. Why is this practice important? What does this practice look like when students are doing it? What can a teacher do to model this practice? What questions could a teacher ask to encourage the use of this practice? What does proficiency look like in this practice? What actions might the teacher make that inhibit the students’ use of this practice? Students are better able to understand new math concepts when they are familiar with the terminology that is being used. Students can understand how to solve real world problems. Students can express themselves to the teacher and to each other using the correct math vocabulary. Students use correct labels with word problems. Make sure to use correct vocabulary terms when speaking with students. Ask students to provide a label when describing word problems. Encourage discussions and explanations and use probing questions. How could you describe this problem in your own words? What are some non-examples of this word? What mathematical term could be used to describe this process. Students are precise in their descriptions. They use mathematical definitions in their reasoning and in discussions. They state the meaning of symbols consistently and appropriately. Teaching students “trick names” for symbols (i.e. the alligator eats the big number) Not using proper terminology in the classroom. Allowing students to use the word “it” to describe symbols or other concepts. Page 8 of 162 CCSSM Practice 7: Look for and make use of structure. Why is this practice important? What does this practice look like when students are doing it? What can a teacher do to model this practice? What questions could a teacher ask to encourage the use of this practice? What does proficiency look like in this practice? What actions might the teacher make that inhibit the students’ use of this practice? When students can see patterns or connections, they are more easily able to solve problems Students look for connections between properties. Students look for patterns in numbers, operations, attributes of figures, etc. Students apply a variety of strategies to solve the same problem. Ask students to explain or show how they solved a problem. Ask students to describe how one repeated operation relates to another (addition vs. multiplication). How could you solve the problem using a different operation? What pattern do you notice? Students look closely to discern a pattern or structure. Provide students with pattern before allowing them to discern it for themselves. Page 9 of 162 CCSSM Practice 8: Look for and express regularity in repeated reasoning Why is this practice important? What does this practice look like when students are doing it? What can a teacher do to model this practice? What questions could a teacher ask to encourage the use of this practice? What does proficiency look like in this practice? What actions might the teacher make that inhibit the students’ use of this practice? When students discover connections or algorithms on their own, they better understand why they work and are more likely to remember and be able to apply them. Students discover connections between procedures and concepts Students discover rules on their own through repeated exposures of a concept. Provide real world problems for students to discover rules and procedures through repeated exposure. Design lessons for students to make connections. Allow time for students to discover the concepts behind rules and procedures. Pose a variety of similar type problems. How would you describe your method? Why does it work? Does this method work all the time? What do you notice when…? Students notice repeated calculations. Students look for general methods and shortcuts. Providing students with formulas or algorithms instead of allowing them to discover it on their own. Not allowing students enough time to discover patterns. Page 10 of 162 RUBRIC – IMPLEMENTING STANDARDS FOR MATHEMATICAL PRACTICE Using the Rubric: 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 Task: Make sense of problems and persevere in solving them. EMERGING NEEDS IMPROVEMENT Is strictly procedural. Does not require students to check solutions for errors. Teacher: (teacher does thinking) Task: Is overly scaffolded or procedurally “obvious”. Requires students to check answers by plugging in numbers. 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. 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. PROFICIENT (teacher mostly models) Task: Is cognitively (students take ownership) Task: Allows for multiple entry points and solution paths. demanding. Requires students to Has more than one entry defend and justify their point. solution by comparing Requires a balance of multiple solution paths. procedural fluency and conceptual Teacher: understanding. Differentiates to keep Requires students to advanced students check solutions for challenged during work errors usingone other time. solution path. Integrates time for explicit meta-cognition. Teacher: Expects students to make Allows ample time for all sense of the task and the students to struggle with proposed solution. task. Institute for Advanced Study/Park City Mathematics Institute Secondary School Teachers Program/Visualizing Functions EXEMPLARY Expects students to evaluate processes implicitly. Models making sense of the task (given situation) and the proposed solution. Summer 2011 Page 11 of 162 PRACTICE Reason abstractly and quantitatively. EMERGING NEEDS IMPROVEMENT Task: Task: 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) 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. Construct viable arguments and critique the reasoning of others. Task: Task: Is either ambiguously stated. Teacher: Does not ask students to present arguments or solutions. Expects students to follow a given solution path without opportunities to make conjectures. Is not at the appropriate level. 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 PROFICIENT EXEMPLARY (teacher mostly models) (students take ownership) Task: Task: Has realistic context. Has relevant realistic context. Requires students to frame solutions in a Teacher: context. Expects students to Has solutions that can be interpret, model, and expressed with multiple connect multiple representations. representations. Prompts students to Teacher: articulate connections Expects students to between algebraic interpret and model procedures and contextual using multiple meaning. representations. Provides structure for students to connect algebraic procedures to contextual meaning. Links mathematical solution with a question’s answer. Task: Teacher: Avoids single steps or Helps students routine algorithms. differentiate between assumptions and logical Teacher: conjectures. Identifies students’ Prompts students to evaluate peer arguments. assumptions. Expects students to Models evaluation of formally justify the validity student arguments. of their conjectures. Asks students to explain their conjectures. Summer 2011 Page 12 of 162 PRACTICE Task: Model with mathematics. EMERGING NEEDS IMPROVEMENT Task: Requires students to identify variables and to perform necessary computations. Teacher: (teacher does thinking) Identifies appropriate variables and procedures for students. Does not discuss Requires students to PROFICIENT (teacher mostly models) Task: Requires students to identify variables and to compute and interpret results. Teacher: Verifies that students have identified appropriate variables and procedures. Explains the appropriateness of model. appropriateness of model. identify variables, compute and interpret results, and report findings using a mixture of representations. Illustrates the relevance of the mathematics involved. Requires students to identify extraneous or missing information. Teacher: EXEMPLARY (students take ownership) Task: Requires students to identify variables, compute and interpret results, report findings, and justify the reasonableness of their results and procedures within context of the task. Teacher: Expects students to justify their choice of variables and procedures. Gives students opportunity to evaluate the appropriateness of model. 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: Does not incorporate additional learning tools. Task: Lends itself to one learning tool. Does not involve mental computations or estimation. Teacher: Demonstrates use of Task: Lends itself to multiple learning tools. Gives students opportunity to develop fluency in mental computations. Teacher: appropriate learning tool. Chooses appropriate learning tools for student use. Models error checking by estimation. Requires multiple learning tools (i.e., graph paper, calculator, manipulative). Requires students to demonstrate fluency in mental computations. Teacher: Allows students to choose appropriate learning tools. Creatively finds appropriate alternatives where tools are not available. Institute for Advanced Study/Park City Mathematics Institute Secondary School Teachers Program/Visualizing Functions Summer 2011 Page 13 of 162 PRACTICE Attend to precision. EMERGING NEEDS IMPROVEMENT Task: Task: (teacher does thinking) 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. 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 PROFICIENT EXEMPLARY (teacher mostly models) (students take ownership) Task: Task: Has precise instructions. Includes assessment Teacher: criteria for communication of ideas. Consistently demands Teacher: precision in communication and in Demands and models precision in mathematical solutions. communication and in Identifies incomplete mathematical solutions. responses and asks Encourages students to student to revise their identify when others are response. not addressing the question completely. Task: 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 162 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. Teacher: (teacher does thinking) 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. future concepts. Teacher: Teacher: Hides or does not draw connections to prior or EXEMPLARY (teacher mostly models) (students take ownership) Task: Task: Reviews prior knowledge Addresses and connects to prior knowledge in a nonand requires cumulative routine way. understanding. Requires recognition of Lends itself to pattern or structure to be developing a completed. pattern or structure. Task: Task: PROFICIENT 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 162 Mathematical Practices: A Walk-Through Protocol *Note: This document should also be used by the teacher for planning and self-evaluation. Mathematical Practices Observations MP.1. Make sense of problems and persevere in solving them 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. MP.2. Reason abstractly and quantitatively. 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. MP.3. Construct viable arguments and critique the reasoning of others. Students are expected to____________________________: Explain their thinking to others and respond to others’ thinking. Participate in mathematical discussions involving questions like “How did you get that?” and “Why is that true?” Construct arguments that utilize prior learning. Question and problem pose. Practice questioning strategies used to generate information. Analyze alternative approaches suggested by others and select better approaches. Justify conclusions, communicate them to others, and respond to the arguments of others. Compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument, explain what it is. CCSSM National Professional Development Page 16 of 162 Mathematical Practices Observations Teachers are expected to______________: Provide opportunities for students to listen to or read the conclusions and arguments of others. MP.4. Model with mathematics. MP 5. Use appropriate tools strategically Students are expected to______________: Apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. Make assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. Experiment with representing problem situations in multiple ways, including numbers, words (mathematical language), drawing pictures, using objects, acting out, making a chart or list, creating equations, etc. Identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts, and formulas. Evaluate their results in the context of the situation and reflect on whether their results make sense. Analyze mathematical relationships to draw conclusions. Teachers are expected to______________: Provide contexts for students to apply the mathematics learned. Students are expected to______________: Use tools when solving a mathematical problem and to deepen their understanding of concepts (e.g., pencil and paper, physical models, geometric construction and measurement devices, graph paper, calculators, computer-based algebra or geometry systems.) Consider available tools when solving a mathematical problem and decide when certain tools might be helpful, recognizing both the insight to be gained and their limitations. Detect possible errors by strategically using estimation and other mathematical knowledge. Teachers are expected to______________: CCSSM National Professional Development Page 17 of 162 Mathematical Practices MP.6. Attend to precision. 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. MP.7. Look for and make use of structure. 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 MP.8. Look for and express regularity in repeated reasoning. 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 162 High School Common Core Math II Curriculum Timeline Topic Intro Unit Similarity Trigonometric Ratios Other Types of Functions Comparing Functions and Different Representations of Quadratic Functions Modeling Unit and Project Quadratic Functions: Solving by Factoring Quadratic Functions: Completing the Square and the Quadratic Formula Probability Geometric Measurement Geometric Modeling Unit and Project Standards Covered G – SRT 1 G – SRT 1a G – SRT 1b G – SRT 6 G – SRT 2 G – SRT 3 G – SRT 4 G – SRT 7 G – SRT 5 Grading Period 1 1 No. of Days 5 20 G – SRT 8 1 20 A – CED 1 A – CED 4 A – REI 1 N – RN 1 N – RN 2 N – RN 3 F – IF 4 F – IF 5 F – IF 6 F – IF 7 F – IF 7a F– IF 9 F – IF 4 F – IF 7b F – IF 7e F – IF 8 F – IF 8b F– BF1 A– CED 1 A– CED 2 F– BF 1 F– BF 1a F – BF 1b F– BF 3 F – BF 1a F – BF 1b F – BF 3 A – SSE 1b N–Q2 2 15 F – LE 3 N– Q 2 S – ID 6a S – ID 6b A – REI 7 2 20 2 10 A – APR 1 A – REI 1 A – REI 4b F – IF 8a A – CED 1 A – SSE 1b A – SSE 3a 3 20 A – REI 1 A – REI 4 A – REI 4a A – REI 4b A – SSE 3b F – IF 8 F – IF 8a A – CED 1 N – CN 1 N – CN 2 N – CN 7 3 20 S – CP 1 S – CP 2 S – CP 3 G – GMD 1 S – CP 4 S – CP 5 S – CP 6 G – GMD 3 S – CP 7 4 20 4 10 G – MG 1 G – MG 2 G – MG 3 4 15 Page 19 of 162 High School Common Core Math II 1st Nine Weeks Scope and Sequence Intro Unit – IO (5 days) Topic 1 – Similarity (20 days) Geometry (G – SRT): 1) Similarity, Right Triangles, and Trigonometry: Understand similarity in terms of similarity transformations. G – SRT 1: Verify experimentally the properties of dilations given by a center and a scale factor. G – SRT 1a: A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. G – SRT 1b: The dilation of a line segment is longer or shorter in the ratio given by the scale factor. G – SRT 2: Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. G – SRT 3: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Geometry (G – SRT): 2) Similarity, Right Triangles, and Trigonometry: Prove theorems involving similarity. G – SRT 4: Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. G – SRT 5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Topic 2 – Trigonometric Ratios (20 days) Geometry (G – SRT): 3) Similarity, Right Triangles, and Trigonometry: Define trigonometric ratios and solve problems involving .right triangles G – SRT 6: Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. G – SRT 7: Explain and use the relationship between the sine and cosine of complementary angles. G – SRT 8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Page 20 of 162 High School Common Core Math II 2nd Nine Weeks Scope and Sequence Topic 3 – Other Types of Functions (15 days) Creating Equations (A – CED): 4) Create equations that describe numbers or relationships A – CED 1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. A – CED 4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R. Reasoning with Equations and Inequalities (A – REI): 5) Understand solving equations as a process of reasoning and explain the reasoning. A – REI 1: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. The Real Number System (N – RN): 6) Extend the properties of exponents to rational exponents. N – RN 1: Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5. N – RN 2: Rewrite expressions involving radicals and rational exponents using the properties of exponents. The Real Number System (N – RN): 7) Use properties of rational and irrational numbers. N – RN 3: Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Interpreting Functions (F – IF): 8) Interpret functions that arise in applications in terms of the context. F – IF 4*: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.* Interpreting Functions (F – IF): 9) Analyze functions using different representations. F – IF 7b: Graph square root, cube root, and absolute value functions. F – IF 7e: Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Page 21 of 162 F – IF 8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. F – IF 8b: Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay. Building Functions (F – BF): 10) Build a function that models a relationship between two quantities. F – BF 1: Write a function that describes a relationship between two quantities. F – BF 1a: Determine an explicit expression, a recursive process, or steps for calculation from a context. F – BF 1b: Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Building Functions (F – BF): 11) Build new functions from existing functions. F – BF 3: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Seeing Structure in Expressions (A – SSE): 12) Interpret the structure of expressions. A – SSE 1b: Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)n as the product of P and a factor not depending on P. Quantities (NQ): 13) Reason quantitatively and use units to solve problems. N – Q 2: Define appropriate quantities for the purpose of descriptive modeling. Topic 4 – Comparing Functions and Different Representations of Quadratic Functions (20 days) Interpreting Functions (F – IF): 14) Interpret functions that arise in applications in terms of the context. F – IF 4*: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.* F – IF 5*: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.* Page 22 of 162 F – IF 6: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Interpreting Functions (F – IF): 15) Analyze functions using different representations. F – IF 7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. F – IF 7a*: Graph linear and quadratic functions and show intercepts, maxima, and minima.* F – IF 9: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Creating Equations (A – CED): 16) Create equations that describe numbers of relationships. A – CED 1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. A – CED 2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Building Functions (F – BF): 17) Build a function that models a relationship between two quantities. F – BF 1: Write a function that describes a relationship between two quantities. F – BF 1a: Determine an explicit expression, a recursive process, or steps for calculation from a context. F – BF 1b: Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Building Functions (F – BF): 18) Build new functions from existing functions. F – BF 3: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Linear and Exponential Models (F – LE): 19) Construct and compare linear and exponential models and solve problems. F- LE 3: Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. Quantities (N-Q): 20) Reason quantitatively and use units to solve problems. N – Q 2: Define appropriate quantities for the purpose of descriptive modeling. Page 23 of 162 Interpreting Categorical and Quantitative Data (S – ID): 21) Summarize, represent, and interpret data on two categorical and quantitative variables. S – ID 6a: Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear and exponential models. S – ID 6b: Informally assess the fit of a function by plotting and analyzing residuals. Reasoning with Equations and Inequalities (A – REI): 22) Solve systems of equations. A – REI 7: Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = -3x and the circle x2 + y2 = 3. Modeling Unit and Project –(10 days) Page 24 of 162 High School Common Core Math II 3rd Nine Weeks Scope and Sequence Topic 5–Quadratic Functions – Solving by factoring (20 days) Arithmetic with Polynomials and Rational Expressions (A – APR): 23) Perform arithmetic operations on polynomials. A – APR 1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Reasoning with Equations and Inequalities (A – REI): 24) Understand solving equations as a process of reasoning and explain the reasoning. A – REI 1: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Reasoning with Equations and Inequalities (A – REI): 25) Solve equations and inequalities in one variable. A – REI 4b: Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Interpreting Functions (F – IF): 26) Analyze functions using different representations. F – IF 8a: Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Creating Equations (A – CED): 27) Create equations that describe numbers of relationships. A – CED 1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Seeing Structure in Expressions (A – SSE): 28) Interpret the structure of expressions. A – SSE 1b: Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)n as the product of P and factor not depending on P. Seeing Structure in Expressions (A – SSE): 29) Write expressions in equivalent forms to solve problems. A – SSE 3a: Factor a quadratic expression to reveal the zeros of the function it defines. Topic 6– Quadratic Functions – Completing the Square/Quadratic Formula (20 days) Reasoning with Equations and Inequalities (A – REI): 30) Understand solving equations as a process of reasoning and explain the reasoning. Page 25 of 162 A – REI 1: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Reasoning with Equations and Inequalities (A – REI): 31) Solve equations and inequalities in one variable. A – REI 4: Solve quadratic equations in one variable. A – REI 4a: Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. A – REI 4b: Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Seeing Structure in Expressions (A – SSE): 32) Write expressions in equivalent forms to solve problems. A – SSE 3b: Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. Interpreting Functions (F – IF): 33) Analyze functions using different representations. F – IF 8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. F – IF 8a: Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Creating Equations (A – CED): 34) Create equations that describe numbers of relationships. A – CED 1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. The Complex Number System (N – CN): 35) Perform arithmetic operations with complex numbers. N – CN 1: Know there is a complex number i such that i 2 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. N – CN 7: Solve quadratic equations with real coefficients that have complex solutions. Page 26 of 162 High School Common Core Math II 4th Nine Weeks Scope and Sequence Topic 7 –Probability (20 days) Conditional Probability and the Rules of Probability (S – CP): 37) Understand independence and conditional probability and use them to interpret data. S – CP 1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”). S – CP 2: Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. S – CP 3: Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. S – CP 4: Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from you school will favor science given that the student is in the tenth grade. Do the same for other subjects and compare the results. S – CP 5: Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. Conditional Probability and the Rules of Probability (S – CP): 38) Use the rules of probability to compute probabilities of compound events in a uniform probability model. S – CP 6: Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. S – CP 7: Apply the Addition Rule, P(A or B) = P(B) – P(A and B), and interpret the answer in terms of the model. Topic 8 – Geometric Measurement (10 days) Geometric Measurement and Dimension (G – GMD): 39) Explain volume formulas and use them to solve problems. G – GMD 1: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. G – GMD 3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Page 27 of 162 Geometric and Modeling Project-(15 days) *Modeling with Geometry (G – MG): 40) Apply geometric concepts in modeling situations. G – MG 1*: Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).* G – MG 2*: Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).* G – MG 3*: Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).* Page 28 of 162 COLUMBUS CITY SCHOOLS HIGH SCHOOL CCSS MATHEMATICS II CURRICULUM GUIDE TOPIC 1 CONCEPTUAL CATEGORY TIME RANGE 20 days Similarity 1, 1a, 1b, 2, 3, 4, Geometry 5 Domain: Geometry: Similarity, Right Triangles, and Trigonometry (G – SRT): Cluster 1) Understand similarity in terms of similarity transformations. 2) Prove theorems involving similarity. GRADING PERIOD 1 Standards 1) Understand similarity in terms of similarity transformations. G – SRT 1: Verify experimentally the properties of dilations given by a center and a scale factor. G – SRT 1a: A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. G – SRT 1b: The dilation of a line segment is longer or shorter in the ratio given by the scale factor. G – SRT 2: Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. G – SRT 3: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. 2) Prove theorems involving similarity. G – SRT 4: Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. G – SRT 5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 29 of 162 TEACHING TOOLS Vocabulary: AA, center of dilation, corresponding parts, cross product, dilation, extremes, figure, image, irregular polygon, means, midsegment, proportion, proportional, ratio, regular polygon, rotational symmetry, scale factor, similar polygons, tessellation, transformations, transversal segments, similarity Teacher Notes Dilations A dilation is a transformation that produces an image that is the same shape as the original, but is a different size. A dilation used to create an image larger than the original is called an enlargement. A dilation used to create an image smaller than the original is called a reduction. The website for the Topic index for dilations at Regents Prep is sited below. It includes lessons, practice and teacher support. http://www.regentsprep.org/Regents/math/geometry/GT3/indexGT3.htm The website below provides a lesson with a warm-up, vocabulary, and examples with solutions for dilations. http://www.chs.riverview.wednet.edu/math/aitken/Integrated-Old/int1-notes/Unit6/Int1_6-6Dilations-notes.pdf At the website below, teachers can look at a tutorial for dilations. There are four different ones: dilating a triangle; invariants in dilation; dilations in the coordinate plane; and problem solving with dilations. http://education.ti.com/en/us/professional-development/pd_onlinegeometry_free/course-outline-andtake-this-course Geometry with Cabri Jr. and the TI-84 Plus Module 9 DILATIONS Lesson 1 - dilating a triangle Lesson 2 - invariants in a dilation Lesson 3 - dilations in the coordinate plane Lesson 4 - problem solving with dilations Similarity Similar polygons are two polygons with congruent corresponding angles and proportional corresponding sides. If the cross product is equal, then the corresponding sides are proportional. Similarity of polygons can be proven in three different ways: Angle-Angle Similarity, Side-Side-Side Similarity, and Side-AngleSide Similarity. A-A Similarity is used when two pairs of corresponding angles are congruent. S-S-S Similarity is used when all three pairs of corresponding sides are proportional. S-A-S Similarity is used when two pairs of corresponding sides are proportional and their included angles are congruent. Below the website listed contains lessons, practice and teacher resources on similarity. http://www.regentsprep.org/Regents/math/geometry/GP11/indexGP11.htm A tutorial on the Pythagorean Theorem and trigonometry can be found at the website below. https://activate.illuminateed.com/playlist/resourcesview/rid/50c56098efea65b540000000/id/50c4c151 efea65fd18000003/bc0/user/bc1/playlist/bc0_id/4fff3767efea650023000698 CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 30 of 162 The website below has cliff notes on AA triangle similarity. http://www.cliffsnotes.com/study_guide/Similar-Triangles.topicArticleId-18851,articleId-18812.html The website below gives examples of SAS, AA, and SSS triangle similarity. http://www.analyzemath.com/Geometry/similar_triangles.html This website offers a teacher resource that includes the warm-up, algebraic review, lecture notes, practice, and hands on activities for similar polygons. http://teachers.henrico.k12.va.us/math/IGO/05Similarity/5_2.html This website offers a teacher resource that includes the warm-up, algebraic review, lecture notes, practice, and hands on activities for similar triangles. http://teachers.henrico.k12.va.us/math/IGO/05Similarity/5_3.html The scale factor is the ratio of lengths of two corresponding sides of similar polygons. The phrase “scale factor” is used in different ways. Example1: If the length of a side of Square A is 4 and the length of a side of Square B is 7, then the scale factor of Square A to Square B is 4/7. Example2: If the length of a side of Square A is 4 and Square A is enlarged by a scale factor of 2, then the length of a side of the new square is 8. Scale factor is used to produce dilations, which can be smaller or larger than the original figure. Real life applications include reading maps, blueprints, and varying recipe sizes. This website offers a teacher resource that includes the warm-up, algebraic review, lecture notes, practice, and hands on activities for using proportions. http://teachers.henrico.k12.va.us/math/IGO/05Similarity/5_1.html This website offers a teacher resource that includes the warm-up, algebraic review, lecture notes, practice, and hands on activities for proportional parts. http://teachers.henrico.k12.va.us/math/IGO/05Similarity/5_4.html The TI-84 and Cabri Jr. can be used for special triangles. An on-line tutorial can be found at the website below. Module 11 SPECIAL TRIANGLES - Lesson 3 - constructing a right triangle http://education.ti.com/en/us/professional-development/pd_onlinegeometry_free/course-outline-andtake-this-course' The TI-84 and Cabri Jr. can be used for special triangles. A tutorial can be found at the website below. Module 14 PROPORTIONS - Lesson 1 - similar triangles http://education.ti.com/en/us/professional-development/pd_onlinegeometry_free/course-outline-andtake-this-course This website offers a teacher resource that includes the warm-up, algebraic review, lecture notes, practice, CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 31 of 162 and hands on activities for Pythagorean Theorem. http://teachers.henrico.k12.va.us/math/IGO/07RightTriangles/7_2.html Below a website is listed for a video tutorial for solving a triangle using SAS. http://patrickjmt.com/solving-a-triangle-sas-example-1/ Below a website is listed for a video tutorial for another example of solving a triangle for SAS. http://patrickjmt.com/solving-a-triangle-sas-example-2/ The website below has a video tutorial to find the missing side and angles of triangle using SAS. http://patrickjmt.com/side-angle-side-for-triangles-finding-missing-sidesangles-example-1/ Another example of finding the missing side and angles of a triangle using SAS can be found at the website below. http://patrickjmt.com/side-angle-side-for-triangles-finding-missing-sidesangles-example-2/ Misconceptions/Challenges: Students do not match up the corresponding sides of figures, and therefore incorrectly set up proportions between similar polygons, which cause them to get the incorrect side lengths or transversal segments. Students believe that adding a particular value to all sides of a polygon will create a similar polygon. Students mix up the possible values of the scale factors for enlargements and reductions. Students do not multiply the scale factor by all sides in the polygon. Students think that all polygons of a particular shape (for example all right traingles, or all rectangles) are similar; they do not recognize that they can have different corresponding angles. CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 32 of 162 Instructional Strategies: SRT 1 Analyzing Congruence Proofs. http://map.mathshell.org.uk/materials/lessons.php?taskid=452 This lesson focuses on the concepts of congruency and similarity, including identifying corresponding sides and corresponding angles within and between triangles. Students will identify and understand the significance of a counter-example, and prove and evaluate proofs in a geometric context Key Visualizations, Geometry: http://ccsstoolbox.agilemind.com/animations/standards_content_visualizations_geometry.html This website has an animation where students can explore dilations of lines by selecting points along the line and thinking about point-by-point dilations. Students make a connection between dilations and ratios. Photocopy Faux Pas http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig_te_04091 3.pdf The website below provides a lesson on the essential features of dilation. It is Classroom Task: 6.1 found on pages 4 – 10. SRT 1a Dialations http://psdsm2ccss.pbworks.com/w/page/56495542/GSRT1%20Verify%20experimentally%20properti es%20of%20dilations Create a dilation of a line segment AB through point C with a scale factor of 2:1 to create segment EF. Find lengths of all segments, EF, AB, BC, CE and CF. Dilate and Reflect http://education.ti.com/xchange/US/Math/AlgebraII/16008/Transformations_Dilating_Functions_Teac her.pdf Students will use the Nspire Handheld to dilate and reflect different types of functions by grabbing points. Students will understand the effect of the coefficient on the vertical stretch or shrink of the function Properties of Dilations http://education.ti.com/en/us/activity/detail?id=0C732215F7EC479AB1A6350A64B161B2 Students explore the properties of dilations and the relationships between the original and image figures. Playing with Dilations http://www.cpalms.org/RESOURCES/URLresourcebar.aspx?ResourceID=SSGunMzEork=D Students explore dilations and rotations using Virtual Manipulates Dilation and Scale Factor http://www.illustrativemathematics.org/illustrations/602 Give student a copy of the picture so they can draw the points A’, B’ and C’. Provide extra space below the picture. This task enables the students to verify that a dilation takes a line that does not pass through the CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 33 of 162 center of a line parallel to the original line and the dilation of the line segment is longer or shorter by a scale factor. Rulers may be useful for duplication of lengths without formal constructions Properties of Dilations https://www.cohs.com/editor/userUploads/file/Meyn/321%20Ch%205a%20Student%20WB.pdf Lesson 5-1 page 5. Activity to investigate properties of dilations using geometry software. Properties include: dilations preserve angle measure, betweenness, collinearity, maps a lines not passing through the center of the dilation to a parallel line and leaves a line passing through the center unchanged, and a dilation of a line segment is longer or shorter in the ratio given by the scale factor. Analogy of Dilation to zoom http://www.geogebra.org/cms/ Draws an analogy of dilation to zoom-in and zoom-out of a camera, a document camera, an iPad, or using geometry software programs such as Geogebra. SRT 1b It is beneficial to use real-life data to discuss ratios with students. You can ask students to compare the number of male students to female students, the number of students in tennis shoes to students not in tennis shoes, and the number of students with homework to students without homework. Have students complete the activity “Are You Golden?” (Included in this Curriculum Guide). Divide students into groups of 3-4. This activity will allow students to discover the golden ratio by finding the ratios of various body parts. Take two triangles that are congruent. The sides can be 3, 4, and 5 units long. Set up ratios comparing 1 . Introduce similar triangles. All congruent triangles are corresponding sides. The ratios all reduce to1 = 1 similar triangles with a scale factor of 1:1. The corresponding sides of similar triangles are proportional and corresponding angles are congruent. Take two similar triangles. One has sides 6, 8, and 12. The other has sides 9, 12, and 18. Each ratio of the corresponding sides reduces to 2:3. Next, we can present situations with similar triangles where the length of one side is missing. We can demonstrate how we can set up ratios comparing corresponding sides and use properties of proportions to calculate the missing side. Similarity and Triangles. https://www.cohs.com/editor/userUploads/file/Meyn/321%20Ch%205a%20Student%20WB.pdf Practice work on applying similarity to triangles. Lesson 5-4 page 17. Students use dilations and rigid motions to map the image of triangle ABC to triangle DEF ( This lesson can also be found in SRT 3) Discuss what a blueprint is and the purpose it serves. Have students do the following activity in small 1 cooperative learning groups. Ask them to make a blueprint of the classroom. Use the scale: inch = 1 foot. 4 Use quarter-inch graph paper for this activity. Have students measure the length and width of the room. Point out those decisions that will need to be made, such as where doors and windows should be located on the scale drawing. As an extension, a scale drawing of the building or the cafeteria could be done. Ask students if the same scale should be used. Ask them to explain why or why not. Discuss options. The link below contains an explanation about dilations. http://www.frapanthers.com/teachers/zab/Geometry(H)/GeometryinaNutshell/GeometryNutshell2005/ Text/Dilations.pdf CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 34 of 162 The website below has a practice sheet for dilations. http://mathematicsburns.cmswiki.wikispaces.net/file/view/DilationsTranslations+activity+worksheet+ for+2-20.pdf Have students complete the activities “The Gumps” and “The Gumps and Similar Figures” (included in this Curriculum Guide) to lead students into discovering that mathematically similar figures have congruent angles and proportional The Gumps sides. Divide students into groups of 3-5. Each group should create one set of figures based on the coordinates given in the chart. Graph paper is required and some figures may require more than one sheet. The sample figures drawn in this Curriculum Guide use a scale factor of 2 in order for each figure to fit on one sheet of paper. Transparencies can be made of the figures to overlay them in order to show that the angles of Giggles, Higgles, and Ziggles are congruent. Are They Similar? http://www.illustrativemathematics.org/illustrations/603 The activity includes a picture of two triangles that appear to be similar but to prove similarity they need further information. Ask students to provide a sequence of similarity transformations that map one triangle to the other one. Remind students that all parts of one triangle get mapped to the corresponding parts of the other one. An additional task includes asking the students to prove or disprove that the triangles are similar in each problem using properties of parallel lines and the definition of similarity. Transformations and Similarity https://www.cohs.com/editor/userUploads/file/Meyn/321%20Ch%205a%20Student%20WB.pdf Similarity practice. Lesson 5-3. Students find that two plane figures are similar if and only if one can be obtained from the other by transformations. Geometry Problems: Circles and Triangles, http://map.mathshell.org/materials/lessons.php?taskid=222 Students solve problems by determining the lengths of the sides in right triangles. They also determine the measurements of shapes by decomposing complex shapes into simpler ones. Scale (or Grid) Drawings and Dilations. http://www.regentsprep.org/Regents/math/geometry/GT3/DActiv.htm Students work with scale (or grid) drawing to reinforce the concept of scalar factor. Angles and Similarity http://education.ti.com/calculators/downloads/US/Activities/Detail?id=13153 Students use technology (TI-Nspire or Nspire CAS) to experiment with the measures of the angles of similar triangles to determine conditions necessary for two triangles to be similar. Corresponding Parts of Similar Triangles http://education.ti.com/calculators/downloads/US/Activities/Detail?id=13150 Students use technology(TI-Nspire or Nspire CAS) to change the scale factor (r) between similar triangles, identify the corresponding parts, and establish relationships between them. Nested Similar Triangles http://education.ti.com/calculators/downloads/US/Activities/Detail?id=13152 Students use technology (TI-Nspire or Nspire CAS) to discover the conditions that make triangles similar by CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 35 of 162 moving the sides opposite the common angles. Demonstrate to students the properties of similarity. Draw a triangle and ask students “How would you draw a triangle similar to the triangle shown?” Include in your discussion that angle measures are the same and sides are proportional. Have students draw two triangles one that is similar to and larger than the original and one that is similar to and smaller than the original. Follow up this introduction to similarity with the “Draw Similar Triangles” activity (included in this Curriculum Guide). Students will need a protractor, straightedge, calculator, and a copy of the worksheet. Students can do this activity in partners or individually. Quilts are a beautiful, practical, and historically significant use of geometric shapes. Students will work with triangles in historic quilt patterns by creating triangles similar to those in a quilt block and then creating their own pattern with the new triangles as described in the activity “Similar Quilt Blocks” (included in this Curriculum Guide). Students may work individually or in groups. Students will need a copy of the “Similar Quilt Blocks” sheet, the “Quilt Calculations” sheet (included in this Curriculum Guide), the “Quilt Design #1” sheet (included in this Curriculum Guide), the “Quilt Design #2” sheet (included in this Curriculum Guide), a ruler, a protractor, and materials to make their quilt design (e.g., construction paper, scissors, etc.). After students have completed this activity, have students share their creations and any challenges they may have had in creating their new pattern. Students who express an interest in this art form may find additional information by searching the web using the keyword “quilt”. Scale Factor Area Perimeter http://education.ti.com/calculators/downloads/US/Activities/Detail?id=13154 Students use technology (TI-Nspire or Nspire CAS) to explore the relationship of perimeter and area in similar triangles when the scale factor is changed. Transformations with Lists, http://education.ti.com/calculators/downloads/US/Activities/Detail?id=10278 Students use list operations to perform reflections, rotations, translations, and dilations on a figure and graph the resulting image using a scatter plot.. Dilations. http://www.frapanthers.com/teachers/zab/Geometry(H)/ClassNotes/14.6Dilations.pdf The website below has practice for dilations. SRT 2 Triangle Similarity. https://ccgps.org/G-SRT_9DRF.html This website offers internet resources for triangle similarity. Investigating Triangles with Two Pairs of Congruent Angles (AA similarity): Have students complete the activity “Investigating Triangles with Two Pairs of Congruent Angles” (included in this Curriculum Guide). Students should discover the AA Similarity Theorem from this activity. Students will need protractors and straightedges to complete this activity. Draw a triangle on the chalkboard. Label the vertices of the triangle A, B, and C. Double the length of AB from point A. Label the resulting endpoint B'. Double the length of AC from point A. Label the resulting endpoint C'. Connect B' and C'. Compare ABC and AB'C'. Discuss with students whether or not the triangles are similar. (They are similar because of SAS for ~ ’s.) CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 36 of 162 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. Have students complete the activity “Similar Triangles Application” (included in this Curriculum Guide) to use their skills with similar triangles in a real life situation. Meter sticks and mirrors are required materials. Students should stand several feet away from an object, placing the mirror on the ground between themselves and the object. The student should place themselves or the mirror in such a way that s/he can spot the top of the object in the mirror. A partner should take three measurements: the distance the student is standing from the mirror, the distance from the mirror to the base of the object,, and the distance from the students line of sight to the ground. Using proportions and similar triangles, the students should be able to indirectly calculate the height of the object. Have students do the activity “Find the Scale Factor” (included in this Curriculum Guide) for more practice in using scale factor to solve similarity problems. Before doing this activity, discuss scale factors with the students. For example, discuss with the students how the scale factor is 5:1 not 4:1 in the figure below. 24 6 Falling Down a Rabbit Hole Can Lead to a King Sized Experience - Exploring Similar Figures Using Proportions,” http://alex.state.al.us/lesson_view.php?id=30067 Students explore similarity. They simplify ratios, solve proportions using cross products, and use properties of proportions to solve real-world problems. Similarity Transformation https://www.cohs.com/editor/userUploads/file/Meyn/321%20Ch%205a%20Student%20WB.pdf: Students find that two plane figures are similar if and only if one can be obtained from the other by transformations (reflections, translations, rotations, and/or dilations Lesson 5-3, page 13 (This lesson is also found in SRT 1b) Triangle Dilations http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig_te_04091 3.pdf Students examine relationships of proportions in triangles that are known to be similar to each other based on dilations. Classroom Task 6.2 pages 11-20 (This lesson is also found in SRT 5.) Similar Triangles and Other Figures http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig_te_04091 3.pdf Students compare the definitions of similarity based on dilations and relationships between corresponding sides and angles. Classroom Task 6.3 pages 21-23 (This lesson is also found in SRT 3.) CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 37 of 162 SRT3 The website below contains lessons for SRT3. https://ccgps.org/G-SRT_AVKU.html Similarity and Triangles. https://www.cohs.com/editor/userUploads/file/Meyn/321%20Ch%205a%20Student%20WB.pdf Practice work on applying similarity to triangles. Lesson 5-4 page 17. Students use dilations and rigid motions to map the image of triangle ABC to triangle DEF. (This lesson was also provided in SRT 1b) Practice with Similarity Proofs, http://www.regentsprep.org/Regents/math/geometry/GP11/PracSimPfs.htm Eight formative assessment questions are provided. Similar Triangles and Other Figures http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig_te_04091 3.pdf Students compare the definitions of similarity based on dilations and relationships between corresponding sides and angles. Classroom Task 6.3 pages 21-23 (This lesson is also found in SRT 3.) SRT4 Pythagorean Theorem https://ccgps.org/G-SRT_G6QQ.html A power point presentation on the Pythagorean Theorem. A Proportionality Theorem. https://www.cohs.com/editor/userUploads/file/Meyn/321%20Ch%205a%20Student%20WB.pdf Students find what happens when a line that is parallel to one side of a triangle “splits” the other two sides. The sides are dived proportionally. It is known as the Side-Splitting Theorem. (This lesson is also found in SRT 5.) Proving the Pythagorean Theorem https://www.cohs.com/editor/userUploads/file/Meyn/321%20Ch%205a%20Student%20WB.pdf Students will use their knowledge about similar triangles to prove the Pythagorean Theorem. Applying Angle Theorems http://map.mathshell.org/materials/lessons.php?taskid=214 Students use geometric mean properties to solve problems using the measures of interior and exterior angles of polygons Have students complete the activity “Let’s Prove the Pythagorean Theorem” (included in this Curriculum Guide) to construct a proof of the Pythagorean Theorem. Have students complete the activity “Proving the Pythagorean Theorem, Again!”(included in this Curriculum Guide) to reinforce the proof of the Pythagorean Theorem. Proofs of the Pythagorean Theorem http://map.mathshell.org/materials/lessons.php?taskid=419&subpage=concept Below link: Students interpret diagrams, link visual and algebraic representations, and produce a mathematical argument CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 38 of 162 Cut by a Transversal http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig_te_04091 3.pdf Students examine proportional relationships of segments when two transversals intersect sets of parallel lines. Classroom Task: 6.4 pages 30-37 Measured Reasoning http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig_te_04091 3.pdf Students apply theorems about lines, angles, and proportional relationships when parallel lines are crossed by multiple transversals. Classroom Task6.5 pages 38-45. (This lesson can also been found at SRT 5.) Pythagoras by Proportions http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig_te_04091 3.pdf Students use similar triangles to prove the Pythagorean Theorem and theorems about geometric means in right triangles. Classroom Task 6.6 pages 36-52(This lesson can also been found at SRT 5.) Finding the Value of a Relationship http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig_te_04091 3.pdf Students solve for unknown values in right triangles using trigonometric ratios. Classroom Task 6.9 pages 67-74 (This lesson can also been found at SRT 5.) SRT5 Proving the Pythagorean Theorem https://www.cohs.com/editor/userUploads/file/Meyn/321%20Ch%205a%20Student%20WB.pdf Students will use their knowledge about similar triangles to prove the Pythagorean Theorem. (Lesson 5-7, page 27) Measured Reasoning, http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig_te_04091 3.pdf Students apply theorems about lines, angles, and proportional relationships when parallel lines are crossed by multiple transversals. Classroom Task 6.5 pages 38-45 (This lesson can also been found at SRT 4.) Finding the Value of a Relationship http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig_te_04091 3.pdf Students solve for unknown values in right triangles using trigonometric ratios. Classroom Task 6.9 pages 67-74 (This lesson can also been found at SRT 4.) Pythagoras by Proportions http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig_te_04091 3.pdf Students use similar triangles to prove the Pythagorean Theorem and theorems about geometric means in CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 39 of 162 right triangles. Classroom Task 6.6 pages 36-52 (This lesson can also been found at SRT 4.) How Tall is the School’s Flagpole https://ccgps.org/G-SRT_AVKU.html Students will apply math concepts concerning similar triangles and trigonometric functions to real life situations. The students will find measurements of objects when they are unable to use conventional measurement. (This lesson can also be found at SRT 3) A Proportionality Theorem. https://www.cohs.com/editor/userUploads/file/Meyn/321%20Ch%205a%20Student%20WB.pdf Students find what happens when a line that is parallel to one side of a triangle “splits” the other two sides. The sides are dived proportionally. It is known as the Side-Splitting Theorem. (This lesson is also found in SRT 4) Solving Problems Using Similarity. https://www.cohs.com/editor/userUploads/file/Meyn/321%20Ch%205a%20Student%20WB.pdf Students use proportionality of corresponding sides to find side lengths of two similar polygons. Lesson 5-5 Solving Geometry Problems: Floodlights http://map.mathshell.org/materials/lessons.php?taskid=429&subpage=problem Students make models, draw diagrams, and identify similar triangles to solve problems. .https://www.georgiastandards.org/Frameworks/GSO%20Frameworks/MathII_Unit2_%20Student_E dition_revised_8-10-09.pdf This website contains a set of lessons on right triangle trigonometry. These lessons include discovering special right triangles, discovering trigonometric ratio relationships, and determining side or angle measures using trigonometry How Far Can You Go in a New York Minute? http://illuminations.nctm.org/LessonDetail.aspx?id=L848 Students use proportions and similar figures to adjust the size of the New York City Subway Map so that it is drawn to scale. http://education.ti.com/en/us/activity/detail?id=A760474813204FBB944031327521B742&ref=/en/us/ac tivity/search/subject?d=6B854F0B5CB6499F8207E81D1F3A25E6&s=B843CE852FC5447C8DD8 8F6D1020EC61&sa=71A40A9FD9E84937B8C6A8A4B4195B58&t=3CC394B76E4347CF8C EFCADAACAE9754 Students will explore the ratio of perimeter, area, surface area, and volume of similar figures in twodimensional figures using graphing technology. Triangle Dilations http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig_te_04091 3.pdf Students examine relationships of proportions in triangles that are known to be similar to each other based on dilations. Classroom Task 6.2 pages 11-20 (This lesson is also found in SRT 2.) CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 40 of 162 Reteach: Construct ABC with sides 5, 8, and 10 units. Tell students that the scale factor of ABC to DEF is 1 2 . Ask the students to construct DEF. Construct ABC with sides 5, 8, and 10 units. Tell students that the scale factor of ABC to RST is 2. Ask the students to construct RST. Construct two equilateral triangles of different sizes on the chalkboard. Ask students to determine if they are congruent or similar. Ask the students to justify their answers. (The equilateral triangles that were drawn are not congruent because the sides do not have the same length. They are similar because the angles all have a measure of 60° and the ratios of the lengths of the corresponding sides are the same.) Extensions: Use coordinate geometry and graph paper to draw the dilation of a figure. Use construction tools to construct the dilation of a figure. Take a map of Ohio or the United States. Make a transparency of the map. Place it over a coordinate plane. Write the coordinates of many of the border points. Have groups multiply each coordinate by a 1 scale factor. Have some groups use a scale factor of 3. Have others use a scale factor of 3 . Tape pages of graph paper together. Have students graph the new image. Discussion: Are the maps proportional? Textbook References: Textbook: Geometry, Glencoe (2005): pp. 282-287, 288 Supplemental: Geometry, Glencoe (2005): Chapter 6 Resource Masters Study Guide and Intervention, pp. 295-296 Skills Practice, p. 297 Practice, p. 298 Reading to Learn Mathematics, pp. vii-viii, 299 Enrichment, p. 300 Textbook: Geometry, Glencoe (2005): pp. 289-297 Supplemental: Geometry, Glencoe (2005): Chapter 6 Resource Masters Study Guide and Intervention, pp. 301-302 Skills Practice, p. 303 Practice, p. 304 Reading to Learn Mathematics, pp. vii-viii, 305 Textbook: Geometry, Glencoe (2005): pp. 298-306, 307-315, 316-323 CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 41 of 162 Supplemental: Geometry, Glencoe (2005): Chapter 6 Resource Masters Learning to Read Mathematics, pp. ix-x Study Guide and Intervention, pp. 307-308, 313-314, 319-320 Skills Practice, pp. 309, 315, 321 Practice, pp. 310, 316, 322 Reading to Learn Mathematics, pp. 311, 317, 323 Enrichment, pp. 312, 318, 324 Textbook: Geometry, Glencoe (2005): pp. 490 – 493 Textbook: Algebra 1, Algebra 1 (2005): pp. 197 – 203 CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 42 of 162 G –SRT 1b Name ______________________________ Date ____________ Period ________ Are You Golden? Materials: meter stick or tape measure, calculator For each group members, measure the length from the shoulder to the tip of the fingers and the length from the elbow to the tip of the fingers. Record the data below. Column A Column B TABLE 1 Column C Column D Group Members’ Names Length from shoulder to tip of fingers (cm) Length from Elbow to tip of fingers (cm) Find the Ratio of Column B to Column C Column E Decimal form of Column D Round to 2 decimal places. 1. Examine Column E. What do you notice about all of the decimals? 2. Find the average of all the decimals in Column E. Round to two decimal places. Now, measure each group member’s height and the height of the navel from the ground (make sure to take off your shoes). Record the data below. Column A Column B Group Members’ Names Height (cm) TABLE 2 Column C Column D Height of Navel from the Ground (cm) Find the Ratio of Column B to Column C Column E Decimal form of Column D Round to 2 decimals places. 3. Examine Column E. What do you notice about all of the decimals? 4. Find the average of all the decimals in Column E. Round to two decimal places. CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 43 of 162 5. How do the decimal averages in #2 and #4 compare? The ratios that you found are very close to what is known as the 1 5 “Golden Ratio”, which is . The decimal approximation 2 of the “Golden Ratio” is 1.618033989… Set up proportions to answer the following questions based on the “Golden Ratio”. 6. If a person’s arm (length of shoulder to tip of fingers) is 68 cm long, what is the length of this person’s elbow to the tip of the fingers? 7. If the height of a person’s navel from the ground is 105 cm tall, how tall is this person? CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 44 of 162 G –SRT 1b Name ______________________________ Date ____________ Period ________ Are You Golden? Answer Key Materials: meter stick or tape measure, calculator For each group members, measure the length from the shoulder to the tip of the fingers and the length from the elbow to the tip of the fingers. Record the data below. Column A Column B TABLE 1 Column C Column D Group Members’ Names Length from shoulder to tip of fingers (cm) Length from elbow to tip of fingers (cm) Find the ratio of Column B to Column C Column E Decimal form of Column D Round to 2 decimal places. 1. Examine Column E. What do you notice about all of the decimals? Answers Will Vary. 2. Find the average of all the decimals in Column E. Round to two decimal places. Answers Will Vary. Now, measure each group member’s height and the height of the navel from the ground (make sure to take off your shoes). Record the data below. TABLE 2 Column A Column B Column C Column D Column E Decimal form of Height of navel Find the ratio of Group Members’ Column D Height (cm) from the ground Column B to Names Round to 2 (cm) Column C decimal places. 3. Examine Column E. What do you notice about all of the decimals? Answers Will Vary. 4. Find the average of all the decimals in Column E. Round to two decimal places. Answers Will Vary. CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 45 of 162 5. How do the decimal averages in #2 and #4 compare? Answers Will Vary Students could conclude that the decimal average is very close to 1.6. The ratios that you found are very close to what is known as the 1 5 “Golden Ratio”, which is . The decimal approximation 2 of the “Golden Ratio” is 1.618033989… Set up proportions to answer the following questions based on the “Golden Ratio”. 6. If a person’s arm (length of shoulder to tip of fingers) is 68 cm long, what is the length of this person’s elbow to the tip of the fingers? 68 1.618 x x 42.03 cm 7. If the height of a person’s navel from the ground is 105 cm tall, how tall is this person? x = 1.618 105 x 169.89 cm CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 46 of 162 G –SRT 1b Name ______________________________ Date ____________ Period ________ The Gumps There are imposters lurking among the family of Gumps. Using the following criteria, you will create a set of characters. They will all look somewhat alike but only some of them are considered to be mathematically similar. Each group should create a set of characters in order to answer the questions that follow. Every graph within the group should be drawn using the same scale in order to see the changes between the Gumps. More than one piece of graph paper may be needed for a particular character. Plot each point on graph paper. For the points in SET 1 and SET 3, connect them in order and connect the last point to the first point. For SET 2, connect the points in order but do not connect the last point to the first point. For SET 4, make a dot at each point. CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 47 of 162 G –SRT 1b Name ______________________________ Date ____________ Period ________ The Gumps Giggles (x,y) SET 1 (4,0) (4,6) (2,4) (0,4) (4,8) (2,10) (2,14) (4,16) (5,18) (6,16) (8,16) (9,18) (10,16) (12,14) (12,10) (10,8) (14,4) (12,4) (10,6) (10,0) (8,0) (8,4) (6,4) (6,0) SET 2 (4,11) (6,10) (8,10) (10,11) SET 3 (6,11) (6,12) (8,12) (8,11) SET 4 (5,14) (9,14) Higgles (2x,2y) SET 1 Wiggles (3x,y) SET 1 Ziggles (3x,3y) SET 1 Miggles (x,3y) SET 1 SET 2 SET 2 SET 2 SET 2 SET 3 SET 3 SET 3 SET 3 SET 4 SET 4 SET 4 SET 4 CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 48 of 162 The Gumps Answer Key Giggles (x,y) SET 1 (4,0) (4,6) (2,4) (0,4) (4,8) (2,10) (2,14) (4,16) (5,18) (6,16) (8,16) (9,18) (10,16) (12,14) (12,10) (10,8) (14,4) (12,4) (10,6) (10,0) (8,0) (8,4) (6,4) (6,0) SET 2 (4,11) (6,10) (8,10) (10,11) SET 3 (6,11) (6,12) (8,12) (8,11) SET 4 (5,14) (9,14) Higgles (2x,2y) SET 1 (8,0) (8,12) (4,8) (0,8) (8,16) (4,20) (4,28) (8,32) (10,36) (12,32) (16,32) (18,36) (20,32) (24,28) (24,20) (20,16) (28,8) (24,8) (20,12) (20,0) (16,0) (16,8) (12,8) (12,0) SET 2 (8,22) (12,20) (16,20) (20,22) SET 3 (12,22) (12,24) (16,24) (16,22) SET 4 (10,28) (18,28) Wiggles (3x,y) SET 1 (12,0) (12,6) (6,4) (0,4) (12,8) (6,10) (6,14) (12,16) (15,18) (18,16) (24,16) (27,18) (30,16) (36,14) (36,10) (30,8) (42,4) (36,4) (30,6) (30,0) (24,0) (24,4) (18,4) (18,0) SET 2 (12,11) (18,10) (24,10) (30,11) SET 3 (18,11) (18,12) (24,12) (24,11) SET 4 (15,14) (27,14) CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Ziggles (3x,3y) SET 1 (12,0) (12,18) (6,12) (0,12) (12,24) (6,30) (6,42) (12,48) (15,54) (18,48) (24,48) (27,54) (30,48) (36,42) (36,30) (30,24) (42,12) (36,12) (30,18) (30,0) (24,0) (24,12) (18,12) (18,0) SET 2 (12,33) (18,30) (24,30) (30,33) SET 3 (18,33) (18,36) (24,36) (24,33) SET 4 (15,42) (27,42) Miggles (x,3y) SET 1 (4,0) (4,18) (2,12) (0,12) (4,24) (2,30) (2,42) (4,48) (5,54) (6,48) (8,48) (9,54) (10,48) (12,42) (12,30) (10,24) (14,12) (12,12) (10,18) (10,0) (8,0) (8,12) (6,12) (6,0) SET 2 (4,33) (6,30) (8,30) (10,33) SET 3 (6,33) (6,36) (8,36) (8,33) SET 4 (5,42) (9,42) Columbus City Schools 6/28/13 Page 49 of 162 Giggles Higgles CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 50 of 162 CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 51 of 162 Wiggles CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 52 of 162 Ziggles CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 53 of 162 Miggles CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 54 of 162 G –SRT 1b Name ______________________________ Date ____________ Period ________ The Gumps and Similar Figures 1. Use a protractor to measure the following angles of the Gumps’ bodies. Giggles Higgles Wiggles Ziggles Miggles Top of Ear Under Arm Neck Smile Do you notice anything about the above measurements? If so, explain. Count the length of the following sides of the Gumps’ bodies. Giggles Higgles Wiggles Ziggles Miggles Width of Head Length of Leg Width of Hand Width of Waist Total Height Compare each Gump’s measurements to Giggles’ measurements. Describe any patterns that you notice. Giggles and Higgles are mathematically similar. Describe what you think it means for two figures to be mathematically similar. What other Gump(s) fit this description. Why? Complete the following table. CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 55 of 162 Nose Width Nose Length Width Length Nose Perimeter Nose Area Giggles (Gump 1) Higgles (Gump 2) Ziggles (Gump 3) Prediction for Gump 4 Prediction for Gump 5 . . . Prediction for Gump 10 Prediction for Gump 20 Prediction for Gump 100 Wiggles Miggles Make ratios using the nose perimeter for the following figures: Gump 2:Gump 1 Gump 3:Gump 1 Gump 4:Gump 1 Gump 5:Gump 1 CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 56 of 162 Make a comparison between the scale factor of objects and the ratio of their perimeters. Make ratios using the nose area for the following figures: Gump 2:Gump 1 Gump 3:Gump 1 Gump 4:Gump 1 Gump 5:Gump 1 Make a comparison between the scale factor of objects and the ratio of their areas. Look at Gump 10, Gump 20 and Gump 100. Using your answers to #9 and #11, show the relationship between scale factor of objects and the ratio of their perimeters and areas. CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 57 of 162 G –SRT 1b Name ______________________________ Date ____________ Period ________ The Gumps and Similar Figures Answer Key 1. Use a protractor to measure the following angles of the Gumps’ bodies. Giggles Higgles Wiggles Ziggles o o o Top of Ear 53 53 112 53o Under Arm 45o 45o 72o 45o o o o Neck 90 90 37 90o Smile 153o 153o 171o 153o Miggles 19o 18o 143o 124o Do you notice anything about the above measurements? If so, explain. Giggles, Higgles and Ziggles have the same angle measurements. They are the same shape just different sizes which preserves their angle measurements. The other two figures are stretched because only one of their dimensions was changed. Count the length of the following sides of the Gumps’ bodies. (Remember to count by 2 on the sample drawings since the scale is 2!) Giggles Higgles Wiggles Ziggles Width of Head 10 20 30 30 Length of Leg 4 8 4 12 Width of Hand 2 4 6 6 Width of Waist 6 12 18 18 Total Height 18 36 18 54 Miggles 10 12 2 6 54 Compare each Gump’s measurements to Giggles’ measurements. Describe any patterns that you notice. All of Higgles’ measurements are two times that of Giggles’. All of Ziggles’ measurements are three times that of Giggles’. Wiggles’ widths only are three times larger than Giggles’ widths because only the x-values were multiplied by 3. Miggles’ lengths only are three times larger than Giggles’ lengths because only the y-values were multiplied by 3. Giggles and Higgles are mathematically similar. Describe what you think it means for two figures to be mathematically similar. Two figures are mathematically similar if their angle measures are the same and all of their dimensions are proportional. What other Gump(s) fit this description. Why? Ziggles is also mathematically similar to Giggles and Higgles because they have the same angle measurements and their sides are all proportional. CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 58 of 162 Complete the following table. Nose Width Nose Length Width Length Nose Perimeter Nose Area Giggles (Gump 1) 1 cm 2 cm 1 2 6 cm 2 cm2 Higgles (Gump 2) 2 cm 4 cm 2 1 = 4 2 12 cm 8 cm2 Ziggles (Gump 3) 3 cm 6 cm 3 1 = 6 2 18 cm 18 cm2 4 cm 8 cm 4 1 = 8 2 24 cm 32 cm2 5 cm 10 cm 5 1 = 10 2 30 cm 50 cm2 10 cm 20 cm 10 1 = 20 2 60 cm 200 cm2 20 cm 40 cm 20 1 = 40 2 120 cm 800 cm2 100 cm 200 cm 100 1 = 200 2 600 cm 20,000 cm2 1 cm 6 cm 1 1 = 6 2 14 cm 6 cm2 2 cm 3 cm 2 1 = 3 2 10 cm 6 cm2 Prediction for Gump 4 Prediction for Gump 5 . . . Prediction for Gump 10 Prediction for Gump 20 Prediction for Gump 100 Wiggles Miggles Make ratios using the nose perimeter for the following figures: Gump 2:Gump 1 Gump 3:Gump 1 12 2 6 1 Gump 4:Gump 1 24 4 6 1 18 3 6 1 Gump 5:Gump 1 30 5 6 1 Make a comparison between the scale factor of objects and the ratio of their perimeters. The ratio of the perimeters of two objects is the same as the scale factor. CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 59 of 162 Make ratios using the nose area for the following figures: Gump 2:Gump 1 Gump 3:Gump 1 18 9 2 1 8 4 2 1 Gump 4:Gump 1 Gump 5:Gump 1 32 16 2 1 50 25 2 1 Make a comparison between the scale factor of objects and the ratio of their areas. The ratios of the areas of two object is equal to the square of the scale factor. 12. Look at Gump 10, Gump 20 and Gump 100. Using your answers to #9 and #11, show the relationship between scale factor of objects and the ratio of their perimeters and areas. Perimeter of Gump 10 10 Perimeter of Gump 1 1 Area of Gump 10 102 2 Area of Gump 1 1 Perimeter of Gump 20 20 Perimeter of Gump 1 1 Area of Gump 20 202 2 Area of Gump 1 1 Perimeter of Gump 100 100 Perimeter of Gump 1 1 Area of Gump 100 1002 2 Area of Gump 1 1 x 10 6 1 x 100 2 1 x 20 6 1 x 400 2 1 x 100 6 1 x 10,000 2 1 CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Perimeter of Gump 10 = 60 cm Area of Gump 10 = 200 cm2 Perimeter of Gump 20 = 120 cm Area of Gump 20 = 800 cm2 Perimeter of Gump 100 = 600 cm Area of Gump 100 = 20,000 cm2 Columbus City Schools 6/28/13 Page 60 of 162 G –SRT 1b Name ______________________________ Date ____________ Period ________ Draw Similar Triangles Instructions: In each problem, draw a triangle similar to the one shown. Remember, corresponding angles of similar triangles have the same measure. Sides of similar triangles are proportional. Show all calculations that verify the triangles are similar. 1. 2. 3. 4. 5. _________________________________________________________________________ CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 61 of 162 6. 7. 8. 9. 10. CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 62 of 162 G –SRT 1b Name ______________________________ Date ____________ Period ________ Similar Quilt Blocks Quilts are a beautiful, practical, and historically significant use of geometric shapes. Create a quilt block using triangles that are similar to the triangles in the quilt block you selected. Select one of the quilt blocks shown on the “Quilt Design” pages. Your quilt block may be a replica of the given quilt block or it may be of your own design. Verify that your triangles are similar and show calculations on the “Quilt Calculations” page. Draw your quilt design in the space below. CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 63 of 162 G –SRT 1b Name ______________________________ Date ____________ Period ________ Quilt Calculations Measure the sides and angles of each triangle in the quilt block. Record these values in the “Original Triangle” section of the chart. Draw a triangle similar to the one you just measured. Measure the sides and angles and record these values in the “New Triangle” section of the chart. Verify that the sides of the similar triangles are proportional and place those calculations in the “Calculations” area. Original Triangle angle A angle B angle C side a side b side c New Triangle angle A angle B angle C side a side b side c Calculations Original Triangle angle A angle B angle C side a side b side c New Triangle angle A angle B angle C side a side b side c Calculations Original Triangle angle A angle B angle C side a side b side c New Triangle angle A angle B angle C side a side b side c Calculations Original Triangle angle A angle B angle C side a side b side c New Triangle angle A angle B angle C side a side b side c Calculations CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 64 of 162 Quilt Design #1 Hopscotch Grandma’s is from the Quilt Pattern Collection of the Camden-Carrol Library, Morehead State University. CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 65 of 162 Quilt Design #2 Laced Star is from the Quilt Pattern Collection of the Camden-Carrol Library, Morehead State University. G –SRT 2 CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 66 of 162 Name ______________________________ Date ____________ Period ________ Investigating Triangles with Two Pairs of Congruent Angles Given: a triangle with one angle measure of 40o and another angle measure of 50o: 1. Construct a triangle with the given angle measures. Label the 40o angle A, the 50o angle B, and the third angle C. 2. Use a ruler to find the length of each side of triangle ABC to the nearest tenth of a centimeter. AB= BC= AC= 3. Draw a second triangle that has the same angle measurements but is not congruent to triangle ABC. Label this triangle A'B'C'. 4. Use a ruler to find the length of each side of triangle A’B’C’ to the nearest tenth of a centimeter. A'B'= B'C'= A'C'= 5. How do the sides of triangle A'B'C' compare to the sides of triangle ABC? 6. How does the measurement of angle C compare to the measurement of angle C'? CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 67 of 162 7. What conclusion can be drawn about triangle ABC compared to triangle A'B'C'? Given: a triangle with one angle measure of 80o and another angle measure of 60o: 8. Construct a triangle with the given angle measures. Label the 80o angle M, the 60o angle N, and the third angle O. 9. Use a ruler to find the length of each side of triangle MNO to the nearest tenth of a centimeter. MN= NO= MO= 10. Draw a second triangle that has the same angle measurements but is not congruent to triangle MNO. Label this triangle M'N'O'. 11. Use a ruler find the length of each side of triangle M'N'O' to the nearest tenth of a centimeter. M'N'= N'O'= 12. M'O'= How do the sides of triangle M'N'O' compare to the sides of triangle MNO? 13. How does the measurement of angle O compare to the measurement of angle O'? 14. What conclusion can be drawn about triangle MNO compared to triangle M'N'O'? 15. What can you conclude about two triangles given two pair of congruent angles? CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 68 of 162 Name ___________________________________ Date __________________ Period ________ Investigating Triangles with Two Pairs of Congruent Angles Answer Key Given a triangle with one angle measure of 40o and another angle measure of 50o. 1. Construct a triangle with the given angle measures. Label the 40o angle A, the 50o angle B, and the third angle C. Answers may vary. 2. Use a ruler to find the length of each side of triangle ABC to the nearest tenth of a centimeter. AB= Answers may vary. BC= Answers may vary. AC= Answers may vary. 3. Draw a second triangle that has the same angle measurements but is not congruent to ABC . Label this triangle A'B'C'. Answers may vary. 4. Use a ruler to find the length of each side of triangle A'B'C' to the nearest tenth of a centimeter. A'B'= Answers may vary. B'C'= Answers may vary. A'C'= Answers may vary. 5. How do the sides of triangle A'B'C' compare to the sides of triangle ABC? They are proportional. 6. How does the measurement of angle C compare to the measurement of angle C'? They are congruent. 7. What conclusion can be drawn about triangle ABC compared to triangle A'B'C'? They are similar Given a triangle with one angle measure of 80o and another angle measure of 60o. 8. Construct a triangle with the given angle measures. Label the 80o angle M, the 60o angle N, and the third angle O. Answers may vary. 9. Use a ruler to find the length of each side of triangle MNO to the nearest tenth of a centimeter. MN= Answers may vary. NO= Answers may vary. MO= Answers may vary. 10. Draw a second triangle that has the same angle measurements but is not congruent to triangle MNO. Label this triangle M'N'O'. Answers may vary. 11. Use a ruler to find the length of each side of triangle M'N'O' to the nearest tenth of a centimeter. CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 69 of 162 M'N'= Answers may vary. N'O'= Answers may vary. M'O'= Answers may vary. 12. How do the sides of triangle M'N'O' compare to the sides of triangle MNO? They are proportional. 13. How does the measurement of angle O compare to the measurement of angle O'? They are congruent. 14. What conclusion can be drawn about triangle MNO compared to triangle M'N'O'? They are similar. 15. What can you conclude about two triangles given two pair of congruent angles? They are similar. CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 70 of 162 G –SRT 2 Name ______________________________ Date ____________ Period ________ Similar Triangles Application Use a mirror, a meter stick, and similar triangles to calculate the height of three objects in the room. Think about what information will be needed and how to accurately collect it. Describe the object, its location, and the measurements taken. Description of Object Distance From Student To Mirror Distance From The Mirror To The Base Of The Object Distance From Line Of Sight To Ground Draw a sketch of each situation and explain why this scenario involves similar s. Label your picture with your measurements and use proportions or scale factor to calculate the height of each object. Record your calculated heights below. Now, measure the actual height of each object. Record the actual (measured) heights below. Describe how well the calculated height matches the actual height. If there is a significant discrepancy, explain where any error may have occurred and if it can be corrected. CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 71 of 162 G –SRT 2 Name ______________________________ Date ____________ Period ________ Find the Scale Factor For each exercise, find the scale factor of figure A to figure B and solve for x. 1. 24 A A B 6 3 x 2. 5 6 A x x+4 B 3. A CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 72 of 162 20 B 15 x 4 4. A 8 10 B x x+1 CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 73 of 162 Name ___________________________________ Date __________________ Period ________ Find the Scale Factor Answer Key For each exercise, find the scale factor of figure A to figure B and solve for x. 1. Scale Factor = 5; x = 12 24 A 6 B 3 x 2. Scale Factor = 5 x 1 ; x = 20 5 6 x+4 A B 3. Scale Factor = 6; x = 3 CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 74 of 162 A 20 B 15 4 x 4. Scale Factor = 3; x = 4 A 8 10 B x x+1 CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 75 of 162 G –SRT 4 Name ______________________________ Date ____________ Period ________ Let’s Prove the Pythagorean Theorem Given the square below, mark a point, E, on AB (not a midpoint). Next, mark a point, F, on DA such that DF = AE. Now, mark a point, G, on CD such that CG = AE. Again, mark a point, H, on BC such that BH = AE. Once you have marked all the new points, connect them to create another square that is inscribed in square ABCD. Label each side of the new smaller square x. A B D C Examine AE and BE . Decide which segment is shorter, s, and which segment is longer, l. Label each segment either s or l accordingly. Do the same thing for DF and AF ; CG and DG ; BH and CH . How many right triangles do you see? Name all of them. In each right triangle, what are the s, l and x (i.e. is it the leg or hypotenuse of the right triangle)? Represent the area of square ABCD in terms of s and l. Simplify the expression. Represent the combined area of all the triangles in terms of s and l. Simplify the expression. Represent the area of the smaller square in terms of x. CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 76 of 162 Write an expression for the total area of all the right triangles and the smaller square. What should this total area be equal to and why? Write an equation that relates part E to Part C. Identify and eliminate any common terms on each side of the equation. Explain what each variable in the new equation represents. You have just proven the Pythagorean Theorem! CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 77 of 162 Name ___________________________________ Date __________________ Period ________ Let’s Prove the Pythagorean Theorem Answer Key Given the square below, mark a point, E, on AB (not a midpoint). Next, mark a point, F, on DA such that DF = AE. Now, mark a point, G, on CD such that CG = AE. Again, mark a point, H, on BC such that BH = AE. Once you have marked all the new points, connect them to create another square that is inscribed in square ABCD. Label each side of the new smaller square x. Examine AE and BE . Decide which segment is shorter, s, and which segment is longer, l. Label each segment either s or l accordingly. Do the same thing for DF and AF ; CG and DG ; BH and CH . How many right triangles do you see? Name all of them. Four triangles - AEF, BEH, CGH, DFG (students could label these differently) In each right triangle, what are the s, l and x (i.e. is it the leg or hypotenuse of the right triangle)? s is a leg, l is a leg and x is the hypotenuse Represent the area of square ABCD in terms of s and l. Simplify the expression. Area = (s + l)2 = s2 + 2sl + l2 Represent the combined area of all the triangles in terms of s and l. Simplify the expression. Area = 4(½)sl = 2sl Represent the area of the smaller square in terms of x. Area = x2 Write an expression for the total area of all the right triangles and the smaller square. What should this total area be equal to and why? Total area = 2sl + x2 CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 78 of 162 This total area should equal to the area of the original square ABCD because the original square consists of the 4 right triangle and the inscribed square. Write an equation that relates part E to Part C. Identify and eliminate any common terms on each side of the equation. Explain what each variable in the new equation represents. s2 + 2sl + l2 = 2sl + x2 s2 + l2 = x2 s and l are the legs of the right triangle and x is the hypotenuse. You have just proven the Pythagorean Theorem! CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 79 of 162 G –SRT 4 Name ______________________________ Date ____________ Period ________ Proving the Pythagorean Theorem, Again! a 2 b c 3 c a 1 b (base1 base2 )height If the formula for finding the area of a trapezoid is , find the area of the above 2 trapezoid. Simplify the expression. If the formula for finding the area of a triangle is base height , find the areas of each triangle in the 2 picture above. Write an equation relating #1 and #2. Using your algebra skills, try to manipulate the equation so that only the Pythagorean Theorem remains. CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 80 of 162 Name ___________________________________ Date __________________ Period ________ Proving the Pythagorean Theorem, Again! Answer Key a 2 b c 3 c a 1 b (base1 base2 )height If the formula for finding the area of a trapezoid is , find the area of the above 2 trapezoid. Simplify the expression. (a b)(a b) a 2 ab ab b 2 a 2 2ab b 2 base1 = a Area of trapezoid = 2 2 2 base2 = b height = a + b If the formula for finding the area of a triangle is picture above. Area of triangle1 = ½ ab base height , find the areas of each triangle in the 2 Area of triangle2 = ½ ab Area of triangle3 = ½ c2 Write an equation relating #1 and #2. Using your algebra skills, try to manipulate the equation so that only the Pythagorean Theorem remains. Area of triangle1 + Area of triangle2 + Area of triangle3 = Area of trapezoid 2 2 1 1 1 2 a +2ab+b ab + ab + c = 2 2 2 2 2 2 ab + ab + c = a + 2ab + b2 2ab + c2 = a2 + 2ab + b2 c2 = a2 + b2 CCSSM II Similarity G-SRT 1, 1a, 1b, 2, 3, 4, 5 Quarter 1 Columbus City Schools 6/28/13 Page 81 of 162 COLUMBUS CITY SCHOOLS HIGH SCHOOL CCSS MATHEMATICS II CURRICULUM GUIDE TOPIC 2 CONCEPTUAL CATEGORY TIME RANGE GRADING 20 days Trigonometric Ratios Geometry PERIOD G-SRT 6, 7, 8 1 Domain: Similarity, Right Triangles, and Trigonometry (G – SRT): Cluster 3) Define Trigonometric ratios and solve problems involving right triangles. Standards 3) Define Trigonometric ratios and solve problems involving similarity. G – SRT 6: Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. G – SRT 7: Explain and use the relationship between the sine and cosine of complementary angles. G – SRT 8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 82 of 162 TEACHING TOOL Vocabulary: acute angle, adjacent, angle of depression, angle of elevation, complementary angles, corresponding sides, cosine, geometric mean, hypotenuse, opposite, proportion, Pythagorean Theorem, Pythagorean triple, ratio, right triangle, similar triangles, sine, solving a triangle, special right triangles, tangent, trigonometric ratios, trigonometry Teacher Notes Properties of Radicals n An expression that contains a radical sign is called a radical expression, a . The expression under the radical sign is the radicand and the numeric value, n is the index. We read this as “the nth root of 5 a. Looking at the radical expression 3x , 3x is the radicand and 5 is the index. 1. c is a square root of a, if c2 = a, e.g., 2 is a square root of 4 because 22 = 4 and -2 is a square root of 4 because (-2)2 = 4. Because there are two values that satisfy the equation x2 = 4, we take the term square root to mean the principal square root which has a non-negative value. In this case 4 2 is the principal square root. Mathematically, we express this as: a2 a 2. c is a cube root of a if c3 = a, e.g., 3 is a cube root of 27 because 33 = 27 and -3 is a cube root of 27 because (-3)3 = -27. The cube root of a negative number is negative. The cube root of a positive number is positive. 3. c is an nth root of a if cn = a. Note that if the index is odd and the radicand is negative then the principal root is negative. For example, 32 2 because (-2)5 = -32. The following are general rules for taking the roots of positive and negative numbers. 5 The answer is the principal root. The answer is the opposite of the principal root. The answer is both roots, the positive and the negative root. odd number even number odd number even number negative number negative number positive number The answer is a negative number. There is no real solution. The answer is the principal root. positive number The answer is the principal root. For any value of x and any even number n, = -5, then 8 n xn x . For example, if x = 4, then 6 46 4 4 . If x (5)8 5 5 . For any value of x and any odd number n greater than 1, example, if x = 4, then . If x = -5, then 4. product rule: n 9 n x n x . For (5)9 5 . a n b n ab , e.g. 3 7 3 5 3 7 5 3 35 , and CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 83 of 162 ( x 6) ( x 6) ( x 6)( x 6) x 2 36 The product rule can be used for factoring to simplify radical expressions as shown below. 50 25 2 25 2 5 2 and 72 2 2 2 3 3 2 2 2 3 3 2 3 2 6 2 19 19 72 can be written as the product of prime factors, and then simplified, but 19 is a prime number so it is already in its simplest form. 5. quotient rule: given n a and n b,b 0 , n a na , e.g. b nb x2 x2 x 16 16 4 6. principle of powers: if a = b then an = bn This website offers a teacher resource that includes a power point presentation for operations with radical expressions. http://teachers.henrico.k12.va.us/math/hcpsalgebra1/module11-3.html This website has an on-line explanation of radicals. http://www.regentsprep.org/Regents/math/algtrig/ATO3/simpradlesson.htm The website below is a teacher resource that has lessons, practice and a tutorial. http://www.regentsprep.org/Regents/math/ALGEBRA/AO1/indexAO1.htm The two following websites have practice with operations with radicals. http://www.algebralab.org/practice/practice.aspx?file=Algebra1_13-2.xml http://www.algebralab.org/practice/practice.aspx?file=Algebra1_13-3.xml Right Triangles Remind students that it is better to remember the Pythagorean Theorem as leg2 + leg2 = hypotenuse2 rather than a2 + b2 = c2, since there is no guarantee that c is always the hypotenuse. There are two special right triangles. The first is a 45-45-90 triangle. The special ratio is 1:1: 2 . The second is a 30-60-90 triangle. The special ratio is 1: 3 : 2 . Solving special right triangles http://www.youtube.com/watch?v=nVTtSE5nv7c http://www.youtube.com/watch?v=NsNaYwHtowA Trigonometry is based on similar right triangles. The sine (sin) of an angle is the ratio of the opposite side to the hypotenuse. The cosine (cos) of an angle is the ratio of the adjacent side to the hypotenuse. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 84 of 162 The cosine of an angle is the ratio of the adjacent side to the hypotenuse. The tangent (tan) of an angle is the ratio of the opposite side to the adjacent side. There are many different ways to help your students remember the sine, cosine, and tangent functions. Use the old Indian chief SOH CAH TOA. Tell a story of how a great Indian chief was also a great mathematician. And he developed sine, cosine, and tangent to match his name. SOH (sin = opp / hyp) CAH (cos = adj / hyp) TOA (tan = opp / adj) The following phrase could also be used. Some Caught Taking Old Horse Another Horse Oats Away The geometric mean is the square root of the product of two numbers. In right triangles, an altitude drawn to the hypotenuse is the geometric mean of the measures of the two segments of the hypotenuse. Each leg of a right triangle is the geometric mean of the measure of the adjacent segment of the hypotenuse and the total measure of the hypotenuse. angle of depression The angle of elevation is the angle between the line of sight and the horizontal when looking up. The angle of depression is the angle between the line of sight and the horizontal when looking down. It is helpful to remember that the angle of elevation and the angle of depression are alternate interior angles to each other. angle of elevation Real life applications are architecture and engineering. Right triangle trigonometry is one of the more practical day-to-day applications of mathematics. Used to find lengths and angles, it is a necessity in construction and home improvement. For example, if you wish to build a deck that is a regular polygon, you only need the length of one side to find the area using trigonometry and simple geometry. The three trigonometric functions, sine, cosine, and tangent are simply ratios of the sides of right triangles. These values can be found in a table, in a calculator, or in a textbook. By the Angle-Angle Similarity Theorem, if the measures of two of the angles of a pair of triangles are equal, then the triangles are similar. Since we are working with right triangles only, all triangles with a second angle of the same measure are similar and their sides are proportional. The given angle is called “theta” and is represented by the symbol . The side of the triangle across from is the “opposite side”. The side of the triangle next to is the “adjacent side”. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 85 of 162 hypotenuse opposite side adjacent side The trigonometric ratios are: = sin opposite =o hypotenuse h θ= cos Words Symbol Trigonometric sine θ sin θ Ratios cosine θ cos θ tangent θ tan θ adjacent =a hypotenuse h θ= tan opposite o = adjacent a Definition opposite sin hypotenuse adjacent cos hypotenuse opposite tan adjacent If the angle measure is 30°, 45° or 60° in a right triangle, special trigonometric relationships exist. θ sin θ 1 2 cos θ tan θ csc θ sec θ cot θ 3 3 2 3 2 3 2 3 3 2 2 45˚ 1 1 2 2 2 2 1 3 2 3 3 60˚ 2 3 2 2 3 3 2 2 Remind students that it is better to remember the Pythagorean Theorem as leg + leg = hypotenuse2 rather than a2 + b2 = c2, since there is no guarantee that c is always the hypotenuse. 30˚ There are two special right triangles. The first is a 45-45-90 triangle. The special ratio is 1 :1 : 2 . The second is a 30-60-90 triangle. The special ratio is 1: 3 : 2 . 45o 30o x 2 x 2x x 3 45 o x x x CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 60o Columbus City Schools 6/28/13 Page 86 of 162 m∠A + m∠B = 90o B sin A= opposite side of A a sin B= opposite side of B b cos A= adjacent side of A b cos B= opposite side of B a hypotenuse c a c hypotenuse C b a2 + b2 = c2 c A tan A= opposite side of A a adjacent side of A b hypotenuse hypotenuse c c tan B= opposite side of B b adjacent side of B a Students must understand that triangles with congruent angles are similar triangles. Students must understand that the ratio of two sides in one triangle is equal to the ratio of the corresponding two sides of all other similar triangles. Right Triangles Right Triangle Trigonometry http://patrickjmt.com/right-triangles-and-trigonometry/ A website video tutorial on right triangle trigonometry. Evaluating Trigonometric functions http://patrickjmt.com/evaluating-trigonometric-functions-for-an-unknown-angle-given-a-pointon-the-angle-ex-1/ Evaluating trigonometric functions for an unknown angle given a point on the angle. Right Triangle Trigonometry http://teachers.henrico.k12.va.us/math/IGO/07RightTriangles/7_4.html Teacher resource that includes the warm-up, algebraic review, lecture notes, practice, and hands on activities for right triangle trigonometry. Special Right Triangles http://teachers.henrico.k12.va.us/math/IGO/07RightTriangles/7_3.html Teacher resource that includes the warm-up, algebraic review, lecture notes, practice, and hands on activities for special right triangles. Basic Trigonometry http://education.ti.com/en/us/activity/detail?id=469426FC7D1542A9B54240E5C87A8593 Students define basic terms relating to trigonometry and use trigonometric ratios using their TI-84 calculator. Module 16 Trigonometric Ratios http://education.ti.com/en/us/professional-development/pd_onlinegeometry_free/course-outlineand-take-this-course Using TI – 84 and Cabri Jr for special triangles Sine and Cosine of Complementary Angles http://learni.st/users/60/boards/3370-sine-and-cosineCCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 87 of 162 of-complementary-angles-common-core-standard-9-12-g-srt-7#/users/60/boards/3370-sine-andcosine-of-complementary-angles-common-core-standard-9-12-g-srt-7 Tutorials to explain the relationship between the sine and cosine of complementary angles. Co-Functions http://www.regentsprep.org/Regents/math/algtrig/ATT6/cofunctions.htm Practice and warm ups to explain co-functions Finding Height Using Trigonometry http://patrickjmt.com/finding-the-height-of-an-object-using-trigonometry-example-1/ Tutorial on finding the height of an object using trigonometry example 1 Finding Height Using Trigonometry http://patrickjmt.com/finding-the-height-of-an-object-using-trigonometry-example-2/ Example 2 Finding Height Using Trigonometry http://patrickjmt.com/finding-the-height-of-an-object-using-trigonometry-example-3/ Example 3 Finding Height Using Trigonometry http://patrickjmt.com/trigonometry-word-problem-finding-the-height-of-a-building-example-1/ Word Problem 1 Finding Height Using Trigonometry http://patrickjmt.com/trigonometry-word-problem-example-2/ Word Problem 2 Misconceptions/Challenges: SRT 6 Students struggle labeling the opposite, adjacent and hypotenuse. Sometimes they use the shortest leg as the opposite leg or confuse adjacent and hypotenuse. Students get confused of where the angle of depression is located. Students confuse the difference on how to use the calculator when finding values of a missing side or missing angle. Students may apply the ratios of the special right triangles to all right triangles. Once trigonometry is taught, students like to use that instead of the ratios of special triangles. But to get exact values, they must use the ratios. SRT 8 Students may not substitute the hypotenuse in for ‘c’ in the Pythagorean Theorem. Angle of depression is often mislabeled as the angle between the vertical and hypotenuse Students incorrectly identify corresponding legs when using hypotenuse-leg congruence for right triangles. Students do not understand that equilateral triangles are also equiangular and vice versa. Students do not realize that congruent angles in an isosceles triangle are opposite the congruent sides. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 88 of 162 Instructional Strategies: This is an entire unit that covers all three standards. There are many references to everyday objects in the lessons. https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_912_AccelCoorAlgebraAnalyticGeom_Unit8SE.pdf This link has instructional strategies and sample formative assessment tasks as well as key concepts and vocabulary. http://www.schools.utah.gov/CURR/mathsec/Core/Secondary-II/II-5-G-SRT-6.aspx project ideas http://ccss.performanceassessment.org/taxonomy/term/1045 The following website has practice on simplifying radical expressions. http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Simplifying%20Radicals.pd f SRT 6 Stay in Shape http://www.teachengineering.org/view_activity.php?url=http://www.teachengineering.org/col lection/cub_/activities/cub_navigation/cub_navigation_lesson03_activity1.xml Lesson on how triangles and The Pythagorean Theorem are used in measuring distance. Fit by Design https://access.bridges.com/usa/en_US/choices/pro/content/applied/topic/aom14CX.html lesson relates actual and calculated measures of right triangles to objects created by mechanical drafters or designers Calculating Volumes of Compound Objects http://map.mathshell.org/materials/lessons.php?taskid=216 Decomposing shapes into simpler ones and using right triangles to solve real-world problems. Geometry Problems: Circles and Triangles http://map.mathshell.org/materials/lessons.php?taskid=222 Students determine the lengths of sides in right triangles to solve problems. Hopewell Geometry. http://map.mathshell.org/materials/tasks.php?taskid=127&subpage=apprentice How the Hopewell people constructed earthworks using right triangles. Have students complete the activity “Exploring Special Right Triangles 45-45-90” (included in this Curriculum Guide) to reinforce the properties of 45-45-90 triangles. Have students complete the activity “Exploring Special Right Triangles 30-60-90” (included in this Curriculum Guide) to reinforce the properties of 30-60-90 triangles. Have students complete the activity “Discovering Trigonometric Ratios” (included in this curriculum guide) to develop their understanding of trigonometry. Students will need centimeter rulers and protractors to measure the parts of the given triangles. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 89 of 162 Have students complete the activity “Make a Model: Trigonometric Ratios” (included in Curriculum Guide) to discover that the trigonometric ratios of any right triangle with specific acute angles are the same regardless of the lengths of the sides. Calculators may be helpful for this activity. Eratosthenes Finds the Circumference of the Earth https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_9-12_AnalyticGeo_Unit2SE.pdf Students examine a diagram and verify the two triangles are similar. Page 12 Discovering Special Triangles https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_9-12_AnalyticGeo_Unit2SE.pdf Students use real-world situations to discover special right triangles. Page 16 Finding Right Triangles in Your Environment https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_9-12_AnalyticGeo_Unit2SE.pdf Students find right triangles such as a ramp. Page 20 Create Your Own Triangles https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_9-12_AnalyticGeo_Unit2SE.pdf Students use paper, compass, straight edge and protractor to create right triangles and verify the measurements. Page 22 The Tangent Ratio https://www.cohs.com/editor/userUploads/file/Meyn/321%20Ch%206a%20Student%20WB. pdf Students solve real-world problems using the tangent ratio Are Relationships Predictable .http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig _te_040913.pdf Students develop and use right triangle relationships based on similar triangles. Classroom Task: 6. pages 53-59. (This strategy can also be found in SRT8.) Relationships with Meaning http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig _te_040913.pdf Students find relationships between the sine and cosine ratios for right triangles, including the Pythagorean identity. Classroom Task: 6.8 pages 60-66 (This strategy can also be found in SRT7.) Solving Right Triangles Using Trigonometric Relationships http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig _te_040913.pdf Students set up and solve right triangles modeling real world context. Classroom Task: 6.10 found on pages 75-81 (This strategy can also be found in SRT7.) CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 90 of 162 SRT 7 Relationships with Meaning http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig _te_040913.pdf Students find relationships between the sine and cosine ratios for right triangles, including the Pythagorean identity. Classroom Task: 6.8 pages 60-66 (This strategy can also be found in SRT 6.) Solving Right Triangles Using Trigonometric Relationships http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig _te_040913.pdf Students set up and solve right triangles modeling real world context. Classroom Task: 6.10 found on pages 75-81 (This strategy can also be found in SRT 6.) Create Your Own Right Triangles https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_9-12_AnalyticGeo_Unit2SE.pdf Students use paper, compass, straight edge and protractor to create right triangles and verify the measurements. Page 22. Discovering Trigonometric Ratio Relationships https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_9-12_AnalyticGeo_Unit2SE.pdf Students use degree measurements of acute angles from right triangles to determine trigonometric ratios. Page 27 Finding Right Triangles in Your Environment https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_9-12_AnalyticGeo_Unit2SE.pdf Students find right triangles such as a ramp. Page 20 The Sine and Cosine Ratios https://www.cohs.com/editor/userUploads/file/Meyn/321 Ch 6a Student WB.pdf Students use sine and cosine ratios of acute angles of right triangles to solve real-world problems. Page 7 Special Right Triangles https://www.cohs.com/editor/userUploads/file/Meyn/321 Ch 6a Student WB.pdf Students investigate special right triangles. Page 11 SRT 8 Have students complete the activity “Application of Trigonometry” (included in this Curriculum Guide) to practice using trigonometry to solve indirect measurement questions. Horizons https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_9-12_AnalyticGeo_Unit2SE.pdf Students use trigonometric ratios to determine distance to the horizon from different locations. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 91 of 162 Page 11 Finding Right Triangles in Your Environment https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_9-12_AnalyticGeo_Unit2SE.pdf Students find right triangles such as a ramp. Page 20 Create Your Own Triangles https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGP S_Math_9-12_AnalyticGeo_Unit2SE.pdf Students use paper, compass, straight edge and protractor to create right triangles and verify the measurements. Page 22. Find that Side or Angle https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_9-12_AnalyticGeo_Unit2SE.pdf Students use graphing technology to find the values of sine and cosine in real-world situations. Access Ramp http://www.achieve.org/files/CCSS-CTE-Task-AccessRamp-FINAL.pdf Students design an access ramp which complies with the Americans with Disabilities Act (ADA) requirements and includes pricing based on local costs. Land Surveying Project http://alex.state.al.us/lesson_view.php?id=25108 Students learn the basics of civil engineering in land surveying. The Clock Tower http://alex.state.al.us/lesson_view.php?id=25107 Students use the Pythagorean Theorem, and Sine, Cosine, and Tangent to find unknown heights of objects. Solving Right Triangles. https://www.cohs.com/editor/userUploads/file/Meyn/321%20Ch%206a%20Student%20WB. pdf Students determine the angle measures in right triangles. Lesson 6-4 Page 15 Determine the Missing Sides of Special Right Triangles http://www.kutasoftware.com/FreeWorksheets/GeoWorksheets/8Special%20Right%20Triangles.pdf Students practice finding missing sides of special right triangles Applied Trigonometry http://learni.st/users/60/boards/3453-trig-ratios-and-the-pythagorean-theorem-common-corestandard-9-12-g-srt-8 Several tutorials on trigonometry Solving Right Triangles https://www.cohs.com/editor/userUploads/file/Meyn/321%20Ch%206a%20Student%20WB. pdf Students determine how to find unknown angle measures of a right triangle. Page 15 CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 92 of 162 Have students complete the activity “Let’s Measure the Height of a Flagpole” (included in this Curriculum Guide) to practice solving problems involving indirect measurement. A clinometer can be built from a piece of cardboard, a drinking straw, a piece of string, a washer, and a protractor as shown on the first page of the activity record sheet. Templates for protractors are included “Grids and Graphics” (included in this Curriculum Guide). Note: Answers will vary. Have students practice finding the missing side or angle of a right triangle with the “Find the Missing Side or Angle” activity (included in this Curriculum Guide). Students will need a copy of the activity and a scientific or graphing calculator. Remind students of the legend of Soh Cah Toa, the “great trigonometry leader”. Embellish the story yourself or ask students to spin their own tall tale, write a poem or create a rap that includes a description of the trigonometric ratios: sine = opposite/hypotenuse, cosine =adjacent/hypotenuse, and tangent = opposite/adjacent. Review how the trigonometric ratios can be used to find missing angle measures or side lengths in right triangles. Arrange students in groups of three to play “paper football”. After each group has made its football by folding a sheet of paper, the group will assign duties and measure its field (the distance along the ground from where the ball is kicked to the uprights). One student will be the kicker, one will hold up their hands as the uprights, and one will measure the height of the “football” from the ground as it crosses the uprights. Each group will draw its field, record the measurements, and calculate the angle of elevation the ball makes with the ground for each kick, for at least 5 kicks. Follow up with the “Between the Uprights” activity (included in this Curriculum Guide). Remind students that the angle the goal post makes with the ground is 90. Students will need a copy of the activity, a sheet of paper to use to make a football, and a calculator. Discuss with students the effect of a five or ten yard penalty on the results for each situation. How significantly would the angle or distances be changed? Arrange students into groups of two. Student will practice their right triangle solving skills with the exercise “Solve the Triangle” (included in this Curriculum Guide). Students will need a copy of the activity, a calculator, a ruler, and a protractor. Have students take turns explaining to their partner how they solved one of the problems on the sheet. Students design their ideal city park in the activity “Right Triangle Park” (included in this Curriculum Guide). The catch is that their “ideal” park can be made up of only right triangles. Students will then measure two parts of each triangle and calculate the remaining parts using trigonometry. Students will need a calculator, ruler, protractor, and a copy of the worksheet. Allow students time to share with the class their design. Ask students to point out several of the right triangles in their design and to explain how they calculated the lengths and/or angles for a few of the triangles. Have students complete the activity “Applications of the Pythagorean Theorem” (included in this Curriculum Guide) to explore real-life applications of the Pythagorean Theorem. Students can reinforce their similar triangle skills by using the properties of similar triangles to measure objects around school. Have the students select three objects they want to measure and use a mirror to create a pair of similar triangles as instructed in the “Find the Height” activity (included in this Curriculum Guide). Each group of two or three students will need a tape measure, mirror, and a copy of the activity instructions. Each student will need a copy of the activity data CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 93 of 162 sheet. After students have completed the activity, discuss as a class their strategies for finding the height of each object. Close Enough http://www.teachengineering.org/view_activity.php?url=http%3A%2F%2Fwww.teachengin eering.org%2Fcollection%2Fcub_%2Factivities%2Fcub_navigation%2Fcub_navigation_les son04_activity1.xml Hands-on activity shows how accurate measurement is important as students use right triangle trigonometry and angle measurements to calculate distances Six Trigonometric Ratio Values of Special Acute http://illuminations.nctm.org/LessonDetail.aspx?id=L383 A puzzle for practicing knowledge of all six trigonometric ratios. Two activities involve angle of elevation and angle of declination. Solving Problems Using Trigonometry http://education.ti.com/en/us/activity/detail?id=EB3E2581FFEC4FDA8FC94C3AA51F3D31 Students use TI-84 calculator to find the angle of elevation or the angle of depression. Basic Trigonometry http://education.ti.com/en/us/activity/detail?id=469426FC7D1542A9B54240E5C87A8593 Students define basic terms relating to trigonometry and use trigonometric ratios. Are Relationships Predictable http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec2_mod6_simrighttrig _te_040913.pdf Students develop the right triangle trigonometric relationships based on similar triangles. (This strategy can also be found in SRT8.) Reteach: Have students complete the activity “Memory Match” (included in this Curriculum Guide) to reinforce right triangle terminology. Students can work in groups of three or four. First place all cards facedown and then have each student take turns drawing two cards. If the two cards drawn go together as a pair the student will keep it as a match. Students take turns drawing. The student with the most pairs or matches wins. Students will need a scientific calculator. Additional practice in solving proportions, using the properties of similar triangles and right triangle trigonometry is available in the “Similar Right Triangles and Trigonometric Ratios” activity (included in this Curriculum Guide). Have students complete the activity “Hey, All These Formulas Look Alike” (included in this Curriculum Guide) to investigate the tangent relationship. Note: For the visual learner, the use of highlighted notes may lead to greater understanding. Highlighting (x2 – x1 ) x and ( y2 – y1) y in yellow and pink, respectively, may aide the visual learner in the formula comparisons. Extensions: Have students complete the activity “Similar Right Triangles and Trigonometric Ratios” CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 94 of 162 (included in Curriculum Guide) to make connections between similar triangles and trigonometric ratios Have students complete the activity “Problem Solving: Trigonometric Ratios” (included in Curriculum Guide) to apply their knowledge of trigonometric ratios. Given the sides of a right triangle inscribed in a circle and circumscribed about another circle, students find the radii of each circle. Also, students analyze sample solutions and compare their own solutions to those given. http://map.mathshell.org/materials/lessons.php?taskid=403#task403 Textbook References: Textbook: Geometry, Glencoe (2005): pp. 349, 350-356 Supplemental: Geometry, Glencoe (2005): Chapter 7 Resource Masters Study Guide and Intervention, pp. 357-358 Skills Practice, p. 359 Practice, p. 360 Enrichment, p. 362 Textbook: Geometry, Glencoe (2005): pp. 357-363 Supplemental: Geometry, Glencoe (2005): Chapter 7 Resource Masters Study Guide and Intervention, pp. 363-364 Skills Practice, p. 365 Practice, p. 366 Enrichment, p. 368 Textbook: Geometry, Glencoe (2005): pp. 364-370 Supplemental: Geometry, Glencoe (2005): Chapter 7 Resource Masters Study Guide and Intervention, pp. 369-370 Skills Practice, p. 371 Practice, p. 372 Enrichment, p. 374 Textbook: Geometry, Glencoe (2005): pp. 342-348 371-376 Supplemental: Geometry, Glencoe (2005): Chapter 7 Resource Masters Study Guide and Intervention, pp. 351-352, 375-376 Skills Practice, pp. 353, 377 Practice, pp. 354, 378 Enrichment, pp. 356, 380 Supplemental: Integrated Mathematics: Course 3, McDougal Littell (2002): Teacher’s Resources for Transfer Students, pp. 39-40 Supplemental: Integrated Mathematics: Course 3, McDougal Littell (2002): Skills Bank, p. 104 Overhead Visuals, folders A, 10 CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 95 of 162 Textbook: Algebra 1, Glencoe (2005): pp. 622 – 630 Textbook: Algebra 1, Glencoe (2005): pp. 698 – 708 CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 96 of 162 G-SRT 6 SRT- 6 Name ___________________________________ Date __________________ Period ________ Exploring Special Right Triangles (45-45-90) Given the isosceles right triangle below. l h l 1. What is the measure of each acute angle? Explain. 2. a) If the length of each leg is 1 unit, find the length of the hypotenuse. Leave answer exact and simplified. b) What are the side-length ratios of leg: leg: hypotenuse? 3. a) If the length of each leg is 2 units, find the length of the hypotenuse. Leave answer exact and simplified. b) How many times longer is the hypotenuse than the leg? c) What are the side-length ratios of leg: leg: hypotenuse? Simplify the ratio. 4. a) If the length of each leg is 5 units, find the length of the hypotenuse. Leave answer exact and simplified. b) How many times longer is the hypotenuse than the leg? c) What are the side-length ratios of leg: leg: hypotenuse? Simplify the ratio. 5. What can you conclude about the side-length ratios of leg: leg: hypotenuse of any isosceles right triangle? CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 97 of 162 Name ___________________________________ Date __________________ Period ________ Exploring Special Right Triangles (45-45-90) Answer Key Given the isosceles right triangle below. l h 1. What is the measurel of each acute angle? Explain how you know. Each acute angle is 45o. Since this is an isosceles right triangle, each angle opposite the legs are congruent. Since there’s a total of 90o for both acute angles and they are congruent, they must be 45o each. 2. a) If the length of each leg is 1 unit, find the length of the hypotenuse. Leave answer exact and simplified. Hypotenuse = 2 b) What are the side-length ratios of leg: leg: hypotenuse? 1: 1: 2 3. a) If the length of each leg is 2 units, find the length of the hypotenuse. Leave answer exact and simplified. Hypotenuse = 8 = 2 2 b) How many times longer is the hypotenuse than the leg? The hypotenuse is 2 times longer than the leg. c) What are the side-length ratios of leg: leg: hypotenuse? Simplify the ratio. 2: 2: 2 2 which is really 1: 1: 2 4. a) If the length of each leg is 5 units, find the length of the hypotenuse. Leave answer exact and simplified. Hypotenuse = 50 = 5 2 b) How many times longer is the hypotenuse than the leg? The hypotenuse is 2 times longer than the leg. c) What are the side-length ratios of leg: leg: hypotenuse? Simplify the ratio. 5: 5: 5 2 which is really 1: 1: 2 5. What can you conclude about the side-length ratios of leg: leg: hypotenuse of any isosceles right triangle? They will always be 1: 1: 2 . CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 98 of 162 G-SRT 6 Name ___________________________________ Date __________________ Period ________ Exploring Special Right Triangles (30-60-90) 1. Given the equilateral triangle below whose sides are 2 units long. A 2 2 B 2 a) What is the angle measure of each acute angle? C b) Fold the triangle along vertex A so that vertex B maps onto vertex C. Draw a segment along the crease. Label point D where the new segment intersects segment BC . What is special about AD ? What is the length of BD (label it in the diagram above)? Explain your reasoning. c) What does AD do to A ? d) Examine ABD . What is the measure of BAD and BDA (label it in the diagram above)? e) Find the length of AD . Leave exact and simplified. f) In ABD , how many times longer is the hypotenuse than the shorter leg? How many times longer is the longer leg than the shorter leg? g) In this 30o-60o-90o triangle, what are the side length ratios of short leg: long leg: hypotenuse? CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 99 of 162 2. Given the equilateral triangle below whose sides are 4 units long. A 4 4 B C 4 a) Fold the triangle along vertex A so that vertex B maps onto vertex C. Draw a segment along the crease. Label point D where the new segment intersects BC . What is the length of BD (label it in the diagram above)? Explain your reasoning. b) Examine ABD . What is the measure of BAD and BDA (label it in the diagram above)? c) Find the length of AD . Leave exact and simplified. d) In ABD , how many times longer is the hypotenuse than the shorter leg? How many times longer is the longer leg than the shorter leg? CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 100 of 162 e) In this 30o-60o-90o triangle, what are the side length ratios of short leg: long leg: hypotenuse? 3. Repeat the steps above using a different number for the length of the side of the equilateral triangle. What can you conclude about the side-length ratios of short leg: long leg: hypotenuse for any 30o-60o-90o triangle? CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 101 of 162 Name ___________________________________ Date __________________ Period ________ Exploring Special Right Triangles (30-60-90) Answer Key 1. Given the equilateral triangle below whose sides are 2 units long. A 2 B 2 D C a) What is the angle measure of each acute angle? 2 60o b) Fold the triangle along vertex A so that vertex B maps onto vertex C. Draw a segment along the crease. Label point D where the new segment intersects BC . What is special about AD ? What is the length of BD (label it in the diagram above)? Explain your reasoning. AD is the altitude, bisector, bisector, and median of ABC . BD = 1 because AD bisects BC c) What does AD do to A ? It bisects A d) Examine ABD . What is the measure of BAD and BDA (label it in the diagram above)? m BAD = 30o and m BDA = 90o e) Find the length of AD . Leave exact and simplified. AD = 3 f) In ABD, how much longer is the hypotenuse than the shorter leg? How much longer is the longer leg than the shorter leg? The hypotenuse is twice as long as the shorter leg and the longer leg is 3 times as long as the shorter leg. g) In this 30o-60o-90o triangle, what are the side-length ratios of short leg: long leg: hypotenuse? 1: 3:2 CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 102 of 162 2. Given the equilateral triangle below whose sides are 4 units long. A 4 4 B C D 4 a) Fold the triangle along vertex A so that vertex B maps onto vertex C. Draw a segment along the crease. Label point D where the new segment intersects BC . What is the length of BD (label it in the diagram above)? Explain your reasoning. BD = 2 because AD bisects BC b) Examine ABD . What is the measure of BAD and BDA (label it in the diagram above)? m BAD = 30o and m BDA = 90o c) Find the length of AD . Leave exact and simplified. AD = 12 = 2 3 d) In ABD , how many times longer is the hypotenuse than the shorter leg? How many times longer is the longer leg than the shorter leg? The hypotenuse is twice as long as the shorter leg and the longer leg is 3 times as long as the shorter leg. e) In this 30o-60o-90o triangle, what are the side-length ratios of short leg: long leg: hypotenuse? 1: 3 : 2 3. Repeat the steps above using a different number for the length of the side of the equilateral triangle. What can you conclude about the side-length ratios of short leg: long leg: hypotenuse for any 30o-60o-90o triangle? They will always be 1: 3 : 2 . CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 103 of 162 G-SRT 6 Name ___________________________________ Date __________________ Period ________ Discovering Trigonometric Ratios For the following right triangles, find the indicated ratios. C C' A' B' B A Find each length to the nearest quarter of an inch and after division round the quotient to three decimal places. 1. Length of AB = Length of AC Length of AB = Length of AC 2. Length of BC = Length of AC Length of B / C / = Length of A/ C / 3. Length of AB = Length of BC Length of A/ B / = Length of B / C / Triangles ABC and A/B/C/ are similar triangles. 4. From the above experiment, what can you conclude about these ratios? Find the measures of C and C/ to the nearest tenth of a degree. 5. m C = 6. m C/ = Using your calculator, find the following using the value of C from above. (#5) 7. sin C = 8. cos C = 9. tan C = 10. What do you notice about the values in #7 - #9 as compares to the ratios in #1 - #3? 11. Match sine, cosine, and tangent to the three ratios in #1 - #3. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 104 of 162 Find the measures of A and A/. 12. A = 13. A/ = Using your calculator, find the following using the value of A from above. (#12) 14. sin A = 15. cos A = 16. tan A = 17. Did this change how sine, cosine, and tangent match with the ratios in #1 - #3? If so, how and why? 18. Write a general equation for sine, cosine, and tangent that could be used with any right triangle. 19. Make your own triangles to test the above equations. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 105 of 162 G-SRT 6 Name ___________________________________ Date __________________ Period ________ Discovering Trigonometric Ratios Answer Key For the following right triangles, find the indicated ratios. C Find each length to the nearest quarter of an inch and after division round the quotient to three decimal places. 2.5 1. Length of AB 2 = 0.8 1.5 = Length of AC 2.5 2. Length of BC 1.5 B = 2.5 = 0.6 Length of AC 2 Length of A/ B / 1 = 0.8 C' = 1.25 Length of A/ C / 1.25 .75 / / Length of B C .75 = 0.6 A' = B' A 1.25 Length of A/ C / 1 3. Length of AB 2 = 1.333 = Length of BC 1.5 Length of A/ B / 1 = 1.333 = .75 Length of B / C / DUE TO HUMAN AND ROUNDING ERRORS THESE MAY BE CLOSE BUT NOT BE EXACT. Triangles ABC and A/B/C/ are similar triangles. 4. From the above experiment, what can you conclude about these ratios? Both ratios in #1 are close to being the same. Both ratios in #2 are close to being the same. Both ratios in #3 are close to being the same. Find the measures of C and C/ to the nearest tenth of a degree. 5. m C = 53.1o 6. m C/ = 53.1o Using your calculator, find the following using the value of C from above. (#5) 7. sin C = 0.7997 8. cos C = 0.6004 9. tan C = 1.3319 10. What do you notice about the values in #7 - #9 as compares to the ratios in #1 - #3? The value in #7 is close to the value in #1 The value in #8 is close to the value in #2 The value in #9 is close to the value in #3 CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 106 of 162 11. Match sine, cosine, and tangent to the three ratios in #1 - #3. Sine matches with the ratio in #1 Cosine matches the ratio in #2 Tangent matches the ratio in #3 Find the measures of A and A/. 12. A = 36.9o 13. A/ = 36.9o Using your calculator, find the following using the value of A from above. (#12) 14. sin A = 0.6004 15. cos A = 0.7997 16. tan A = 0.7508 17. Did this change how sine, cosine, and tangent match with the ratios in #1 - #3? If so, how and why? Yes this changed. When looking at #14 - #16, sine matches with #2, cosine matches with #1, and tangent matches with the reciprocal of #3. 18. Write a general equation for sine, cosine, and tangent that could be used with any right triangle. sin = opposite hypotenuse cos = adjacent hypotenuse tan = opposite adjacent 19. Make your own triangles to test the above equations. Answers may vary. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 107 of 162 G-SRT 6 Name ___________________________________ Date __________________ Period ________ Make a Model: Trigonometric Ratios Materials: protractor, metric ruler, compass, plain or graph paper, and scientific calculator for each person in the group. Directions: Everyone in the group should do Steps 1 - 5 individually. Steps 6 – 8 should be done collectively. Step 1: On a sheet of graph or plain paper, use a protractor to make as large a right triangle ABC as possible with mB = 90°, mA = 20°, and mC = 70°. Label the vertices appropriately. Step 2: Use your ruler to measure sides AB, AC, and BC to the nearest millimeter. AB = ______ mm AC = ______ mm BC = ______ mm Step 3: Recall by definition: r hypotenuse y leg opposite sin = length of leg opposite θ y = length of hypotenuse r cos = length of leg adjacent θ x = length of hypotenuse r tan = length of leg opposite θ y = length of leg adjacent θ x x leg adjacent to Step 4: Use the information obtained in Step 2 to complete the following statements. Write the following ratios in fraction form and decimal form to the nearest thousandth. Fraction Decimal sin 20° = length of leg opposite A length of hypotenuse cos 20° = length of leg adjacent to A = __________ = __________ length of hypotenuse tan 20° = length of leg opposite A = __________ = __________ length of leg adjacent to A = __________ = __________ CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 108 of 162 sin 70° = length of leg opposite C = __________ = __________ length of hypotenuse cos 70° = length of leg adjacent C = __________ = __________ length of hypotenuse tan 70° = length of leg opposite C = __________ = __________ length of leg adjacent C Step 5: In the tables below, record your ratios in decimal form to the nearest thousandth in the appropriate boxes (Individual) under sin A, cos A, tan A, sin C, cos C, and tan C. Step 6: Compare the ratios obtained by the members of your group. Calculate the average of each of the ratios found by the members of your group. In the tables below, record the ratios in the appropriate boxes (Group Averages) under sin A, cos A, tan A, sin C, cos C, and tan C. Step 7: Use a calculator to check your group’s results. Calculate sin 20, cos 20, tan 20, sin 70, cos 70, and tan 70. Record the results in your tables. How do the trigonometric ratios that were found by measuring the sides compare with the trigonometric ratios that were found by using a calculator? ______________________________________________________________ m A = 20 Ratios (Individual) sin A cos A tan A sin C cos C tan C Ratios (Group Averages) Ratios (Calculator) mC = 70 Ratios (Individual) Ratios (Group Averages) Ratios (Calculator) CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 109 of 162 Step 8: Questions for Discussion A. Are all right triangles with acute angles measuring 20° and 70° similar? Explain. __________________________________________________________________ B. For any two right triangles with acute angles measuring 20° and 70°: The sin 20°, cos 20°, and tan 20° are ____________________________ the same. sometimes, always, or never The sin 70°, cos 70°, and tan 70° are ____________________________ the same. sometimes, always, or never C. Why are the trigonometric ratios of any right triangle with acute angles measuring 20° and 70° the same regardless of the lengths of the sides? __________________________________________________________________ CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 110 of 162 Name ___________________________________ Date __________________ Period ________ Make a Model: Trigonometric Ratios Answer Key The responses given in Steps 1-6 are based on the lengths of the sides of the different triangles that are drawn by individual students. The responses in Steps 1-6 may vary. Step 7: Use a calculator to check your group’s results. Calculate sin 20, cos 20, tan 20, sin 70, cos 70, and tan 70. Record the results in your tables. How do the trigonometric ratios that were found by measuring the sides compare with the trigonometric ratios that were found by using a calculator? They are the equal or approximately equal to each other. sin A cos A tan A mA = 20 Ratios (Individual) May Vary May Vary May Vary Ratios (Group Averages) May Vary May Vary May Vary Ratios (Calculator) .342 .940 .364 mC = 70 Ratios (Individual) Ratios (Group Averages) Ratios (Calculator) sin C May Vary May Vary .940 cos C May Vary May Vary .342 tan C May Vary May Vary 2.747 Step 8: Questions for Discussion A. Are all right triangles with acute angles measuring 20° and 70° similar? Explain. Yes. Two triangles are similar if their corresponding angles are congruent. B. For any two right triangles with acute angles measuring 20° and 70°: The sin 20°, cos 20°, and tan 20° are always the same. The sin 70°, cos 70°, and tan 70° are always the same. C. Why are the trigonometric ratios of any right triangle with acute angles measuring 20° and 70° the same regardless of the lengths of the sides? A trigonometric ratio is a ratio of the lengths of two sides of a right triangle. All right triangles with acute angles measuring 20° and 70° are similar; therefore, the ratio of any two sides of one triangle will equal the ratio of the corresponding two sides of another. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 111 of 162 G-SRT 8 Name ___________________________________ Date __________________ Period ________ Let’s Measure the Height of the Flagpole 50 60 40 0 13 0 12 0 14 30 0 15 20 10 0 16 0 17 The clinometer is used to measure the heights of objects. It is a simplified version of the quadrant, an important instrument in the Middle Ages, and the sextant, an instrument for locating the positions of ships. Each of these devices has arcs which are graduated in degrees for measuring angles of elevation. The arc of the clinometer is marked from 0 to 90 degrees. When an object is sighted through the straw, the number of degrees in angle BXY can be read from the arc. Angle BAC is the angle of elevation of the clinometer. Angle BXY on the clinometer is equal to the angle of elevation, angle BAC. 70 0 11 X 20 30 40 50 10 0 17 0 16 0 15 0 13 0 12 60 70 0 14 0 11 0 10 80 90 0 10 80 Drinking Straw B A Y C Objective: You will use your skills of right triangle trigonometry to measure the height of the school’s flagpole. Materials: - clinometer, meter stick, calculator CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 112 of 162 Procedures & Questions: - Pick a certain distance (in meters) that you want to stand from the flagpole. Record it below. _____________ meters - Use the clinometer and look through the straw to locate the top of the flagpole. Record the angle measure that is created from the string below. ______________ degrees - Draw a picture of this situation and label all parts clearly. - Use your knowledge of right triangle trigonometry to find the height of the flagpole. Show algebraic work. Round answer to two decimal places. - Can you think of another method to find the height of the flagpole? Explain clearly and be very specific. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 113 of 162 G-SRT 8 Name _____________________________ Date ______________ Period ________ Applications of Trigonometry Using Indirect Measurement 1. ODOT (Ohio Department of Transportation) uses an electronic measurement device to measure distances by recording the time required for a signal to reflect off the object. They use the equipment to survey a portion of the Hocking Hills as below. How much taller is the left part of the Hocking Hills than the right part? T M 950 ft 880 ft 60o 50o B C A 2. You are designing a jet plane as shown. In preparing the documentation for your design, you are required to find the measures of RPQ and PQR in the wing (triangle PQR). What are the measures? P 30 ft 12 ft R Q 3. The first flight of a biplane (doubled-winged plane) was the historic flight of the Wright brothers in 1903. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 114 of 162 A B C E D G F Use the diagram to find the measure of the indicated segment or angle. Given that ADGE is a rectangle, BFC is equilateral, AEF DGF, EF = 15, and BC = 9. Round your answers to two decimal places. a) BF b) AE c) AF d) AB e) AFE f) g) ABF h) FBC FAB In #3h, you can find m FBC in two ways. Describe the two ways. Do they yield the same value? CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 115 of 162 4. You are standing beside Alum Creek to survey the structure of Hoover Reservoir. Using an electronic measuring device, you find the angle of elevation to the top of the dam to be 55 o, and the distance to the top of the dam to be 922 feet. 922 ft 55º 500 ft x ft a) Use the diagram to find the height of the dam. b) If you are standing 500 feet from the base of the dam, find x. 5. You are standing 382.5 feet away from the center of the Eiffel Tower and the angle of elevation is 70o. Find the height of the Eiffel Tower. 70º 382.5 ft CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 116 of 162 6. A yacht is sailing toward the lighthouse and a airplane is flying toward the lighthouse as well. The lighthouse is 250 feet tall. The yacht is 400 feet from the lighthouse and the airplane is 300 feet from the lighthouse and has the same height as the top of the lighthouse. 300 ft y 250 ft x 400 ft Find the angle of elevation of the yacht and the angle of depression of the airplane. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 117 of 162 Name ___________________________________ Date __________________ Period ________ Application of Trigonometry Using Indirect Measurement Answer Key 1. ODOT (Ohio Department of Transportation) uses an electronic measurement device to measure distances by recording the time required for a signal to reflect off the object. They use the equipment to survey a portion of the Hocking Hills as below. How much taller is the left part of the Hocking Hills than the right part? T M 950 ft 880 ft 60o 50o B C A In the right triangle ∆ATB, you can use the sine ratio to find the length of TB . TB TB sin TAB = sin 60o = 950(sin 60o) = TB 822.72 TB 950 TA Use the same procedure to find the length of MC in AMC. MC MC sin MAC = sin 50o = 880(sin 50o) = MC 674.12 MC MA 880 From these two approximations, you can conclude that the difference in the heights is: 822.72 – 674.12 = 148.6 feet. 2. You are designing a jet plane as shown. In preparing the documentation for your design, you are required to find the measures of RPQ and PQR in the wing (triangle PQR). What are the measures? P 30 ft 12 ft R Q To find the measure of RPQ, you can use the tangent ratio. RQ 30 = 2.5 tan P = tan P = m P 68.2o RP 12 Because P and Q are complementary, you can determine the measure of Q to be m Q = 90o – 68.2o = 21.8o 3. The first flight of a biplane (doubled-winged plane) was the historic flight of the Wright brothers in 1903. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 118 of 162 A B C E D G F Use the diagram to find the measure of the indicated segment or angle. Given that ADGE is a rectangle, BFC is equilateral, AEF DGF, EF = 15, and BC = 9. Round your answers to two decimal places. a) BF b) AE 9 7.79 or 4.5 3 c) AF 16.90 d) AB 10.5 e) AFE 27.46o f) FAB 27.46o g) ABF 120o h) FBC 60o In #3h, you can find m FBC in two ways. Describe the two ways. Do they yield the same value? Method 1: Each angle of equilateral FBC is 60o. 4.5 = 1 Method 2: cos FBC = 9 2 So m FBC = 60o; yes. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 119 of 162 4. You are standing beside Alum Creek to survey the structure of Hoover Reservoir. Using an electronic measuring device, you find the angle of elevation to the top of the dam to be 55 o, and the distance to the top of the dam to be 922 feet. 922 ft 55º 500 ft x ft a) Use the diagram to find the height of the dam. opp hyp opp sin 55o = 922 sin 55o = 922(sin 55o) = opp 922(.819) opp 755.12 opp = the height of the dam b) If you are standing 500 feet from the base of the dam, find x. adj hyp 500 + x cos 55o = 922 cos 55o = 922(cos 55o) = 500 + x 922(.573) 500 + x 528.31 500 + x 28.31 x CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 120 of 162 5. You are standing 382.5 feet away from the center of the Eiffel Tower and the angle of elevation is 70o. Find the height of the Eiffel Tower. opp adj opp tan 70o = 382.5 tan 70o = 382.5(tan 70o) = opp 382.5(2.747) opp 1050.7 ft opp = height of Eiffel Tower 70º 382.5 ft 6. A yacht is sailing toward the lighthouse and an airplane is flying toward the lighthouse as well. The lighthouse is 250 feet tall. The yacht is 400 feet from the lighthouse and the airplane is 300 feet from the lighthouse and has the same height as the top of the lighthouse. 300 ft y 250 ft x 400 ft Find the angle of elevation of the yacht and the angle of depression of the airplane. Angle of Elevation: tan x = Angle of Depression: opp adj tan y = opp adj tan x = 250 400 tan y = 250 300 tan x = .625 tan y .833 x 32o y 39.81o CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 121 of 162 G-SRT 8 Name ___________________________________ Date __________________ Period ________ Find the Missing Side or Angle Instructions: Find the missing side or angle as indicated in each of the right triangles below. 1. 2. = ___________ x = ___________ 10 23 28 x 18 3. 4. 45 9 x 30 x = ___________ c c = ___________ 70 5. x 55 6. = ___________ x = __________ 25 11 2 38 17 7. 8. 8 a 65 a = ___________ b = __________ b 9. Describe a situation when you would use sine. Use illustrations to support your answer. 10. Describe a situation when you would use cos-1. Use illustrations to support your answer. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 122 of 162 Name ___________________________________ Date __________________ Period ________ Find the Missing Side or Angle Answer Key Instructions: Find the missing side or angle as indicated in each of the right triangles below. 1. 2. 10 x= 18.81 = 23 28 18 x 51.5o 3. 4. 9 x x= 45 8.46 30 70 c= 42.42 c 5. 6. 11 = x 55 10.30 x= 14.34 25 2 7. 8. 17 a 65 a= 38 7.93 b= 6.25 8 b 9. Describe a situation when you would use sine. Use illustrations to support your answer. When you know the measure of an angle and the measure of either the opposite side or the hypotenuse. x 15 25 10. Describe a situation when you would use cos-1. Use illustrations to support your answer. When you know the measure of the adjacent side and the hypotenuse and want to find the measure of the angle. 10 x 5 CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 123 of 162 G-SRT 8 Name ___________________________________ Date __________________ Period ________ Between the Uprights Using the picture below: Find the angle of elevation the ball makes with the ground when it is kicked. Find the length of the most direct path from where the ball is kicked to where it crosses the uprights (hypotenuse). 24 feet 60 50 40 30 20 10 yards Using the picture below: Find the angle of elevation the ball makes with the ground when it is kicked. Find the length of the most direct path from where the ball is kicked to where it crosses the uprights (hypotenuse). 20 feet 40 30 20 10 yards Using the picture below: CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 124 of 162 Find the angle of elevation the ball makes with the ground when it is kicked. Find the length of the most direct path from where the ball is kicked to where it hits the uprights (hypotenuse). 10 feet 50 40 30 yards 10 20 Using the picture below: If the angle of elevation the ball makes with the ground when it is kicked is 27o, at what distance from the ground will it cross the uprights? Find the length of the most direct path from where the ball is kicked to where it crosses the uprights (hypotenuse). ? 30 CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 20 yards 10 Columbus City Schools 6/28/13 Page 125 of 162 Write your own problem for the picture below. Label all parts. Solve the problem showing all calculations. ___ feet 30 20 yards CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 10 Columbus City Schools 6/28/13 Page 126 of 162 G-SRT 8 Name ___________________________________ Date __________________ Period ________ Between the Uprights Answer Key Using the picture below: Find the angle of elevation the ball makes with the ground when it is kicked. 7.59 Find the length of the most direct path from where the ball is kicked to where it crosses the uprights (hypotenuse). 60.53 yds 24 feet 60 50 40 20 30 10 yards Using the picture below: Find the angle of elevation the ball makes with the ground when it is kicked. 9.46 Find the length of the most direct path from where the ball is kicked to where it crosses the uprights (hypotenuse). 40.55 yds 20 feet 40 30 CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 20 10 yards Columbus City Schools 6/28/13 Page 127 of 162 Using the picture below: Find the angle of elevation the ball makes with the ground when it is kicked. 3.81 Find the length of the most direct path from where the ball is kicked to where it hits the uprights (hypotenuse). 50.11 yds 10 feet 50 20 30 yards 40 10 Using the picture below: If the angle of elevation the ball makes with the ground when it is kicked is 27o, at what distance from the ground will it cross the uprights? 45.86 ft = 15.29 yds Find the length of the most direct path from where the ball is kicked to where it crosses the uprights. 33.67 yds ? 30 20 yards 10 Write your own problem for the picture below. Label all parts. Solve the problem showing all calculations. Answers will vary. ___ feet 30 20 yards CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 10 Columbus City Schools 6/28/13 Page 128 of 162 G-SRT 8 Name ___________________________________ Date __________________ Period ________ Solve the Triangle Instructions: Measure the sides (centimeters) and/or angle (degrees) listed in the given column for the right triangles shown below. Example: A b C c a B Given Side or Angle A a Measure Calculated Side or Angle B b c Measure Once you have completed your measurements, solve each triangle (find all missing sides and angles), placing values in the table. Trade papers with your partner and check each other's completed triangles using the following checklist: ____ all calculations are correct ____ the sum of all angles of each triangle is 180o, accuracy within 1o ____ the Pythagorean Theorem holds true for your values of the legs and hypotenuse, i.e., a2 + b2 = c2 1. A c b C a B Given Side or Angle B c CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Measure Calculated Side or Angle Measure Columbus City Schools 6/28/13 Page 129 of 162 2. A c b a C 3. B A Given Side or Angle b c c b C Measure Measure Calculated Side or Angle Calculated Side or Angle Measure Measure B a 4. A Given Side or Angle A b c b B Given Side or Angle A c C a Measure Calculated Side or Angle Measure 5. C a B b c A Given Side or Angle B b CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Measure Calculated Side or Angle Measure Columbus City Schools 6/28/13 Page 130 of 162 6. b A C a c Given Side or Angle A a Measure Calculated Side or Angle Measure B 7. A c b C 8. B a C Given Side or Angle B a Measure Calculated Side or Angle Measure B a b c Given Side or Angle a b Measure Calculated Side or Angle Measure A CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 131 of 162 Name ___________________________________ Date __________________ Period ________ Solve the Triangle Answer Key Instructions: Measure the sides (centimeters) and/or angle (degrees) listed in the given column for the right triangles shown below. Example: A c b C B a Given Side or Angle A a Measure o 30 1.5 cm Calculated Side or Angle B b c Measure 60o 2.6 cm 3 cm Once you have completed your measurements, solve each triangle (find all missing sides and angles), placing values in the table. Trade papers with your partner and check each other’s completed triangles using the following checklist: ____ all calculations are correct ____ the sum of all angles of each triangle is 180o, accuracy within 1o ____ the Pythagorean Theorem holds true for your values of the legs and hypotenuse, i.e., a2 + b2 = c2 1. A c b C a B Given Side or Angle B c CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Measure 27o 2.8 cm Calculated Side or Angle A a b Measure 63o 2.50 cm 1.27 cm Columbus City Schools 6/28/13 Page 132 of 162 A 2. a C 3. Given Side or Angle A c c b B Measure o 50 3.3 cm Calculated Side or Angle B a b Measure 40o 2.53 cm 2.12 cm A Given Side or Angle b c c b C B a Measure 2.5 cm 2.7 cm Calculated Side or Angle A B a Measure 22.19o 67.81o 1.02 cm A 4. c b B C a Given Side or Angle A b Measure o 65 1.9 cm Calculated Side or Angle B a c Measure 25o 4.07 cm 1.72 cm 5. a C B b c Given Side or Angle B b Measure 21o 1.6 cm A Calculated Side or Angle A a c Measure 69o 4.17 cm 4.46 cm 6. b A c C Given Side or Angle A a a Measure 42o 2.5 cm Calculated Side or Angle B b c Measure 48o 2.78 cm 3.74 cm B CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 133 of 162 7. A b 8. c C a B C a B b c Given Side or Angle B a Given Side or Angle a b Measure 29o 2.6 cm Measure 2.5 cm 4 cm Calculated Side or Angle A b c Calculated Side or Angle A B c Measure 61o 1.44 cm 2.97 cm Measure 32o 58o 4.72 cm A CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 134 of 162 G-SRT 8 Name ___________________________________ Date __________________ Period ________ Right Triangle Park Because of your reputation for drawing and your keen mathematical ability, you have been selected to design a very special park for your neighborhood! This park will be designed using only right triangles! Instructions: Design a city park using only right triangles. Your park must include at least 4 different components such as picnic tables, swing sets, slides, gardens, skating ramps, etc. Draw your design in the area provided below. Measure one side and one acute angle of each triangle in your design. Solve and label each triangle, using trigonometry to find the missing sides and angles. Right Triangle Park CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 135 of 162 G-SRT 8 Name ___________________________________ Date __________________ Period ________ Find the Height When you see an image in a mirror, the angle your line of sight makes with the ground is the same as the angle the top of the object being reflected makes with the ground as shown below. You and your work group will use this fact and your knowledge of similar triangles to find the heights of structures in your school yard. Instructions: Select 3 tall objects you wish to measure (tree, flagpole, smokestack, school, goalpost, etc). Place a mirror on the ground between yourself and the object whose height you are calculating. Stand so you can see the top of the object in the mirror. While you stand, your partner will measure the ground distance from you to the mirror and from the mirror to the object. Record the measurements on the “Find the Height” data sheet. Record the mirror watcher's height on the data sheet. Set up your proportion and find the height of the object. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 136 of 162 G-SRT 8 Name ___________________________________ Date __________________ Period ________ Find the Height Data Sheet Object measured Sketch your reflection experiment in the box below. Label all measurements. Proportion _____________________________ Height of Object _________________________ Object measured Sketch your reflection experiment in the box below. Label all measurements. Proportion _____________________________ Height of Object _________________________ Object measured Sketch your reflection experiment in the box below. Label all measurements. Proportion _____________________________ Height of Object _________________________ CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 137 of 162 G-SRT 8 Name ___________________________________ Date __________________ Period ________ Applications of the Pythagorean Theorem For each of the following word problems, draw a picture to represent the situation, write an equation and solve for the missing parts. A 25-ft ladder leans against the side of a house. If you place the ladder 15 ft from the base of the house, how high up will the ladder reach? A broadcast antenna needs a support wire replaced. If the support wire is attached to the ground 58 ft from the antenna base and is attached to the antenna 125 ft from the ground, how long is the support wire? Ralph purchased a 7 m slide and it covers a 4.3 m distance on the ground. How tall is the slide’s ladder? The bases on a baseball diamond are 90 ft apart. If the catcher stands at home plate and throws to second base, how far does the catcher throw? CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 138 of 162 Name ___________________________________ Date __________________ Period ________ Applications of the Pythagorean Theorem Answer Key For each of the following word problems, draw a picture to represent the situation, write an equation and solve for the missing parts. A 25-ft ladder leans against the side of a house. If you place the ladder 15 ft from the base of the house, how high up will the ladder reach? x2 + 152 = 252 x = 20 ft 25 ft x ft 15 ft A broadcast antenna needs a support wire replaced. If the support wire is attached to the ground 58 ft from the antenna base and is attached to the antenna 125 ft from the ground, how long is the support wire? 2 2 125 ft 2 x ft 125 + 58 = x x = 137.8 58 ft ft Ralph purchased a 7 m slide and it covers a 4.3 m distance on the ground. How tall is the slide’s ladder? 4.32 + x 2 = 72 x = 5.5 xm 7m 4.3 m The bases on a baseball diamond are 90 ft apart. If the catcher stands at home plate and throws to second base, how far does the catcher throw? 902 + 902 = x 2 x = 127.3 ft 90 ft x ft 90 ft CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 139 of 162 Reteach Name ___________________________________ Date __________________ Period ________ Memory Match – Up Students can be put into groups of 3 – 4. First place all cards face down and have each student take turns drawing two cards. If the two cards drawn go together as a pair, then the student will keep it as a match. The student with the most matches wins. Note: There are 3 cards that say “1”. There are 2 cards that say “ 3 ”. There are 2 cards that say 3 ”. There are 2 cards that say “ 1 ”. Make sure that students know that two cards with the exact “ 2 2 same expression on them are not considered a match. For example: A card with a “1” on it does not match a card with a “1” on it. A card with a “1” on it is a match with a card that has “tan 45º” on it. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 140 of 162 Memory Match – Up Cards Pythagorean Theorem 2 45o 1 ? 45o 30o 45o ? ? 60o 45o ? 1 3 CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 141 of 162 3 1 2 2 2 1 tan 45º sin 45º sin 30º cos 30º CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 142 of 162 30o 30o ? 60o 60o ? It can be used to solve for an acute angle in a right triangle. 3 2 sin 1 2 3 2 3 3 -1 CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 143 of 162 sin cos tan opposite hypotenuse adjacent hypotenuse opposite adjacent 3 1 2 leg2 + leg2 = hypotenuse2 o tan 45 o sin 30 CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Page 144 of 162 Columbus City Schools 6/28/13 oo 45 sin 30 cos tan 30º sin 60º cos 60º tan 60º CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 145 of 162 Memory Match-Up Answer Key leg2 + leg2 = hypotenuse2 Pythagorean Theorem opposite hypotenuse adjacent hypotenuse opposite adjacent sin cos tan 1 2 3 2 3 3 2 2 sin 30º cos 30º tan 30º sin 45º tan 45º 1 3 2 1 2 sin 60º cos 60º 3 tan 60º It can be used to solve for an acute angle in a right triangle. sin -1 ? 45 º 1 45º ? ? 45º 2 45º ? 30º 2 60º 1 30º 60º ? ? 3 30º 60º CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 146 of 162 Reteach Name ___________________________________ Date __________________ Period ________ Similar Right Triangles and Trigonometric Ratios Draw a right triangle. Label it ABC, with C being the right angle. Measure the sides in centimeters and the angles in degrees. Complete this chart by filling in the measurements for each angle and each side. Side or Angle Measure A B A B C Remember that the trigonometric ratios are defined as shown below. sin = length of opposite leg length of hypotenuse cos = length of adjacent leg length of hypotenuse tan = length of opposite leg length of adjacent leg Complete the chart from your measurements. Use the trigonometric functions on your calculator to find the values. If the two sets of values are not about the same, measure and compute again. Trigonometric Value From Measurement From Calculator sin A cos A tan A sin B cos B tan B CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 147 of 162 Draw a right triangle, DEF, whose angles are the same as those in triangle ABC, but whose sides are twice as long. Complete the chart as you did for triangle ABC. Side or Measure Angle D E d e f Trigonometric Value From Measurement From Calculator sin D cos D tan D sin E cos E tan E Make a triangle GHI, that is similar to the other two triangles, with side GH measuring 20 cm long. Show how you find the length of the other two sides. What do you know about the sine, cosine, and tangents of angles G and H? CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 148 of 162 Extension Name ___________________________________ Date __________________ Period ________ Similar Right Triangles and Trigonometric Ratios Given: rt. ABC ~ rt. DEF A D b e c f d E a F B the fact that the triangles are similar C to find the missing term (?). Write the missing term Part A: Use in the space provided. __________ 1. a d b ? __________ 2. ? f b e __________ 3. e ? f c __________ 4. d f ? c __________ 5. a ? c f __________ 6. b e ? f __________ 7. ? b d a __________ 8. f c d ? CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 149 of 162 Part B: Describe two ways that similarity proportions can be formed. ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ Part C: Each of the proportions below are true for the two similar triangles given. The ratios that form the proportions can be written as trigonometric ratios. Complete the statements below that correspond to the given proportions to make them true. A D b e c f d E a 1. 3. 5. Bd a = b e sin A = sin _____ C c f 2. = b e sin C = sin _____ c f = b e cos A = cos _____ a d = c f tan A = tan _____ F 4. 6. a d = b e cos C = cos _____ c f = a d tan C = tan _____ CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 150 of 162 Part D: Use the information in Part C to complete the following statements. 1. sin C = cos _____ 2. cos A = sin _____ 3. sin A = cos _____ 4. cos C = sin _____ 5. Describe the relationship that exists between angles A and C? ________________________________________________________________________ 6. sin D = cos _____ 7. cos F = sin _____ 8. sin F = cos _____ 9. cos D = sin _____ 10. Describe the relationship that exists between angles D and F? ________________________________________________________________________ Conclusion: ________________________________________________________________________ ________________________________________________________________________ Part E: Complete the following statements. 1. sin 20° = cos _____ 2. cos 35° = sin _____ 3. sin 10° = cos _____ 4. cos 45° = sin _____ 5. sin 30° = cos _____ 6. cos 60° = sin _____ 7. sin x° = cos _____ 8. cos y° = sin _____ CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 151 of 162 Name ___________________________________ Date __________________ Period ________ Similar Right Triangles and Trigonometric Ratios Given: rt. ABC ~ rt. DEF Answer Key A D b e c f d E a B F C Part A: Use the fact that the triangles are similar to find the missing term (?). Write the missing term in the space provided. e 1. a d b ? c 2. ? f b e b 3. e ? f c a 4. d f ? c d 5. a ? c f c 6. b e ? f e 7. ? b d a a 8. f c d ? CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 152 of 162 Part B: Describe two ways that similarity proportions can be formed. 1. One way: Each ratio of the proportion compares a side of one triangle with the corresponding side of the other triangle. 2. Another way: Each ratio of the proportion compares two sides from the same triangle with two corresponding sides of the other triangle. Part C: Each of the proportions below are true for the two similar triangles given. The ratios that form the proportions can be written as trigonometric ratios. Complete the statements below that correspond to the given proportions to make them true. A D b e c f d E a B a d 1. = b e sin A = sin D C 2. c f = b e cos A = cos D 3. 5. F 4. a d = c f tan A = tan D 6. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 c f = b e sin C = sin F a d = b e cos C = cos F c f = a d tan C = tan F Columbus City Schools 6/28/13 Page 153 of 162 Part D: Use the information in Part C to complete the following statements. 1. sin C = cos A 2. cos A = sin C 3. sin A = cos C 4. cos C = sin A 5. Describe the relationship that exists between angles A and C? Angles A and C are complementary angles. The sum of their measures equals 90°. 6. sin D = cos F 7. cos F = sin D 8. sin F = cos D 9. cos D = sin F 10. Describe the relationship that exists between angles D and F? Angles D and F are complementary angles. The sum of their measures equals 90°. Conclusion: In a right triangle, the sine of one of the acute angles equals the cosine of the other acute angle (complement of the angle). Part E: Complete the following statements. 1. sin 20° = cos 70° 2. cos 35° = sin 55° 3. sin 10° = cos 80° 4. cos 45° = sin 45° 5. sin 30° = cos 60° 6. cos 60° = sin 30° 7. sin x° = cos (90 – x)° 8. cos y° = sin (90 – y)° CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 154 of 162 Reteach Name ___________________________________ Date __________________ Period ________ Hey, All These Formulas Look Alike! Show your work. Include formulas in your explanations. Consider the ABC. Show thatCBA is a right angle. C (3, 4) A (0, 0) B (3, 0) Complete the chart. Slope, AC Distance, AC Pythagorean Theorem, ABC Tan CAB Write Formulas Substitute Values and Simplify Compare the expressions and the values for the slope of AC and tan CAB. Are the formulas the same? Are the values equal? Support your answer by showing your work. This exercise utilized the first quadrant only. Predict if your conclusions will hold if the triangle is rotated 90, about the origin, counterclockwise. Test your prediction. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 155 of 162 Name ___________________________________ Date __________________ Period ________ Hey, All These Formulas Look Alike! Answer Key Show your work. Include formulas in your explanations. Consider the ABC. Show thatCBA is a right angle. Solution: If CBA is a right angle, then CB BA . If CB BA , then m1 * m2 = -1. This is a special case in which the slope of one of the perpendicular lines is undefined and the slope of the other line is zero. 40 4 00 0 Slope BC = Slope BA = 33 0 30 3 C (3, 4) A (0, 0) B (3, 0) 2. Complete the chart. Slope, AC Write Formulas m y2 y1 x2 x1 Pythagorean Theorem, ABC (x2-x1)2+(y2-y1)2=AC2 Distance, AC 3 4 AC 2 Substitute Values and Simplif 40 4 30 3 length of opp. side length of adj. side ( x2 x1 )2 ( y2 y1 )2 (3 0)2 (4 0)2 AC 2 9 16 AC 25 AC 5 = AC Tan CAB (3-0)2+(4-0)2=(AC)2 32 + 42 = (AC)2 9 + 16 = (AC)2 25 = (AC)2 40 4 30 3 25 ( AC )2 5 = AC 3. Compare the expressions and the values for the slope of AC and tan CAB. Are the formulas the same? Are the values equal? Support your answer by showing your work. The formula for the slope of AC and the equation for tan CAB are the same. Slope of AC = y2 y1 x2 x1 Tan CAB = length of opposite side y2 y1 x2 x1 length of adjacent side The values are the same. The slope AC of and tan CAB both equal 40 4 30 3 4. This exercise utilized the first quadrant only. Predict if your conclusions will hold if the triangle is rotated 90, about the origin, counterclockwise. Test your prediction. Answers will vary. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 156 of 162 Extension Name ___________________________________ Date __________________ Period ________ Problem Solving: Trigonometric Ratios Materials: scientific calculator Use the information given in the figure below to determine the sine, cosine, and tangent of . Explain your answer. Sin = _______ Cos = _______ Tan = _______ (0,5) B (3,4) A C (5,0) Use the information given in the figure below to determine the perimeter of rectangle ABCD. Support your answer by showing your work. B C 100 cm 35 A D Perimeter = ____________ 3. Which of the following trigonometric ratios: sine, cosine, or tangent of an angle can have a value greater than 1? Why is it that the values of the other two trigonometric ratios can never be greater than 1? Explain. 4. John, an employee of the U.S. Forestry Service has been asked to determine the height of a tall tree in Wayne National Forest. He uses an angle measuring device to determine the angle of CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 157 of 162 elevation (angle formed by the line of sight to the top of the tree and a horizontal) to be about 33. He walks off 40 paces to the base of the tree. If each pace is .6 meters, how tall is the tree to the nearest meter? Support your answer by showing your work and including a diagram. 5. Determine the perimeter to the nearest centimeter and the area to the nearest square centimeter of the triangle shown below. Support your answer by showing your work and giving an explanation. B 10 cm A C 6. Use what you know about the side lengths of special right triangles to complete the following table. Express your answers in simplified radical form. 30 45 45 30 45 60 60 Sin Cos Tan CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 158 of 162 Name ___________________________________ Date __________________ Period ________ Problem Solving: Trigonometric Ratios Answer Key Use the information given in the figure below to determine the sine, cosine, and tangent of . 4 3 4 Explain your answer. Sin = Cos = Tan = (0,5) 5 5 3 B (3,4) Solution: The lengths of AC and BC can be determined by using the coordinates of point B(3,4). The length of AB can be determined by using the fact that it is a radius of a circle. AB = 5, BC = 4, and AC = 3. By definition: A C (5,0) BC 4 AC 3 BC 4 sin θ = = ; cos θ = = ; and tan θ = = AB 5 AB 5 AC 3 Use the information given in the figure below to determine the perimeter of rectangle ABCD. Support your answer by showing your work. Solution: By definition: 57.36 cm B C AB AB sin 35° = ; .5736 ; AB 57.36 100 100 BD BD cos 35° = ; .8192 ; BD 81.92 100 100 81.92 cm 100 cm 35 A D The perimeter of the rectangle = 2(57.36) +2(81.92) = 278.56 cm.; Perimeter = 278.56 cm Which of the following trigonometric ratios: sine, cosine, or tangent of an angle can have a value greater than 1? Why is it that the values of the other two trigonometric ratios can never be greater than 1? Explain. Solution: The tangent of an angle can be greater than 1. The sine of an acute angle of a right triangle length of leg opposite the angle is defined as and the cosine of an acute angle of a right triangle is length of hypotenuse length of leg adjacent to the angle defined as . The length of a leg of a right triangle will always length of hypotenuse be less than the length of the hypotenuse. If the numerator of a fraction is less than the denominator, the fraction is always less than 1. Therefore, the sine and cosine of an angle will never be greater than 1 by definition of the sine and cosine ratios. CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 159 of 162 4. John, an employee of the U.S. Forestry Service has been asked to determine the height of a tall tree in Wayne National Forest. He uses an angle measuring device to determine the angle of elevation (angle formed by the line of sight to the top of the tree and a horizontal) to be about 33. He walks off 40 paces to the base of the tree. If each pace is .6 meters, how tall is the tree to the nearest meter? Support your answer by showing your work and including a diagram. Solution: h tan 33° = 24 h h .6494 24 h 16 m 33 24 m 5. Determine the perimeter to the nearest centimeter and the area to the nearest square centimeter of the triangle shown below. Support your answer by showing your work and giving an explanation. Sample Solution: Triangle ABC is an isosceles right triangle. The legs have equal lengths, therefore the acute angles each have a measure of 45. The ratio of the sides of a 45- 45- 90 triangle is 10 10 2 B 1:1: 2 . The length of each leg is = = 5 2. 2 2 The perimeter of the triangle is 45 10 + 5 2 + 5 2 = 10 + 10 2 24 cm. 10 cm The area of the triangle is 1 1 • 5 2 • 5 2 = • 25 • 2 25 cm 2 . 2 2 45 A C Sample Solution: Triangle ABC is an isosceles right triangle. The legs have equal lengths, therefore the acute angles each have a measure of 45. The lengths of the legs can be found by using the sine and cosine ratios. B AC 10 AC = sin 45° •10 .707 •10 7.07 cm AB cosB = cos 45° = 10 AB = cos 45° •10 .707 •10 7.07 cm sinB = sin 45° = 45 10 cm 45 A C The perimeter of the triangle is 7.07 + 7.07 + 10 = 24.14 24 cm. The area of the triangle is (.5)(7.07)(7.07) = 24.99 25 cm2 CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 Columbus City Schools 6/28/13 Page 160 of 162 6. Use what you know about the side lengths of special right triangles to complete the following table. Express your answers in simplified radical form. 30 45 45 60 30 Sin 1 2 Cos Tan 3 2 1 3 = 3 3 45 1 2 = 2 2 1 2 = 2 2 1 =1 1 CCSSM II Trigonometric Ratios G-SRT 6, 7, 8 Quarter 1 60 3 2 1 2 3 = 3 1 Columbus City Schools 6/28/13 Page 161 of 162 Grids and Graphics Addendum CCSSM II Comparing Functions and Different Representations of Quadratic Functions FIF 3, 4, 5, 6, 7, 7a, 9, F-BF 1, 1a, 1b, A-CED 1, 2, , F-LE 3, , N-NQ 2, , S-ID 6a, 6b, A-REI 7 Quarter 2 Page 162 of 162 Columbus City Schools 6/28/13 Algebra Tiles Template Grids and Graphics Page 1 of 18 Columbus Public Schools 6/27/13 10 by 10 Grids Grids and Graphics Page 2 of 18 Columbus Public Schools 6/27/13 20 by 20 Grids Grids and Graphics Page 3 of 18 Columbus Public Schools 6/27/13 Small Coordinate Grids Grids and Graphics Page 4 of 18 Columbus Public Schools 6/27/13 Dot Paper Grids and Graphics Page 5 of 18 Columbus Public Schools 6/27/13 Isometric Dot Paper Grids and Graphics Page 6 of 18 Columbus Public Schools 6/27/13 Quarter-Inch Grid Grids and Graphics Page 7 of 18 Columbus Public Schools 6/27/13 Half-Inch Graph Paper Grids and Graphics Page 8 of 18 Columbus Public Schools 6/27/13 One-Inch Grid Paper Grids and Graphics Page 9 of 18 Columbus Public Schools 6/27/13 Centimeter Grid Grids and Graphics Page 10 of 18 Columbus Public Schools 6/27/13 Pascal’s Triangle Template Grids and Graphics Page 11 of 18 Columbus Public Schools 6/27/13 Probability Spinners Grids and Graphics Page 12 of 18 Columbus Public Schools 6/27/13 Protractor 80 70 60 50 120 130 140 30 150 20 160 10 170 110 100 90 100 110 80 70 120 130 140 30 150 20 160 10 170 40 110 80 100 90 100 80 110 70 60 Grids and Graphics 50 120 130 140 30 150 20 160 10 170 140 150 20 160 10 170 110 90 100 100 80 110 70 130 50 120 40 130 140 30 150 20 160 10 170 140 40 150 30 20 160 10 170 110 100 90 100 80 110 70 130 50 120 130 140 30 150 20 160 10 170 140 40 40 150 30 20 100 80 110 70 120 160 10 170 Page 13 of 18 130 60 140 150 30 20 160 10 170 110 80 90 100 100 80 110 70 120 130 60 140 50 40 150 30 20 160 10 170 70 50 100 90 40 60 120 60 80 50 70 50 80 110 60 120 60 110 120 80 70 130 60 140 50 40 150 30 160 20 40 30 70 120 130 140 30 150 20 160 10 170 130 100 10 170 70 40 80 110 60 120 50 60 40 160 10 170 70 50 150 20 60 120 40 130 140 30 150 20 160 10 170 120 130 140 30 150 20 160 10 170 100 90 40 30 40 50 50 140 50 70 50 130 60 40 60 80 70 60 120 110 80 100 90 100 80 110 70 120 130 60 140 50 40 150 30 20 160 10 170 Columbus Public Schools 6/27/13 Tangram Template Grids and Graphics Page 14 of 18 Columbus Public Schools 6/27/13 Blank 11 Grids and Graphics 11 Geoboards Page 15 of 18 Columbus Public Schools 6/27/13 Blank Number Lines Grids and Graphics Page 16 of 18 Columbus Public Schools 6/27/13 Rulers mm 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 mm mm mm mm mm Grids and Graphics Page 17 of 18 Columbus Public Schools 6/27/13 Websites for Graph Paper and More! Below you will find great web sites to visit for graph paper and other things to use in your math activities. http://www.mathematicshelpcentral.com/graph_paper.htm (requires Adobe Acrobat Reader version 5.0 or higher to view or print graphs) This is a wonderful collection of all different kinds of graphs from full-page format to several per page for multiple problems. You will also find a page set up specifically for proofs and graph paper for 3-space, polar coordinates, and logarithms. http://mathpc04.plymouth.edu/gpaper.html At this site you will find several versions of coordinate, semi-logarithmic, full logarithmic, polar, and triangular graph paper. http://mason.gmu.edu/~mmankus/Handson/manipulatives.htm This is site to go to if you need to make math manipulatives. Cutouts are available for pattern blocks, geometric shapes, base-ten and base-five blocks, xy blocks, attribute blocks, rods, and color tiles. Graph paper can be printed as well. http://www.handygraph.com/free_graphs.htm Several forms of coordinate graphs and number lines sized just right for homework and tests. http://donnayoung.org/frm/spepaper.htm Not only does this site have graph paper it contains notebook paper, Lego design paper, music paper, and award certificates. http://www.lib.utexas.edu/maps/map_sites/outline_sites.html#W Outline maps for states, countries, regions, and the world. Grids and Graphics Page 18 of 18 Columbus Public Schools 6/27/13