High School CCSS Mathematics I Curriculum Guide -Quarter 3- Columbus City Schools. Page 1 of 207

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1 High School CCSS Mathematics I Curriculum Guide -- Columbus City Schools Page 1 of 207

2 Table of Contents Math Practices Rationale... 4 RUBRIC IMPLEMENTING STANDARDS FOR MATHEMATICAL PRACTICE Mathematical Practices: A Walk-Through Protocol Curriculum Timeline Scope and Sequence Constructions G CO 12, Teacher Notes Copying an Angle Bisecting an Angle Perpendicular Lines Perpendicular Lines Parallel Lines Constructing a Square Constructing an Equilateral Triangle and a Regular Hexagon Inscribed in a Circle Constructing a Square Inscribed in a Circle Basic Definitions and Rigid Motions G CO 1,2,3,4,5,6,7, Teacher Notes Partitions The Gumps The Gumps and Similar Figures Geometric Relationships and Properties G CO 9, 10, 11, G C Teacher Notes Exploring Parallel Lines Cut by a Transversal Transversals and Special Angles Proof Cards Activity Proving Theorems How Far Up the Hill, Do We Drill? Writing and Solving Linear Equations in Geometric Situations Memory Cards Match-Up Writing Linear Equations: Parallel and Perpendicular Solving Systems of Equations and Finding Equations of Parallel and Perpendicular Lines Page 2 of 207

3 Investigating Centers of Triangles Centers of Triangles Writing and Solving Systems of Linear Equations in Geometric Situations: Centroids of a Triangle Coordinate Geometry: Is the Given Quadrilateral a Parallelogram? TI-92/Cabri Exploration: Properties of Parallelograms Polygon Concept Map Circles: Expanding Our Vocabulary Circles: Expanding Our Vocabulary Discovery Lesson: Properties of Circles Lesson 1(Part 1) Discovery Lesson: Properties of Circles Lesson 1(Part 2) Discovery Lesson:Properties of Circles Lesson 2 (Part 1) Discovery Lesson:Properties of Circles Lesson 2 (Part 2) Discovery Lesson: Properties of Circles Lesson Discovery Lesson: Properties of Circles Lesson Discovery Lesson: Properties of Circles Lesson Discovery Lesson: Properties of Circles Lesson Discovery Lesson:Properties of Circles Lesson 7 (Part 1) Discovery Lesson:Properties of Circles Lesson 7 (Part 2) Discovery Lesson: Properties of Circles Lesson Discovery Lesson: Properties of Circles Lesson Parts of a Circle: Round Robin Pairs Check: Properties of Circles Numbered Noggins: Properties of Circles Page 3 of 207

4 Math Practices Rationale CCSSM Practice 1: Make sense of problems and persevere in solving them. Why is this practice important? Helps students to develop critical thinking 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? skills. Teaches students to think for themselves. Helps students to see there are multiple approaches to solving a problem. Students immediately begin looking for methods to solve a problem based on previous knowledge instead of waiting for teacher to show them the process/algorithm. Students can explain what problem is asking as well as explain, using correct mathematical terms, the process used to solve the problem. Frame mathematical questions/challenges so they are clear and explicit. Check with students repeatedly to help them clarify their thinking and processes. How would you go about solving this problem? What do you need to know in order to solve this problem? What methods have we studied that you can use to find the information you need? Students can explain the relationships between equations, verbal descriptions, tables, and graphs. Students check their answer using a different method and continually ask themselves, Does this make sense? They understand others approaches to solving complex problems and can see the similarities between different approaches. Showing the students shortcuts/tricks to solve problems (without making sure the students understand why they work). Not giving students an adequate amount of think time to come up with solutions or processes to solve a problem. Giving students the answer to their questions instead of asking guiding questions to lead them to the discovery of their own question. Page 4 of 207

5 CCSSM Practice 2: Reason abstractly and quantitatively. Why is this practice important? 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. 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? rounding) Students develop different ways of seeing a problem and methods of solving it. Students are able to translate a problem situation into a number sentence or algebraic expression. Students can use symbols to represent problems. Students can visualize what a problem is asking. Ask students questions about the types of answers they should get. Use appropriate terminology when discussing types of numbers/answers. Provide story problems and real world problems for students to solve. Monitor the thinking of students. What is your unknown in this problem? What patterns do you see in this problem and how might that help you to solve it? What does proficiency look like in this practice? What actions might the teacher make that inhibit the students use of this practice? Students can recognize the connections between the elements in their mathematical sentence/expression and the original problem. Students can explain what their answer means, as well as how they arrived at it. Giving students the equation for a word or visual problem instead of letting them figure it out on their own. Page 5 of 207

6 CCSSM Practice 3: Construct viable arguments and critique the reasoning of others Why is this practice important? Students better understand and remember concepts when they can 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? 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 where the answers make sense. Explain flaws in arguments of others. Not listening to students justify their solutions or giving adequate time to critique flaws in their thinking or reasoning. Page 6 of 207

7 CCSSM Practice 4: Model with mathematics Why is this practice important? Helps students to see the connections between math symbols and real world 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? 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 if the solution is reasonable. How could you represent what the problem was asking? How does your equation relate to the problems? How does your strategy help you to solve the problem? Students can write an equation to represent a problem. Students can analyze their solutions and determine if their answer makes sense. Students can use assumptions and approximations to simplify complex situations. Not give students any problem with real world applications. Page 7 of 207

8 CCSSM Practice 5: Use appropriate tools strategically Why is this practice important? Helps students to understand the uses and limitations of different mathematical and technological tools as well as which ones can be applied to 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? 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 8 of 207

9 CCSSM Practice 6: Attend to precision. Why is this practice important? Students are better able to understand new math concepts when they are familiar with the terminology that is 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? being used. Students can understand how to solve real world problems. Students can express themselves to the teacher and to each other using the correct math vocabulary. Students use correct labels with word problems. Make sure to use correct vocabulary terms when speaking with students. Ask students to provide a label when describing word problems. Encourage discussions and explanations and use probing questions. How could you describe this problem in your own words? What are some non-examples of this word? What mathematical term could be used to describe this process. Students are precise in their descriptions. They use mathematical definitions in their reasoning and in discussions. They state the meaning of symbols consistently and appropriately. Teaching students trick names for symbols (i.e. the alligator eats the big number) Not using proper terminology in the classroom. Allowing students to use the word it to describe symbols or other concepts. Page 9 of 207

10 CCSSM Practice 7: Look for and make use of structure. Why is this practice important? When students can see patterns or connections, they are more easily able 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? 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. What actions might the teacher make that inhibit the students use of this practice? Provide students with pattern before allowing them to discern it for themselves. Page 10 of 207

11 CCSSM Practice 8: Look for and express regularity in repeated reasoning Why is this practice important? 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 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? apply them. Students discover connections between procedures and concepts Students discover rules on their own through repeated exposures of a concept. Provide real world problems for students to discover rules and procedures through repeated exposure. Design lessons for students to make connections. Allow time for students to discover the concepts behind rules and procedures. Pose a variety of similar type problems. How would you describe your method? Why does it work? Does this method work all the time? What do you notice when? Students notice repeated calculations. Students look for general methods and shortcuts. Providing students with formulas or algorithms instead of allowing them to discover it on their own. Not allowing students enough time to discover patterns. Page 11 of 207

12 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 Make sense of problems and persevere in solving them. NEEDS IMPROVEMENT Task: Is strictly procedural. Does not require students to check solutions for errors. 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. Task: EMERGING (teacher does thinking) Is overly scaffolded or procedurally obvious. Requires students to check answers by plugging in numbers. Teacher: Allots too much or too little time to complete task. Encourages students to individually complete tasks, but does not ask them to evaluate the processes used. Explains the reasons behind procedural steps. Does not check errors CITYly. Task: PROFICIENT (teacher mostly models) Is cognitively demanding. Has more than one entry point. Requires a balance of procedural fluency and conceptual understanding. Requires students to check solutions for errors usingone other solution path. Teacher: Allows ample time for all students to struggle with task. Expects students to evaluate processes implicitly. Models making sense of the task (given situation) and the proposed solution. Task: EXEMPLARY (students take ownership) Allows for multiple entry points and solution paths. Requires students to defend and justify their solution by comparing multiple solution paths. Teacher: Differentiates to keep advanced students challenged during work time. Integrates time for explicit meta-cognition. Expects students to make sense of the task and the proposed solution. Institute for Advanced Study/Park City Mathematics Institute Summer 2011 Page 12 of 207 Secondary School Teachers Program/Visualizing Functions

13 PRACTICE Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. NEEDS IMPROVEMENT Task: Lacks context. Does not make use of multiple representations or solution paths. Teacher: Does not expect students to interpret representations. Expects students to memorize procedures with no connection to meaning. 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. Task: EMERGING (teacher does thinking) Is embedded in a contrived context. Teacher: Expects students to model and interpret tasks using a single representation. Explains connections between procedures and meaning. Task: PROFICIENT (teacher mostly models) Has realistic context. Requires students to frame solutions in a context. Has solutions that can be expressed with multiple representations. Teacher: Expects students to interpret and model using multiple representations. Provides structure for students to connect algebraic procedures to contextual meaning. Links mathematical solution with a question s answer. Task: Task: Teacher: Is not at the appropriate Avoids single steps or level. routine algorithms. 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. Teacher: Identifies students assumptions. Models evaluation of student arguments. Asks students to explain their conjectures. EXEMPLARY (students take ownership) Task: Has relevant realistic context. Teacher: Expects students to interpret, model, and connect multiple representations. Prompts students to articulate connections between algebraic procedures and contextual meaning. Helps students differentiate between assumptions and logical conjectures. Prompts students to evaluate peer arguments. Expects students to formally justify the validity of their conjectures. Page 13 of 207

14 PRACTICE Model with mathematics NEEDS IMPROVEMENT Task Requires students to identify variables and to perform necessary computations. Teacher: Identifies appropriate variables and procedures for students. Does not discuss appropriateness of model. Task: EMERGING (teacher does thinking) 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. Task: PROFICIENT (teacher mostly models) Requires students to identify variables, compute and interpret results, and report findings using a mixture of representations. Illustrates the relevance of the mathematics involved. Requires students to identify extraneous or missing information. Teacher: Asks questions to help students identify appropriate variables and procedures. Facilitates discussions in evaluating the appropriateness of model. 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. Use appropriate tools strategically 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 appropriate learning tool. Page 14 of 207 Task: Lends itself to multiple learning tools. Gives students opportunity to develop fluency in mental computations. Teacher: Chooses appropriate learning tools for student use. Models error checking by estimation. Task: Requires multiple learning tools (i.e., graph paper, calculator, manipulatives). 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 Summer 2011 Secondary School Teachers Program/Visualizing Functions

15 PRACTICE Attend to precision. Look for and make use of structure. NEEDS IMPROVEMENT Task: Gives imprecise instructions. Teacher: Does not intervene when students are being imprecise. Does not point out instances when students fail to address the question completely or directly. Task: Requires students to automatically apply an algorithm to a task without evaluating its appropriateness. Teacher: Does not recognize students for developing efficient approaches to the task. Requires students to apply the same algorithm to a task although there may be other approaches. Task: EMERGING (teacher does thinking) Has overly detailed or wordy instructions. Teacher: Inconsistently intervenes when students are imprecise. Identifies incomplete responses but does not require student to formulate further response. Task: PROFICIENT (teacher mostly models) Teacher: Has precise instructions. Consistently demands precision in communication and in mathematical solutions. Identifies incomplete responses and asks student to revise their response. Task: EXEMPLARY (students take ownership) Includes assessment criteria for communication of ideas. Teacher: Demands and models precision in communication and in mathematical solutions. Encourages students to identify when others are not addressing the question completely. Task: Task: Task: Requires students to Requires students to Requires students to analyze a task before analyze a task and identify the most efficient automatically applying identify more than one solution to the task. an algorithm. approach Teacher: to the problem. Teacher: Identifies individual Prompts students to students efficient Teacher: identify mathematical approaches, but does Facilitates all students structure of the task in not expand in developing order to identify the most understanding to reasonable and effective solution path. the rest of the class. efficient ways to Encourages students to Demonstrates the same accurately perform justify their choice of algorithm to all related basic operations. algorithm or solution path. tasks although there Continuously questions may be other more students about the effective reasonableness of their approaches. intermediate results. Institute for Advanced Study/Park City Mathematics Institute Summer 2011 Secondary School Teachers Program/Visualizing Functions Page 15 of 207

16 PRACTICE Look for and express regularity in repeated reasoning. NEEDS IMPROVEMENT Task: Is disconnected from prior and future concepts. Has no logical progression that leads to pattern recognition. Teacher: Does not show evidence of understanding the hierarchy within concepts. Presents or examines task in isolation. Task: EMERGING PROFICIENT EXEMPLARY (teacher does thinking) (teacher mostly models) (students take ownership) Task: Task: Is overly repetitive or Reviews prior Addresses and connects to has gaps that do not knowledge and requires prior knowledge in a nonroutine allow for development cumulative way. of a pattern. understanding. Teacher: Hides or does not draw connections to prior or future concepts. Lends itself to developing a pattern or structure. Teacher: Connects concept to prior and future concepts to help students develop an understanding of procedural shortcuts. Demonstrates connections between tasks. Requires recognition of pattern or structure to be completed. Teacher: Encourages students to connect task to prior concepts and tasks. Prompts students to generate exploratory questions based on the current task. Encourages students to monitor each other s intermediate results. Institute for Advanced Study/Park City Mathematics Institute Summer 2011 Secondary School Teachers Program/Visualizing Functions Page 16 of 207

17 Mathematical Practices: A Walk-Through Protocol Mathematical Practices MP.1. Make use of Problems and persevere in solving them Observations Students are expected to: Engage in solving problems Explain the meaning of a problem an restate it in their own words. Analyze given information to develop possible strategies for solving the problem. Identify and execute appropriate strategies to solve the problem. Check their answers using a different method and continually ask Does this make sense? MP.2. Reason abstractly and quantitatively. Teachers are expected to: Provide time for students to discuss problem solving. Students are expected to: Connect quantity to number and symbols (decontextualize the problem). Recognize that a number represents a specific quantity (contextualize the problem). Contextualize and decontextualize within the process of solving a problem. *Note: This document should also be used by the teacher for planning and self-evaluation CCSSM National Professional Development Teachers are expected to: Provide appropriate representation of problems. Page 17 of 207

18 .3. Construct viable arguments and critique the reasoning of others. MP.4. Model with mathematics Students are expected to: Explain their thinking to others and respond to others thinking. Participate in mathematical discussions involving questions like How did you get that? and Why is that true? Construct arguments that utilize prior learning. Question and problem pose. Practice questioning strategies used to generate information. Analyze alternative approaches suggested by others and select better approaches. -Justify conclusions, communicate them to others, and respond to the arguments of others. Compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument, explain what it is. Teachers are expected to : Provide opportunities for students to listen to or read the conclusions and arguments of others. 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, CCSSM National Professional Development Page 18 of 207

19 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 geometry systems.) MP 5. Use appropriate tools strategically 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. MP.6. Attend to precision. CCSSM National Professional Development Detect possible errors by strategically using estimation and other mathematical knowledge. Teachers are expected to: Model the use of appropriate tools (e.g. manipulatives) instructionally. 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. Page 19 of 207

20 MP.7. Look for and make use of structure. Teachers are expected to: Emphasize the importance of precise communication. Students are expected to: Describe a pattern or structure. Look for, develop, generalize, and describe a pattern orally, symbolically, graphically and in written form. Relate numerical patterns to a rule or graphical representation Apply and discuss properties. MP.8. Look for and express regularity in repeated reasoning. Teachers are expected to: Provide time for applying and discussing properties. Students are expected to: Describe repetitive actions in computation Look for mathematically sound shortcuts. Use repeated applications to generalize properties. Use models to explain calculations and describe how algorithms work. Use models to examine patterns and generate their own algorithms. Check the reasonableness of their results. Teachers are expected to: Encourage students to look for and discuss regularity in reasoning. CCSSM National Professional Development Page 20 of 207

21 High School Common Core Math I Curriculum Timeline Topic Grading Standards Covered Period Intro Unit 1 5 Modeling with 1 15 Functions Linear and Exponential Functions N Q 1 N Q 2 N Q 3 F IF 1 F LE 1, 1a, 1b, 1c F LE 2 F LE 3 F IF 2 F IF 3 F IF 4 F IF 5 F LE 5 F BF 1, 1a F BF 2 F IF 6 F IF 7, 7a F IF 9 F BF 4, 4a 1 No. of Days 20 Linear Equations and Inequalities in One Variable and Exponentials Linear Equations and Inequalities in Two Variables A REI 1 A REI 3 A REI 11 A CED 2 A CED 3 A CED 4 A CED 1 A CED 3 A CED 4 A REI 5 A REI 6 A REI 10 A SSE 1, 1a, 1b A SSE 3, 3c 2 15 A REI Modeling Unit 2 4 Project 2 5 Tools and Constructions G CO 12 G CO Basic Definitions and Rigid Motions Geometric Relationships and Properties G CO 1 G CO 2 G CO 3 G CO 9 G CO 10 G CO 4 G CO 5 G CO 6 G CO 11 G C 3 G CO 7 G CO Modeling Unit 4 4 Statistics S ID 1 S ID 2 S ID 3 S ID 5 S ID 6, 6a, 6c S ID 7 S ID 8 S ID Project 4 5 Page 21 of 207

22 Intro Unit (5 days) High School Common Core Math I 1 st Nine Weeks Scope and Sequence Topic 1 Modeling with Functions (15 days) Quantities (N Q): 1) Reason quantitatively and use units to solve problems. N Q 1: Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. N Q 2: Define appropriate quantities for the purpose of descriptive modeling. N Q 3: Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. Interpreting Functions (F IF): 2) Understand the concept of a function and use function notation. F IF 1: Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). F IF 2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. F IF 3: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n + 1) = f(n) + f(n 1) for n 1. Interpreting Functions (F IF): 3) Interpret functions that arise in applications in terms of the context. F IF 4*: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. F IF 5*: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number Page 22 of 207

23 of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Interpreting Functions (F IF): 4) Interpret functions that arise in applications in terms of the context. 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): 5) 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 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. Topic 2 Linear and Exponential Functions (20 days) Linear, Quadratic, and Exponential Models (F LE): 6) Construct and compare linear, simple quadratic, and simple exponential models and solve problems. F LE 1: Distinguish between situations that can be modeled with linear functions and with exponential functions. F LE 1a: Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors of equal intervals. F LE 1b: Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. F LE 1c: Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. F LE 2: Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). 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. Linear, Quadratic, and Exponential Models (F LE): Page 23 of 207

24 7) Interpret expressions for functions in terms of the situation they model. F LE 5: Interpret the parameters in a linear or exponential function in terms of a context. Building Functions (F BF): 8) Build a function that models a relationship between two quantities. F BF 1: Write a linear 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 2: Write arithmetic sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Building Functions (F BF): 9) Build new functions from existing functions. F BF 4: Find inverse functions. F BF 4a: Solve a linear equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. Page 24 of 207

25 High School Common Core Math I 2 nd Nine Weeks Scope and Sequence Topic 3 Linear Equations and Inequalities in One Variable and Exponentials (15 days) Reasoning with Equations and Inequalities (A REI): 10) 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): 11) Solve equations and inequalities in one variable. A REI 3: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Reasoning with Equations and Inequalities (A REI): 12) Represent and solve equations and inequalities graphically. A REI 11: Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make the tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. Creating Equations (A CED): 13) 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 3: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. 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 the resistance R. Seeing Structure in Expressions (A SSE): 14) Interpret the structure of expressions. A SSE 1: Interpret expressions that represent a quantity in terms of its context. Page 25 of 207

26 A SSE 1a: Interpret parts of an expression, such as terms, factors, and coefficients. A SSE 1b: Interpret complicated expressions by viewing one or more of their parts a single entity. For example, interpret P(1 + r) n as the product of P and a factor not depending on P. Seeing Structure in Expressions (A SSE): 15) Write expressions in equivalent forms to solve problems. A SSE 3: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. A SSE 3c: Use the properties of exponents to transform expressions for exponential functions. For example, the expression 1.15 t can be rewritten as ( ) 12t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. Topic 4 Linear Equations and Inequalities in Two Variables (15 days) Creating Equations (A CED): 16) Create equations that describe numbers or relationships. A CED 2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. A CED 3: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. 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 the resistance R. Reasoning with Equations and Inequalities (A REI): 17) Solve systems of equations. A REI 5: Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. A REI 6: Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Reasoning with Equations and Inequalities (A REI): 18) Represent and solve equations and inequalities graphically. A REI 10: Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Page 26 of 207

27 A REI 12: Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. Modeling Unit 4 days) Project (5 days) Page 27 of 207

28 High School Common Core Math I 3 rd Nine Weeks Scope and Sequence Topic 5 Tools and Construction (10 days) Congruence (G CO): 19) Make geometric constructions. G CO 12: Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; constructing perpendicular lines, including the perpendicular bisector of a line segment and constructing a line parallel to a given line through a point not on the line. G CO 13: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Topic 6 Basic Definitions and Rigid Motions (20 days) Congruence (G CO): 20) Experiment with transformations in the plane. G CO 1: Know the precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of a point, line, distance along a line, and distance around a circular arc. G CO 2: Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translations versus horizontal stretch). G CO 3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. G CO 4: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. G CO 5: Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. G CO 6: Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Page 28 of 207

29 G CO 7: Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. G CO 8: Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Topic 7 Geometric Relationships and Properties (15 days) Congruence (G CO): 21) Prove geometric theorems. G CO 9: Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment s endpoints. G CO 10: Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 ; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. G CO 11: Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Circles (G C): 22) Understand and apply theorems about circles. G C 3: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Page 29 of 207

30 Modeling Unit (4 days) Topic 8 Statistics (30 days) High School Common Core Math I 4 th Nine Weeks Scope and Sequence Interpreting Categorical and Quantitative Data (S ID): 23) Summarize, represent, and interpret data on a single count of measurement variable. S ID 1: Represent data with plots on the real number line (dot plots, histograms, and box plots). S ID 2: Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. S ID 3: Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). Interpreting Categorical and Quantitative Data (S ID): 24) Summarize, represent, and interpret data on two categorical and quantitative variables. S ID 5: Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. S ID 6: Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. 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, quadratic, and exponential models. S ID 6c: Fit a linear function for a scatter plot that suggests a linear association. Interpreting Categorical and Quantitative Data (S ID): 25) Interpret linear models. S ID 7: Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of data. S ID 8: Compute (using technology) and interpret the correlation coefficient of a linear fit. Page 30 of 207

31 S ID 9: Distinguish between correlation and causation. Project (5 days) Page 31 of 207

32 COLUMBUS CITY SCHOOLS HIGH SCHOOL CCSS MATHEMATICS I CURRICULUM GUIDE Topic 5 Constructions G CO 12,13 Domain: Congruence (G CO) CONCEPTUAL CATEGORY Geometry Cluster 19) Making geometric constructions. TIME RANGE 10 days GRADING PERIOD 3 Standards 19) Making geometric constructions. G CO 12: Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. G CO 13: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Constructions G CO 12,13 12/1/13 Page 32 of 207

33 TEACHING TOOLS Vocabulary: arc, bisector, circle, circumscribed, compass, congruent, construction, diameter, equilateral triangle, inscribe, parallel, perpendicular, radius, regular hexagon, regular polygon, square, straightedge, triangle Teacher Notes: A SMART Board presentation has been created to teach all of the topics in this unit covering constructions with a compass. Teachers can print the presentation as full-page handouts and then have all the worksheets necessary for the entire unit. The worksheets are also presented in this guide. Students are taught how to use a compass to create geometric constructions. While the lessons in the presentation focus on using a standard compass, teachers and students can also use safety compasses. Constructions can also be performed using patty paper. Links to helpful videos are provided in the Instructional Strategies section of this curriculum guide. Technology also plays a part in making constructions. Links are provided for a lesson on the TI-nspire calculator and Geometer s Sketchpad. The entire compass construction presentation can be found at the link below: Misconceptions/Challenges: One main challenge to using a compass is to make sure students know not to use them in an unsafe manner: the points can be very sharp. Safety compasses can be used if desired. When making a construction, remind students to keep the compass open to the same setting throughout the entire construction, unless told to re-adjust it. To ensure that students are correctly making constructions and not just estimating a parallel line or the bisector of an angle, remind students that you will be looking for the marks made by the sharp point of the compass. There also should be arcs made on the drawing and it should be clear where arcs cross each other or the circle. Observe the folds on patty paper to ensure students actually performed the constructions properly. Constructions G CO 12,13 12/1/13 Page 33 of 207

34 Instructional Strategies: Download the SMART board presentation and print full-page handouts. Each section shows how to make a construction and then has 4-8 problems for the students to complete before moving on to another construction. There is a video enclosed within the presentation on how to create a circle with six arcs for making inscribed figures. Geometer s Sketchpad Tutorial This website contains information on creating constructions as well as transformations. Constructions Texas Instruments activity to teach constructions on the TI-nspire. For patty paper (tracing paper) constructions, the links below are very helpful. Most of the videos are just one to two minutes in length. Teachers can show the videos to students and have them complete each activity after viewing. Patty paper videos - how to construct an angle bisector: Patty paper video - how to construct a line parallel to a given line: Patty paper video - how to construct a line perpendicular to a point on the line: Patty paper video - how to construct a perpendicular bisector: Patty paper videos - how to bisect an angle: Constructions G CO 12,13 12/1/13 Page 34 of 207

35 Reteach/Extension Reteach Constructions This website shows step-by-step drawings of many constructions, including many great extension opportunities. Creation of constructions An animated website Extension Included in the presentation is how to create a square from two points on a line (slide 16). This is optional. The last slide encourages students to try to figure out a way to construct an octagon inscribed in a circle. Have students create their own creative drawing using nothing more than a compass and straightedge. Students can create stars inside a circle, dodecagons, etc. Students can also calculate the area of each image inside a circle. Constructions G CO 12,13 12/1/13 Page 35 of 207

36 Textbook References Textbook: Pearson Integrated High School Common Core Mathematics I (Volume 2) G CO 12: Chapter 10 Reasoning and Proof (pp. 566) Lesson 1 Basic Constructions (pp ) Chapter 11 Proving Theorems About Lines and Angles (pp. 630) Lesson 6 Constructing Parallel and Perpendicular Lines (pp ) Chapter 12 Congruent Triangles (pp. 688) Lesson 4 Using Corresponding Parts of Congruent Triangles (pp ) 4 Activity Lab: Paper-Folding Conjectures (pp. 723) Chapter 13 Proving Theorems About Triangles (pp. 766) Lesson 2 Perpendicular and Angle Bisectors (pp ) G CO 13: Chapter 11 Proving Theorems About Lines and Angles (pp. 630) Lesson 6 Constructing Parallel and Perpendicular Lines (pp ) Chapter 12 Congruent Triangles (pp. 688) Lesson 5 Isosceles and Equilateral Triangles (pp ) Constructions G CO 12,13 12/1/13 Page 36 of 207

37 G CO 12 Name Date Period Constructions Constructions G CO 12,13 12/1/13 Page 37 of 207

38 Name Date Period Copying a Line Segment G CO 12 Constructions G CO 12,13 12/1/13 Page 38 of 207

39 Name Date Period Constructions G CO 12,13 12/1/13 Page 39 of 207

40 G CO 12 Name Date Period Copying an Angle Constructions G CO 12,13 12/1/13 Page 40 of 207

41 Name Date Period Constructions G CO 12,13 12/1/13 Page 41 of 207

42 Name Date Period Bisecting an Angle G CO 12 Constructions G CO 12,13 12/1/13 Page 42 of 207

43 Name Date Period Constructions G CO 12,13 12/1/13 Page 43 of 207

44 G CO 12 Name Date Period Perpendicular Lines Constructions G CO 12,13 12/1/13 Page 44 of 207

45 Name Date Period Constructions G CO 12,13 12/1/13 Page 45 of 207

46 G CO 12 Name Date Period Perpendicular Lines Constructions G CO 12,13 12/1/13 Page 46 of 207

47 Name Date Period G CO 12 Constructions G CO 12,13 12/1/13 Page 47 of 207

48 G CO 12 Name Date Period Bisect A Segment Constructions G CO 12,13 12/1/13 Page 48 of 207

49 G CO 12 Name Date Period Constructions G CO 12,13 12/1/13 Page 49 of 207

50 G CO 12 Name Date Period Parallel Lines Constructions G CO 12,13 12/1/13 Page 50 of 207

51 Name Date Period G CO 12 Constructions G CO 12,13 12/1/13 Page 51 of 207

52 G CO 12 Name Date Period Constructing a Square Constructions G CO 12,13 12/1/13 Page 52 of 207

53 G CO 13 Name Date Period Constructing an Equilateral Triangle and a Regular Hexagon Inscribed in a Circle Constructions G CO 12,13 12/1/13 Page 53 of 207

54 G CO 13 Name Date Period Constructions G CO 12,13 12/1/13 Page 54 of 207

55 G CO 13 Name Date Period Constructing a Square Inscribed in a Circle Constructions G CO 12,13 12/1/13 Page 55 of 207

56 G CO 13 Name Date Period Constructions G CO 12,13 12/1/13 Page 56 of 207

57 COLUMBUS CITY SCHOOLS HIGH SCHOOL CCSS MATHEMATICS I CURRICULUM GUIDE Topic 6 Basic Definitions and Rigid Motions G CO 1,2,3,4,5,6,7,8 CONCEPTUAL CATEGORY Geometry Domain: Congruence (G CO) Cluster 20) Experiment with transformations in the plane. 21) Understand congruence in terms of rigid motions. TIME RANGE 20 days GRADING PERIOD 3 Standards 20) Experiment with transformations in the plane. G CO 1: Know the precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of a point, line, distance along a line, and distance around a circular arc. G CO 2: Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translations versus horizontal stretch). G CO 3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. G CO 4: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. G CO 5: Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. 21) Understand congruence in terms of rigid motions. G CO 6: Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. G CO 7: Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. G CO 8: Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Basic Definitions and Rigid Motions G CO 1,2,3,4,5,6,7,8 12/1/13 Page 57 of 207

58 TEACHING TOOLS Vocabulary: angle, arc, circle, congruence, coordinate distance, dilation, line, line segment, parallel lines, perpendicular lines, point, reflection, regular polygon, rigid motion, rotation, transformation, translation, Teacher Notes: The concepts of congruence, similarity and symmetry can be understood from the perspective of geometric transformation. Fundamental are the rigid motions: translations, rotations, reflections, and combinations of these, all of which are here assumed to preserve distance and angles (therefore shapes generally). Reflections and rotations each explain a particular type of symmetry, and the symmetries of an object offer insight into its attributes as when the reflective symmetry of an isosceles triangle assures that its base angles are congruent. In the approach taken here, two geometric figures are defined to be congruent if there is a sequence of rigid motions that carries one onto the other. This is the principle of superposition. For triangles, congruence means the equality of all corresponding pairs of sides and all corresponding pairs of angles. During the middle grades, through experiences drawing triangles, students notice ways to specify enough measures in a triangle to ensure that all triangles drawn with those measures are congruent. Once these triangle congruence criteria (ASA, SAS, and SSS) are established using rigid motions, they can be used to prove theorems about triangles, quadrilaterals, and other geometric figures. Similarity transformations (rigid motions followed by dilations) define similarity in the same way that rigid motions define congruence, thereby formalizing the similarity ideas of same shape and scale factor developed in the middle grades. Misconceptions/Challenges: Students need to know how to graph/recognize lines and equations of lines to perform reflections. Students often assume rotations are about the origin, when they can be about any specified point. Students think SSA is a congruence criterion for triangles Basic Definitions and Rigid Motions G CO 1,2,3,4,5,6,7,8 12/1/13 Page 58 of 207

59 Instructional Strategies: Transformation Tease (G - CO 2, 4, 5, 6) This activity is good to present to pupils who have just started work with transformations in the coordinate plane. It can also be used in a revision situation. Transformation Game (G - CO 2, 5, 6, 7, 8) This game provides an opportunity to practice a lot of transformations in the context of a game. The scoring system encourages the use of combinations of transformations. Opponents usually watch for cheating and errors, so there is built-in feedback for students. The structure of a game encourages students to play strategically. Representing and Combining Transformations (G - CO 4, 5, 6) This lesson unit is intended to help you assess how well students are able to: Recognize and visualize transformations of 2D shapes. Translate, reflect and rotate shapes, and combine these transformations. It also aims to encourage discussion on some common misconceptions about transformations. 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 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. (G CO 2) Analyzing Congruence Proofs (G - CO 6, 7, 8) This lesson unit is intended to help you assess how well students are able to: Work with concepts of congruency and similarity, including identifying corresponding sides and corresponding angles within and between triangles. Identify and understand the significance of a counter-example. Prove, and evaluate proofs in a geometric context. Basic Definitions and Rigid Motions G CO 1,2,3,4,5,6,7,8 12/1/13 Page 59 of 207

60 Circles in Triangles (G - CO 2, 3) %20Circles%20in%20Triangles.pdf This task challenges a student to use geometric properties of circles and triangles to prove that two triangles are congruent. A student must be able to use congruency and corresponding parts to reason about lengths of sides. A student must be able to construct lines to make sense of a diagram. A student needs to use geometric properties to find the radius of a circle inscribed in a right triangle. Quadrilaterals (G - CO 1) This is an inquiry lesson used to review Algebra objectives by applying them to geometry concepts. Students explore the properties of quadrilaterals and classify them by definition. Trapezoids: What s Right or Equal About Them (G - CO 1) This lesson examines the properties of two trapezoids - isosceles and right. The properties are used to solve deeper analytical problems. Have students engage in the activity Partitions (included in this curriculum guide) to allow students to use the definition of congruence as it relates to rigid motions to prove that the shapes they partitioned satisfy the conditions given in the directions of the activity. (G CO 3, 5, 6) Transformations and Frieze Patterns (G - CO 3, 5, 6) This unit introduces students to the world of symmetry and rotation in figures and patterns. Students learn to recognize and classify symmetry in decorative figures and frieze patterns, and also get the chance to create and classify their own figures and patterns using JavaSketchpad applets. Math nspired: Points, Lines, and Angles: Points, Lines, and Planes (G - CO 1) Students explore the relationships between points, lines, and planes using the TI-nspire CAS or TI-nspire. The Circle (G - CO 1) Students examine the definitions and formulas for radius, diameter, circumference, and area. Basic Definitions and Rigid Motions G CO 1,2,3,4,5,6,7,8 12/1/13 Page 60 of 207

61 Students also solve practice problems involving the circumference and area of a circle. Congruence Theorems Students investigate congruence by manipulating the sides and angles of a triangle. Reteach/Extension Reteach Functions notation, composition of functions, functions transformations Extension Inscribing Circles in Triangles This lesson assesses how well students are able to use geometric properties to solve problems. In particular, it will help you identify and help students who have difficulty: Decomposing complex shapes into simpler ones in order to solve a problem. Bringing together several geometric concepts to solve a problem. Finding the relationship between radii of inscribed and circumscribed circles of right triangles Textbook References Textbook: Pearson Integrated High School Common Core Mathematics I (Volume 1) G CO 1: Chapter 7 Tools of Geometry (pp. 412) Lesson 1 Nets and Drawings for Visualizing Geometry (pp ) 2 Points, Lines, and Planes (pp ) 3 Measuring Segments (pp ) 4 Measuring Angles (pp ) 5 Measuring Angle Pairs (pp ) G CO 2: Chapter 8 Transformations (pp. 474) 1 Activity Lab: Tracing Paper Transformations (pp. 477) Lesson 1 Translations (pp ) 2 Reflections (pp ) 3 Rotations (pp ) 3 Activity Lab: Exploring Multiple Transformations (pp ) 4 Composition of Isometries (pp ) G CO 3: Chapter 7 Tools of Geometry (pp. 412) Lesson 6 Midpoint and Distance in the Coordinate Plane (pp ) Chapter 8 Transformations (pp. 474) Basic Definitions and Rigid Motions G CO 1,2,3,4,5,6,7,8 12/1/13 Page 61 of 207

62 Lesson 3 Activity Lab: Symmetry (pp ) G CO 4: Chapter 8 Transformations (pp. 474) Lesson 1 Translations (pp ) 2 Reflections (pp ) 3 Rotations (pp ) G CO 5: Chapter 8 Transformations (pp. 474) Lesson 1 Translations (pp ) 1 Activity Lab: Paper Folding and Reflections (pp. 487) 2 Reflections (pp ) 3 Rotations (pp ) 3 Activity Lab: Exploring Multiple Transformations (pp ) 4 Composition of Isometries (pp ) G CO 6: Chapter 12 Congruent Triangles (pp. 688) Lesson 7 Activity Lab: Review of Transformations (pp ) 8 Congruence Transformations (pp ) G CO 7: Chapter 12 Congruent Triangles (pp. 688) Lesson 1 Congruent Figures (pp ) 8 Congruence Transformations (pp ) G CO 8: Chapter 12 Congruent Triangles (pp. 688) Lesson 8 Congruence Transformations (pp ) Basic Definitions and Rigid Motions G CO 1,2,3,4,5,6,7,8 12/1/13 Page 62 of 207

63 G CO 3, 5, 6 Name Date Period Partitions 1. Partition a square into two congruent figures. Explain why the figures are congruent. 2. Partition a square into two congruent figures that are different from those you created in question one. Explain why the figures are congruent. 3. Partition an equilateral triangle into three congruent figures. Explain why the figures are congruent. 4. Partition an equilateral triangle into three congruent figures different from those in question three. Explain why the figures are congruent. 5. Partition a square into four congruent figures. Explain why the figures are congruent. 6. Partition a square into four congruent figures different from those in question 5. Explain why the figures are congruent. Basic Definitions and Rigid Motions G CO 1,2,3,4,5,6,7,8 12/1/13 Page 63 of 207

64 7. Partition a square into 3 congruent figures. Explain why the figures are congruent. 8. Explain why the two figures you created in question one are congruent using another transformation. 9. Explain why the three figures you created in question three are congruent using another transformation. 10. Explain why the four figures you created in question five are congruent using another transformation. Basic Definitions and Rigid Motions G CO 1,2,3,4,5,6,7,8 12/1/13 Page 64 of 207

65 G CO 3, 5, 6 Name Date Period Partitions Answer Key The possibilities for these partitions are endless. Here are some sample answers: The two figures are congruent because they can be carried to each other by - reflecting over the drawn segment. - translating horizontally a distance of half the length of one side. -rotating 180 degreed about the center of the square. The two figures are congruent because they can be carried to each other by - rotating 60 degrees clockwise about the center of the triangle. - rotating 60 degrees counterclockwise about the center of the triangle. - rotating 120 degrees (either direction) about the center of the triangle. The two figures are congruent because they can be carried to each other by - rotating 90 degrees either direction about the center of the square. - translating horizontally and/or vertically a distance half the length of one side of the square. - reflecting over the vertical, horizontal, and/or diagonal segments. The two figures are congruent because they can be carried to each other by - translating horizontally a distance one third the length of one side of the square. - reflecting over the vertical line of symmetry and then translating horizontally a distance one third the length of one side of the square. Basic Definitions and Rigid Motions G CO 1,2,3,4,5,6,7,8 12/1/13 Page 65 of 207

66 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. Basic Definitions and Rigid Motions G CO 1,2,3,4,5,6,7,8 12/1/13 Page 66 of 207

67 G CO 2 Name Date Period The Gumps Giggles Higgles Wiggles Ziggles Miggles (x,y) (2x,2y) (3x,y) (3x,3y) (x,3y) SET 1 SET 1 SET 1 SET 1 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 SET 2 SET 2 SET 2 SET 2 (4,11) (6,10) (8,10) (10,11) SET 3 SET 3 SET 3 SET 3 SET 3 (6,11) (6,12) (8,12) (8,11) SET 4 SET 4 SET 4 SET 4 SET 4 (5,14) (9,14) Basic Definitions and Rigid Motions G CO 1,2,3,4,5,6,7,8 12/1/13 Page 67 of 207

68 Name Date Period The Gumps Answer Key Giggles Higgles Wiggles Ziggles Miggles (x,y) (2x,2y) (3x,y) (3x,3y) (x,3y) SET 1 SET 1 SET 1 SET 1 SET 1 (4,0) (8,0) (12,0) (12,0) (4,0) (4,6) (8,12) (12,6) (12,18) (4,18) (2,4) (4,8) (6,4) (6,12) (2,12) (0,4) (0,8) (0,4) (0,12) (0,12) (4,8) (8,16) (12,8) (12,24) (4,24) (2,10) (4,20) (6,10) (6,30) (2,30) (2,14) (4,28) (6,14) (6,42) (2,42) (4,16) (8,32) (12,16) (12,48) (4,48) (5,18) (10,36) (15,18) (15,54) (5,54) (6,16) (12,32) (18,16) (18,48) (6,48) (8,16) (16,32) (24,16) (24,48) (8,48) (9,18) (18,36) (27,18) (27,54) (9,54) (10,16) (20,32) (30,16) (30,48) (10,48) (12,14) (24,28) (36,14) (36,42) (12,42) (12,10) (24,20) (36,10) (36,30) (12,30) (10,8) (20,16) (30,8) (30,24) (10,24) (14,4) (28,8) (42,4) (42,12) (14,12) (12,4) (24,8) (36,4) (36,12) (12,12) (10,6) (20,12) (30,6) (30,18) (10,18) (10,0) (20,0) (30,0) (30,0) (10,0) (8,0) (16,0) (24,0) (24,0) (8,0) (8,4) (16,8) (24,4) (24,12) (8,12) (6,4) (12,8) (18,4) (18,12) (6,12) (6,0) (12,0) (18,0) (18,0) (6,0) SET 2 SET 2 SET 2 SET 2 SET 2 (4,11) (8,22) (12,11) (12,33) (4,33) (6,10) (12,20) (18,10) (18,30) (6,30) (8,10) (16,20) (24,10) (24,30) (8,30) (10,11) (20,22) (30,11) (30,33) (10,33) SET 3 SET 3 SET 3 SET 3 SET 3 (6,11) (12,22) (18,11) (18,33) (6,33) (6,12) (12,24) (18,12) (18,36) (6,36) (8,12) (16,24) (24,12) (24,36) (8,36) (8,11) (16,22) (24,11) (24,33) (8,33) SET 4 SET 4 SET 4 SET 4 SET 4 (5,14) (10,28) (15,14) (15,42) (5,42) (9,14) (18,28) (27,14) (27,42) (9,42) Basic Definitions and Rigid Motions G CO 1,2,3,4,5,6,7,8 12/1/13 Page 68 of 207

69 Giggles Basic Definitions and Rigid Motions G CO 1,2,3,4,5,6,7,8 12/1/13 Page 69 of 207

70 Higgles Basic Definitions and Rigid Motions G CO 1,2,3,4,5,6,7,8 12/1/13 Page 70 of 207

71 Wiggles Basic Definitions and Rigid Motions G CO 1,2,3,4,5,6,7,8 12/1/13 Page 71 of 207

72 Ziggles Basic Definitions and Rigid Motions G CO 1,2,3,4,5,6,7,8 12/1/13 Page 72 of 207

73 Miggles Basic Definitions and Rigid Motions G CO 1,2,3,4,5,6,7,8 12/1/13 Page 73 of 207

74 G CO 2 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 2. Do you notice anything about the above measurements? If so, explain. 3. 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 Basic Definitions and Rigid Motions G CO 1,2,3,4,5,6,7,8 12/1/13 Page 74 of 207

75 4. Compare each Gump s measurements to Giggles measurements. Describe any patterns that you notice. 5. Giggles and Higgles are mathematically similar. Describe what you think it means for two figures to be mathematically similar. 6. What other Gump(s) fit this description. Why? Basic Definitions and Rigid Motions G CO 1,2,3,4,5,6,7,8 12/1/13 Page 75 of 207

76 7. Complete the following table. 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 Basic Definitions and Rigid Motions G CO 1,2,3,4,5,6,7,8 12/1/13 Page 76 of 207

77 8. 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 9. Make a comparison between the scale factor of objects and the ratio of their perimeters. 10. 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 Make a comparison between the scale factor of objects and the ratio of their areas. Basic Definitions and Rigid Motions G CO 1,2,3,4,5,6,7,8 12/1/13 Page 77 of 207

78 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. Basic Definitions and Rigid Motions G CO 1,2,3,4,5,6,7,8 12/1/13 Page 78 of 207

79 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 Miggles Top of Ear 53 o 53 o 112 o 53 o 19 o Under Arm 45 o 45 o 72 o 45 o 18 o Neck 90 o 90 o 37 o 90 o 143 o Smile 153 o 153 o 171 o 153 o 124 o 2. 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. 3. 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 Miggles Width of Head Length of Leg Width of Hand Width of Waist Total Height G CO 2 4. 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. Basic Definitions and Rigid Motions G CO 1,2,3,4,5,6,7,8 12/1/13 Page 79 of 207

80 5. 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. 6. 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. 7. Complete the following table. Giggles (Gump 1) Nose Width Nose Length 1 cm 2 cm Width Length 1 2 Nose Perimeter Nose Area 6 cm 2 cm 2 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 2 cm 4 cm 3 cm 6 cm 4 cm 8 cm 5 cm 10 cm 10 cm 20 cm 20 cm 40 cm 100 cm 200 cm 1 cm 6 cm cm 8 cm cm 18 cm cm 32 cm cm 50 cm cm 200 cm cm 800 cm cm 20,000 cm cm 6 cm Miggles 2 cm 3 cm cm 6 cm Basic Definitions and Rigid Motions G CO 1,2,3,4,5,6,7,8 12/1/13 Page 80 of 207

81 8. Make ratios using the nose perimeter for the following figures: Gump 2:Gump 1 Gump 3:Gump Gump 4:Gump 1 Gump 5:Gump 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. 9. Make ratios using the nose area for the following figures: Gump 2:Gump 1 Gump 3:Gump Gump 4:Gump 1 Gump 5:Gump 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. Basic Definitions and Rigid Motions G CO 1,2,3,4,5,6,7,8 12/1/13 Page 81 of 207

82 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 = 60 cm Area of Gump 10 = 200 cm 2 Perimeter of Gump 20 = 120 cm Area of Gump 20 = 800 cm 2 Perimeter of Gump 100 = 600 cm Area of Gump 100 = 20,000 cm 2 Basic Definitions and Rigid Motions G CO 1,2,3,4,5,6,7,8 12/1/13 Page 82 of 207

83 COLUMBUS CITY SCHOOLS HIGH SCHOOL CCSS MATHEMATICS I CURRICULUM GUIDE Topic 7 Geometric Relationships and Properties G CO 9, 10, 11, G C 3 Domain: Congruence (G CO) Cluster 22) Prove geometric theorems. CONCEPTUAL CATEGORY Geometry Domain: Circles (G C) Cluster 23) Understand and apply theorems about circles. TIME RANGE 15 days GRADING PERIOD 3 Standards 22) Prove geometric theorems. G CO 9: Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment s endpoints. G CO 10: Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 ; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. G CO 11: Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. 23) Understand and apply theorems about circles. G C 3: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Page 83 of 207

84 TEACHING TOOLS Vocabulary: acute triangle, angles, alternate interior angles, alternate exterior angles, base, base angles, bisect, bisector, centroid, circumcenter, circumscribe, complementary angles, concurrent, consecutive interior angles, corresponding angles, diagonal, equiangular triangle, equidistant, equilateral triangle, exterior angle, hypotenuse, incenter, inscribe, inscribed arc, inscribed angle, inscribed quadrilateral, interior angle, isosceles triangle, leg, linear pair, lines, midsegment, obtuse triangle, orthocenter, parallel lines, parallelogram, perpendicular, perpendicular bisector, quadrilateral, rectangle, remote angle, rhombus, right angles, scalene triangle, square, supplementary angles, transversal line, vertex angle, vertical angles Teacher Notes: Lines and Angles Parallel lines and the angle relationships created by the transversal will also have a significant impact on Similarity and proofs of special quadrilaterals. The concepts of transversals will significantly impact problem solving strategies on nested similar triangles, such as the picture below. When examining solutions of systems of equations, if two distinct lines intersect, then their intersection is exactly one point. If two lines are parallel, then they will not intersect and there will be no solutions. If two lines are coincident, then there will be an infinite number of solutions. Students should know that parallel lines will have the same slope but with different y-intercepts, where coincident lines have the same slope and the same y-intercept and intersecting lines will have different slopes. Students should be reminded that an equation in the form y = mx + b, m represents the slope and b represents the y-intercept. Students should also be reminded that 2 1 given two points the slope could be found by using m y y. Students will need to know that x x 2 1 given a slope and a point, an equation of a line can be found by using the Point-Slope Formula, which is y y1 = m (x x1). Page 84 of 207

85 Triangles Triangles can be classified by their sides and their angles. Side classifications are scalene, isosceles, and equilateral. Angle classifications are acute, obtuse, right, and equiangular. Combinations of side and angle classifications can also be used. In triangles, an emphasis should be placed on the fact that the sum of the interior angles of a triangle must always equal 180 degrees. There are five different congruent triangle proof strategies: side-side-side, side-angle-side, angleside-angle, angle-angle-side, and hypotenuse-leg. There are four special segments of a triangle. An altitude is a segment drawn from a vertex of the triangle perpendicular to the opposite side of the triangle. An angle bisector is a ray drawn from a vertex of the triangle that divides an angle into two equal angles. A median is a segment drawn from a vertex of the triangle to the midpoint of the opposite side. A perpendicular bisector is a line, segment, or ray drawn through the midpoint of a side of the triangle that is also perpendicular to that side. When the same special segment is drawn from the three angles or sides of a triangle, the three lines, segments, or rays intersect at one point. The point of intersection is called the point of concurrency. The three altitudes of a triangle intersect at the orthocenter. The three angle bisectors of a triangle intersect to form the incenter. The incenter is the center of a circle that is inside the triangle and tangent to the sides of the triangle. The three medians of a triangle intersect to form the centroid. The three perpendicular bisectors of a triangle intersect to form the circumcenter. The circumcenter is the center of the circle that is outside the triangle attached to the vertices of the triangle. Students will also explore the Triangle Midsegment Theorem in this topic. Later in the year, these concepts will be necessary for Polygons. The Triangle Sum Theorem will also serve as the basis for the formula development of interior sums of polygons. Polygons Polygons are closed figures whose sides are segments. They have no intersections (other than endpoints). Polygons are named by the number of sides in their shape. The chart below gives the names of common polygons. Page 85 of 207

86 3 sides triangle 4 sides quadrilateral 5 sides pentagon 6 sides hexagon 7 sides heptagon 8 sides octagon 9 sides nonagon 10 sides decagon 12 sides dodecagon n sides n - gon Special attention is given to the quadrilaterals. Some quadrilaterals can be classified in subsets of parallelograms, rectangles, rhombi, squares, kites, trapezoids, and isosceles trapezoids. Parallelograms are the largest of these groups. There are five tests for parallelograms. Only one of the tests must be verified to determine if a shape is a parallelogram. 1) Both pairs of opposite sides are parallel. 2) Both pairs of opposite sides are congruent. 3) Both pairs of opposite angles are congruent. 4) The diagonals bisect each other. 5) One pair of sides is both parallel and congruent. The previous concept of parallel lines and the angle relationships created by a transversal will be utilized in the quadrilateral tests. The Triangle Sum Theorem will be the basis of the polygon interior sum formula. The distance and slope formulas can also be used to verify the type of quadrilateral. Later in the year, these concepts will be applied in the topic: Area and Volume. Real life applications are found in engineering, architecture, and design. Circles A circle is the set of all points in a plane which are all the same distance from a point called the center of the circle. A circle is named by its center. If the center is point Q, then the name of the circle is Q. Page 86 of 207

87 In circles, there are special relationships between angles, arcs, and segments that can intersect in, on or outside the circle. These relationships can be used to find the measures of angles, arcs, and segments associated with circles. tangent chord radius Q secant diameter Circle Q The illustration above gives a circle and various lines and segments connected with circles. A tangent is a line in the plane of a circle that intersects the circle in exactly one point called the point of tangency. A chord is a line segment whose end points lie on the circle. A diameter of a circle is a chord that contains the center of the circle. A radius is a segment or distance from the center of a circle to a point on the circle. A secant is a line that intersects a circle in two points. The diameter of a circle is twice the radius of the circle. Pi ( ) is the ratio of circumference to diameter of a circle. Pi is an irrational number that is (approx) The area of a circle is given by the equation: A = r 2. The circumference of a circle is given by the equation: C = d or C = 2r. Circumferences and arc lengths can be used to solve real life problems, such as finding the number of revolutions of a tire. Students need to understand the difference between the measure of an arc, which is given in degrees, and the length of an arc, which is given in linear units such as inches. Emphasize to students that arc length is a fraction of the circumference. Page 87 of 207

88 Arcs are measured by their corresponding central angles. The measure of a semicircle is 180º and is designated by three letters. The measure of a minor arc is the measure of its central angle. The measure of a minor arc is less than 180º and is designated by two letters. The measure of a major arc is the difference between 360º and the measure of its associated minor arc. The measure of a major arc is greater than 180º and is designated by three letters. minor arc C A R major arc B A circle is inscribed in a polygon if each side of the polygon is tangent to the circle, so an inscribed circle touches each side of the polygon at exactly one point. Inscribed Circle A circle is circumscribed about a polygon if each vertex of the polygon lies on the circle. A circumscribed circle passes through each vertex of the polygon. Circumscribed Circle Students need to understand that when a circle is inscribed in a polygon, then the polygon is circumscribed about the circle and when a circle is circumscribed about a polygon, then the polygon is inscribed in the circle. Page 88 of 207

89 Misconceptions/Challenges: Remembering which point of concurrency is created by the four special triangle segments. The medians make the centroid, the perpendicular bisectors make the circumcenter, the angle bisectors make the incenter, and the altitudes make the orthocenter Classifying quadrilaterals as more specific polygons may be lacking proper evidence to support the classification. For example, calling a quadrilateral a rectangle, but failing to prove that it is a parallelogram by showing the slopes of opposite sides are the same, thus proving they are parallel. Then needing to prove that two adjacent sides are perpendicular, and thus must have opposite reciprocal slopes, which would prove that it has a right angle, thus proving the quadrilateral is a rectangle. Instructional Strategies: G CO 9 Use the Exploring Parallel Lines Cut by a Transversal (included in this Curriculum Guide) activity to locate and analyze angle relationships. Use the Transversals and Special Angles (included in this Curriculum Guide) activity to analyze angle pair relationships. Use the Proof Cards Activity (included in this Curriculum Guide) activity to prove angle relationships using two-column proofs. Use the How Far Up the Hill, Do We Drill? (included in this Curriculum Guide) activity to analyze slope and find perpendicular slopes and equations of perpendicular lines. Use the Writing and Solving Linear Equations in Geometric Situations (included in this Curriculum Guide) activity to use solving equations to find missing angle measurements based on angle relationships formed by parallel lines cut by a transversal. Use the Memory Cards Match-Up (included in this Curriculum Guide) activity to matchup geometric terms with their corresponding figures and algebraic statements. Common Core State Standards Math Resource Page ents/geometry/congruence/g_co_9/ Vocabulary and I can statements for the G CO 9 Points equidistant from two points in the plane Lesson Activity Page 89 of 207

90 G CO 10 This task gives the important characterization of the perpendicular bisector of a line segment as the set of points equidistant from the endpoints of the segment. Use the Investigating Centers of Triangles (included in this Curriculum Guide) activity to draw triangles and investigate the different characteristics of the centers of triangles created by medians, perpendicular bisectors, altitudes, and angle bisectors. Use the Centers of Triangles (included in this Curriculum Guide) activity to investigate relationships between the centers of triangles and draw logical conclusions and apply these properties to various applications. Use the Writing and Solving Systems of Linear Equations in Geometric Situations: Centroids of a Triangle (included in this Curriculum Guide) activity to graph and label a triangle on the coordinate plane, construct the midpoint of each segment so that the medians can be constructed, and then determine the point of intersection that represents the centroid. Common Core State Standards Math Resource Page ents/geometry/congruence/g_co_10/ Vocabulary and I can statements for the G CO 10 Classifying Triangles Lesson Activity The goal of this task is to help students synthesize their knowledge of triangles. They will need to know that a triangle inscribed in a circle, with a diameter of the circle as one side, is a right triangle. Circumcenter of a triangle Lesson Activity This task shows that the three perpendicular bisectors of the sides of a triangle all meet in a point, using the characterization of the perpendicular bisector of a line segment as the set of points equidistant from the two ends of the segment. The point so constructed is called the circumcenter of the triangle. G CO 11 Page 90 of 207

91 Use the Coordinate Geometry: Is the Given Quadrilateral a Parallelogram? (included in this Curriculum Guide) activity to prove if the given quadrilateral graphed on the coordinate plane is a parallelogram. Use the TI-92/Cabri Exploration: Properties of Parallelograms (included in this Curriculum Guide) activity to help students make observations about the properties of parallelograms so that students make their own conjectures about the theorem about opposite sides of a parallelograms. Common Core State Standards Math Resource Page ents/geometry/congruence/g_co_11/ Vocabulary and I can statements for the G CO 11 Is this a Parallelogram? Lesson Activity This task develops an alternative characterization of a parallelogram in terms of congruence of opposite sides. Midpoints of the Sides of a Parallelogram Lesson Activity This is a reasonably direct task aimed at having students use previously-derived results to learn new facts about parallelograms, as opposed to deriving them from first principles. Quadrilaterals Performance Assessment Task (MARS) %20Quadrilaterals.pdf This task challenges a student to use geometric properties to find and prove relationships about an inscribed quadrilateral. G C 3 Use the Circles: Expanding Our Vocabulary (included in this Curriculum Guide) activity to learn circle vocabulary, summarize that vocabulary into their own words, and provide figures that model the vocabulary. Use the Discovery Lesson: Properties of Circles (Lessons 1-9) (included in this Curriculum Guide) activities construct properties of circles and make conjectures based on their observations. Page 91 of 207

92 Use the Parts of a Circle: Round Robin (included in this Curriculum Guide) activity to illustrate the different properties of circles. Use the Pairs Check: Properties of Circles (included in this Curriculum Guide) activity to use the properties of circles to find missing angle measurements in circles. Use the Numbered Noggins: Properties of Circles (Parts 1-5) (included in this Curriculum Guide) activity to use the properties of circles to find missing angle measurements in circles. Common Core State Standards Math Resource Page ents/geometry/circles/g_c_3/ Vocabulary and I can statements for the G C 3 Locating Warehouse Lesson Activity This task can be implemented in a variety of ways. For a class with previous exposure to the incenter or angle bisectors, part (a) could be a quick exercise in geometric constructions. Placing a Fire Hydrant Lesson Activity This task can be implemented in a variety of ways. For a class with previous exposure to properties of perpendicular bisectors, part (a) could be a quick exercise in geometric constructions, and an application of the result. Inscribing a triangle in a triangle Lesson Activity This problem introduces the circumcenter of a triangle and shows how it can be used to inscribe the triangle in a circle. It also shows that there cannot be more than one circumcenter. Note that while the construction of the circumcenter (using, e.g., compass and straightedge or dynamic geometry software) was not the focus of the task, the method for doing so is easily seen in the second half of the solution. This task could be used as a lead-in to an exercise performing such a construction. Inscribing a circle in a triangle I Lesson Activity Page 92 of 207

93 This task shows how to inscribe a circle in a triangle using angle bisectors. Inscribing a circle in a triangle II Lesson Activity This task focuses on a remarkable fact which comes out of the construction of the inscribed circle in a triangle: the angle bisectors of the three angles of triangle ABC all meet in a point. Circles in Triangles Performance Assessment Task (MARS) %20Circles%20in%20Triangles.pdf This task challenges a student to use geometric properties of circles and triangles to prove that two triangles are congruent. Reteach/Extension Reteach G CO 9 Use the Writing Linear Equations: Parallel and Perpendicular (included in this Curriculum Guide) activity to review the characteristics of parallel and perpendicular lines, and how to find the parallel and perpendicular equations. Use the Solving Systems of Equations and Finding Equations of Parallel and Perpendicular Lines (included in this Curriculum Guide) activity to review solving systems of equations and writing equations for parallel and perpendicular lines. G CO 11 Use the Polygon Concept Map (included in this Curriculum Guide) activity to help students categorize and organize the properties of different quadrilaterals. Extension G CO 11 Page 93 of 207

94 Use the Coordinate Geometry: Is the Given Quadrilateral a Parallelogram? (included in this Curriculum Guide) activity to prove if the given quadrilateral graphed on the coordinate plane is a parallelogram, but extend that proof to determine if the parallelogram is a rectangle. Textbook References Textbook: Pearson Integrated High School Common Core Mathematics I (Volume 2) Prior to G CO 9, 10, 11: Chapter 10 Reasoning and Proof (pp. 566) Lesson 2 Patterns and Inductive Reasoning (pp ) 3 Conditional Statements (pp ) 4 Biconditionals and Definitions (pp ) 5 Deductive Reasoning (pp ) 6 Reasoning in Algebra and Geometry (pp ) G CO 9: Chapter 10 Reasoning and Proof (pp. 566) Lesson 7 Proving Angles Congruent (pp ) Chapter 11 Proving Theorems About Lines and Angles (pp. 630) Lesson 2 Properties of Parallel Lines (pp ) 3 Proving Lines Parallel (pp ) 4 Parallel and Perpendicular Lines (pp ) 5 Parallel Lines and Triangles (pp ) Chapter 13 Proving Theorems About Triangles (pp. 766) Lesson 2 Perpendicular and Angle Bisectors (pp ) Chapter 14 Proving Theorems About Quadrilaterals (pp. 834) Lesson 1 The Polygon Angle-Sum Theorems (pp ) 6 Trapezoids and Kites (pp ) G CO 10: Chapter 11 Proving Theorems About Lines and Angles (pp. 630) Lesson 5 Parallel Lines and Triangles (pp ) Chapter 12 Congruent Triangles (pp. 688) Lesson 5 Isosceles and Equilateral Triangles (pp ) 6 Congruence in Right Triangles (pp ) Page 94 of 207

95 7 Congruence in Overlapping Triangles (pp ) Chapter 13 Proving Theorems About Triangles (pp. 766) Lesson 1 Midsegments of Triangles (pp ) 3 Bisectors in Triangles (pp ) 4 Medians and Altitudes (pp ) 5 Indirect Proof (pp ) 6 Inequalities in One Triangle (pp ) 7 Inequalities in Two Triangles (pp ) Chapter 14 Proving Theorems About Quadrilaterals (pp. 834) Lesson 7 Applying Coordinate Geometry (pp ) 8 Proofs Using Coordinate Geometry (pp ) G CO 11: Chapter 14 Proving Theorems About Quadrilaterals (pp. 834) Lesson 2 Properties of Parallelograms (pp ) 3 Proving That a Quadrilateral Is a Parallelogram (pp ) 4 Properties of Rhombuses, Rectangles, and Squares (pp ) 5 Conditions for Rhombuses, Rectangles, and Squares (pp ) 7 Applying Coordinate Geometry (pp ) 8 Proofs Using Coordinate Geometry (pp ) G C 3: Chapter 13 Proving Theorems About Triangles (pp. 766) Lesson 3 Bisectors in Triangles (pp ) Page 95 of 207

96 Name Date Period Exploring Parallel Lines Cut by a Transversal Materials: a cut up index card (right triangle shape), ruler, protractor Procedures & Questions: - Use a straight edge to draw line k on your paper. Place your triangle on your paper so leg a lies along line k. Trace along the hypotenuse to draw PQ. Label 1, the acute angle formed by line k and PQ. - Measure 1 using the protractor. Record it. m 1: - Now, slide the triangle to the right along line k to a new position. Trace the hypotenuse to draw RS. Label 2, the acute angle formed by line k and RS. - Measure 2 using the protractor. Record it. m 2 : - How do PQ and RS appear to be related? k P 1 Q R a S b - What do you notice about the relationship between m 1 and m 2? - Locate the angle that is adjacent to 1. Label it 3. - Locate the angle that is adjacent to 2. Label it 4. - How are 1 and 3 related? How are 2 and 4 related? - Find the measure of 3 algebraically. Explain how you found it. Check your measurement using the protractor. - Find the measure of 4. Explain how you found it. Check your measurement using the protractor. - What relationship do you see between m 3 and m 4? - 1 and 2 are corresponding angles and so are 3 and 4. What can you conjecture about corresponding angles? Page 96 of 207

97 Name Date Period Exploring Parallel Lines Cut by a Transversal Answer Key Materials: a cut up index card (right triangle shape), ruler, protractor Procedures & Questions: - Use a straight edge to draw line k on your paper. Place your triangle on your paper so leg a lies along line k. Trace along the hypotenuse to draw PQ. Label 1, the acute angle formed by line k and PQ. - Measure 1 using the protractor. Record it. m 1: varies - Now, slide the triangle to the right along line k to a new position. Trace the hypotenuse to draw RS. Label 2, the acute angle formed by line k and RS. - Measure 2 using the protractor. Record it. m 2 : varies - How do PQ and RS appear to be related? They are parallel. k P Q 1 2 R a S b - What do you notice about the relationship between m 1 and m 2? They are the same measure. - Locate the angle that is adjacent to 1. Label it 3. - Locate the angle that is adjacent to 2. Label it 4. - How are 1 and 3 related? How are 2 and 4 related? They are supplementary. - Find the measure of 3 algebraically. Explain how you found it. Check your measurement using the protractor. Answers will vary. 180 o m 1 = m 3. - Find the measure of 4. Explain how you found it. Check your measurement using the protractor. Answers will vary. 180 o m 2 = m 4. - What relationship do you see between m 3 and m 4? They are congruent. - 1 and 2 are corresponding angles and so are 3 and 4. What can you conjecture about corresponding angles? They are congruent only if the lines are parallel. Page 97 of 207

98 Name Date Period Transversals and Special Angles 1. When a transversal intersects two lines, some special pairs of angles are formed. Special Pairs of Angles Formed by Two Lines and a Transversal A. Angles 2 and 6 are corresponding angles. Name 3 other pairs of corresponding angles.,, and B. Angles 3 and 6 are alternate interior angles. Name another pair of alternate interior angles Page 98 of 207

99 C. Angles 2 and 7 are alternate exterior angles. Name another pair of alternate exterior angles In your own words, describe what is meant by each of the following: A. Corresponding Angles B. Alternate Interior Angles C. Alternate Exterior Angles Page 99 of 207

100 3. Identify each given pair of angles as alternate interior, corresponding, alternate exterior, or none of these A. 1 and 8 B. 3 and 7 C. 4 and 5 D. 2 and 8 E. 3 and 5 Page 100 of 207

101 Name Date Period Transversals and Special Angles Answer Key 1. A. s 4 and 8, s 1 and 5, and s 3 and 7 are corresponding angles. B. s 4 and 5 are alternate interior angles. C. s 1 and 8 are alternate exterior angles. 2. A. Corresponding Angles When a transversal intersects two lines, corresponding angles are on the same side of the transversal and on the same side of the given lines. B. Alternate Interior Angles When a transversal intersects two lines, alternate interior angles are on opposite sides of the transversal and on the inside of the given lines. C. Alternate Exterior Angles When a transversal intersects two lines, alternate exterior angles are on opposite sides of the transversal and on the outside of the given lines. 3. A. alternate exterior angles B. corresponding angles C. alternate interior angles D. none of these E. none of these Page 101 of 207

102 Name Date Period Directions: Proof Cards Activity Cut cards apart. (Note: The statements and reasons are given in a logical sequence. The master can serve as the answer key.) Divide the class into small groups. Give each group a set of cards. Each set of cards has the problem, statements, and reasons that complete a two-column proof. (Note: There may more than one viable sequence of statements and reasons.) Have each group arrange the cards so that they are in a logical sequence from the Given to the Prove. A transparency set can be made, cut apart, and used to facilitate class discussion. Groups can use the transparency set to show their sequencing. A transparency of the master may also be useful. Note: Several sets are included. It may be helpful to make each set a different color so that the statements and reasons for one set do not get mixed up with those of another. Page 102 of 207

103 Proof Cards (Set 1) Given: a b l1 l2, 1 2 Given 1 3 Corresponding Angles Postulate 2 3 Transitive Property of Congruence a b Corresponding Angles Converse Page 103 of 207

104 Proof Cards (Set 2) Given: a b l1 l2, 1 2 Given 1 3 Corresponding Angles Postulate 3 2 Transitive Property of Congruence a b Corresponding Angles Converse Page 104 of 207

105 Proof Cards (Set 3) Given: l m 2 1 n p l m, 1 4 Given m1 + m2 = 180 Linear Pair Postulate 2 3 Alternate Interior Angles Theorem m2 = m3 Definition of Congruence m1 + m3 = 180 Substitution Property of Equality m1 = m4 Definition of Congruence m4 + m3 = 180 Substitution Property of Equality n p Consecutive Interior Angles Converse Page 105 of 207

106 Proof Cards (Set 4) Given: l 3 m n p Given l m Corresponding Angles Converse 4 5 Alternate Interior Angles Theorem 3 5 Transitive Property of Congruence n p Corresponding Angles Converse Page 106 of 207

107 Proof Cards (Set 5) Given: m1 = m3 = b c m1 = 50 m3 = 130 Given m1 + m2 = 180 Linear Pair Postulate 50 + m2 = 180 Substitution Property of Equality m2 = 130 Subtraction Property of Equality m2 = m3 Transitive Property of Equality 2 3 Definition of Congruence b c Alternate Exterior Angles Converse Page 107 of 207

108 Proof Cards (Set 6) a b Given: 1 and 2 are supplementary angles l1 l2, 1 and 2 are supplementary angles. Given m1 + m2 = 180 Definition of Supplementary Angles 1 3 Alternate Interior Angles Theorem m1 = m3 Definition of Congruence m3 + m2 = 180 Substitution Property of Equality a b Consecutive Interior Angles Converse Page 108 of 207

109 Name Date Period Proving Theorems Directions: For each of the following theorems, draw a picture, state the given, state what is to be proven, and write a two-column proof. 1. If two parallel lines are cut by a transversal, then alternate interior angles are congruent. 2. If two parallel lines are cut by a transversal, then alternate exterior angles are congruent. Page 109 of 207

110 3. If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the second. 4. If two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel. Page 110 of 207

111 5. If two lines are cut by a transversal so that consecutive interior angles are supplementary, then the lines are parallel. 6. If two lines are cut by a transversal so that alternate exterior angles are congruent, then the lines are parallel. Page 111 of 207

112 Name Date Period Proving Theorems Answer Key Note: Answers will vary based on student s illustration. 1. Given: a b p is a transversal of a and b Prove: p b a Statement Reason 1. a b 1. Given 2. p is a transversal of a and b 2. Given Corresponding Angles Postulate Corresponding Angles Postulate Vertical Angles Theorem Vertical Angles Theorem Transitive Property of Congruence Transitive Property of Congruence Page 112 of 207

113 2. Given: a b p is a transversal of a and b Prove: p b a Statement Reason 1. a b 1. Given 2. p is a transversal of a and b 2. Given Corresponding Angles Postulate Corresponding Angles Postulate Vertical Angles Theorem Vertical Angles Theorem Transitive Property of Congruence Transitive Property of Congruence Page 113 of 207

114 3. Given: m l n is a transversal of m and l n l Prove: n m n 1 l 3 m Statement Reason 1. m l 2. n l 3. n is a transversal of l and m 4. 1 is a right angle 5. m 1 = m 1 = m 3 8. m 3 = is a right angle 10. n m 1. Given 2. Given 3. Given 4. Definition of Perpendicular 5. Definition of right angles 6. Corresponding Angles Postulate 7. Definition of congruence 8. Transitive Property of Equality 9. Definition of right angle 10. Definition of perpendicular. Page 114 of 207

115 4. Given: 2 7 p is a transversal of a and b Prove: a b p b a Statements p is a transversal of a and b Reasons 1. Given 2. Given Vertical Angles Theorem Transitive Property of Congruence 5. a b 5. If corresponding s are, lines are. Page 115 of 207

116 5. Given: 2 and 3 are supplementary p is a transversal of a and b Prove: a b p Statements Reasons 1. 2 and 3 are supplementary 1. Given 2. p is a transversal of a and b 2. Given 3. 1 and 2 are supplementary 3. Supplement Theorem Supplements of the same are. 5. a b 5. If corresponding s are, lines are. Page 116 of 207

117 6. Given: 1 8 p is a transversal of a and b Prove: a b Statement Reason Given 2. p is a transversal of a and b 2. Given Vertical Angles Theorem Transitive Property of Congruence 5. a b 5. If corresponding s are, lines are. Page 117 of 207

118 G CO Name Date Period How Far Up the Hill, Do We Drill? After seeing a graphed cross-section of a hill (see picture below), you find that there is a diamond at point (12,1). The hill has a relatively constant slope and crosses the point (14,7). The drill you will use to excavate the diamond drills perpendicular to the ground it rests on. How far up the hill will you have to push the drill? Slope of hill: Slope of perpendicular to hill: Equation of perpendicular line: Page 118 of 207

119 Equation of line perpendicular that passes through diamond: Point of intersection: How far must you push the drill up the hill? Page 119 of 207

120 G CO 9 Name Date Period How Far Up the Hill, Do We Drill? Answer Key After seeing a graphed cross-section of a hill (see picture below), you find that there is a diamond at point (12,1). The hill has a relatively constant slope and crosses the point (14,7). The drill you will use to excavate the diamond drills perpendicular to the ground it rests on. How far up the hill will you have to push the drill? Slope of hill: 7 = = 0.5 Slope of perpendicular to hill: -2 Equation of perpendicular line: y = -2x +? Equation of line perpendicular that passes through diamond: y 1 = -2(x 12) y 1 = -2x + 24 y = -2x + 25 Page 120 of 207

121 Point of intersection: y = -2x x = -2 x + 25 x = 10 and y = y = x x = 25 Point of intersection: (10, 5) 2 x = 10 How far must you push the drill up the hill? = d = d = d 2 d = units Page 121 of 207

122 G CO 9 Name Date Period Writing and Solving Linear Equations in Geometric Situations Parallel Lines t m n 1. Given: Parallel lines m and n cut by transversal t. m3 = (3x + 30) and m6 = (7x 6) Write and solve a linear equation that can be used to determine the measures of the following angles. A. m1 = E. m2 = B. m3 = F. m4 = C. m5 = G. m6 = D. m7 = H. m8 = Page 122 of 207

123 2. Given: Parallel lines m and n cut by transversal t. m2 = x. Determine the measures of the following angles. A. m1 = E. m2 = B. m3 = F. m4 = C. m5 = G. m6 = D. m7 = H. m8 = Page 123 of 207

124 G CO 9 Name Date Period Writing and Solving Linear Equations in Geometric Situations Parallel Lines Answer Key t m n 1. Given: Parallel lines m and n cut by transversal t. m3 = (3x + 30) and m6 = (7x 6). Write and solve a linear equation that can be used to determine the measures of the following angles. m3 = m6 (Alternate Interior Angles) 3x + 30 = 7x 6 m3 = 3(9) = 4x 6 = = 4x = 57 4x = 36 m6 = 7(9) - 6 x = 9 m6 = 57 Page 124 of 207

125 A. m1 = 123 E. m2 = 57 B. m3 = 57 F. m4 = 123 C. m5 = 123 G. m6 = 57 D. m7 = 57 H. m8 = Given: Parallel lines m and n cut by transversal t. m2 = x. Determine the measures of the following angles. A. m1 = (180 x) E. m2 = x B. m3 = x F. m4 = (180 x) C. m5 = (180 x) G. m6 = x D. m7 = x H. m8 = (180 x) Page 125 of 207

126 G-CO 9 Name Date Period Memory Cards Match-Up Form groups of 2-3 students. Give each group a set of cards (cut apart). There are three cards that go together for each term. Each set has 1 card with the term, 1 card with a figure, and 1 card with an algebraic statement(s). Have students turn all cards face down and mix them up. Students should take turns, choosing three cards at a time. They must match all three to make a set. The student with the most sets at the end wins the game. It may be beneficial to play more than once. If students have trouble making matches, it may be helpful to allow them to make matches with all the cards facing up. This will familiarize them with the cards so that the game may go more smoothly. Page 126 of 207

127 Memory Cards oblique lines m1 + m2 = 180 mx + my = 180 alternate interior angles parallel lines 1 2 consecutive interior angles Page 127 of 207

128 Memory Cards a b x y x y y3x 2 4 y= x linear pair corresponding angles Page 128 of 207

129 Memory Cards mp = mq 1 2 a b vertical angles perpendicular lines alternate exterior angles y = 3x 4 y = 1 + 3x Page 129 of 207

130 Memory Cards y = 4x 6 y = -4x x y Page 130 of 207

131 G CO 9 Name Date Period Memory Cards Match-Up Answer Key vertical angle a b a b parallel lines y = 3x 4 y = 1 + 3x alternate exterior angles p q mp = mq consecutive interior angles 1 2 m1 + m2 = 180 corresponding angles linear pair x y mx + my = 180 perpendicular lines y = y = 3 4 x x 2 alternate interior angles x y x y oblique lines y = 4x 6 y = -4x + 3 Page 131 of 207

132 G CO 9 Name Date Period Writing Linear Equations: Parallel and Perpendicular 1. Complete each of the following statements. A. If two lines are parallel, their slopes are. B. If two lines are perpendicular, their slopes are. 2. Determine the slopes of the lines represented below. Equation of the Line Slope of Given Line Slope of Line to Slope of Line to Given Line Given Line A. y = 3 x B. y = 2 C. 3x + 5y = 9 D. x = 7 E. -4y + 2x 3 = 0 Page 132 of 207

133 3. Determine the equations of the lines with the given characteristics: (Write equations in standard form Ax + By = C.) A. Has a slope of 7 and y-intercept -2. B. Contains the points (1, -2) and (3, 6). C. Contains the point (3, -4) and has slope of 1 2. D. Is parallel to the line represented by y = 3x + 7 and contains the point (0, -9). E. Is perpendicular to the line represented by 2x + 3y = 9 and contains the point (2, 5). F. Is parallel to the line represented by 5x + 2y = 6 and contains the point (6, 4). Page 133 of 207

134 G-CO 9 Name Date Period Writing Linear Equations: Parallel and Perpendicular Answer Key 1. Complete each of the following statements. A. If two lines are parallel, their slopes are equal to each other. B. If two lines are perpendicular, their slopes are negative reciprocals of each other. 2. Determine the slopes of the lines represented below. Equation of the Line Slope of Given Line Slope of Line to Slope of Line to Given Line Given Line A. y = 3 4 x B. y = undefined C. 3x + 5y = D. x = 7 undefined undefined 0 E. -4y + 2x 3 = Page 134 of 207

135 3. Determine the equations of the lines with the given characteristics: (Write equations in standard form Ax + By = C.) A. Has a slope of 7 and y-intercept -2. 7x y = 2 B. Contains the points (1, -2) and (3, 6). 4x y = 6 C. Contains the point (3, -4) and has slope of 1 2. x 2y = 11 D. Is parallel to the line represented by y = 3x + 7 and contains the point (0, -9). 3x y = 9 E. Is perpendicular to the line represented by 2x + 3y = 9 and contains the point (2, 5). 3x 2y = -4 F. Is parallel to the line represented by 5x + 2y = 6 and contains the point (6, 4). 5x + 2y = 38 Page 135 of 207

136 G CO 9 Name Date Period Solving Systems of Equations and Finding Equations of Parallel and Perpendicular Lines Use the algebraic method of your choice to solve the following systems of equations and describe the graphs of the lines in each system of equations. 1. 2x + 3y = 5 3x 4y = x y = 2-2x 2y = 4 3. x y = -5-3x + 3y = 4 4. Find the equation of the line parallel to y = -3x 2 that goes through the point (7,1). Page 136 of 207

137 5. Find the equation of the line parallel to y = 2 x + 1 that goes through (6,4) Find the equation of the line perpendicular to y = 4x + 2 that goes through (12, -2). 7. Find the equation of the line that is perpendicular to y = x that goes through (9, 4). Page 137 of 207

138 G CO 9 Name Date Period Solving Systems of Equations and Finding Equations of Parallel and Perpendicular Lines Answer Key In exercises 1 3, use the algebraic method of your choice to solve the following systems of equations. Describe the graphs of the lines in each system of equations. 1. 2x + 3y = 5 3x 4y = 3 Solution: 29 9, The lines intersect at 29 9, x y = 2-2x 2y = 4 Solution: There are infinitely many solutions. The lines coincide. 3. x y = -5-3x + 3y = 4 Solution: There is no solution. The lines are parallel. 4. Find the equation of the line parallel to y = -3x 2 that goes through the point (7,1). y 1 = -3(x 7) y 1 = -3x + 21 y = -3x Find the equation of the line parallel to y = 2 x + 1 that goes through (6,4). 3 2 y 4 ( x 6) 3 2 y 4 x y x 3 Page 138 of 207

139 6. Find the equation of the line perpendicular to y = 4x + 2 that goes through (12, -2). 1 y 2 ( x12) 4 1 y 2 x y x Find the equation of the line that is perpendicular to y = x that goes through (9, 4). 4 y 4 ( x 9) 3 4 y 4 x y x8 3 Page 139 of 207

140 G CO 10 Name Date Period Investigating Centers of Triangles I. Medians and the Centroid 1. Draw ABC 2. Label the midpoints of each side of ABC so that D is between A and B, E is between B and C, and F is between A and C. 3. Draw segments connecting each midpoint to its opposite vertex. AE, BF, and CD are called medians. 4. What do you notice about these three segments? 5. Label the intersection of the medians G. This intersection point is called the centroid. 6. Fill in the table with your measurements as well as measurements from your classmates. student AE AG GE BF BG GF CD CG GD Page 140 of 207

141 7. Find a relationship between the measurements of each median and its parts. 8. Summarize the relationship of medians and the centroid. 9. Describe the position of the centroid in a triangle. Extension: Find the area of ABG, ACG, and CBG Page 141 of 207

142 II. Perpendicular Bisectors, Circumcenter, and Circumcircle 1. Draw XYZ. 2. Label the midpoints of each side of XYZ so that R is between X and Y, S is between Y and Z, and T is between X and Z. 3. Draw a line through each midpoint that is perpendicular to the side of the triangle. These lines are called perpendicular bisectors. 4. What do you notice about these three lines? 5. Label the intersection of the perpendicular bisectors C. This intersection point is called the circumcenter. 6. Fill in the table with your measurements as well as measurements from your classmates. Student XC YC ZC 7. What do you notice about the measurements from the circumcenter to each vertex of the triangle? Page 142 of 207

143 8. Draw a circle with center C and radius XC. This circle is called the circumcircle. 9. What do you notice about the circumcircle? III. Altitudes and the Orthocenter 1. Draw HIJ. 2. Draw a line through each vertex that is perpendicular to its opposite side. These lines are called altitudes. 3. What do you notice about these three segments? 4. Label the intersection of the altitudes O. This intersection point is called the orthocenter. 5. Drag a vertex of the triangle. Observe the orthocenter as HIJ changes shape. (If you are using paper and pencil, compare each different type of triangle.) 6. Summarize the position of the orthocenter in various triangles. Page 143 of 207

144 IV. Angle Bisectors, Incenter, and Incircle 1. Draw LMO 2. Bisect the angles at each vertex. These lines are called angle bisectors. 3. What do you notice about these three lines? 4. Label the intersection of the angle bisectors I. This intersection point is called the incenter. 5. Draw a line through I that is perpendicular to LO. Label the intersection of this line and LO, N. 7. Draw a circle with center I and radius IN. This circle is called the incircle. 8. What do you notice about the incircle? 8. What can you conclude about the position of the incenter in the triangle? Page 144 of 207

145 G CO 10 Name Date Period Investigating Centers of Triangles Answer Key I. Medians and the Centroid 1. Draw ABC 2. Label the midpoints of each side of ABC so that D is between A and B, E is between B and C, and F is between A and C. 3. Draw segments connecting each midpoint to its opposite vertex. AE, BF, and CD are called medians. 4. What do you notice about these three segments? All three segments intersect at the same point. 5. Label the intersection of the medians G. This intersection point is called the centroid. 6. Fill in the table with your measurements as well as measurements from your classmates. Answers will vary. Student AE AG GE BF BG GF CD CG GD 7. Find a relationship between the measurements of each median and its parts. Some students may only develop one of these relationships, but through class discussion, all students should see all relationships. AE = AG+GE AG=2(GE) AG 2 AE 3 GE 1 AE 3 Page 145 of 207

146 8. Summarize the relationship of medians and the centroid. The centroid is 2 of the distance from the vertex along the median Describe the position of the centroid in a triangle. The centroid is the intersection of the medians. Extension: Find the area of ABG, ACG, and CBG The areas of these three triangles are equal. The centroid is the center of gravity. Page 146 of 207

147 II. Perpendicular Bisectors, Circumcenter, and Circumcircle 1. Draw XYZ. 2. Label the midpoints of each side of XYZ so that R is between X and Y, S is between Y and Z, and T is between X and Z. 3. Draw a line through each midpoint that is perpendicular to the side of the triangle. These lines are called perpendicular bisectors. 4. What do you notice about these three lines? These three lines intersect at the same point. 5. Label the intersection of the perpendicular bisectors C. This intersection point is called the circumcenter. 6. Fill in the table with your measurements as well as measurements from your classmates. Answers may vary. Student XC YC ZC 7. What do you notice about the measurements from the circumcenter to each vertex of the triangle? For each triangle, these three measurements are equal. 8. Draw a circle with center C and radius XC. This circle is called the circumcircle. 9. What do you notice about the circumcircle? The circumcircle goes through each vertex of the triangle. Page 147 of 207

148 Triangles C III. Altitudes and the Orthocenter 1. Draw HIJ 2. Draw a line through each vertex that is perpendicular to its opposite side. These lines are called altitudes. 3. What do you notice about these three lines? These three lines all intersect at the same point. 4. Label the intersection of the altitudes O. This intersection point is called the orthocenter. 5. Drag a vertex of the triangle. Observe the orthocenter as HIJ changes shape. (If you are using paper and pencil, compare each different type of triangle.) 6. Summarize the position of the orthocenter in various triangles. When the triangle is acute, the orthocenter is inside the triangle. When the triangle is a right triangle, the orthocenter is on the hypotenuse. When the triangle is obtuse, the orthocenter lies outside the triangle. IV. Angle Bisectors, Incenter, and Incircle 1. Draw LMO. 2. Bisect the angles at each vertex. These lines are called angle bisectors. 3. What do you notice about these three lines? These three lines all intersect at the same point. 4. Label the intersection of the angle bisectors I. This intersection point is called the incenter. Page 148 of 207

149 5. Draw a line through I that is perpendicular to LO. Label the intersection of this line and LO, N. 6. Draw a circle with center I and radius IN. This circle is called the incircle. 7. What do you notice about the incircle? The incircle is inscribed in the triangle because it intersects each side of the triangle in exactly one point. 8. What can you conclude about the position of the incenter in the triangle? The incenter is equidistant from the sides of the triangle. Page 149 of 207

150 G CO 10 Name Date Period Centers of Triangles 1. Is it possible for an altitude to be the same line as a perpendicular bisector? Explain. 2. Is it possible for an angle bisector to be the same line as a median? Explain. 3. Is it possible for an incenter to be the same point as an orthocenter? Explain. 4. Is it possible for a centroid to be the same point as a circumcenter? Explain. Page 150 of 207

151 5. Which centers of triangles can lie outside of the triangle? 6. A new power plant is going to be built servicing three nearby towns. Where should the power plant be built in order for it to be the same distance from each city and how is this point located? (The towns are not collinear.) 7. You are the owner of a triangular piece of land that you would like to sell. In order to increase your profit, you decide to sell it off in three equal parts. Each new piece of land must be a triangle with one of the original boundary lines as a side. Where should you draw the new boundary lines for the three new pieces of land? 8. Jack has four new lion cubs at his zoo that are playing very well together in their acute triangular pen, but there is one problem. They are not getting equal nutritional value because some of the lions are stealing the other lions food. Jack decides to build a fence that touches each side of the pen in only one place and separates it into four sections to be used only during feedings. How can he minimize the amount of fence to be used? 9. Bruce is organizing a race on a triangular island. He created a starting point in the interior of the island and three routes of equal distance. Each route leads to the water on a different side of the island and returns to the starting line. How did Bruce create these three courses of equal distance in order to be fair? Where must he have put the starting point? Page 151 of 207

152 G CO 10 Name Date Period Centers of Triangles Answer Key 1. Is it possible for an altitude to be the same line as a perpendicular bisector? Explain. Yes, if they are drawn in an isosceles triangle. If the altitude is drawn from the angle included by the legs so that it is perpendicular to the base and the perpendicular bisector is drawn to the base. 2. Is it possible for an angle bisector to be the same line as a median? Explain. Yes, if they are drawn in an isosceles triangle. The angle bisector must bisect the angle included by the legs and the median must connect the midpoint of the base to the vertex of the angle included by the legs. 3. Is it possible for an incenter to be the same point as an orthocenter? Explain. Yes, if the triangle is equilateral. 4. Is it possible for a centroid to be the same point as a circumcenter? Explain. Yes, if the triangle is equilateral. 5. Which centers of triangles can lie outside of the triangle? The orthocenter and the circumcenter. 6. A new power plant is going to be built servicing three nearby towns. Where should the power plant be built in order for it to be the same distance from each city and how is this point located? (The towns are not collinear.) The power plant should be built at the circumcenter because it is equidistant from the vertices of a triangle. In order to find this point, one must find the intersection of the perpendicular bisectors. Page 152 of 207

153 7. You are the owner of a triangular piece of land that you would like to sell. In order to increase your profit, you decide to sell it off in three equal parts. Each new piece of land must be a triangle with one of the original boundary lines as a side. Where should you draw the new boundary lines for the three new pieces of land? The boundary lines should be drawn from each vertex to the centroid. The centroid divides a triangle into equal areas and is found by the intersection of the medians. 8. Jack has four new lion cubs at his zoo that are playing very well together in their acute triangular pen, but there is one problem. They are not getting equal nutritional value because some of the lions are stealing the other lions food. Jack decides to build a fence that touches each side of the pen in only one place and separates it into four sections to be used only during feedings. How can he minimize the amount of fence to be used? This exercise is difficult to investigate without the use of technology. The new fence should connect the three points where the altitudes intersect the side of the triangle. 9. Bruce is organizing a race on a triangular island. He created a starting point in the interior of the island and three routes of equal distance. Each route leads to the water on a different side of the island and returns to the starting line. How did Bruce create these three courses of equal distance in order to be fair? Where must he have put the starting point? The starting point should be at the incenter and the route should be drawn perpendicular to each side of the island. Page 153 of 207

154 G CO 10 Name Date Period Writing and Solving Systems of Linear Equations in Geometric Situations: Centroids of a Triangle The centroid of a triangle is the point where the three medians of a triangle meet. The centroid of a triangle divides each median into two parts so that the distance from the centroid to the vertex is twice the distance from the centroid to the midpoint. The centroid is a very useful point of concurrency, especially in physics. If you had a thin, solid object of uniform density in the shape of a triangle, the centroid would be the center of mass of the triangle. This is sometimes called the center of gravity. You could balance the triangle at the centroid. 1. Graph triangle ABC with coordinates A(-2,-3), B(2,4) and C(6,-1) in the grid below. Page 154 of 207

155 2. Construct the midpoint of each side. A. Label the midpoint of side AB: X B. Label the midpoint of side BC: Y C. Label the midpoint of side AC: Z 3. Use the Midpoint Formula to find the coordinates of the midpoint of each side of the triangle. A. The coordinates of the midpoint of side AB are. B. The coordinates of the midpoint of side BC are. C. The coordinates of the midpoint of side AC are. 4. Construct the three medians in each triangle. 5. Locate the centroid and estimate its coordinates. 6. Determine the equations of the following lines: A. The line containing the median drawn from vertex A (Line AY) B. The line containing the median drawn from vertex B (Line BZ) C. The line containing the median drawn from vertex C (Line CX) 7. Use a system of equations to determine where the following medians intersect: A. Line Segment AY and Line Segment BZ B. Line Segment BZ and Line Segment CX C. Line Segment AY and Line Segment BZ 8. Based on the information gathered in #7, where do the three medians of triangle ABC intersect? What is the name of this point? Point of Intersection: Name of Point of Intersection: Page 155 of 207

156 9. How do the coordinates of the centroid determined algebraically in #s 7 and 8 above compare to the estimation made in # 5 above? Page 156 of 207

157 Name Date Period Writing and Solving Systems of Linear Equations in Geometric Situations: Centroids of a Triangle Answer Key The centroid of a triangle is the point where the three medians of a triangle meet. The centroid of a triangle divides each median into two parts so that the distance from the centroid to the vertex is twice the distance from the centroid to the midpoint. The centroid is a very useful point of concurrency, especially in physics. If you had a thin, solid object of uniform density in the shape of a triangle, the centroid would be the center of mass of the triangle. This is sometimes called the center of gravity. You could balance the triangle at the centroid. 1. Graph triangle ABC with coordinates A(-2,-3), B(2,4) and C(6,-1) in the grid below. G CO Construct the midpoint of each side. A. Label the midpoint of side AB: X B. Label the midpoint of side BC: Y C. Label the midpoint of side AC: Z Page 157 of 207

158 3. Use the Midpoint Formula to find the coordinates of the midpoint of each side of the triangle. A. The coordinates of the midpoint of side AB are B. The coordinates of the midpoint of side BC are 1 0, , 2. C. The coordinates of the midpoint of side AC are (2, -2). 4. Construct the three medians in each triangle. 5. Locate the centroid and estimate its coordinates. Answers will vary. Approximately (2, 0) 6. Determine the equations of the following lines: A. The line containing the median drawn from vertex A (Line AY) 3x 4y = 6 B. The line containing the median drawn from vertex B (Line BZ) x = 2 C. The line containing the median drawn from vertex C (Line CX) x + 4y = 2 7. Use a system of equations to determine where the following medians intersect: A. Line Segment AY and Line Segment BZ (2,0) B. Line Segment BZ and Line Segment CX (2,0) C. Line Segment AY and Line Segment BZ (2,0) 8. Based on the information gathered in #7, where do the three medians of triangle ABC intersect? What is the name of this point? Point of Intersection: (2,0) Name of the Point of Intersection: centroid Page 158 of 207

159 9. How do the coordinates of the centroid determined algebraically in #s 7 and 8 above compare to the estimation made in # 5 above? Ideally, the coordinates of the centroid obtained algebraically should be the same as the estimated coordinates of the centroid. Page 159 of 207

160 G CO 11 Name Date Period Coordinate Geometry: Is the Given Quadrilateral a Parallelogram? Directions: Use the coordinates of the vertices of the quadrilateral to prove that it is a parallelogram. Write your justification below. y 9 A 8 7 B D C -7-8 x Page 160 of 207

161 G CO 11 Name Date Period Coordinate Geometry: Is the Given Quadrilateral a Parallelogram? Answer Key A quadrilateral is a parallelogram if one pair of opposite sides is both parallel and congruent. Use the slope formula to determine that one pair of opposite sides is parallel. The slope of AB. 2 ( 3) 5 The slope of CD 4 ( 1) Use the Distance Formula to determine if those same sides are congruent. The length of The length of AB 2 ( 3) (8 5) CD ( 1) ( 5) ( 3) The given quadrilateral has one pair of opposite sides that are parallel (slopes are equal) and congruent (have equal lengths), therefore the quadrilateral is a parallelogram. OR Use the slope formula to determine that one pair of opposite sides is parallel. The slope of 5 ( 4) 9 9 BC The slope of 8 ( 1) 9 9 AD Use the Distance Formula to determine if those same sides are congruent. The length of The length of 2 BC ( 3 2) 5 ( 4) ( 5) AD ( 1) ( 5) (9) The given quadrilateral has one pair of opposite sides that are parallel (slopes are equal) and congruent (have equal lengths), therefore the quadrilateral is a parallelogram. Page 161 of 207

162 G CO 11 Name Date Period TI-92/Cabri Exploration: Properties of Parallelograms Explorations Observations Opposites Sides 1. Draw a line near the bottom of your screen (F2,4) 2. Construct a line parallel to the first (F4,2) 3. Construct a transversal towards the left-hand side of the screen that intersects your two parallel lines (F2, 4) at a slant. 4. Construct another transversal on the right hand side of the screen that is parallel to the 1 st transversal. (F4, 2) 5. Make sure you have intersection points at the 4 vertices of your new quadrilateral. (F2, 3) 6. Measure 1 pair of opposite sides. (Measure the length of each side and compare the two.) (F6, 1) 7. Now do the same thing with the other pair of opposite sides and record your results. (F6, 1) From the observations above (not from your textbook), make a conjecture for the wording of the theorem that discusses opposite sides of a parallelogram. Page 162 of 207

163 Explorations Observations 1. Clear Screen (F8,8) Opposite angles 2. Follow steps 1-5 from exploration #1 and create another parallelogram. 3. Find the measure of the angles in the upper right hand corner and lower left-hand corner and compare. (F6, 3) 4. Find the measure of the angles in the upper left-hand corner and lower right hand corner and compare. (F6, 3) Consecutive angles (Use above picture for the observation) 1. Add the measures of the angles in the upper left-hand corner and the upper right hand corner and record your results. (F6,6) 2. Add the measures of the angles in the lower right hand corner and lower left-hand corner and record your results. (F6,6) 3. Continue this process until you have measured the four pairs of consecutive angles inside the parallelogram. (F6,6) From the observations above, write a conjecture for the wording of the part of the theorem that discusses opposite angles and consecutive angles of a parallelogram. Page 163 of 207

164 Explorations Observations Diagonals of a Parallelogram 1. Clear screen (F8,8) 2. Follow steps 1-5 from observation #1 to create another parallelogram. 3. Construct both diagonals (F2, 5) 4. Show the intersection point of the two diagonals. (F2, 3) 5. Measure all 4 segments formed by the intersection of the two diagonals. (F6,1) 6. Make observations From the observations above, write a conjecture for the wording of the part of the theorem that discusses diagonals of a parallelogram. Class Discussion Page 164 of 207

165 G CO 11 Name Date Period Polygon Concept Map Page 165 of 207

166 G CO 11 Name Date Period Polygon Concept Map Answer Key Polygon Triangle Octagon Pentagon Hexagon Quadrilateral Kite no parallel sides Parallelogram two pair of parallel sides 1. Opposite sides are congruent. 2. Opposite angles are congruent. 3. Consecutive angles are supplementary. Trapezoid one pair of parallel sides 1. Midsegment is half the sum of the bases. 1. Diagonals are perpendicular. 2. Exactly one pair of opposite angles are congruent. Rhombus-4 congruent sides Rectangle 4 congruent angles Isosceles Trapezoid nonparallel sides are congruent 1. Diagonals are perpendicular. 2. Diagonals bisect vertices. 1. Diagonals are congruent. 1. Diagonals are congruent. 2. Base angles are congruent. Square Page 166 of 207

167 G CO 3 Name Date Period Circles: Expanding Our Vocabulary 1. Each of the following is a radius of circle O. Each of the following is not a : radius of circle O: CD, MN, and XY PO, QO, and RO P O Q Y R X Use your observations to write a definition of a radius. Draw and label a radius in the circle provided. M C O D N A radius of a circle is 2. Each of the following is a diameter of circle O Each of the following is not a RS, XY, and AB diameter of circle O: R B C D CD, PO, and FG X A O S Y Use your observations to write a definition of a diameter. Draw and label a diameter in the circle provided. A diameter of a circle is P O F G Page 167 of 207

168 3. Each of the following is a chordeach of the following is not a of circle O: chord of circle O: XY, AB, and RS X S CD, OP, and VW C D P A R O Y B V O W Use your observations to write a definition of a chord. Draw and label a chord in the circle provided. A chord of a circle is 4. Each of the following is a secanteach of the following is not a of circle O: secant of circle O: AB, CD, and XY PQ, MN, and VW C A Q V Y P N W O B M O X D Use your observations to write a definition of a secant. Draw and label a secant in the circle provided. A secant of a circle is Page 168 of 207

169 5. Each of the following is a tangent Each of the following is not a of circle O: tangent of circle O: A B R O X S Y AB, XY, and RS P C N PQ, CD, NO, FG and VW W V Q G O F D Use your observations to write a definition of a tangent. Draw and label a tangent in the circle provided. A tangent of a circle is 6. Each of the following is an Each of the following is not an inscribed angle of circle O: inscribed angle of circle O: ABC, RST, and XYZ JKL, NOP, and FGH R X Z B T S O Y A C J L K N O P G H F Use your observations to write a definition of an inscribed angle. Draw and label an inscribed angle in the circle provided. An inscribed angle of a circle is Page 169 of 207

170 7. Each of the following is a central Each of the following is not a angle of circle O: central angle of circle O: R X Y S O A B ROS, AOB, and XOY CDE, UVW, and FGH D U V C O E F W G H Use your observations to write a definition of a central angle. Draw and label a central angle in the circle provided. A central angle of a circle is Page 170 of 207

171 G C 3 Name Date Period Circles: Expanding Our Vocabulary Answer Key Use your observations to write a definition of a radius. Draw and label a radius in the circle provided. A radius of a circle is a segment that has the center as one endpoint and a point on the circle as the other endpoint. radius Use your observations to write a definition of a diameter. Draw and label a diameter in the circle provided. A diameter of a circle is a chord (segment whose endpoints are on the circle) that passes through the center. diameter Use your observations to write a definition of a chord. Draw and label a chord in the circle provided. A chord of a circle is a segment whose endpoints are on the circle chord Use your observations to write a definition of a secant. Draw and label a secant in the circle provided. secant A secant of a circle is a line that intersects the circle in two points. Page 171 of 207

172 Use your observations to write a definition of a tangent. Draw and label a tangent in the circle provided. tangent A tangent of a circle is a line that intersects the circle at exactly one point. Use your observations to write a definition of an inscribed angle. Draw and label an inscribed angle in the circle provided. An inscribed angle of a circle is an angle whose vertex is on the circle and its sides are chords of the circle. inscribed angle Use your observations to write a definition of a central angle. Draw and label a central angle in the circle provided. A central angle of a circle is an angle whose vertex is the center of the circle. central angle Page 172 of 207

173 G C 3 Name Date Period Discovery Lesson: Properties of Circles Lesson 1(Part 1) O Step 1: In circle O above, use your compass to construct two congruent chords. Label the chords AB and CD. Step 2: Construct radii OA, OB, OC, and OD. Step 3: With your protractor, measure BOA and COD. Compare your results with the results of others in your group. Make a conjecture based on your observations. Conjecture: If two chords in a circle are congruent, then they determine two central angles that are. Page 173 of 207

174 Discovery Lesson: Properties of Circles Lesson 1(Part 2) The measure of a minor arc is defined as the measure of its central angle. Therefore, if two central angles are congruent, their intercepted arcs are congruent. The conjecture made in Lesson 1 (Part 1) and the definition of arc measure lead directly to the following conjecture. Conjecture: If two chords in a circle are congruent, then their are congruent. Page 174 of 207

175 G C 3 Name Date Period Discovery Lesson:Properties of Circles Lesson 2 (Part 1) O Step 1:In circle O above, use your compass to construct two nonparallel congruent chords that are not diameters. Step 2: Construct the perpendiculars from the center of the circle to each chord. Think About It: How does a perpendicular from the center of a circle divide a chord? Compare your results with the results of others in your group. Make a conjecture based on your observations. Conjecture: The perpendicular from the center of a circle to a chord is the of the chord. Page 175 of 207

176 Discovery Lesson:Properties of Circles Lesson 2 (Part 2) Step 1: Refer back to the constructions you made in Lesson 2 (Part 1). With your compass, compare the distances (measured along the perpendiculars) from the center to the chords. Conjecture: Two congruent chords in a circle are from the center of the circle. Page 176 of 207

177 G C 3 Name Date Period Discovery Lesson: Properties of Circles Lesson 3 O Step 1: In circle O above, construct two nonparallel chords that are not diameters. Step 2: Construct the perpendicular bisector of each chord and extend the bisectors until they intersect. Think About It: What is special about the point of intersection? Compare your results with the results of others in your group. Make a conjecture based on your observations. Conjecture: The perpendicular bisector of a chord. Page 177 of 207

178 G C 3 Name Date Period Discovery Lesson: Properties of Circles Lesson 4 O Step 1:In circle O above, use your straightedge to draw a line that appears to touch the circle at only one point. This line is called a tangent of circle O. Label the point T. Construct OT. Step 2: Use your protractor to measure the angle at T. Step 3: In circle O above, use your straightedge to draw a different line that appears to touch the circle at only one point. Label this point W. Construct OW. Step 4: Use your protractor to measure the angle at W. Think about it: How does the measure of T compare with the measure of W? Compare your results with the results of others in your group. Make a conjecture based on your observations. Conjecture: A tangent to a circle is to the radius drawn to the point of tangency. Page 178 of 207

179 G C 3 Name Date Period Discovery Lesson: Properties of Circles Lesson 5 O Step 1: Choose a point outside the circle and label it P. Step 2: Draw two lines through point P that appear to be tangent to the circle. Mark the points where these lines appear to touch the circle and label them X and Y. Step 3: Use your compass to compare PX and PY (tangent segments). Compare your results with the results of others in your group. Make a conjecture based on your observations. Conjecture: Tangent segments to a circle from the same point outside the circle are. Page 179 of 207

180 G C 3 Name Date Period Discovery Lesson: Properties of Circles Lesson 6 D E O Step 1: Measure DOF with your protractor and determine m DF. F mdof = m DF = Step 2: Measure DEF. How does mdef compare with m DF? mdef = The mdef is m DF. Step 3: In the space to the right of circle O above, construct a circle of your own with an inscribed angle and its corresponding central angle. Step 4: What is the measure of the central angle? What is the measure of the intercepted arc? Step 5: What is the measure the inscribed angle? Think About It: How does the measure of the inscribed angle compare with the measure of its intercepted arc? Compare your results with the results of others in your group. Make a conjecture based on your observations. Conjecture: The measure of an inscribed angle in a circle. Page 180 of 207

181 G C 3 Name Date Period Discovery Lesson:Properties of Circles Lesson 7 (Part 1) O Step 1: Select two points on circle O above. Label them R and S. Step 2: Select a point A on the major arc and construct inscribed RAS. Step 3: What is the measure of RAS? Step 4: Select another point B on the major arc and construct inscribed RBS. Step 5: What is the measure of RBS? Think About It: How does the measure of RBS compare with the measure of RAS? Think About It: Do you think you can find an angle inscribed in RAS that is not to RAS? Explain. congruent Step 6: Complete Lesson 7 (Part 2) on the next page. Page 181 of 207

182 G C 3 Name Date Period Discovery Lesson:Properties of Circles Lesson 7 (Part 2) O Step 1: Select two points on circle O above. Label them X and Y. Step 2: Select a point L on the minor arc XY and construct inscribed XLY. Step 3: What is the measure of XLY? Step 4: Select another point M on the minor arc XY and construct inscribed XMY. Step 5: What is the measure of XMY? Think About It: How does the measure of XMY compare with the measure of XLY? Think About It: Do you think you can find an angle inscribed in XY that is not congruent to XLY? Explain. Compare your results with the results of others in your group. Make a conjecture based on your observations. Consider Part 1 also. Conjecture: Inscribed angles that intercept the same arc are. Page 182 of 207

183 G C 3 Name Date Period Discovery Lesson: Properties of Circles Lesson 8 O Step 1: In circle O above, construct a diameter. Step 2: Inscribe three angles in the same semicircle. Step 3: Measure each angle with your protractor. Think About It: How do the angles compare? Compare your results with the results of others in your group. Make a conjecture based on your observations. Conjecture: Angles inscribed in a semicircle are. Page 183 of 207

184 Name Date Period Discovery Lesson: Properties of Circles Lesson 9 G C 3 O Step 1: In circle O above, construct an inscribed quadrilateral. Step 2: Determine the measure of each interior angle (inscribed angle) of the quadrilateral. Write the measure in each angle. Step 3: In the space to the right of circle O above, construct a circle of your own and inscribe a quadrilateral in it. Step 4: Measure each interior angle of the quadrilateral. Write the measure in each angle. Think About It: Is there a special relationship between some pairs of angles in each quadrilateral? Compare your results with the results of others in your group. Make a conjecture based on your observations. Conjecture: The angles of a quadrilateral inscribed in a circle are. Page 184 of 207

185 Name Date Period Discovery Lesson: Properties of Circles Lessons 1-9 Answer Key Please note: Sample constructions are given. Discovery Lesson: Properties of Circles: Chords Lesson 1 (Part 1) G C 3 C A Conjecture: D O B If two chords in a circle are congruent, then they determine two central angles that are congruent. Discovery Lesson: Properties of Circles: Chords Lesson 1 (Part 2) *Students use the figure constructed in Lesson 1 (Part 1) to make the conjecture. Conjecture: If two chords in a circle are congruent, then their intercepted arcs are congruent. Discovery Lesson: Properties of Circles: Chords Lesson 2 (Part 1) O Conjecture: The perpendicular from the center of a circle to a chord is the perpendicular bisector of the chord. Page 185 of 207

186 Discovery Lesson: Properties of Circles: Chords Lesson 2 (Part 2) *Students use the figure constructed in Lesson 2 (Part 1) to make the conjecture. Conjecture: Two congruent chords in a circle are equally distant from the center of the circle. Discovery Lesson: Properties of Circles: Chords Lesson 3 O Conjecture: The perpendicular bisector of a chord passes through the center of the circle. Discovery Lesson: Properties of Circles: Tangents Lesson 4 W O T Conjecture: A tangent to a circle is perpendicular to the radius drawn to the point of tangency. Page 186 of 207

187 Circles A Discovery Lesson: Properties of Circles: Tangents Lesson 5 X O P Conjecture: Y Tangent segments to a circle from the same point outside the circle are congruent. Discovery Lesson: Properties of Circles: Inscribed Angles Lesson 6 Step 1: mdof = 105 m DF = 105 Step 2: mdef = 52.5 The mdef is one-half m DF. Step 4: Answers will vary. Step 5: Answers will vary. X Z O Conjecture: Y The measure of an inscribed angle in a circle is half the measure of the arc it intercepts. Page 187 of 207

188 Circles A Discovery Lesson: Properties of Circles: Inscribed Angles Lesson 7 (Parts 1 and 2) Step 3: Answers will vary. Step 5: Answers will vary. B O R O X L M Conjecture: A S Y Inscribed angles that intercept the same arc are congruent. Discovery Lesson: Properties of Circles: Inscribed Angles Lesson 8 O Conjecture: Angles inscribed in a semicircle are right angles Discovery Lesson: Properties of Circles: Inscribed Quadrilaterals Lesson O Conjecture: 70 The opposite angles of a quadrilateral inscribed in a circle are supplementary. Page 188 of 207

189 G C 3 Name Date Period Parts of a Circle: Round Robin This activity is designed to be done in groups of 4. Each person in the group takes a number from 1 to 4. Students take turns illustrating the parts of a circle that are indicated in the following exercises. Person #1 completes problem #1 and passes it on to person #2. Person #2 checks the work done by person #1 and gives an okay or provides assistance if needed. Person #2 completes problem #2 and passes it on to person #3. Person #3 checks the work done by person #2 and gives an okay or provides assistance if needed. Person #3 can also check the problem done by #1. If he or she disagrees with the response, he or she must communicate with Person #2 (since Person #2 agreed with Person #1) and Person #2 should therefore communicate with Person #1. This process continues until all of the exercises have been completed. 1. Illustrate any radius. Label it AB. 2. Illustrate any diameter. Label it CD. Done by: Done by: Checked by: Checked by: 3. Illustrate any chord. Label it EF. 4. Illustrate any minor arc. Label it GH. Done by: Done by: Checked by: Checked by: 5. Illustrate any major arc. Label it IJK. 6. Illustrate any central angle. Label it LMN. Done by: Done by: Geometric Relationships and Properties G CO 9, 10, 11, G C 3 12/1/13 Checked by: Page 189 of 207 Checked by:

190 7. Illustrate an angle inscribed in a minor arc. Label it OPQ. Done by: 8. Illustrate an angle inscribed in a major arc. Label it RST. Done by: Checked by: Checked by: 9. Illustrate two inscribed angles intersecting the same arc. Label them UVW and UYW. Done by: Checked by: 10. Illustrate an inscribed angle with the center of the circle on the angle. Label it ABC. Done by: 11. Illustrate an inscribed angle with the the center of the circle in the exterior of 12. Illustrate an inscribed angle that is a right angle. Label it GHI. Done by: angle. Label it DEF. Done by: Checked by: 13. Illustrate any tangent. Label it JK. 14. Illustrate any secant. Label it LM. Done by: Done by: Geometric Relationships and Properties Checked G CO by: 9, 10, 11, G C 3 Checked by: 12/1/13 Page 190 of 207

191 15. Illustrate two chords that appear to be parallel. Label them NO and PQ. Done by: 16. Illustrate two chords that have a common endpoint. Label them RS and ST. Done by: Checked by: Checked by: 17. Illustrate a tangent and a chord 18. Illustrate any quadrilateral inscribed in Label intersecting at a point on a circle. them UV and WX. Done by: a circle. Label it ABCD. Done by: Checked by: 19. Illustrate two chords intersecting in the interior of a circle. Label them EF and GH. Done by: 20. Illustrate a tangent and a secant intersecting in the exterior of a circle. Label them IJ and KL. Done by: Checked by: Checked by: 21. Illustrate two tangents intersecting in the exterior of a circle. Label them MN and OP. Done by: 22. Illustrate two secants intersecting in the exterior of a circle. Label them QR and ST. Done by: Columbus Checked City by: Schools Checked by: Page 191 of 207

192 G C 3 Name Date Period Parts of a Circle: Round Robin Answer Key This activity is designed to be done in groups of 4. Each person in the group takes a number from 1 to 4. Students take turns illustrating the parts of a circle that are indicated in the following exercises. Person #1 completes problem #1 and passes it on to person #2. Person #2 checks the work done by person #1 and gives an okay or provides assistance if needed. Person #2 completes problem #2 and passes it on to person #3. Person #3 checks the work done by person #2 and gives an okay or provides assistance if needed. Person #3 can also check the problem done by #1. If he or she disagrees with the response, he or she must communicate with Person #2 (since Person #2 agreed with Person #1) and Person #2 should therefore communicate with Person #1. This process continues until all of the exercises have been completed. 1. Illustrate any radius. Label it AB. B 2. Illustrate any diameter. Label it CD. C A Done by: Checked by: Done by: D Checked by: 3. Illustrate any chord. Label it EF. E Done by: F 4. Illustrate any minor arc. Label it GH. G H Done by: Checked by: Checked by: 5. Illustrate any major arc. Label it IJK. J 6. Illustrate any central angle. Label it LMN. L N I K Done by: Done by: M Checked by: Checked by: Page 192 of 207

193 7. Illustrate an angle inscribed in a minor arc. Label it OPQ. O P Q 9. Illustrate two inscribed angles Done by: Checked by: 8. Illustrate an angle inscribed in a major arc. Label it RST. R Done by: T S Checked by: 10. Illustrate an inscribed angle with the intersecting the same arc. Label them U UVW and W UYW. Done by: Label A center of the circle on the angle. C it ABC. Y V Checked by: B Done by: 11. Illustrate an inscribed angle with the 12. Illustrate an inscribed angle that is a the center of the circle in the exterior of D angle. Label it DEF. F Done by: E G right angle. Label it GHI. H Done by: I Checked by: 13. Illustrate any tangent. Label it JK. J K Done by: 14. Illustrate any secant. Label it LM. L Done by: Checked by: Checked by: Page 193 of 207 M

194 15. Illustrate two chords that appear to be parallel. Label them NO and PQ. N P Done by: 16. Illustrate two chords that have a common endpoint. Label them RS and ST. R Done by: O Q Checked by: S T Checked by: 17. Illustrate a tangent and a chord 18. Illustrate any quadrilateral inscribed in intersecting at a point on a circle. Label them UV and WX. A a circle. Label it ABCD. B Done by: U X V W Done by: C D Checked by: 19. Illustrate two chords intersecting in the interior of a circle. Label them EF and GH. G E F H Done by: Checked by: 21. Illustrate two tangents intersecting 20. Illustrate a tangent and a secant intersecting in the exterior of a circle. Label them IJ and KL. J K Done by: I L Checked by: 22. Illustrate two secants intersecting in the in the exterior of a circle. Label them MN and OP. O N M P Done by: Checked by: exterior of a circle. Label them QR and ST. Q Done by: Checked by: Page 194 of 207 S R T

195 G C 3 Name Date Period Pairs Check: Properties of Circles Given: Circle P with diameter EB, mab 70, mbc 55, mcd 85, and mfa 30. F 30 A E P B D C Work in pairs and take turns to find the measures of the following angles in the order that they are given. (One person in the pair solves the problem, while the second person in the pair functions as coach by observing carefully, giving hints or pointing out errors as needed and giving positive feedback to the solver.) Switch roles for the next problem. Continue this process until all of the problems have been solved. When you have solved all the problems, check your responses with the responses of the other pair within your group to see if you agree. 1. m1 = 2. m2 = 3. m3 = 4. m4 = 5. m5 = 6. m6 = 7. m7 = 8.m8 = 9. m9 = 10. m10 = 11. m11 = 12. m12 = 13. m13 = 14. m14 = 15. m15 = 16. m16 = 17. m17 = 18. m18 = 19. m19 = 20. m20 = 21. m21 = 22. m22 = 23. m23 = 24. m24 = 25. m25 = 26. m26 = 27. m27 = 28. m28 = 29. m29 = 30. m30 = Page 195 of 207

196 Name Date Period Pairs Check: Properties of Circles Answer Key Given: Circle P with diameter EB, mab 70, mbc 55, mcd 85, and mfa 30. E 4 F A P B D C 14 Work in pairs and take turns to find the measures of the following angles in the order that they are given. (One person in the pair solves the problem, while the second person in the pair functions as coach by observing carefully, giving hints or pointing out errors as needed and giving positive feedback to the solver.) Switch roles for the next problem. Continue this process until all of the problems have been solved. When you have solved all the problems, check your responses with the responses of the other pair within your group to see if you agree. 1. m1 = m2 = m3 = m4 = m5 = 85 6.m6 = m7 = m8 = m9 = m10 = m11 = m12 = m13 = m14 = m15 = m16 = m17 = m18 = m19 = m20 = m21 = m22 = m23 = m24 = m25 = m26 = m27 = m28 = m29 = m30 = 40 Page 196 of 207

197 Name Date Period Teacher Instructions Numbered Noggins: Properties of Circles G C 3 This activity can be done in groups of four. Each person in the group is given a number from one to four. Present the following problems one at a time. Give the students the opportunity to discuss each problem within the group and prepare to respond. The teacher then calls upon students by number to represent the group. Make one or two copies of the master below for each group of four. Cut into strips so that you can distribute the problems one at a time. For convenience, make overheads of the problems to facilitate class discussion. Each sheet has a header that includes a place for students to write their names if you would like to assign it to individuals. Page 197 of 207

198 Numbered Noggins: Properties of Circles (Part 1) 1A Determine the value of x. Show your work in the space below. x B 100 x 40 Determine the value of x. Show your work in the space below. 1C 95 Determine the value of x. Show your work in the space below x 1D 75 x y Determine the value of x and y. Show your work in the space below. 5y 5y Page 198 of 207 4y

199 Numbered Noggins: Properties of Circles (Part 2) 2A 6y Determine the value of x and y. Show your work in the space below. x 3y 2B Determine the value of x. Show your work in the space below. 250 x 2C Determine the value of x. Show your work in the space below. x 300 2D 230 Determine the value of x. Show your work in the space below. x Page 199 of 207

200 Numbered Noggins: Properties of Circles (Part 3) 3A x 110 Determine the value of x. Show your work in the space below. 3B Determine the value of x. Show your work in the space below. x 140 3C Determine the value of x. Show your work in the space below. 76 x 3D Determine the value of x. Show your work in the space below. 3x x Page 200 of 207

201 Numbered Noggins: Properties of Circles (Part 4) 4A Determine the value of x. Show your work in the space below. x 4B 3x Determine the value of x. Show your work in the space below. 4x 2x 4C x Determine the value of x and y. Show your work in the space below. 3y 4D 5y 140 Determine the value of x. Show your work in the space below. x Page 201 of 207

202 Numbered Noggins: Properties of Circles (Part 5) 5A 3x Determine the value of x. Show your work in the space below. 95 5B x + 6 Determine the value of x. Show your work in the space below. 90 x 40 5C Determine the value of x. Show your work in the space below. 120 x 130 5D x Determine the value of x. Show your work in the space below. CCSSM 85 I Page 202 of 207

203 Numbered Noggins: Properties of Circles (Part 1) Answer Key 1A SOLUTION 50 1B SOLUTION x x Determine the value of x. Show your work in the space below. x = ( ) = (120) = 60 Determine the value of x. Show your work in the space below. x = = 95 = = C SOLUTION x Determine the value of x. Show your work in the space below. (170 + x) = x = 250 x = 80 1D SOLUTION x y Determine the value of x and y. Show your work in the space below. y + 5y + 4y + 5y = y = 360 y = 24 5y 5y x = (4y y) = (3y) = (3 24) = (72) Geometric Relationships and Properties G CO 9, 10, 11, x = G 36 4y C 3 12/1/13 Page 203 of 207