PAPUA NEW GUINEA Nauru Solomon Islands Tuvalu AUSTRALIA Fiji Islands Samoa Islands Phoenix Islands Gilbert Islands Vanuatu B Howland Island PACIFIC OCEAN CORAL SEA Nikumaroro Island C E F A D 0 625 mi N S W E Lae Name Date Class Situation A, Amelia Earhart 2.2B LEARNING AID © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. Problem 2.2 The Bee Dance and Amelia Earhart: Measuring Angles and Distance 89 SAMPLE
At a Glance The objective of this problem is to develop student understanding and skill in working with the vertical, supplementary, and complementary angles. In the Initial Challenge, students will investigate the relationship among these angles. In the What If . . . ? situations, they will write generalizations and use algebraic expressions about these relationships. PROBLEM 2.3 Vertical, Supplementary, and Complementary Angles Arc of Learning Exploration Analysis Analysis NOW WHAT DO YOU KNOW? Describe what you know about angles formed by intersecting lines. Include parallel lines and the angles formed by a line that intersects the parallel lines in your description. Key Terms Materials parallel vertical angles supplementary angles complementary angles adjacent angles For each student • Learning Aid 2.3A: Initial Challenge • Learning Aid 2.3B: What If . . . ? Situation A • Learning Aid 2.3C: What If . . . ? Situations B and C For the class • Teaching Aid 2.3A: Parallel and Nonparallel Lines • Teaching Aid 2.3B: Complimentary Angles (optional) Pacing 1 day Groups Think, Pair, Share A 13–19 C 26–27 E 32–34 Note: If you have a Grade 7 Classroom Materials Kit, please refer to A Guide to Connected Mathematics® 4 for a detailed list of materials included or items you will need to prepare ahead of time. For more Teacher Moves, refer to the General Pedagogical Strategies and the Attending to Individual Learning Needs Framework in A Guide to Connected Mathematics® 4. Facilitating Discourse Teacher Moves LAUNCH CONNECTING TO PRIOR KNOWLEDGE Use Teaching Aid 2.3: Parallel and Nonparallel Lines to talk about parallel and nonparallel lines. Students were first introduced to parallel lines in elementary school. It will be helpful to briefly review the definitions. PRESENTING THE CHALLENGE Use the introduction in the Student Edition to introduce the setting of parallel lines in our lives. Launch the problem with the suggested question “How do we determine or construct parallel lines?” Language Have a brief conversation with students after they’ve worked on the Initial Challenge. 90 Investigation 2 Designing with Angles SAMPLE
Facilitating Discourse Teacher Moves EXPLORE PROVIDING FOR INDIVIDUAL NEEDS Having students stand up and work on a vertical surface during this problem can increase engagement and collaboration between students as they work out the relationships with these angles. Suggested Questions • Do any of the angles have the same measurement? How do you know? • Could any of the angles help you in finding the measurement of another angle? • Are there any “special” angles that could help you find relationships? • Does measuring the angles help you to see any new relationships • What relationships can help you find the measurements of the angles in What If . . . ? Situation A? • In What If . . . ? Situation C, how did we write and solve equations in Variables and Patterns and Bits of Rational in grade 6? PLANNING FOR THE SUMMARY Listen for how students are making sense of vertical, supplementary, and complementary angles. Look for students who are using the relationships formed by a line that intersects the parallel lines versus actually measuring the angles to lead the discussions in the summary. Going Vertical SUMMARIZE DISCUSSING SOLUTIONS AND STRATEGIES Have students share the relationships they found in the Initial Challenge before they work on the What If . . . ? situations. Suggested Question • What relationships among the vertical angles did you find in the Initial Challenge? Have students share their evidence to prove or disprove their conjectures from their work on the What If . . . ? situations. MAKING THE MATHEMATICS EXPLICIT Suggested Questions • What are some relationships you noticed in the angle measures in What If . . . ? Situation A? • Do you agree with Jamal’s claim in What If . . . ? Situation B? Why or why not? • How did you find the value of x for each figure in What If . . . ? Situation C? The variety of relationships among angles formed by parallel lines is useful to know. Be sure students have those facts figured out (not necessarily by their familiar names of alternate interior, alternate exterior, and corresponding angles). A good way to deal with this in the Summarize might be to present a different sketch other than those offered in the problem and ask students to start from the measure of just one angle to deduce all the other measures. Have them explain how they know the measure of an angle. As you finish the mathematical discussions, have students reflect on the Now What Do You Know? question(s). Compare Thinking Language 2.3 Problem 2.3 Vertical, Supplementary, and Complementary Angles 91 SAMPLE
Problem Overview The objective of this problem is to develop student understanding and skill in working with vertical, supplementary, and complementary angles. Students will investigate the relationship among these angles. They will write generalizations about these relationships. Implementation Note: Have a brief conversation with students after they’ve worked on the Initial Challenge before having them work on the What If . . . ? section. Mathematical Note: This problem will lay the foundation for work in grade 8 in the Flip, Spin, Slide, and Stretch unit, where students formally look at two parallel lines intersected by a transversal and the congruent angles that are created. Launch (Getting Started) Connecting to Prior Knowledge Use Teaching Aid 2.3: Parallel and Nonparallel Lines to talk about parallel and nonparallel lines. Students were first introduced to parallel lines in elementary school. They may have had brief encounters with parallel lines when they classified shapes by sides and angles and when they developed area formulas for parallelograms in grade 6. It will be helpful to briefly review the definitions. (Language) In Teaching Aid 2.3: Parallel and Nonparallel Lines, one pair is parallel, one pair is not parallel but not quite intersecting yet, and the third pair is not parallel. Suggested Question • What are parallel lines? (Parallel lines are lines that never intersect, no matter how far they are extended in the plane.) Plane is a term that students might have heard before, but they do not need to know its formal definition. The term, however, is important to ensure that the definition of parallel lines is mathematically correct. Presenting the Challenge Use the introduction in the Student Edition to introduce the setting of parallel lines in our lives. Launch the problem with the suggested question “How do we determine or construct parallel lines?” EXTENDED LAUNCH—EXPLORE—SUMMARIZE 92 Investigation 2 Designing with Angles SAMPLE
Introduce the new vocabulary words of vertical angles, adjacent angles, supplementary, and complementary using the Student Edition and/or page 2 of Teaching Aid 2.3: Parallel and Nonparallel Lines. Pass out Learning Aid 2.3A: Initial Challenge. Have students begin the work independently and then share their ideas with a partner. Have a brief discussion about this section before having students work on the What If . . . ? with Learning Aid 2.3B: What If . . . ? Situation A and Learning Aid 2.3B: What If . . . ? Situations B and C. Explore (Digging In) Providing for Individual Needs Having students stand up and work on a vertical surface during this problem can increase engagement and collaboration between students as they work out the relationships with these angles. Suggested Questions • Do any of the angles have the same measurement? How do you know? (As students share angles with the same measurement help them to connect measurements to the vocabulary word and the relationships in the angles.) • Could any of the angles help you in finding the measurement of another angle? (As students are looking help them to use today’s vocabulary word to help them find measurements) • Are there any “special” angles that could help you find relationships? (Supplementary, complementary, right angles, etc.) • Does measuring the angles help you to see any new relationships? (Answers will vary. This might be when some see the congruence in vertical angles and/or the supplementary relationships.) • What relationships can help you find the measurements of the angles in What If . . . ? Situation A? (Supplementary angles, vertical angles) • In What If . . . ? Situation C, how did we write equations in Variables and Patterns in grade 6? (Answers will vary. They will most likely discuss writing out generalizations to describe a situation in words and then shortening those words using letters into equations.) • In What If . . . ? Situation C, how did we solve equations in Bits of Rational in grade 6? (Answers will vary. Students don’t have a formal process at this time. This will get developed in Moving Straight Ahead later this year. They will most likely use fact families, guess and check, or “undoing what was done to x.”) LES Extended Lanuch—Explore—Summarize 93 SAMPLE
Planning for the Summary What evidence will you use in the summary to clarify and deepen understanding of the Now What Do You Know? question? What will you do if you do not have evidence? NOW WHAT DO YOU KNOW? Describe what you know about angles formed by intersecting lines. Include parallel lines and the angles formed by a line that intersects the parallel lines in your description. (Listen for how students are making sense of vertical, supplementary, and complementary angles. Look for students who are using the relationships formed by a line that intersects the parallel lines versus actually measuring the angles to lead the discussions in the summary.) Summarize (Orchestrating the Discussion) Discussing Solutions and Strategies Have students share the relationships they found in the Initial Challenge before they work on the What If . . . ? Suggested Question • What relationships among the vertical angles did you find in the Initial Challenge? (Vertical angles are equal to each other.) Have students share their evidence to prove or disprove their conjectures from their work on the What If . . . ? situations. Making the Mathematics Explicit The variety of relationships among angles formed by parallel lines is very useful to know. Be sure students have those facts figured out (not necessarily by their familiar names of alternate interior, alternate exterior, and corresponding angles). A good way to deal with this in the summary might be to present a different sketch other than those offered in the problem and ask students to start from the measure of just one angle to deduce all the other measures. Have them explain how they know the measure of an angle. (Language) Suggested Questions • What are some relationships you noticed in the angle measures in What If . . . ? Situation A? (Focus the conversation around vertical, supplementary, and adjacent angles.) • Do you agree with Jamal’s claim in What If . . . ? Situation B? Why or why not? (Jamal is correct. We can use the relationships between supplementary angles, vertical angles, and adjacent angles to find the opposite angles are the same measures.) LES 94 Investigation 2 Designing with Angles SAMPLE
• How did you find the value of x for each figure in What If . . . ? Situation C? (Answers will vary. Students don’t have a formal structure for solving equations, so they will use informal language. You can script what the students say.). Here is an example of a conversation that occurred in a classroom when discussing What If . . . ? Situation C, Card 4. Teacher: How did you find the value of x for Card 4? Vince: We knew that angle x with angle 5 + x made a straight line so they were supplementary angles, but then we just kept trying to guess what would work to get 180° and we gave up. Teacher: What did you try before giving up? Tommi: I was Vince’s partner. We tried 10° first. So 10° + 5° and then another 10° was only 25°. That was way too low, so we tried 100°. Then 100° + 5° and then another 100° was 205°, which was way too much. Joan: Wait, before you go on, I don’t get why you added your guess to 5° and then added that guess on again. Vince: (Points to the picture displayed on the board.) L1 L2 5 + x x See how if you have 5 + x and you put it with the x, it would be 180°? So every guess we tried we’d first add 5 and then add that guess on it to get the full 180°. Joan: Okay. Tommi: We only tried one more, and that was 50°. That was too low, so we knew it was between 50° and 100°, but we wanted to look at the other ones, so we stopped there. Teacher: Did anyone else use guess and check or another strategy and think they have found the value for x? Nikki: We also said that x + 5 and x had to equal 180° when put together. We wrote this equation: x + 5 + x = 180°. We said LES Extended Lanuch—Explore—Summarize 95 SAMPLE
that meant that really there were two xs and then 5 added on to get 180°. We just kept guessing like Vince and Tommi and got a little closer with 87°. Because 87° + 87° + 5 = 179°. We said it had to be about 87°. Teacher: Anyone do it a different way? Brett: We started like Nikki with x + x + 5 = 180° and said it was two xs and then adding 5 to get 180° just like he said. But we figured it was easier to just get rid of the 5 that was in the way, so we took 5 degrees off the 180°, which gave us 175°. Then since two xs had to add to get 175°, that was like multiplying the x by 2. So, 2 times x had to be 175°. To get it we used a fact family idea and did 175° ÷ 2 = 87.5°. Vince: Does that really work? (He goes up to the board and writes x + 5 + x = 180°.) If Brett is right, you are saying x is 87.5°. So 87.5 + 5 + 87.5 = 180°. (He grabs a calculator to add them.) It is right! The teacher continued with this way of questioning student thinking about finding x for the other two figures. The focus is on thinking about the numbers and operations so that in the Moving Straight Ahead unit, students will have this to draw from when they formalize a structure for solving for x. (Time) As an amusing way to help students remember that complementary angles form a right angle, end the class with Teaching Aid 2.3B: Compliment Angles (optional). Now What Do Students Know? Ask students to reflect on the discussion and answer the Now What Do You Know? questions REFLECTING ON STUDENT LEARNING Use the following questions to assess student understanding at the end of the lesson. • What evidence do I have that students understand the Now What Do You Know? question? • Where did my students get stuck? • What strategies did they use? • What breakthroughs did my students have today? • How will I use this to plan for tomorrow? For the next time I teach this lesson? • Where will I have the opportunity to reinforce these ideas as I continue through this unit? The next unit? LES 96 Investigation 2 Designing with Angles SAMPLE
INITIAL CHALLENGE In the following figure, lines L1 , L2, and L3 are parallel lines that are intersected (cut) by another line. Look at the angles formed where the single line cuts across each parallel line. m n po q t r s w x y z L1 L2 L3 135° 135° 135° 135° 135° 135° 45° 45° 45° 45° 45° 45° • What do you notice about the angles? What do you wonder? • Measure the angles. • Describe any relationships you notice in the angle measures. Do they agree with your prediction? • Are any angles vertical, supplementary, adjacent, or complementary? Is so, name some. Vertical Supplementary Adjacent Complementary m-p m-n p-n m-n n-o o-n m-o p-o n-p n-r q-t m-r p-r p-o n-s s-r m-s p-s o-m n-x w-z m-x p-x q-r n-y y-x m-y p-y r-t o-r q-n t-n t-s o-s q-o t-o s-q o-x q-r t-r w-x o-y q-s t-s x-z r-s q-x t-x z-y r-x q-y t-y y-w r-y w-n z-n s-x Answers Embedded in Student Edition Problems Vertical, Supplementary, and Complementary Angles PROBLEM 2.3 Answers Problem 2.3 Vertical, Supplementary, and Complementary Angles 97 SAMPLE
Vertical Supplementary Adjacent Complementary w-o z-o s-y w-r z-r x-y w-s z-s w-x z-x w-y z-y WHAT IF . . . ? Situation A. Testing Your Conjectures In the following two figures, two lines are intersected by another line. Are any of the relationships you found in the Initial Challenge true for each set of lines? Explain. Figure 1 Line 1 and line 2 are parallel lines. They are intersected by line 3. L1 L2 L3 a b c g d f e 150° Figure 2 Line 1 and line 2 are not parallel lines. They are intersected by line 3. m n p o q r t s L2 L1 L3 L1 L2 L3 a b c g d f e 30° 150° 30° 30° 30° 150° 150° 150° ∠a + 150° = 180°, so ∠a = 30°. ∠a and ∠c are vertical angles, so ∠c = 30°. If L1 and L2 are parallel, then ∠g and ∠e are in the same relative position as (are corresponding to) ∠a and ∠c, so ∠g = 30° and ∠e = 30°. ∠b = 150° because it is a vertical angle to ∠150°. If L1 and L2 are parallel, then ∠f and ∠d are in the same relative position as (are corresponding to) ∠150° and ∠b, so ∠f = 150° and ∠d = 150°. Vertical angles have the same measure. The vertical angles ∠m = 80° and ∠o = 80°. The vertical angles ∠p = 100° and ∠n = 100°. The vertical angles ∠q and ∠s = 120°. The vertical angles ∠r and ∠t = 60°. Because L1 and L2 are not parallel, there are no angles that are in the same relative position. Even if we thought of them that way, they are not equal measure. 98 Investigation 2 Designing with Angles 2.3 Answers SAMPLE
Supplementary Pairs of supplementary angles include ∠a and ∠b, ∠b and ∠c, ∠c and ∠150°, ∠150° and ∠a, ∠g and ∠d, ∠d and ∠e, ∠e and ∠f, ∠f and ∠g, ∠a and ∠f, ∠b and ∠e, ∠c and ∠d, ∠150° and ∠g, ∠a and ∠d, ∠b and ∠g, ∠c and ∠f, ∠150° and ∠e. Vertical Pairs of vertical angles include ∠a and ∠c, ∠g and ∠e ∠150° and ∠b, ∠f and ∠d. L1 L2 L3 a b c g d f e 150° Adjacent Pairs of adjacent angles include ∠a and ∠b, ∠b and ∠c, ∠c and ∠150°, ∠150° and ∠a, ∠g and ∠d, ∠d and ∠e, ∠e and ∠f, ∠f and ∠g. Another relationship is that the measure of all the angles around a point add to 360°. The measure of any two adjacent angles adds up to 180°. They are supplementary. For example, ∠m and ∠n are supplementary. Situation B. Jamal’s Claim About Parallelograms Is Jamal’s claim correct? Explain. Jamal’s Claim I added a line to Figure 1 in Situation A. I put in L4 parallel to L3. I was trying to find the angle measures. I think that • the four lines create a parallelogram; • the opposite angles of a parallelogram have the same angle measures; and • the sum of the interior angles of the parallelogram is 360º. L4 L1 L3 L2 m l n o k j h i g d f e a b c 150° Problem 2.3 Vertical, Supplementary, and Complementary Angles 99 2.3 Answers SAMPLE
Yes. We know from Situation A that lines 1 and 2 are parallel. Now, lines 3 and 4 are parallel. This makes the shape a parallelogram. We can use the relationships between supplementary angles, vertical angles, and adjacent angles to find that the opposite angles are the same measures. ∠o and ∠g are both 30°, and ∠150° and ∠h are both 150°. For example, ∠b and ∠150° are vertical angles, so ∠b is 150°. If the lines are parallel, then ∠b and ∠h are in the same relative position (are corresponding). So ∠h must also be 150°. L4 L1 L3 L2 m l n o k j h i g d f e a b c 150° 150° 150° 150° 150° 150° 150° 150° 30° 30° 30° 30° 30° 30° 30° 30° Situation C. Hank Again: Angles and the Algebra Connection Hank added cards to show what he knows about supplementary and vertical angles. Write an equation that will help you find a value of x. Note: x is a rational number. Card 4 x is a rational number. What is x? L1 L2 x 5 + x Card 5 x is a rational number. What is x? L1 L2 x 5x Card 6 x is a rational number. What is x? Lines L1 amd L2 are parallel. L1 L2 L3 x 3x 5 + ∠x + ∠x = 180 ∠x + ∠x = 175 ∠x = 87.5° ∠x + ∠5x = 180 6x = 180 x = 30° and 5x = 150° ∠x + ∠3x = 180 4x = 180 x = 45° and 4x = 135° 100 Investigation 2 Designing with Angles 2.3 Answers SAMPLE
NOW WHAT DO YOU KNOW? Describe what you know about angles formed by intersecting lines. Include parallel lines and the angles formed by a line that intersects the parallel lines in your description. At the point where two lines intersect, the vertical angles are equal. The sum of the two adjacent angles on a straight line is 180 degrees. They are supplementary. When two parallel lines are cut by a third line, the angles that occupy the same relative position (corresponding angles) are equal. The corresponding angles are not equal for two nonparallel lines. When the measurements of two angles add to 90°, the angles are called complementary. We can use the relationships among vertical, supplementary, complementary, and corresponding angles to find missing angles. Knowing the relationships between angles can eliminate having to measure every angle. Problem 2.3 Vertical, Supplementary, and Complementary Angles 101 2.3 Answers SAMPLE
Parallel Lines Examples Nonparallel Lines Examples Parallel and Nonparallel Lines 2.3A TEACHING AID © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. 102 Investigation 2 Designing with Angles SAMPLE
Vertical Angles Angles on opposite sides of a vertex of intersecting lines Vertical Angle Examples 4 1 3 2 4 1 3 2 ∠4 and ∠2 ∠1 and ∠3 Adjacent Angles Angles with a common side and a common vertex and don’t overlap (angles that are next to each other) Adjacent Angle Examples 4 1 3 2 4 1 3 2 ∠1 and ∠2 ∠2 and ∠3 Supplementary Angles Two angles with a sum of 180º angles that are separate 150° 30° angles formed by intersecting lines 60° 60° 30° 30° 150° 30° adjacent angles that form a straight line 60° 60° 30° 30° 150° 30° Complementary Angles Two angles with a sum of 90º angles that are separate 60° 30° angles formed by intersecting lines 60° 60° 30° 30° 150° 30° adjacent angles that form a right angle 60° 60° 30° 30° 150° 30° © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. Problem 2.3 Vertical, Supplementary, and Complementary Angles 103 SAMPLE
B You are the best! Thanks! Right back at you, my friend! A When your friends make it all right . . . Complimentary Complimentary Angles Complimentary Angles 2.3B TEACHING AID © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. 104 Investigation 2 Designing with Angles SAMPLE
Name Date Class m n o p q t r s w x y z L1 L2 L3 Initial Challenge 2.3A LEARNING AID © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. Problem 2.3 Vertical, Supplementary, and Complementary Angles 105 SAMPLE
Name Date Class Figure 1 Line 1 and line 2 are parallel lines. They are intersected by line 3. L1 L2 L3 a b c g d f e 150° Figure 2 Line 1 and line 2 are not parallel lines. They are intersected by line 3. m n p o q r t s L2 L1 L3 What If . . . ? Situation A 2.3B LEARNING AID © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. 106 Investigation 2 Designing with Angles SAMPLE
Name Date Class Situation B L4 L1 L3 L2 m l n o k j h i g d f e a b c 150° Situation C What If . . . ? Situations B and C 2.3C LEARNING AID Card 4 x is a rational number. What is x? L1 L2 x 5 + x Card 5 x is a rational number. What is x? L1 L2 x 5x Card 6 x is a rational number. What is x? Lines L1 amd L2 are parallel. L1 L2 L3 x 3x © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. Problem 2.3 Vertical, Supplementary, and Complementary Angles 107 SAMPLE
Facilitating Discourse Teacher Moves EXPLORE Having students refer to their notes from the Now What Do You Know? in each problem in the investigation can help them to synthesize all their ideas around geometric shapes. As a class, discuss the Mathematical Reflection. Use an idea like those on the next page to have students synthesize and record their thinking. Suggested Questions • After this investigation, what do we know about angles and angle measurements? • What did we learn in each problem of this investigation? • How might we describe the “big mathematical idea(s)” of the investigation? Time Anchor Chart Portrayal Language At a Glance NOW WHAT DO YOU KNOW? What do you know about geometric shapes? The Mathematical Reflection provides an opportunity to discuss the goals of the investigation. Students can pull together their reasoning from the Now What Do You Know? questions to summarize their learning over the time. Students can record their responses to the Mathematical Reflection to create a record of their current understandings of the big ideas of the unit. The Mathematical Reflection can provide a self-assessment for students. Each student can have checkpoints of their understanding of the mathematics after each investigation. Students can use the Mathematical Reflection to consolidate their mathematical thinking, take notes, and provide evidence of what they know and can do. A teacher can gain an understanding of student thinking during a discussion of the reflection question. Then one can assess individual understanding based on each student’s written work. For more Teacher Moves listed here, refer to the General Pedagogical Strategies section in A Guide to Connected Mathematics® 4. MR Mathematical Reflection Arc of Learning Exploration Analysis Pacing __1 2 day 108 Investigation 2 Designing with Angles SAMPLE
Student Responses At the beginning of the year, students will need more collaboration to outline and summarize the important ideas. They may need examples of writing, diagrams, and/or justifications from other students to help build their vision of what is expected when answering a Mathematical Reflection. Early in the year, you may want to start writing Mathematical Reflections as a whole group. Then as the year progresses, move to small groups, pairs, and finally individuals. Each investigation contributes to students’ conceptual understandings of the ideas in the unit. Students’ explanations at the beginning of a unit might be just forming. As you progress through the unit, students can use the contexts, representation, and connections to express a more solid understanding. By the end of the unit, students can create a complete picture of understanding. Example Strategies for Student Participation Here are a few creative strategies teachers use to encourage students’ ownership of their learning. Anchor Charts • After a discussion, chart the emerging understanding, and post it in the classroom. This can be done on poster paper or electronically. • Work with students throughout the unit to reference, add to, or refine their understandings. Note: For teachers who move classrooms or have multiple classes of the same grade level, create the chart in all classes, but keep just one to represent all of your classes. Post this one in the room, or bring it out when needed. Note Organization • Some teachers use the Mathematical Reflections as an organizer for note-taking during the investigation. • As part of the Summarize section of the problems, students record key ideas to the Now What Do You Know? reflection questions on a separate paper. • At the close of the investigation, students synthesize their notes into responses that summarize their emerging understandings of the ideas in the unit. Word Bank • As a class, create a word bank of terms from the Investigation. • Have groups of students write three or four statements using the words from the bank. • After formatively assessing their statements, you may choose to have a class discussion to refine the statements. Chalk Talk With a chalk talk, your writing does “the talking” instead of talking aloud. • Students post the question(s) on sheets of chart paper or on sections of your board. • Small groups record responses while collaborating in “chalk talk” format. • Students move to others’ work and add their thinking in the form of new ideas and connections. Final Reflection Presentation Teachers sometimes use the Mathematical Reflection after the last investigation as a summary of students’ learning. • Students consolidate their learning from the unit. • Teachers choose from various ways to present their ideas. Presentation choices might include creating a poster, written paper, presentation, or song/rap. Partner Write • Students create a written response to the reflection question with a partner. • Students discuss the reflection question with a partner. • Students create and write a response with a partner. Mathematical Reflection 109 SAMPLE
In this unit, we are investigating some general properties of geometric shapes (figures), including angles and polygons, and using this information to design shapes and solve problems. At the end of this investigation, ask yourself: What do you know about geometric shapes? Students should discuss key angle concepts and skills for problem-solving, including being able to • read, sketch, estimate, and use benchmark angles described by the number of degrees or the amount of turn; • understand measurement limitations and possible effects of estimation and/or measurement error; • apply angle concepts and skills to both mathematical problem contexts such as intersecting lines (including parallel lines) and real-world contexts beyond geometric shapes; • recognize, describe, and use angle relationships such as complementary and supplementary angle pairs, adjacent angles, vertical angles, and straight angles; and • begin to use angle relationships to write equations and solve multistep problems involving angles. MR Answers Embedded in Student Edition Problems Mathematical Reflection 110 Investigation 2 Designing with Angles SAMPLE
INVESTIGATION 2 APPLICATIONS—CONNECTIONS—EXTENSIONS (ACE) APPLICATIONS 1. An angle whose measure is less than 90° is called an acute angle. An angle whose measure is greater than 90° and less than 180° is called an obtuse angle. Which of these angles are acute, which are obtuse, and which are right? Answers Embedded in a. d. b. e. c. f. Acute angles are c and e; right angles are b and d; obtuse angles are a and f. 2. Snowboarders use angle measures to describe their flips and spins. Explain what a snowboarder would mean by each statement. a. I did a 720. two complete rotations b. I did a 540. one and one-half complete rotations c. I did a 180. one-half of a complete turn (essentially reversing direction) Applications—Connections—Extensions (ACE) 111 SAMPLE
3. Write an equation, and find the measure of the angle labeled x, without measuring. a. x 30° c. x 27° b. x 125° d. 35° a. x = 180 − 30; x = 150° b. x = 180 − 125; x = 55° c. x = 90 − 27; x = 63° d. x = 360 − 35; x = 225° 4. Decide whether each angle is closest to 30°, 60°, 90°, 120°, 150°, 180°, 270°, or 360° without measuring. Explain your reasoning. a. c. e. b. d. f. ACE 112 Investigation 2 Designing with Angles SAMPLE
a. 180°; half a rotation b. 90°; __1 4of a rotation c. 150°; more than 120º but less that 180º d. 60°; __ 2 3 of a right angle e. 270°; __3 4of a rotation f. 360°; almost a rotation g. 120°; about 30º past a 90º rotation h. 30°; __1 3 of a right angle i. Classify each angle in ACE 4a–h as right, acute, or obtuse. right angle: b; acute angles: d and h; obtuse angles: c and g 5. A full turn is 360°. Find the fraction of a turn or number of turns for the given measurement. a. 90° __1 4 of a turn b. 270° __3 4 of a turn c. 720° 2 full turns d. How many degrees is ____ 25 360of a full turn? 25° 6. You can think of a right angle as one-quarter of a complete rotation. a. How many degrees is one-third of a quarter-rotation? 30° b. How many degrees is two times a quarter-rotation? 180° c. How many degrees is two and one-third times a quarter-rotation? 210° 7. A test question asked students to choose the larger angle. In one class, most students chose angle 2. Do you agree? Why or why not? angle 1 angle 2 No. Although the rays that make up the drawing of angle 2 are longer, the angle measure of angle 1 indicates a greater turn and thus a larger angle. g. h. AACECE Applications—Connections—Extensions (ACE) 113 SAMPLE
8. Estimate the measure of each angle. Then check your answers with an angle ruler or a protractor. a. b. c. d. e. a. 50° b. 130° c. 21° d. 210° e. 170° 9. Draw an angle for each measure. Include an arc indicating the turn. a. 45° b. 25° c. 180° d. 200° a. 45° b. 25° ACE 114 Investigation 2 Designing with Angles SAMPLE
c. 180° d. 200° 10. Without measuring, decide whether the angles in each pair have the same measure. If they do not, tell which angle has the greater measure. Then, find the measure of the angles with an angle ruler or protractor to check your work. a. 1 2 b. 1 2 c. 1 2 a. They do not have the same measure. Angle 1 is larger; angle 1 measures 60°, and angle two measures 30°. b. Angle 1 and 2 have the same measure, 135°. c. They do not have the same measure. Angle 1 is larger; angle 1 measures 90°, and angle 2 measures 45°. AACECE Applications—Connections—Extensions (ACE) 115 SAMPLE
a. ∠BVA = 45° and ∠AVB = 315° b. ∠LKJ = 75° and ∠JKL = 285° c. ∠RQP = 120° and ∠PQR = 240° d. ∠ZYX = 160° and ∠XYZ = 200° 12. Ms. Cosgrove asked her students to estimate the measure of the angle shown. Deshawn thought 150° would be a good estimate. Sofia said it should be 210°. Who is closer to the exact measurement? Explain. Both students have given reasonable answers. However, when no direction of rotation is indicated, it is customary to focus on the angle as a union of two rays with common endpoint and measure between 0 and 180 degrees. 11. For each pair of angles, estimate the measure of each angle. Then, check your estimates by measuring with an angle ruler or a protractor. a. A V B c. Q P R b. K J L d. X Y Z ACE 116 Investigation 2 Designing with Angles SAMPLE
13. Use the diagram of the protractor below. Angle 1 and angle 2 are called adjacent angles because they have a common vertex and a common side. Find the angle measures. 90 1 2 3 4 180 0 a. ∠1 b. ∠1 + ∠2 c. ∠1 + ∠2 + ∠3 d. ∠2 e. ∠2 + ∠3 f. ∠3 g. the complement of ∠1 h. the supplement of ∠1 i. the complement of ∠3 j. the supplement of ∠1 combined with ∠2 a. 60° b. 110° c. 150° d. 50° e. 90° f. 40° g. 30°; 90 − 60 = 30 h. 120°; 180 − 60 = 120 i. 50°; 90 − 40 = 50 j. 70°; 180 − 110 = 70 14. Use the diagram to answer. Write an equation using the angle measures shown. Then, find the measures of ∠A and ∠B. 2x = B A = 3x 2x + 3x = 180 so 5x = 180; x = 36° The measure of ∠A is 108° (3 × 36). The measure of ∠B is 72° (2 × 36). AACECE Applications—Connections—Extensions (ACE) 117 SAMPLE
15. Use what you know about supplemental and vertical angles to find the measure of each angle. Show the equation you could use to find the value of y. 3y y y + 3y = 180; 4y = 180, so y = 45°. Angle y and the angle vertical to it are 45°. Angle 3y (3 × 45) is 135°, as is the angle vertical to it. 16. Lines 1 and 2 are perpendicular; they meet at 90° angles. w x y z t v L1 L3 L2 a. Name all the pairs of complementary angles. ∠v and ∠t; ∠x and ∠y b. If the measure of ∠t is twice the measure of ∠v, what are the measures of all of the angles? Explain. Angles v and t add to 90°. Angle t is 2v. So v + 2v = 90. 3v = 90, so v = 30. ∠v = 30°. ∠t = 60° because it is 2v. ∠w = 90° because it is a right angle. ∠x = 60° because it is vertical to ∠t. ∠y= 30° because it is vertical to ∠v. ∠z = 90° because it is a right angle. ACE 118 Investigation 2 Designing with Angles SAMPLE
17. In the diagram, lines L1 and L2 are parallel. L1 L2 T 35° g d f e a b c a. Find the degree measures of angles labeled a–g. Angles b, e, and g measure 35°; angles a, c, d, and f measure 145°. b. Name the pairs of opposite or vertical angles in the figure. Vertical angle pairs: ∠a and ∠c; ∠b and the 35° angle; ∠e and ∠g; and ∠d and ∠f. c. Name three pairs of supplementary angles in the figure. There are many pairs of supplementary angles, including ∠a and 35°, ∠a and ∠b, ∠b and ∠c, ∠c and 35°, ∠e and ∠d, ∠d and ∠g, ∠g and ∠f, ∠f and ∠e, ∠a and ∠e, ∠a and ∠g, ∠b and ∠d, and ∠b and ∠f. 18. L1 and L2 are not parallel lines. They are intersected by L3. Explain what you know about the measure of each angle. 105° L2 L1 L3 g e f h a b d ∠a is vertical to the 105° angle, which makes ∠a = 105°. ∠b is supplementary to ∠a, which makes it 180 − 105 = 75°. ∠d is vertical to ∠b, which makes ∠d = 75°. Because L1 and L2 are not parallel, we cannot find the measures of angles e, f, g, and h without measuring. If we measure one of the angles, we could use vertical and supplemental relationships to find the other three measures. AACECE Applications—Connections—Extensions (ACE) 119 SAMPLE
19. In the diagram, L1 and L2 are parallel lines. Without measuring, find the measure of each angle, and explain how you know. L1 L2 50° b a c e f d g a = 130° because the 50° angle and angle a are supplementary angles and their measures equal 180° when added together. b = 50° because it is vertical to the 50° angle and vertical angles have the same measure. c = 130° because it is vertical to angle a, which is 130°. d = 130° because L1 and L2 are parallel so when they are intersected by a transversal, the two sets of four angles formed will be equivalent. e = 50° because it is equal to angle b. f = 130° because it is equal to angle c. g = 50° because it is vertical to e so therefore equal to e. CONNECTIONS 20. The number 360 has many factors. This may be why it was chosen for the number of degrees in a full turn. a. List all of the factors of 360. 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 360 b. Find the prime factorization of 360. 360 = 23 × 32 × 5 21. At the start of each hour, the minute hand of a clock points straight up at 12. In parts (a)–(f), determine the angle between the minute hand at the start of an hour and the minute hand after the given amount of time passes. For each situation, sketch the angle, and indicate the rotation of the minute hand. a. 15 minutes 90° b. 30 minutes 180° c. 20 minutes 120° d. one hour 360° e. 5 minutes 30° f. one and one-half hours 540° ACE 120 Investigation 2 Designing with Angles SAMPLE
22. Write each fraction as a decimal. a. __ 2 5 b. __3 4 c. __3 10 d. __1 4 e. __7 10 f. ___7 20 g. __ 4 5 h. __7 8 i. ___ 15 20 j. __ 3 5 a. __ 2 5 = __4 10 = 0.4 b. __3 4 = ____ 75 100 = 0.75 c. 0.3 d. __1 4 = ____ 25 100 = 0.25 e. 0.7 f. ___7 20 = ____ 35 100 = 0.35 g. __ 4 5 = __8 10 = 0.8 h. 0.875 i. ___ 15 20 = ____ 75 100 = 0.75 j. __ 3 5 = 0.6 23. Find the value of n. a. __1 2 = ____ n 360 b. __1 10 = ___ 36 n c. __1 n = ____ 40 360 d. __ n 3 = ____ 120 360 a. __1 2 = ____ 180 360 b. __1 10 = ____ 36 360 c. __1 9 = ____ 40 360 d. __1 3 = ____ 120 360 24. The members of the skateboard club were practicing. They wanted to work on increasing the angle rotation each member did when standing and turning their skateboards. They recorded their rotations: 0° 10° 20° 30° 40° 50° 80° 100° a. What is the median of the turns that the club members can make on their boards? AACCEE Applications—Connections—Extensions (ACE) 121 SAMPLE
b. What is the mean of the turns that the club members can make on their boards? Three weeks later, they each tried a stand and turn again. They recorded their rotations again: 0° 10° 20° 30° 40° 50° 80° 100° c. How does their new data compare? Solution: a. Median is 40o. b. Mean is 50o. (20 + 30 + 40 + 40 + 40 + 50 + 80 + 100) _________________________________ 8 = ____ 400 8 = 50. c. The new data has the same median, mean, and range (80o between the min and the max). However, the data is clustered differently with bigger gaps between the clusters of members. 25. Find a value of n that makes the equation true. a. 3 × n = 24 n = 8 b. 5 + n = 60 n = 55 c. 144 = 12 × n n = 12 d. 160 ÷ 8 = n n = 20 e. 2 + n = 50 n = 48 26. Multiple Choice Which of the following shaded regions is not a representation of __4 12 ? A. B. ACE 122 Investigation 2 Designing with Angles SAMPLE
C. D. 0180 20 10 0 170 10 160 20 150 30 140 40 50 130 60 120 70 110 80 100 90 100 80 110 70 120 60 130 50 140 40 150 30 160 170 180 350 340 330 320 310 300 290 280 260 250 240 230 220 210 200 190 190 200 210 220 230 240 250 260 280 270 290 300 310 320 330 340 350 AACCEE 180 0 10 20 30 40 50 60 70 80 100 110 120 130 140 150 160 170 180 170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0 Answer: B is __4 16 or __1 4 27. For each equation or inequality, find the value of x. a. 3x = 180 x = 60 b. 3x > 180 x > 60 c. 3x = 90 x = 30 d. 3x > 90 x > 30 e. x + x = 180 x = 90 f. x + x = 90 x = 45 g. 72 + x = 180 x = 108 h. x + 15 = 90 x = 75 Applications—Connections—Extensions (ACE) 123 SAMPLE
EXTENSIONS 28. Design a new polar coordinate grid for Four in a Row in Problem 1.2. Play your game with a friend or family member. What ideas did you use to design your new grid? Explain. How does playing on your grid compare to playing on the original grids? Variations of the Four in a Row game could take a variety of forms—more concentric circles, different benchmark angle patterns (e.g., multiples of 10°), or others that we haven’t imagined. 29. List all polygons in the Shapes Set from the manipulatives kit or Learning Aid 3.1 Template: Shapes Set Template, that have the following characteristics: a. only right-angle corners Figures B, G, H, and J have only right-angle corners. b. only obtuse-angle corners Figures C, D, E, and F have only obtuse-angle corners. c. only acute-angle corners Figures A and P have only acute-angle corners. d. at least one angle of each type—acute, right, and obtuse Figures Q and S have at least one angle of each type 30. A compass is a tool used in wilderness navigation. On a compass, north is assigned the direction label 0°, east is 90°, south is 180°, and west is 270°. Directions that are between those labels are assigned degree labels such as NE (northeast) at 45°, for example. NW (315°) NE (45°) W (270°) E (90°) SW (225°) SE (135°) N (0°) S (180°) a. What degree measures would you expect for the direction south-southwest? For north-northwest? ∠SSW is 202.5°. ∠NNW is 337.5°. b. A ship at sea is on a heading of 300°. Approximately what direction is it traveling? The ship is traveling in a direction 30° north of due west. ACE 124 Investigation 2 Designing with Angles SAMPLE
31. Major airports label runways with the numbers by the compass heading. For example, a plane on runway 15 is on a compass heading of 150°. A plane on runway 9 is on a compass heading of 90°. (Refer to information in ACE 30 on the design of a compass.) W E N S a. What is the runway number of a plane that is taking off on a heading due west? On a heading due east? The runway heading due west is 27; heading due east is 9. b. What is the compass heading of a plane landing on runway 6? On runway 12? Runway 6 implies a compass heading of 60°. Runway 12 implies a compass heading of 120°. c. Each actual runway has two direction labels. The label depends on the direction in which a landing or taking-off plane is headed. How are those labels related to each other? Labels for runways in opposite directions differ by 18, related to the 180° difference in their directions. 32. Soledad extended the sides of the green parallelogram below. She looked at the vertical angles and supplementary angles. Soledad claims that the opposite angles in a parallelogram will always have the same measures. Do you agree? Explain. L3 L1 L2 L4 Yes. The lines are all parallel, so opposite angles are equal to the same-size vertical angles. In the drawing, we can see that all of the angles indicated by the blue arcs have the same measures and all of the angles indicated by the red arcs have the same measure. The adjacent angles on the parallelogram are supplementary. We can see this because they are the same as the measures of an internal angle and external angle that add to 180°. ACACEE Applications—Connections—Extensions (ACE) 125 SAMPLE
33. In the parallelogram, find the measure of each numbered angle. 4 1 117° 3 2 5 Angles 1, 3, and 5 are all 63°; angles 2 and 4 are both 117°. 34. In the triangle ABC, a line has been drawn through vertex A, parallel to side BC. 4 5 1 2 3 B C A a. What is the sum of the measures of angles 1, 2, and 3? 180° b. Explain why angle 1 has the same measure as angle 4 and why angle 3 has the same measure as angle 5. Both pairs of angles are on opposite sides of a transversal between parallel lines. c. How can you use the results of parts (a) and (b) to show that the angle sum of a triangle is 180°? The angles 1, 2, and 3 have the same measures as angles 6, 2, and 5, respectively, and angles 6, 2, and 5 are the angles of a triangle. Since the sum of the measures of angles 1, 2, and 3 is the measure of a straight angle, the sum of the measures of those angles is 180°. This means that the sum of the measures of angles 6, 2, and 5 is also 180°. ACE 126 Investigation 2 Designing with Angles SAMPLE
Partner Quiz 1. Determine whether each statement is always true, sometimes true, or never true. Circle your response, and then explain your reasoning. a. Exactly one triangle can be formed with side lengths 10 in., 5 in., and 6 in. Always true Sometimes true Never true Explain: b. Exactly one quadrilateral can be formed with side lengths 10, 5, 6, and 7 feet. Always true Sometimes true Never true Explain: c. The sum of two adjacent angles is 90°. Always true Sometimes true Never true Explain: d. Angles x and y are complementary, so the sum of their measures is 180°. Always true Sometimes true Never true Explain: Name Date Class © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. Shapes and Designs 127 SAMPLE
2. This drawing of a building contains many angles and polygons. You will be asked about angles A, B, C, D, E, F, G, H, I, and J below. A B C D E F G H I J Name Date Class a. Which labeled angles appear to be right angles? Which appear to be acute angles? Which appear to be obtuse angles? Organize your answers in the table below. In the columns, list the angles from smallest to largest. Right angles Acute angles Obtuse angles Other angles b. Estimate the degree measure of each angle. Record your estimate by the angle in your table. © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. 128 Partner Quiz SAMPLE
4. The figure below is a rectangle. 1 2 10 9 8 7 6 4 5 11 14 3 12 13 Name Date Class 3. a. Estimate the measure of the angle below. b. Sketch an angle that is 120°. a. Name a pair of supplementary angles. b. Name a pair of complementary angles. c. Name a pair of vertical angles. © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. Partner Quiz 129 SAMPLE
5. For each of the following equations, N is the angle measure of a rotation angle. Find a value for N that makes each statement true. Then sketch the angle on the grid. a. __ 2 3 N = 60° 0° b. 60° + 2N = 360° 0° Name Date Class © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. 130 Partner Quiz SAMPLE
Partner Quiz: Answers 1. Determine whether each statement is always true, sometimes true, or never true. Circle your response, and then explain your reasoning. a. Exactly one triangle can be formed with side lengths 10 in., 5 in., and 6 in. Always true Sometimes true Never true Explain: The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. With three measures of the correct length, there is only one triangle that can be formed. b. Exactly one quadrilateral can be formed with side lengths 10, 5, 6, and 7 feet. Always true Sometimes true Never true Explain: It does form a quadrilateral, but different quadrilaterals can be constructed when given four sides of the quadrilateral. If you arrange the adjacent side lengths differently, you can get different quadrilaterals. Examples: 10 ft 10 ft 5 ft 5 ft 6 ft 6 ft 7 ft 7 ft Note some students may answer “Sometimes true,” thinking that one quadrilateral can be formed and then other quadrilaterals can also be formed, too. c. The sum of two adjacent angles is 90°. Always true Sometimes true Never true Explain: Two adjacent angles might add to 90°, but they could also have many other sums. d. Angles x and y are complementary, so the sum of their measures is 180°. Always true Sometimes true Never true Explain: The sum of complementary angles is 90°. © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. Shapes and Designs 131 SAMPLE
2. This drawing of a building contains many angles and polygons. You will be asked about angles A, B, C, D, E, F, G, H, I, and J below. A B C D E F G H I J a. Which labeled angles appear to be right angles? Which appear to be acute angles? Which appear to be obtuse angles? Organize your answers in the table below. In the columns, list the angles from smallest to largest. Right angles Acute angles Obtuse angles Other angles ∠F ≈ 90° ∠C ≈ 30° ∠D ≈ 30° ∠H ≈ 45° ∠B ≈ 120° ∠E ≈ 120° ∠I ≈ 145° ∠J ≈ 145° ∠G appears to be a straight angle or 180°. ∠A is greater than a straight angle (sometimes referred to as a reflex angle) ≈ 240°. b. Estimate the degree measure of each angle. Record your estimate by the angle in your table. Answers in chart above. © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. 132 Partner Quiz: Answers SAMPLE
4. The figure below is a rectangle. 1 2 10 9 8 7 6 4 5 11 14 3 12 13 3. a. Estimate the measure of the angle below. ≈ 90° b. Sketch an angle that is 120°. a. Name a pair of supplementary angles. Many possible answers include ∠11 and ∠12 ∠10 and ∠3 ∠3 and ∠4 ∠8 and ∠9 ∠12 and ∠13 ∠10 and ∠4 ∠3 and ∠5 ∠8 and ∠5 ∠13 and ∠14 ∠10 and ∠5 ∠3 and ∠8 ∠8 and ∠4 ∠14 and ∠11 ∠10 and ∠8 ∠3 and ∠9 ∠10 and ∠9 b. Name a pair of complementary angles. Possible answers include ∠1 and ∠2 ∠7 and ∠6 ∠1 and ∠7 ∠6 and ∠2 c. Name a pair of vertical angles. Possible answers include ∠12 and ∠14 ∠11 and ∠13 © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. Partner Quiz: Answers 133 SAMPLE
5. For each of the following equations, N is the angle measure of a rotation angle. Find a value for N that makes each statement true. Then sketch the angle on the grid. a. 2_ 3 N = 60° N = 90° 0° 90° b. 60° + 2N = 360° N = 150° 0° 150° © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. 134 Partner Quiz: Answers SAMPLE
Implementation Key Terms Materials Resources Problem 3.1 Think, Pair, Share Pacing 1 day tilings tessellations regular polygon irregular polygon For each pair of students • Learning Aid Template: Shapes Set or plastic Shapes Set • Learning Aid 3.1: Initial Challenge Table • large paper • markers For the class • Teaching Aid 3.1: Honeycombs Problem 3.2 Groups of 2 Pacing 2 days angle sum For each student • Learning Aid 3.2A: Initial Challenge Shapes • Learning Aid 3.2B: Angle Sum Patterns in Regular Polygons • Learning Aid 3.2C: Trevor’s, Casey’s, and Maria’s Strategies • Learning Aid 3.2D: Zane’s Conjectures • scissors (optional) For the class • Teaching Aid 3.2A: Angle Sum of Any Polygon • Teaching Aid 3.2B: Different-Sized Regular Polygons Problem 3.3 Groups of 2 Pacing 1 day interior angles exterior angles For each student • Learning Aid 3.3: Pentagonal Path • polystrips (optional for Launch) Problem 3.4 Groups of 2 Pacing 1 day For each pair of students • Learning Aid 3.4A: Message Cards • Learning Aid 3.4B: Possible Polygon Designs • tools to create shapes: polystrips, angle rulers, dot paper, geoboards, geometry technology, etc. INVESTIGATION 3 PLANNING CHART Designing Polygons: The Angle Connection INVESTIGATION 3 Materials for All Investigations: calculators; student notebooks; colored pens, pencils, or markers (continued) 135 SAMPLE
Implementation Key Terms Materials Resources Mathematical Reflection Whole Class Individual Notes Pacing __1 2 day For the class • large poster paper (optional) Word bank created by students and/or teacher (optional) Assessment Unit Test Individual Pacing 1 day For each student • Unit Test (continued from page 135) 136 Investigation 3 Designing Polygons: The Angle Connection SAMPLE
Note: If you have a Grade 7 Classroom Materials Kit, please refer to A Guide to Connected Mathematics® 4 for a detailed list of materials included or items you will need to prepare ahead of time. For more Teacher Moves, refer to the General Pedagogical Strategies and the Attending to Individual Learning Needs Framework in A Guide to Connected Mathematics® 4. At a Glance This problem uses results about angle measurement in polygons to explain the tiling properties of polygons. The use of tessellations or tiling will help to solidify ideas that students are forming around characteristics of shapes and in particular angle measurements. In the Initial Challenge, students will investigate tiling several polygons. In the What If . . . ? Situations, students look at the angle measures of polygons and tiling with more than one polygon. Back to the Bees: Tiling a Plane Experiment PROBLEM 3.1 Arc of Learning Analysis Synthesis NOW WHAT DO YOU KNOW? Which regular polygons can be used to tile a surface? Explain why they tile. Give some examples of how tiling polygons can be useful. Key Terms Materials tilings tessellations For each pair of students • Learning Aid Template: Shapes Set or plastic Shapes Set • Learning Aid 3.1: Initial Challenge Table • large paper • markers For the class • Teaching Aid 3.1: Honeycombs Pacing 1 day Groups Think, Pair, Share A 1–3 C 22–25 E 38 Facilitating Discourse Teacher Moves LAUNCH CONNECTING TO PRIOR KNOWLEDGE Launch the problem by asking students if they have seen honeycombs made by bees. Ask where they might have seen a similar pattern of hexagons, since it is a very common tile pattern. Suggested Questions • Why do these regular hexagons fit so neatly? • What other polygons do you think can be used to tile a surface? Problem 3.1 Back to the Bees: Tiling a Plane Experiment 137 SAMPLE
Facilitating Discourse (continued) Teacher Moves (continued) LAUNCH PRESENTING THE CHALLENGE Have students begin looking for shapes that will tile independently. Transition them to working with a partner after they’ve had time to experiment on their own. The goal is to find which polygons will tile and why. Have students record their findings on large sheets of paper that show drawings and explanations of their work and/or Learning Aid 3.1: Initial Challenge Table. Portrayal Think, Pair, Share EXPLORE PROVIDING FOR INDIVIDUAL NEEDS It will be helpful for many students to have the plastic shapes. This will give them a tactile and visual sense of what goes wrong with the nontiling figures. Suggested Questions • Could you turn/flip the shape to see if it will fit together and tile? • What do you notice about the shapes that tile versus the shapes that don’t tile? Do regular polygons and irregular polygons tile? • What are some examples of polygons that tile? • What do you notice about where the tiles meet? • What does that tell us about the measures of the angles around the point? Be sure students draw pictures to illustrate their answers. PLANNING FOR THE SUMMARY As you are circulating, look for examples of tiling and nontiling regular polygons for students to share in the summary. Listen for the ways that students are generalizing the relationships they’ve found with the angle measures and tiling. SUMMARIZE DISCUSSING SOLUTIONS AND STRATEGIES Have groups share the shapes and combinations of shapes that they found would tile. Focus the discussion on why those work and others did not. MAKING THE MATHEMATICS EXPLICIT Students should be able to explain why there are only three regular polygons that tile, using angle measure as part of their argument. They should also be able to explain why certain combinations of regular polygons work, using angle measures. Suggested Questions • When you tried just a single shape, did all of your shapes tile? • With the shapes that did tile, what is the sum of the measures of the angles around each vertex point in a tiling? • What happens with the heptagon? • How did finding the shapes that tile help you with thinking about angle measurements? • Since we know that triangles, squares, and hexagons tile, what does this tell us about the measure of the angles if those are regular shapes? • What is the measure of one angle of a regular triangle? A square? A regular hexagon? • What other combinations of shapes tile? As you finish the mathematical discussions, have students reflect on the Now What Do You Know question(s). At the end of this problem, you might want to share videos by Chris Watson about tessellations and why bees “love” hexagons by Zack Patterson and Andy Peterson. Claim, Support, Question 3.1 138 Investigation 3 Designing Polygons: The Angle Connection SAMPLE