Problem Overview 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. Implementation Note: The shapes are a subset of the Shapes Set. If you have Shapes Sets, you can use them. Otherwise, you will have to use copies cut from the learning aid or develop an electronic solution. Launch (Getting Started) Connecting to Prior Knowledge Launch the problem by asking students if they have seen honeycombs made by bees and, if so, why they think the surface has its distinctive pattern of hexagons. You can use Teaching Aid 3.1: Honeycombs. Ask where they might have seen a similar pattern of hexagons, since it is a very common tile pattern. After discussing hexagon patterns, ask the following questions. Suggested Questions • Why do these regular hexagons fit so neatly? (Students may not have a solid explanation yet. Let them conjecture and formalize this during the summary. In a regular hexagon, each angle measures 120°, so when three hexagons meet, there is a total rotation of 360° around the intersection of the three sides.) • What other polygons do you think can be used to tile a surface? (Students will make many conjectures. Encourage them to explain their thinking, and then they will investigate these in this problem.) Presenting the Challenge Have students begin looking for shapes that will tile independently. Distribute Shapes Sets or use Learning Aid Template: Shapes Set. 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) EXTENDED LAUNCH—EXPLORE—SUMMARIZE Extended Lanuch—Explore—Summarize 139 SAMPLE
Explore (Digging In) Providing for Individual Needs It will be helpful for many students to have plastic shapes or cut-out shapes that have been printed on card stock. 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? (Answers will vary depending on the shape they are trying to tile.) • What do you notice about the shapes that tile versus the shapes that don’t tile? (Only triangles, quadrilaterals, and hexagons tile.) • Do regular polygons and irregular polygons tile? (Yes.) • What are some examples of polygons that tile? (Triangles, rectangles, parallelograms, some trapezoids, and hexagons can tile.) • What do you notice about where the tiles meet? (There is a point with no gaps or overlaps.) • What does that tell us about the measures of the angles around the point? (Together, the angles are a full rotation or 360°.) Be sure students draw pictures to illustrate their answers. The following are examples of two irregular polygons and two regular polygons that tile. Regular Quadrilateral (square) Regular Hexagon Each angle of a square is 90°. 4(90) = 360 120° 120° 120° Irregular Quadrilateral Irregular Triangle 4 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3 S S S S S S S S S S S S S S 145° 35° 35° 2 2 145° 25° 65° 90° 25° 65° 90° LES 140 Investigation 3 Designing Polygons: The Angle Connection SAMPLE
During What If . . . ? Situation B, if students need an extra challenge after finding groups of two or more regular polygons that will tile, add shapes such as Shape R from the Shapes Set or rectangular and nonrectangular parallelograms to see if those combinations will tile. (Agency, Identity, Ownership) 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? Which polygons can be used to tile a surface? Explain why they tile. Give some examples of how tiling polygons can be useful. (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 (Orchestrating the Discussion) 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. These are just a few possibilities you may get from your class: Squares Triangles Squares and Triangles 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. These are the only three regular polygons with an angle measure that is a factor of 360. They should also be able to explain why certain combinations of regular polygons work, using angle measures. Be sure to ask them to show why they think their design forms a tiling with no gaps or overlaps. One simple example is octagons and squares: The interior angle in a regular octagon is 135°, so two LES Extended Lanuch—Explore—Summarize 141 SAMPLE
octagons would be 270°, and adding an angle from the square would make the necessary 360°. Suggested Questions • When you tried just a single shape, did all of your shapes tile? (Yes, except the heptagon.) • With the shapes that did tile, what is the sum of the measures of the angles around each vertex point in a tiling? (The sum is 360°.) • What happens with the heptagon? (We cannot fit them in together. There are gaps, or the shapes overlaps. The angle measures do not add to 360°.) • How did finding the shapes that tile help you with thinking about angle measurements? (Answers will vary. Some students have discussed how knowing when a shape tessellated it connected at 360° helped them to have more visuals for some benchmark 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? (Like Maisie, if we know the shapes fit together at a point, there is 360°. Since the shapes are regular, all the angles are equal measure and equally share the 360°.) • What is the measure of one angle in a regular triangle? A square? A regular hexagon? (60°, 90°, and 120°.) • What other combinations of shapes tile? (Answers will vary.) Some teachers use this opportunity to explain to students a shorthand notation for describing the shapes and combinations of shapes used to tessellate. For example, a student would write 4, 4, 4, 4 to describe the tiling of squares and 3, 3, 3, 3, 3, 3 to describe the tiling of triangles. To describe the combination of shapes presented above, you can write 4, 3, 3, 3, 4. The notation identifies the shape by its number of sides. It also tells the number of shapes and the order in which the shapes surround a point. (Portrayal) 3, 3, 3, 3, 3, 3 Number of triangles needed to make 360° around a point Tiling of Triangles 3 3 3 3 3 3 4, 3, 3, 3, 4 Sequence of squares and triangles to make 360° around a point Tiling of Squares and Triangles 4 4 3 3 3 Tiling of Squares 4, 4, 4, 4 Number of squares needed to make 360° around a point 4 4 4 4 You could suggest that the class look for interesting tiling patterns in their homes or in school. Have them make a sketch of any designs they find. LES 142 Investigation 3 Designing Polygons: The Angle Connection SAMPLE
There are eight combinations of regular polygons that will tile. Note that the numbers in parentheses refer to the polygon by side number (8 means a regular octagon, 6 means a regular hexagon, etc.) and the order in which they appear around a vertex of the tiling. • 2 octagons and 1 square (8-8-4) • 1 square, 1 hexagon, and 1 dodecagon (4-6-12) • 4 triangles and 1 hexagon (3-3-3-3-6) • 3 triangles and 2 squares (4-3-4-3-3) • 1 triangle, 2 squares, and 1 hexagon (4-3-4-6) • 1 triangle and 2 dodecagons (3-12-12) • 4 triangles and 2 squares (4-3-3-3-3-4) • 2 triangles and 1 hexagon (3-3-6) Note: There are two arrangements with triangles and squares, but depending on the arrangement, they produce different tile patterns. LES Extended Lanuch—Explore—Summarize 143 n o p q r s t u v w SAMPLE
At the end of this problem, you might want to share researched articles on pentagons, videos from online searches about bees and hexagons, and tessellations by M. C. Escher. 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 144 Investigation 3 Designing Polygons: The Angle Connection SAMPLE
Answers Embedded in Student Edition Problems INITIAL CHALLENGE To answer the question of why the bees choose a hexagon, Ms. Bennet’s class explored other shapes to see if they would tile a plane. They started with the following shapes and recorded their information in a chart. T 1 3 2 B 4 3 1 2 K 3 2 4 1 S 3 2 4 1 D 2 34 5 6 1 E 1 2 3 45 6 7 Make a Prediction • Which shapes will tile a plane? Student predictions will vary based on their past experiences. Conduct the Experiment Back to the Bees: Tiling a Plane Experiment PROBLEM 3.1 Equipment › several copies of the Shape Set Directions (individually or with a partner or small group) › Use the copies of each shape to see if it will tile. › Record the following information in a table. Shape Does the Shape Tile a Plane? The Total Degrees in the Angles Where the Shapes Meet Triangle T 1 3 2 Yes. 25° 65° 90° 25° 65° 90° 2(90 + 65 + 25) = 360 Note: There are complementary and supplementary angles around the point where the shapes meet. Answers Problem 3.1 Back to the Bees: Tiling a Plane Experiment 145 SAMPLE
Shape Does the Shape Tile a Plane? The Total Degrees in the Angles Where the Shapes Meet Quadrilateral B 4 3 1 2 Yes. Each angle of a square is 90°. 4 (90) = 360 Quadrilateral S 3 2 4 1 Yes. S 1 2 4 3 S 1 2 4 3 S 1 2 4 3 S 1 2 4 3 S 1 2 4 3 S 1 2 4 3 S 1 2 4 3 S 1 2 4 3 1 2 4 3 S 1 2 4 3 S 1 2 4 3 S 1 2 4 3 S 1 2 4 3 S 1 2 4 3 S 1 2 4 3 S 1 2 4 3 S Or 4 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3 4 3 S S S S S S S S S S S S S S S S 35° 145° 180° 1 1 2 2 2 1 2 35 + 145 + 180 = 360 or S S 145° 145° 35° 35° 4 3 2 2 1 1 3 4 1 1 2 2 2(145 + 35) = 360 Or 4 4 S S 90° 90° 90° 90° 3 3 33 44 4 (90) = 360 Quadrilateral 3 2 K 4 1 Yes. 120° 120° 60° 60° 2(60 + 120) = 360 Note: vertex angles formed 3.1 Answers 146 Investigation 3 Designing Polygons: The Angle Connection SAMPLE
Shape Does the Shape Tile a Plane? The Total Degrees in the Angles Where the Shapes Meet Hexagon D 2 34 5 6 1 Yes. 120° 120° 120° 3(120) = 360 Heptagon E 1 2 3 45 6 7 No. The shapes overlap or leave a big gap. Showing gap: Showing overlap: ≈129° ≈129° 102° Angles do not add to 360˚. We need about 102˚ more to complete the full turn of 360. Analyze the Data • Which polygons can be used to cover a flat surface without gaps or overlap? Explain why they fit tightly with no gaps around the points where they meet. The triangle, quadrilaterals, and hexagon tile the plane. They fit without gaps or overlaps because the shapes can be arranged so the angles total 360°, which is a complete rotation. 3.1 Answers Problem 3.1 Back to the Bees: Tiling a Plane Experiment 147 SAMPLE
WHAT IF . . . ? Situation A. Angle Measure and Tiling Maisie makes a claim. Is she correct? Explain why or why not. Maisie’s Claim I was thinking about the bees using regular hexagons. I think that the size of an angle has something to do with regular polygons that tile. I measured the angles of a regular hexagon. Each angle is 120º. So three regular hexagons tile around a point since the sum of the vertex angles is 360º. I looked at other regular polygons. I know that the vertex angles in regular polygons are equal. Regular Polygon Number of Sides Measure of One Angle in Degrees Number of Shapes Needed to Tile Triangle 3 60 6 (60 ) = 360 Square 4 90 4 (90 ) = 360 Pentagon 5 Does not tile Hexagon 6 120 3 (120 ) = 360 I don’t need to try regular polygons with more than six sides. If I put more than two together, the sum of the vertex angles is greater than 360º, which means they will not tile. So the hexagon is the last regular polygon that will tile. Maisie is correct. The regular polygons that tile have vertex angles that are factors of 360. Regular polygons with more sides than a hexagon will not tile. If we look at the pattern in the table, the next regular polygon to tile would need to have an angle measure of 180°. 2 (180) = 360 A regular polygon cannot be made up of 180° angles. Note: The interior angles of regular polygons keep increasing in size as the number of sides increases. The measure of each interior angle approaches but never reaches 180°. The angle measure of regular polygons will be looked at in the next problem. 3.1 Answers 148 Investigation 3 Designing Polygons: The Angle Connection SAMPLE
Regular Polygon Number of Sides Measure of One Angle Does the Regular Polygon Tile? Triangle 3 60° (factor of 360) yes Square 4 90° (factor of 360) yes Pentagon 5 108° Hexagon 6 120° (factor of 360) yes Heptagon 7 ≈ 128.6° Octagon 8 135° Nonagon 9 140° Decagon 10 144° 11-agon 11 ≈ 147.3 ° 12-agon 12 150° Situation B. Tiling with More Than One Polygon Luke wondered if he could use more than one polygon to tile a flat surface. Is this possible? Explain why or why not. Semiregular tilings are possible with several different combinations of polygons. Students may have noticed some while they were tiling the polygons. Or they may have seen these patterns on flooring, tiled walls, or other locations. The most common combination is probably regular octagons and squares. As the following sketch shows, the meeting points of the figures will link two octagon angles of 135° and one square angle of 90° for a total of 360°. Another example using a regular hexagon and equilateral (regular) triangle: In any “mixed” tiling with regular shapes, the angles surrounding any point of the grid will add up to 360°. In the octagon and square example above, that sum is 2 • (135°) + 90° = 360°. In the hexagon and triangle example, the sum around the point is 2 • 120° + 2 • 60° = 360°. 3.1 Answers Problem 3.1 Back to the Bees: Tiling a Plane Experiment 149 SAMPLE
NOW WHAT DO YOU KNOW? Which polygons can be used to tile a surface? Explain why they tile. Give some examples of how tiling polygons can be useful. Polygons either of the exact same shape or mixed with other shapes such that they fit together to form 360° can be used to tile a surface. Tiling (or tessellation) of shapes is possible because they fit into each other without any gaps or overlaps. Bees could use triangles, quadrilaterals, or hexagons for their hives. Hexagons give them more area for storage. Some examples of how tiling polygons is useful might include covering a ceiling with tiles, covering a floor with ceramic shapes, and making a repetitive design for an artistic effect on a wall or in the background of a logo. 3.1 Answers 150 Investigation 3 Designing Polygons: The Angle Connection SAMPLE
Problem 3.1 Back to the Bees: Tiling a Plane Experiment 151 Honeycombs 3.1 TEACHING AID © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. SAMPLE
152 Investigation 3 Designing Polygons: The Angle Connection Name Date Class A 1 3 2 B 1 2 4 3 1 5 2 4 3 C D 6 1 5 2 4 3 E 1 7 2 6 5 4 3 F 8 1 5 4 2 3 7 6 H 4 1 3 2 K 4 1 3 2 J 4 3 1 2 Shapes Set LEARNING AID TEMPLATE © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. SAMPLE
Problem 3.1 Back to the Bees: Tiling a Plane Experiment 153 Name Date Class G 4 3 1 2 I 3 1 2 L 2 4 1 3 V 1 3 4 2 M 4 1 3 2 R Q 4 3 2 1 O 4 3 2 1 P 1 3 2 1 2 4 3 T 1 2 3 U 2 1 4 3 N S 3 4 1 3 2 4 1 2 © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. SAMPLE
154 Investigation 3 Designing Polygons: The Angle Connection Name Date Class Shape Does the Shape Tile a Plane? The Total Degrees in the Angles Where the Shapes Meet Triangle T 1 3 2 Quadrilateral B 4 3 1 2 Quadrilateral S 3 2 4 1 Initial Challenge Table LEARNING AID 3.1 © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. SAMPLE
Problem 3.1 Back to the Bees: Tiling a Plane Experiment 155 Name Date Class Shape Does the Shape Tile a Plane? The Total Degrees in the Angles Where the Shapes Meet Quadrilateral K 3 2 4 1 Hexagon D 2 4 3 5 6 1 Heptagon D 1 2 3 45 6 7 © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. SAMPLE
At a Glance The goal of this problem is to add another piece of information that will help students in constructing geometric shapes by developing a formula that predicts the sum of interior angles for a polygon of n sides. Students begin in the Initial Challenge by tearing the angles off of shapes and rearranging them around a point. The problem goes on to offer three different strategies for making that generalization and using equations to represent these generalizations in What If . . . ? Situation A. What If . . . ? Situations B and C apply their ideas in new situations. Relating Angle Measures to Number of Sides of Polygons Experiment PROBLEM 3.2 Arc of Learning Analysis Synthesis Analysis NOW WHAT DO YOU KNOW? What is the relationship between the angle sum S of a polygon with n sides and the number of sides? How can you find the measure of an angle in a regular polygon with n sides? Key Terms Materials 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 Pacing 2 days Groups 2 students A 4–10 C 26–29 E 39–40 Facilitating Discourse Teacher Moves LAUNCH CONNECTING TO PRIOR KNOWLEDGE Review the definition of a regular polygon. Suggested Questions • What is a regular polygon? • What happens to the measures of the angles as the number of sides increases? Divide the class into groups for What If . . . ? Situation A for each strategy. For What If . . . ? 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. 156 Investigation 3 Designing Polygons: The Angle Connection SAMPLE
Facilitating Discourse (continued) Teacher Moves (continued) LAUNCH PRESENTING THE CHALLENGE Demonstrate Devon’s method of “draw and tear.” Suggested Questions • Does Devon’s strategy make sense? • What if we used Devon’s “draw and tear” method for other polygons? Situation B, divide the shapes among groups in the class, and then they can share their findings in the summary. (ProblemSolving Environment) EXPLORE PROVIDING FOR INDIVIDUAL NEEDS This problem takes an experimental approach to the question about angle sums in polygons to give students more information as they become more proficient in constructing geometric shapes. Students are guided to make measurements of the regular polygons and to look for patterns in those measurements. For students who may need an adaptation for tearing the angles off, you could have them try folding the angles. They will need to number the angles on both sides of their triangle. Then, the angles can be folded in to show that the sum of the angles of a triangle create a 180° angle. With Trevor’s method, check that students see how the angles of subdividing triangles actually add up to the angles of the polygon. With Casey’s method, check that students see how the subdivision into triangles actually includes 360° that are not part of the polygon angles (around the center point). PLANNING FOR THE SUMMARY In describing the relationship that relates the angle sum S to the number of sides n, look for the variety of ways that students may describe this. Use these in the summary. Agency, Identity, Ownership Compare Thinking Selecting and Sequencing SUMMARIZE DISCUSSING SOLUTIONS AND STRATEGIES Display Learning Aid 3.2B: Angle Sum Patterns in Regular Polygons. With input from the class, fill in the missing information. Accept, record, and then discuss all answers. During the summary, a person from each group can present the argument for the reasoning in the strategy they explored from What If . . . ? Situation A. MAKING THE MATHEMATICS EXPLICIT Suggested Questions • What patterns do you notice in the relationship between the number of sides and the angle sum? • So for any polygon, how could I find the angle sum? • How are regular polygons different from irregular polygons? • So if we know the sum of the angle measures for a polygon, how can we find the measure of one angle in a regular polygon? • How would knowing about the “inside” angle sums help you in constructing a geometric shape? • How would you describe Trevor’s strategy? Casey’s? Maria’s? Problem-Solving Environment 3.2 (continued) Problem 3.2 Relating Angle Measures to Number of Sides of Polygons Experiment 157 SAMPLE
Facilitating Discourse (continued) Teacher Moves (continued) • Does the relationship about the angles inside a polygon remain the same for Zane’s shapes in Situation B? • What information are Neveah and Amy using to find the measure of the missing angle in triangle ABC? As you finish the mathematical discussions, have students reflect on the Now What Do You Know? question(s). 3.2 (continued from page 157) 158 Investigation 3 Designing Polygons: The Angle Connection SAMPLE
Problem Overview The goal of this problem is to add another piece of information that will help students in constructing geometric shapes by developing a formula that predicts the sum of interior angles for a polygon of n sides, especially the instances of that pattern for triangles (180°) and quadrilaterals (360°). Knowing these angle sums is important for determining missing angles in constructing triangles and quadrilaterals. For example, if you know that two angles of a triangle are 70º and 50º, then the other angle is 180º − 120º, or 60º. Students begin by examining angle sums of any polygon. Then they look specifically at regular polygons and make an important generalization about the angle sum property. The problem offers different strategies for making and confirming that generalization. While you might find it unnecessary to use all the strategies for thinking about angle sums, each approach is an illustration of a very important practice in geometry—cutting a given figure into smaller pieces (preferably triangles) and showing how the properties of those pieces can be used to get results about the whole figure. Implementation Note: Divide the class into three groups for What If . . . ? Situation A. One group will investigate Trevor’s strategy, one group will investigate Casey’s strategy, and one group will investigate Maria’s strategy. For What If . . . ? Situation B, divide the shapes among groups in the class. Groups can share their findings in the summary. (ProblemSolving Environment) Launch (Getting Started) Connecting to Prior Knowledge Begin by reviewing the definition of a regular polygon. (Language) Suggested Question • What is a regular polygon? (A regular polygon is one in which all sides are the same size and all angles are the same size.) In this problem, students will continue working with angles within polygons. They will be investigating the size of the angles within polygons. EXTENDED LAUNCH—EXPLORE—SUMMARIZE Extended Lanuch—Explore—Summarize 159 SAMPLE
• What happens to the measures of the angles as the number of sides increases? (You don’t need a full answer at this time. This is what students will be investigating. Generate predictions.) Students should understand that the angles being discussed are the angles “inside” the polygon, or interior angles. Interior angle is a term that is not necessary at this point and will be formally introduced in Problem 3.3. Presenting the Challenge Demonstrate Devon’s method of “draw and tear.” 2 1 3 3 2 1 Suggested Questions • Does Devon’s strategy make sense? (Leave this rhetorical so students can investigate it in the problem.) • What if we used Devon’s “draw and tear” method for other polygons? (Leave this rhetorical so students can investigate it in the problem.) Distribute Learning Aid 3.2A: Initial Challenge Shapes and Learning Aid 3.2B: Angle Sum Patterns in Regular Polygons. Have students work with partners. Have Learning Aid 3.2C: Trevor’s, Casey’s and Maria’s Strategies and Learning Aid 3.2D: Zane’s Conjectures available for students to use when they are working on the What If . . . ? situations. Explore (Digging In) Providing for Individual Needs This problem takes an experimental approach to the question about angle sums in polygons to give students more information as they become more proficient in constructing geometric shapes. Students are guided to make measurements of the regular polygons and to look for patterns in those measurements. For students who may need an adaptation for tearing the angles off and keeping track matching the angles around one point, you could try two other methods. Example 1 shows folding the shape to match up the angles. Students will need to number the angles on both sides of their triangle. Then, the angles can be folded in to show that the LES 160 Investigation 3 Designing Polygons: The Angle Connection SAMPLE
sum of the angles of a triangle create a 180° angle. Example 2 below shows comparing the angles to a straight angle of 180°. Students will need to number the angles on both sides of their shape. They can combine, compare, and estimate the angles as they compare to a straight angle. Example 1. Alternate to “tear off” angles. 2 1 3 1 2 3 Fold the “top” angle down. 2 1 3 Fold in the angles on the base. 1 3 Example 2. Alternate to “tear off” angles and surround a point. 1 3 2 4 O 2 1 3 4 ∠1 + ∠2 = 180° ∠3 + ∠4 = 180° Suggested Questions • What do you observe about the sum of the angles of the triangle? (Since the three angles form a straight line, the sum of the angles is 180°.) • How can you arrange the angles of the quadrilateral around a point? One example: 3 1 3 2 4 2 1 4 • Can you make a conjecture about the angle sum of any quadrilateral? (The angle sum of any quadrilateral is 360°.) In Trevor’s method, check that students see how the angles of subdividing triangles actually add up to the angles of the polygon. To assist some groups in moving their conversation forward around seeing that the sum of the angles of a polygon is equal to the sum of the angles in (n − 2) triangles, you can suggest numbering the angles of the triangles using Devon’s method. (Portrayal) LES Extended Lanuch—Explore—Summarize 161 SAMPLE
In Casey’s method, check that students see how the subdivision into triangles actually includes 360° that are not part of the polygon angles (around the center point). A similar numbering to Devon’s method may also help with Casey’s method. Some students may need to be reminded that the sum of the angles around a point is 360º. 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? What is the relationship between the angle sum S of a polygon with n sides and the number of sides? How can you find the measure of an angle in a regular polygon with n sides? (In describing the relationship that relates the angles sum S to the number of sides n, look for the variety of ways that students may describe this. Use these in the summary.) Summarize (Orchestrating the Discussion) Discussing Solutions and Strategies The key result of work on this problem is to find the formula for the angle sum of any polygon. Students might not be able to express this in the most compact symbolic form, 180(n − 2), or even in symbolic form at all, but you might be able to help them get to such an expression. Display Learning Aid 3.2B: Angle Sum Patterns in Regular Polygons. With input from the class, fill in the missing information. Accept, record, and then discuss all answers. (Portrayal) It is likely that many students will have angle measures that are close to the actual measures but not exact. Some students may disagree with the measurements that others give. During the summary, a person from each group can present the argument for the reasoning in the strategy they explored from What If . . . ? Situation A. (Problem-Solving Environment) During the summary, have students share their thinking about the shapes in What If . . . ? Situation B. LES 162 Investigation 3 Designing Polygons: The Angle Connection SAMPLE
Making the Mathematics Explicit Suggested Questions • What patterns do you notice in the relationship between the number of sides and the angle sum? (As the number of sides increases by 1, the angle sum increases by 180.) • So for any polygon, how could I find the angle sum? (Some students may use words: “Take the number of sides, subtract 2, and multiply by 180 to get the angle sum.” Some students may use symbols: 180(n − 2) = S.) • How are regular polygons different from irregular polygons? (All the side lengths and angle measures are equal.) • If we know the sum of the angle measures for a polygon, how can we find the measure of one angle in a regular polygon? (Some students may use words: “Find the sum of all the angles, and then divide that by n.” Some students may use symbols: ________ 180(n − 2) n .) • How would knowing about the “inside” angle sums help you in constructing a geometric shape? (Answers will vary.) • Do you have any What If . . . ? questions? Ask the students if–then questions to help deepen their understanding. (You may need to wait until after the What If . . . ? exploration to use these questions.) (Time) • If a regular polygon has 20 sides, what will be the sum of all the angles in that polygon? Explain why your answer makes sense. (The sum of all the angles is 180(20 − 2), which is 3,240°. Using the triangle approach, we see that 18 triangles can be formed in a 20-sided polygon.) • If a regular polygon has 20 sides, each angle must have how many degrees? Explain why your answer makes sense. (Each angle measures _____ 3,240 20 , or 162°. Since we have a lot more sides, the interior angles are getting wider.) • If you gently encourage students to make observations about patterns in the chart, some may look at the angle sums and observe the relationship to the triangle’s 180° angle sum. You may have a student who extends this relationship by noticing that a square contains two triangles (by drawing one diagonal, 180° × 2 = 360°), a pentagon contains three triangles (by drawing two diagonals from one vertex, 180° × 3 = 540°), and so on. You may want to hold on this conversation until after the What If . . . ? exploration or use it as the Launch into the What If . . . ? LES Extended Lanuch—Explore—Summarize 163 SAMPLE
• How would you describe Trevor’s strategy? (Answers will vary. They should include something about dividing into triangles or groups of 180°.) • How does knowing the number of triangles help you? (We found in the Initial Challenge that all triangles have 180°, so knowing how many of those are in a polygon was important.) • How many triangles or groups of 180° did you find in the polygons? (It was always two less than the number of sides, or [n − 2].) • How did this help you generalize a relationship about the angle sum and the number of sides of a polygon? (Answers will vary. Students may describe in words that the number of triangles formed is two less than the number of sides of the polygon. The angles of the triangles piece together to give the angles of the polygons. Since a triangle has an angle sum of 180°, you would take the number of sides minus two and then multiply that by 180.) • How can we describe this generalization in an equation? (It can be described as S = 180(n − 2).) • How would you describe Casey’s strategy? (Answers will vary. They should include something about creating triangles within the polygon around a center point.) • How did Casey’s strategy help you? (These triangles always came together in a circle, and we knew that a circle has 360°.) • How many triangles did you find for each polygon? (The same number of triangles as the side lengths.) • How did this help you generalize a relationship about the angle sum and the number of sides of a polygon? (Answers will vary. Students may describe in words that in this method you always draw the same number of triangles as you have sides. Each triangle has 180°, but there are 360° in the center of the polygon that aren’t part of the polygon’s angles, so those get subtracted.) • How can we describe this generalization in an equation? (It can be described as S = 180n − 360.) • For Trevor, we have the expression 180(n − 2), and for Casey, we have the expression 180n − 360. Are these expressions equal? (Yes. Students might use evidence from the charts or shapes to justify the equivalence of the expressions. Or students might mention the Distributive Property as evidence that the expressions are equivalent.) • How would you describe Maria’s strategy? (Angles 1, 2, and 3 have a sum of 180°. We can visualize this as a way to describe Maria’s strategy.) LES 164 Investigation 3 Designing Polygons: The Angle Connection SAMPLE
• What do you notice about the sides and angles of the shapes in What If . . . ? Situation B? (The shapes have angles that “poke into” the shape. This creates some interesting shapes. And there are angles larger that 180° inside the shape.) What If . . . ? Situation B introduces polygons that are concave. This is not essential vocabulary for students at this time. Familiar figures like triangles, parallelograms, and trapezoids are called convex polygons. Figures like those pictured in What If . . . ? Situation B are called concave polygons. A concave polygon will always have at least one reflex interior angle—that is, an angle with a measure that is between 180 degrees and 360 degrees. • Do the patterns you discovered for that angle sum of convex polygons work for concave polygons? (Yes.) Discuss the equations for triangle ABC in What If . . . ? Situation C. • Are Amy’s and Neveah’s equations equivalent in Situation C? (Yes.) • What information are Neveah and Amy using to find the measure of the missing angle in triangle ABC? (Amy is using the formula that the sum of interior angles of a triangle is equal to 180 degrees. Neveah seems to be using the understanding that in a right triangle, the sum of two non-90° angles is equal to 90 degrees.) Use a culminating question so students can apply their thinking and answer the Now What Do You Know? question. • Does the pattern relating number of sides, measures of angles, and angle sums apply to all of these shapes? Explain your reasoning. • What is the relationship between the angle sum S of a polygon with n sides and the number of sides? • How can you find the measure of an angle in a regular polygon with n sides? LES Extended Lanuch—Explore—Summarize 165 SAMPLE
INITIAL CHALLENGE Devon used the shapes from Problem 3.1. He suggested starting with a triangle. Relating Angle Measures to Number of Sides of Polygons Experiment PROBLEM 3.2 Devon’s Strategy The Sum of the Angles of a Triangle I began by drawing irregular triangles. I tore the corners off the triangle and then rearranged them. The angles form a straight line or a straight angle. The sum of the angles of a triangle is 180º. 1 1 2 3 3 2 Make a Prediction • Does Devon’s strategy make sense? Will it work on other triangles? Other polygons? Answers will vary based on students’ prior experiences. Conduct the Experiment Equipment › a copy of the Shape Set from Problem 3.1 Directions (individually or with a partner or small group) › Tear off each angle, and rearrange them around a point. › Record the following information in a table. Polygon Number of Sides (n) Angle Sum (S) Triangle Square Pentagon Answers Answers Embedded in Student Edition Problems 166 Investigation 3 Designing Polygons: The Angle Connection SAMPLE
Analyze the Data • What patterns do you observe in the table? Do your observations agree with your predictions? As you increase the number of sides by 1, the angle sum increases by 180. Looking at this pattern in the table, we can see that (n − 2)180 = angle sum. Polygon Number of Sides (n) Angle Sum (S) Triangle 3 180° Square 4 360° Pentagon 5 540° Hexagon 6 720° Heptagon 7 900° Octagon 8 1,080° Nonagon 9 1,260° Decagon 10 1,440° n-gon n 180 • two less than the number of sides or 1180(n − 2) This gives a good opportunity to revisit the Distributive Property and early work on equivalent expressions from the Number Connections unit in grade 6. Note: At this point, we do not expect students to have a deductive argument justifying their formula, just an observation about the pattern in the table values. Polygon Number of Sides (n) Angle Sum (S) Hexagon Heptagon Octagon Nonagon Decagon n-gon +1 180 180 180 180 +1 +1 +1 3.2 Answers Problem 3.2 Relating Angle Measures to Number of Sides of Polygons Experiment 167 SAMPLE
(number of sides − 2) • 180 = angle sum or (n − 2)180 = S or 180(n − 2) = S or 180n − 360 = S • What is the sum of the angles of an octagon? Nonagon? Decagon? N-gon? See table above. • What is the angle measure of regular polygons? Octagon? Nonagon? Decagon? N-gon? For regular polygons, we can take the angle sum and divide it by the number of angles, which is the same as the number of sides, to get the measure of one angle. Regular Polygon Number of Sides (n) Angle Sum (S) Measure of an Angle in a Regular Polygon (A) Triangle 3 180° 180 ÷ 3 = 60° Square 4 360° 360 ÷ 4= 90° Pentagon 5 540° 540 ÷ 5 = 108° Hexagon 6 720° 720 ÷ 6 = 120° Heptagon 7 900° 900 ÷ 7 ≈ 128.6° Octagon 8 1,080° 1080 ÷ 8 = 135° Nonagon 9 1,260° 1260 ÷ 9 = 140° Decagon 10 1,440° 1440 ÷ 10 = 144° n-gon n 180(n − 2) Take the angle sum (180(n − 2)), and divide it equally among the angles, or (180(n − 2)) ÷ n. WHAT IF . . . ? Situation A. Trevor’s, Casey’s, and Maria’s Strategies The following are students’ strategies for finding the sum of the angles of a polygon. Are they correct? Explain why. 3.2 Answers 168 Investigation 3 Designing Polygons: The Angle Connection SAMPLE
Trevor’s Strategy The Sum of the Angles of a Polygon I used Devon’s results for a triangle. I divided polygons into smaller triangles by drawing diagonals from one vertex. Casey’s Strategy I used Devon’s results for triangles. I divided polygons into triangles by drawing line segments from a point within the polygon. Maria’s Strategy I found a different way to find the sum of the angles of a triangle. I used parallel lines L1 and L2, which are intersected by two line segments, AB and AC. L1 L2 2 1 4 B C A 5 3 I used Devon’s thinking and angle relationships to find the sum of the angles of other polygons. Yes. Trevor is decomposing each polygon into triangles drawn from one vertex of the polygon to all the other vertices. Trevor can then use what we know about triangle angle sum to find the angle sum of any polygon. Trevor partitions each polygon into n − 2 triangles (where n is the number of sides of the polygon). The angles of the triangles piece together to give the angles of the polygons. If a triangle has an angle sum of 180°, then a polygon of n sides must have an angle sum of 180(n − 2)°. This will help students make the connection to the equation for angle sum: (n − 2)180 = S. Yes. Casey is decomposing each polygon into triangles drawn from a point inside the polygon to each vertex. Casey can then use what we know about triangle angle sum to find the angle sum of any polygon. Casey partitions each polygon into n triangles. The angles of the polygon are part of those triangles, plus there is always a set of angles at the center with measures summing to 360°, which needs to be subtracted. This will help students to make the connection that the equation from this perspective is 180n − 360 = S. Yes. Maria’s strategy uses angle relationships in parallel lines. ∠1 and ∠4 have the same measure because they have the same relative position (corresponding) as vertex angles. ∠3 and ∠5 have the same measure because they have the same relative position (corresponding) as vertex angles. 3.2 Answers Problem 3.2 Relating Angle Measures to Number of Sides of Polygons Experiment 169 SAMPLE
Situation B: Zane Checks His Conjecture Zane wondered about his group’s conjecture about the sum of the angles of a polygon in the Initial Challenge. What is the sum of the angles in these shapes? Do the measures match what you found about the sum of angles in the Initial Challenge? Shape 1 Shape 2 Shape 3 Shape 4 Yes, strategies will work even on these figures. Shape 1 is a quadrilateral with angle sum of 360°. Possible strategies to test the interior angle sum. 2 • 25° + 275° + 35° = 360° 25° 25° 275° 35° 2 triangles 2 • 180° = 360° 180° 180° Shape 2 is a pentagon with angle sum of 540°. Possible strategies to test the interior angle sum. 2 • 90° + 295° + 35° + 30° = 540° 295° 90° 90° 35° 30° Three triangles 3 • 180° = 540° 180° 180° 180° 3.2 Answers 170 Investigation 3 Designing Polygons: The Angle Connection SAMPLE
Shape 3 is an octagon with angle sum of 1,080°. Possible strategies to test the interior angle sum. 6 • 90° + 2 • 270° = 1,080° 270° 90° Three rectangles 3 • 360° = 1,080° 360° 360° 360° Shape 4 is a decagon with angle sum of 1,440°. Possible strategies to test the interior angle sum. 5 • 40° + 5 • 248° = 1,440° 248° 40° Five triangles plus a pentagon 5 • 180° + 540° = 1,440° 540° 180° 180° 180° 180° 180° Situation C. Right Triangles and Algebra Triangle ABC is a right triangle 40° x A B C Neveah’s and Amy’s Strategies Neveah and Amy use different equations to find the measure of the angle whose measure is x°. Amy wrote: 90 + 40 + x = 180. Neveah wrote 40 + x = 90. 3.2 Answers Problem 3.2 Relating Angle Measures to Number of Sides of Polygons Experiment 171 SAMPLE
1. Are these equations equivalent? How is each student thinking about this problem? Yes, these equations are equivalent. Amy is using the formula that the sum of interior angles of a triangle is equal to 180 degrees. Neveah seems to be using the understanding that in a right triangle, the sum of the two non-90° angles is equal to 90 degrees. 2. Neveah claims that her equation shows that the two acute angles in a right triangle are always complementary. Is she correct? Why? Yes. The sum of all the angles in a triangle is 180°. A right triangle has one that is 90°. This means that the measure of the other two acute angles must add to 90°. That makes the angles complementary. NOW WHAT DO YOU KNOW? What is the relationship between the angle sum S of a polygon with n sides and the number of sides? How can you find the measure of an angle in a regular polygon with n sides? To find the angle sum or measure of an angle, you can use an angle ruler, a protractor, or the equation that relates angle sum to number of sides. The sum of the interior angle measures in a polygon can be calculated by partitioning the polygon into triangles and using the relationship that the sum of angles of a triangle is 180°. The equation to represent the relationship can be written as S = 180(n − 2) or S = 180n − 360. If the polygon is regular, where all the angles are the same measure, then the measure of each angle is (n − 2)180 ÷ n = A or 180 − ____ 360 n = A. The angle sum of a polygon does not change when you change the type (regular, convex, or concave) of polygon. 3.2 Answers 172 Investigation 3 Designing Polygons: The Angle Connection SAMPLE
Angle Sum of Any Polygon 3.2A TEACHING AID © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. Problem 3.2 Relating Angle Measures to Number of Sides of Polygons Experiment 173 Devon’s Strategy 2 1 4 1 3 3 2 1 1 3 2 4 2 3 Trevor’s Strategy Casey’s Strategy SAMPLE
174 Investigation 3 Designing Polygons: The Angle Connection Different-Sized Regular Polygons TEACHING AID 3.2B © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. SAMPLE
Problem 3.2 Relating Angle Measures to Number of Sides of Polygons Experiment 175 Name Date Class 4 3 2 2 1 2 3 4 3 1 1 1 2 5 2 6 4 3 4 1 3 2 1 T K D E S B 7 6 3 5 4 Initial Challenge Shapes 3.2A LEARNING AID © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. SAMPLE
176 Investigation 3 Designing Polygons: The Angle Connection Name Date Class Regular Polygons 1 3 3 4 4 3 1 1 7 2 7 8 1 5 4 6 3 2 6 5 4 3 5 5 4 3 2 6 1 2 1 2 2 A B E F C D Initial Challenge Shapes (continued) LEARNING AID 3.2A © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. SAMPLE
Problem 3.2 Relating Angle Measures to Number of Sides of Polygons Experiment 177 Name Date Class Regular Polygon Number of Sides (n) Angle Sum (S) Measure of an Angle in a Regular Polygon (A) Triangle Square Pentagon Hexagon Heptagon Octagon Nonagon Decagon n-gon Angle Sum Patterns in Regular Polygons 3.2B LEARNING AID © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. SAMPLE
178 Investigation 3 Designing Polygons: The Angle Connection Name Date Class Trevor’s Strategy Casey’s Strategy Maria’s Strategy L1 L2 2 1 4 B C A 5 3 Trevor’s, Casey’s, and Maria’s Strategies LEARNING AID 3.2C © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. SAMPLE
Problem 3.2 Relating Angle Measures to Number of Sides of Polygons Experiment 179 Name Date Class Shape 1 Shape 2 Shape 3 Shape 4 Zane’s Conjectures 3.2D LEARNING AID © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. SAMPLE
At a Glance The goal of Problem 3.3 is to add another piece of information that will help students in constructing geometric shapes by developing student understanding of the concept of exterior angle and to apply that idea to reasoning, from another perspective, about interior angles of polygons. In the Initial Challenge, the context of polygonal bike paths has students investigating sums of exterior angles. In the What If . . . ? situations, students will look at the relationships of interior and exterior angles through analyzing student claims and writing/ solving equations to find angle measures. The Ins and Outs of Polygons: Using Supplementary Angles PROBLEM 3.3 Arc of Learning Analysis Synthesis Analysis NOW WHAT DO YOU KNOW? What do you know about the exterior angles of a polygon? How might this knowledge be useful? Key Terms Materials interior angles exterior angles For each student • Learning Aid 3.3: Pentagonal Path • polystrips (optional for Launch) Pacing 1 day Groups 2 students A 11–15 C 30–32 E 41 Facilitating Discourse Teacher Moves LAUNCH CONNECTING TO PRIOR KNOWLEDGE Have students study the figures in which exterior angles have been drawn. Show the class an example of an exterior and interior angle of a polygon. These two angles come in pairs. Tell the class that there are two sets of exterior angles depending on how you extend the sides of a polygon. The important thing is that the sides have to be extended in the same direction—either all clockwise or all counterclockwise. PRESENTING THE CHALLENGE Ask the class if any of them have done any cycling. You might ask them how angles and the language of angles are used in cycling. Tell the class about the cyclist who is cycling around a path in the shape of a pentagon. You might demonstrate this with a pentagon. 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. 180 Investigation 3 Designing Polygons: The Angle Connection SAMPLE
Facilitating Discourse (continued) Teacher Moves (continued) LAUNCH You can also tape a large pentagon on the floor that resembles the pentagon in the Initial Challenge and have students simulate going around it and making the left turns through an exterior angle. Tell the class that their challenge is to find how many degrees the cyclist bikes through as she bikes once around a pentagon-shaped path. Tell the class that the cyclist is going counterclockwise around the track. Portrayal EXPLORE PROVIDING FOR INDIVIDUAL NEEDS Allowing students to use whiteboards, windows, desks, or laminated chart paper to try their ideas before writing them in their notes can alleviate student fears of being wrong. If students are having some trouble realizing that the sum of exterior angles is always 360°, it might be helpful to lay out a triangle or other polygon with rulers or tape on the classroom floor. Then have students walk around the polygon and notice how their direction changed a total of exactly one full turn. PLANNING FOR THE SUMMARY Find examples from partners of polygons they’ve drawn and found the sum of the exterior angles to use to lead the discussions in the summary. Nonpermanent Work Surfaces Selecting and Sequencing SUMMARIZE DISCUSSING SOLUTIONS AND STRATEGIES Have groups share their ideas from the Initial Challenge. Use student drawings and conjectures to drive the conversations. Have groups discuss the claims in What If . . . ? Situation A. Have students share the measures of the angles in What If . . . ? Situation B. MAKING THE MATHEMATICS EXPLICIT Suggested Questions • What is an exterior angle of a triangle? • What is the sum of the exterior angles in any polygon? How do you know? • What pattern do you see in the sizes of the interior angles as the number of sides increases? • How did you find the measures of the angles represented by w, x, y, and z in What If . . . ? Situation B? As you finish the mathematical discussions, have students reflect on the Now What Do You Know? question(s). Compare Thinking 3.3 Problem 3.3 The Ins and Outs of Polygons: Using Supplementary Angles 181 SAMPLE
Problem Overview The goal of Problem 3.3 is to add another piece of information that will help students in constructing geometric shapes by developing student understanding of the concept of exterior angle and to apply that idea to reasoning from another perspective, about interior angles of polygons. Defining interior and exterior angles is a nontrivial task. It may be helpful for students to see that the interior and exterior angles at a vertex are always supplementary. An interesting fact emerges—the sum of the exterior angles of a polygon is always 360º. Intuitively, this may not be surprising. That is, if you start at one vertex and walk completely around the polygon to the starting vertex, you have in essence done a 360º rotation. In all of the preceding work with angles and polygons, we have focused student attention on convex figures, making all interior angles of measure less than 180°. As the star and arrowhead polygons shown in the Student Edition indicate, there are some figures that seem to qualify as polygons but involve interior angles greater than a straight angle. These shapes are briefly visited as an interesting contrast. Launch (Getting Started) Connecting to Prior Knowledge Have students study the figures in which exterior angles have been drawn. By doing this, students should be clear about the concept of exterior angles. Point out that there are two possible exterior angles at any vertex of a polygon. When looking for a pattern in the measures of exterior angles for any specific polygon, it will help to study angles that are drawn in a consistent direction (counterclockwise or clockwise). Although students have been told to measure angles counterclockwise, Figures 1 and 2 in the introduction to Problem 3.1 shows walking around a polygon in a clockwise direction. If students notice this, you can discuss that they can think of these angles as having the same measurement. The important thing is that the sides have to be extended in the same direction—either all clockwise or all counterclockwise. Show the class an example of an exterior and interior angle of a polygon. These two angles come in pairs. Students might note that the sum of their measures is 180º. The interior and exterior angles at a vertex are always supplementary. EXTENDED LAUNCH—EXPLORE—SUMMARIZE 182 Investigation 3 Designing Polygons: The Angle Connection SAMPLE
exterior angle interior angle Tell the class that there are two sets of exterior angles depending on how you extend the sides of a polygon. The important thing is that the sides have to be extended in the same direction—either all clockwise or all counterclockwise. (Language) Presenting the Challenge Ask the class if any of them have done any cycling. You might ask them how angles and the language of angles are used in cycling. Tell the class about the cyclist who is cycling around a path in the shape of a pentagon. You might demonstrate this with a pentagon. • As the cyclist turns the first corner at vertex B, what angle of turn is made? (About 85º. Student measurements may vary slightly but should be around 85º.) You can also tape a large pentagon on the floor that resembles the pentagon in the Initial Challenge and have students simulate going around it and making the left turns through an exterior angle. (Portrayal) Tell the class that their challenge is to find how many degrees the cyclist bikes through as she bikes once around a pentagon-shaped path. Tell the class that the cyclist is going counterclockwise around the track. See the counterclockwise figure above. Have students work on Problem 3.3 in partners. Learning Aid 3.3A: Pentagonal Path may be helpful for some students. Implementation Note: You may want to divide up the three claims in What If . . . ? Situation A among the groups to save time. They can share their findings in the summary. LES Extended Lanuch—Explore—Summarize 183 SAMPLE
Explore (Digging In) Providing for Individual Needs Allowing students to use whiteboards, windows, desks, or laminated chart paper to try their ideas before writing them in their notes can alleviate student fears of being wrong. If students are having some trouble realizing that the sum of exterior angles is always 360°, it might be helpful to lay out a triangle or other polygon with rulers or tape on the classroom floor. Then have students walk around the polygon and notice how their direction changed a total of exactly one full turn. (Problem-Solving Environment) 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? What do you know about the exterior angles of a polygon? How might this knowledge be useful? Find examples from partners of polygons they’ve drawn and found the sum of the exterior angles to use to lead the discussions in the summary. Summarize (Orchestrating the Discussion) Discussing Solutions and Strategies Have groups share their ideas from the Initial Challenge. Use student drawings and conjectures to drive the conversations. Have groups discuss the claims in What If . . . ? Situation A. Have students share the measures of the angles in What If . . . ? Situation B. Making the Mathematics Explicit Suggested Questions • What is an exterior angle of a triangle? (An exterior angle is an angle at a vertex of a triangle made by extending one side of the triangle. The angle between the extended side and the side of the triangle is an exterior angle. An exterior angle is always adjacent to and supplementary to an interior angle.) • What is the sum of the exterior angles in any polygon? How do you know? (360º. The exterior angles of a triangle, whether added clockwise or counterclockwise, make a full circle.) LES 184 Investigation 3 Designing Polygons: The Angle Connection SAMPLE
• What pattern do you see in the sizes of the interior angles as the number of sides increases? (The sizes of the angles are increasing. They are getting closer and closer to 180°.) • Will they ever equal or be greater than 180°? (No. If they are equal to 180°, then the angles would all lie on a straight line and there would be no polygon. Since the polygon is regular, if one angle is greater than 180°, then all of the angles are greater than 180°. In this case, there would be no way to connect the sides to form a polygon.) • What happens to the shape of the polygon? (It becomes more and more like a circle.) • How did you find the measures of the angles represented by w, x, y, and z in What If . . . ? Situation B? (Answers will vary. One solution might be ∠x is vertical to the 70° angle. So ∠x = 70°. ∠w and ∠60° are supplementary. ∠w + ∠60° = 180° That makes ∠w = 120°. We know the measure of angles x and w. We know ∠x + ∠w + ∠y + 80° = 360. So, ∠70° + ∠120° + ∠y + 80° = 360 ∠y = 360 − 70 − 120 − 80 ∠y = 90°) Display some regular polygons from the Shapes Set. Draw extensions on the sides of the polygons you display. Ask for the measure of each interior angle. Then ask for the sum of the exterior angles. Ask for the measure of each exterior angle, and mark these measures on the polygon. Display an irregular polygon, and label the measures of each interior angle. Draw extensions on the sides of the irregular polygon. Ask for the sum of the exterior angles. The result that the sum of the exterior angles of any polygon is 360º is usually an amazing fact for students, except perhaps for those students who do skateboarding or other activities that involve turning. Now What Do Students Know? Ask students to reflect on the discussion and answer the Now What Do You Know? questions. LES Extended Lanuch—Explore—Summarize 185 SAMPLE
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 186 Investigation 3 Designing Polygons: The Angle Connection SAMPLE
INITIAL CHALLENGE Members of the Columbia Triathlon Club train by bicycling around the polygonal path shown. GPS Route Menu E A D C Bike Path B Pentagonal Path They start at vertex A and go on to vertices B, C, D, and E. Then they return to A and start another lap. At each vertex, the cyclists have to make a left turn through an exterior angle of the polygon. • What is the sum of the left-turn exterior angles that the cyclists make on one full lap? Explain how you can determine an answer without measuring. Then, measure the exterior angles to check your thinking. The sum of the turn angles will be equal to 360°. If we walk around a pentagon like the one given, we will end up facing the same direction that we started. We will have turned a full circle. So we need to test if ∠A + ∠B + ∠C + ∠D + ∠E is equal to 360°. ∠A + ∠B + ∠C + ∠D + ∠E 40° + 75° + 85° + 65° + 95° = 360° More Polygonal Paths • Draw several other polygons, including a triangle, quadrilateral, and hexagon. The Ins and Outs of Polygons: Using Supplementary Angles PROBLEM 3.3 Answers Embedded in Student Edition Problems Answers Problem 3.3 The Ins and Outs of Polygons: Using Supplementary Angles 187 SAMPLE
• Is the angle you turn at each point as you walk around these polygons the same as that for a pentagon? Explain. The angle that you turn can be different for each exterior angle of each polygon. • Find the sum of the turn angles if you cycle around each figure and return to your starting point. The sum of the turn angles is always 360° for any triangle, quadrilateral, pentagon, and hexagon. WHAT IF . . . ? Situation A. More Angle Patterns Students were discussing some patterns they noticed in the pentagonal training track. 1. How does their thinking compare to the strategies you and your classmates used in the Initial Challenge? Answers will vary based on how students thought about computing the sums. The claims made by Neveah, Amy, and Nic may help students to generalize their own claims. 2. Will the claims work for other polygons? Explain. Neveah’s will work for other polygons. Neveah uses variables, which let us change the calculations to match any shape. Nic’s is the same idea as Neveah’s, but his is specific to a triangle. Amy’s idea will work for other polygons, but you would have to change the measurement according to the shape. Her idea is not generalized for any shape. 180° 180° 180° 180° 180° D C A B E Amy’s Claim • There are five supplementary or straight angles in the pentagonal diagram. • The total angle measure for the supplementary angles T is T = 5 • 180°. • If we subtract the interior angle sum (540°) of a pentagon, we get 360°. • This is the sum of the five exterior angles of the pentagon. 5 • 180° − 540°= 360° 3.3 Answers 188 Investigation 3 Designing Polygons: The Angle Connection SAMPLE