Name Date Class Diagonals and Rigidity Shapes 1.3B LEARNING AID Triangles Quadrilaterals Pentagons Hexagons Heptagon Octagon Nonagon © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. Problem 1.3 Rigidity Experiment 39 SAMPLE
At a Glance 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 on the 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 Exploration Pacing __1 2 day Facilitating Discourse Teacher Moves 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 presented here to have students synthesize and record their thinking. Suggested Questions • After this investigation, what do we know about triangles and quadrilaterals? • 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 40 Investigation 1 Designing Triangles Experiment: The Side Connection 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 41 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? To make a shape from given lengths, the sum of the lengths of the shortest sides must be greater than the length of the longest side. Only one triangle can be formed when the three side lengths meet to form a triangle. Many different quadrilaterals can be constructed when the four sides meet to form a quadrilateral. Triangles are rigid figures. Quadrilaterals (and all other polygons with more than three sides) are not. This explains the use of triangles in building structures. Supports that form triangles can be put in polygons with more than three sides to make them more rigid. MR Answers Embedded in Student Edition Problems Mathematical Reflection 42 Investigation 1 Designing Triangles Experiment: The Side Connection SAMPLE
INVESTIGATION 1 Answers Embedded in APPLICATIONS—CONNECTIONS—EXTENSIONS (ACE) APPLICATIONS 1. For each set of side lengths, follow these directions. • If possible, build a triangle with the side lengths. Sketch your triangle. • Tell whether your triangle is the only one that is possible. Explain. • If a triangle is not possible, explain why. a. 5 cm, 5 cm, 3 cm These conditions define a unique isosceles triangle. Students might not use the term isosceles but they should recognize that these sides will make one unique triangle. b. 8 cm, 8 cm, 8 cm These conditions define a unique equilateral triangle. Students might not use the term equilateral but they should recognize that these side will make one unique triangle. c. 7 cm, 8 cm, 15 cm These conditions cannot be satisfied by a triangle, since 7 + 8 = 15. d. 5 cm, 6 cm, 10 cm These conditions define a unique scalene triangle. Students might not use the term scalene but they should recognize that these sides will make one unique triangle. e. 3 cm, 4 cm, 5 cm These conditions define a unique right triangle. Students might not use the term right triangle but they should recognize that these sides will make one unique triangle. f. 2 cm, 4 cm, 6 cm These conditions cannot be satisfied by a triangle, since 2 + 4 = 6. 2. From ACE 1, which sets of side lengths can make each of the following shapes? a. an equilateral triangle (all three sides equal length) Letter b is equilateral. b. an isosceles triangle (two sides equal length) Letter a is isosceles. c. a scalene triangle (no two sides equal length) Letter d is scalene. d. a right triangle (having one right angle) Letter e is a right triangle. Applications—Connections—Extensions (ACE) 43 SAMPLE
3. Multiple Choice Which of the following could not represent the side lengths of a triangle? A. 5 cm, 5 cm, 5 cm 5 + 5 > 5 B. 3 cm, 7 cm, 2 cm 3 + 2 < 7 C. 6 cm, 8 cm, 10 cm 6 + 8 > 10 D. 4 cm, 7 cm, 4 cm 4 + 4 > 7 Answer: B 4. Multiple Choice Which of the following could represent the side lengths of a triangle? A. 2 cm, 1 cm, 1 cm 1 + 1 = 2 B. 2 cm, 3 cm, 4 cm 2 + 3 > 4 C. 7 cm, 12 cm, 3 cm 7 + 3 < 12 D. 4 cm, 4 cm, 10 cm 4 + 4 < 10 Answer: B 5. Name a set of three lengths that would not form a triangle. How do you know that these lengths would not work? Multiple answers. Side a + side b < side c, where c is the longest side. The two shorter sides must be longer than the third, or they will not touch to form a triangle. 6. Name a set of lengths that could be used to form a triangle. How do you know that these lengths would work? Multiple answers. Side a + side b > side c, where c is the longest side. If the two shorter sides are longer than the third, than a triangle can be formed. 7. For each set of side lengths, follow these directions. • If possible, build a quadrilateral with the side lengths. Sketch your quadrilateral. • Tell whether your quadrilateral is the only one that is possible. Explain. • If a quadrilateral is not possible, explain why. a. 5 cm, 5 cm, 8 cm, 8 cm It is possible to build a quadrilateral with sides of the given lengths, but the shape is not unique. Rectangles, nonrectangular parallelograms, and kites are all possible. 5 + 5 + 8 > 8 b. 5 cm, 5 cm, 6 cm, 14 cm It is possible to build quadrilaterals with these side lengths, including an isosceles trapezoid. 5 + 5 + 6 > 14 c. 4 cm, 3 cm, 5 cm, 14 cm Not possible since 4 + 3 + 5 < 14. ACE 44 Investigation 1 Designing Triangles Experiment: The Side Connection SAMPLE
AACECE d. 8 cm, 8 cm, 8 cm, 8 cm Many rhombuses are possible, including a square: 8 + 8 + 8 > 8. e. 5 cm, 5 cm, 5 cm, 20 cm Not possible since 5 + 5 + 5 < 20. 8. From ACE 7, which sets of side lengths can make each of the following shapes? a. a square Part (d) could be a square. b. a quadrilateral with all angles the same size parts (a) and (d) c. a parallelogram parts (a) and (d) d. a quadrilateral that is not a parallelogram parts (a) (could be a kite) and (b) 9. A quadrilateral with four equal sides is called a rhombus. Which set(s) of side lengths from ACE 7 can make a rhombus? part (d) 10. A quadrilateral with just one pair of parallel sides is called a trapezoid. Which sets of side lengths from ACE 7 can make a trapezoid? part (b) 11. Multiple Choice Which of the following could not represent the side lengths of a quadrilateral? A. 5 cm, 5 cm, 5 cm, 5 cm 5 + 5 + 5 > 5 B. 3 cm, 7 cm, 2 cm, 1cm 3 + 2 + 1 < 7 C. 6 cm, 8 cm, 10 cm, 4 cm 4 + 6 + 8 > 10 D. 4 cm, 7 cm, 4 cm, 7 cm 4 + 4 + 7 > 7 Answer: B 12. Multiple Choice Which of the following could represent the side lengths of a quadrilateral? A. 3 cm, 1 cm, 1 cm, 1 cm 1 + 1 + 1= 3 B. 2 cm, 3 cm, 4 cm, 5 cm 2 + 3 + 4 > 5 C. 7 cm, 12 cm, 3 cm, 1 cm 7 + 3 + 1 < 12 D. 2 cm, 2 cm, 10 cm, 2 cm 2 + 2 + 2 < 10 Answer: B 13. Draw the polygons described to help you answer the questions. a. To build a square, what must be true of the side lengths? Squares require all sides to be the same length. b. Suppose you want to build a rectangle that is a square. What must be true of the side lengths? Squares require all sides to be the same length. All squares are special rectangles. Applications—Connections—Extensions (ACE) 45 SAMPLE
ACE c. Suppose you want to build a rectangle that is not a square. What must be true of the side lengths? Rectangles that are not squares require opposite sides to be the same length but not all four the same length. 14. Think about your polystrip experiments with triangles and quadrilaterals. What explanations can you now give for the common use of triangular shapes in structures like bridges and towers for transmitting cell phone and television signals? The cross braces that turn quadrilaterals into linked triangles provide rigidity to the structures. CONNECTIONS 15. Copy the fractions. Insert <, >, or = to make a true statement. a. __5 12 __9 12 b. ___ 15 35 ___ 12 20 c. __7 13 ___ 20 41 d. ___ 45 36 ___ 35 28 a. __5 12 < ___ 12 20 b. ___ 15 35 < ___ 12 20 c. __7 13 > ___ 20 41 d. ___ 45 36 = ___ 35 28 46 Investigation 1 Designing Triangles Experiment: The Side Connection SAMPLE
AACCEE 16. Find two equivalent fractions for each fraction. Find one fraction with a denominator less than the one given. Find another fraction with a denominator greater than the one given. a. __4 12 Answers will vary. In some sense, nearest of each type would be __3 9 and __5 15 . b. __9 15 Answers will vary. In some sense, nearest of each type would be __6 10 and ___ 12 20 . c. ___ 15 35 Answers will vary. In some sense, nearest of each type would be ___ 12 28 and ___ 18 42 . d. ___ 20 12 Answers will vary. In some sense, nearest of each type would be __15 9 and ___ 25 15 . 17. Consider the inequality x < 20. Determine whether the following values for x make this inequality true or false. For each value, explain your reasoning. a. 7 True. The value 7 is less than 20. b. 20. False. The value 20 is equal to (not less than) 20. c. 28 False. The value 28 is greater than 20. 18. Consider the inequality x > 50. Determine whether the following values for x make this inequality true or false. For each value, explain your reasoning. a. 7 False. The value 7 is less than 50. b. 50 False. The value 50 is equal to (not less than) 50. c. 51 True. The value 51 is greater than 50. 19. The following diagram shows two 1-acre squares (Sections 18 and 19) of Tupelo Township. Section 18 Section 19 Lapp Bouck Wong Krebs Stewart Fitz Fuentes Gardella Burg Walker Foley Theule a. Each 1-acre section is about a 70-yard by 70-yard square. How many square feet of land does Lapp own? b. What is the average amount of land owned by landowners in Section 18? Applications—Connections—Extensions (ACE) 47 SAMPLE
ACE Solution: a. 1,225 yd2. Lapp owns __1 4of an acre. A full acre square is about 70 × 70 = 4,900 square yards. So, __1 4of 4,900 is 1,225 square yards. b. __1 8acre or 612.5 yd2. The total amount of land in Section 18 is 1 acre, and there are 8 landowners, so the average is __1 8 . 20. Find the area and perimeter of each rectangle. 6 cm 2 cm 2 cm 3 cm 2 cm 3 cm 3 cm L P R M N Q 4 cm 1 cm 2 cm 7 cm 8 cm L: area = 24 square cm, perimeter = 20 cm M: area = 4 square cm, perimeter = 8 cm N: area = 6 square cm, perimeter = 10 cm P: area = 7 square cm, perimeter = 16 cm Q: area = 9 cm2, perimeter = 12 cm R: area = 16 cm2, perimeter = 20 cm EXTENSIONS 21. Compare the three quadrilaterals. How are the quadrilaterals alike and different? Alike: All three are parallelograms. This means that opposite sides are parallel and Quadrilateral 1 Quadrilateral 2 Quadrilateral 3 congruent and that opposite angles are congruent. All three have the same height. Different: Rectangle 2 does not have all equal length sides, making it different from the square; parallelogram 3 does not have four right angles, making it different from both the square and the rectangle. 48 Investigation 1 Designing Triangles Experiment: The Side Connection SAMPLE
ACACEE 22. Which of the following statements are true? Be able to justify your answers. a. All squares are rectangles. True b. No squares are rhombuses. False c. All rectangles are parallelograms. True d. Some rectangles are squares. True e. Some rectangles are trapezoids. False f. No trapezoids are parallelograms. True. Note: By our chosen definition, a trapezoid is a quadrilateral with one and only one pair of parallel sides. g. Every quadrilateral is a parallelogram, a trapezoid, a rectangle, a rhombus, or a square. False 23. Build the following figure from polystrips. The vertical sides are all the same length. The distance from B to C equals the distance from E to D. The distance from B to C is twice the distance from A to B. F E D A B C a. Experiment with holding various strips fixed (one at a time) and moving the other strips. In each case, tell which strip you held fixed, and describe the motion of the other strips. Regardless of the strip held fixed, the figure will move in a way that keeps all parallelism relationships intact. b. Fix a strip between points F and B, and then try to move strip CD. What happens? Explain why this occurs. If you introduce a bracing strip between points F and B, the whole figure will become rigid. Applications—Connections—Extensions (ACE) 49 SAMPLE
Name Date Class Checkup 1 For each set of side lengths, is it possible to make a shape? 1. Side Lengths of 6, 6, 4 a. Is it possible to build a triangle with these side lengths? YES or NO b. Explain why or why not. Include a sketch of the shape. c. If yes, is there more than one triangle? Explain. 2. Side Lengths of 3, 5, 10 a. Is it possible to build a triangle with these side lengths? YES or NO b. Explain why or why not. Include a sketch of the shape. c. If yes, is there more than one triangle? Explain. © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. 50 Shapes and Designs SAMPLE
Name Date Class 3. Side Lengths of 2, 6, 7 a. Is it possible to build a triangle with these side lengths? YES or NO b. Explain why or why not. Include a sketch of the shape. c. If yes, is there more than one triangle? Explain. 4. Side Lengths of 6, 6, 9, 9 a. Is it possible to build a quadrilateral with these side lengths? YES or NO b. Explain why or why not. Include a sketch of the shape. c. If yes, is there more than one quadrilateral? Explain. Checkup 1 51 © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. SAMPLE
Name Date Class 5. Side Lengths of 7, 7, 7, 7 a. Is it possible to build a quadrilateral with these side lengths? YES or NO b. Explain why or why not. Include a sketch of the shape. c. If yes, is there more than one quadrilateral? Explain. 6. Side Lengths of 3, 5, 6, 15 a. Is it possible to build a quadrilateral with these side lengths? YES or NO b. Explain why or why not. Include a sketch of the shape. c. If yes, is there more than one quadrilateral? Explain. © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. 52 Checkup 1 SAMPLE
Checkup 1: Answers For each set of side lengths, is it possible to make a shape? 1. Side Lengths of 6, 6, 4 a. Is it possible to build a triangle with these side lengths? YES or NO b. Explain why or why not. Include a sketch of the shape. Possible Answers: It is an isosceles triangle. 6 units 6 units 4 units c. If yes, is there more than one triangle? Explain. There is only triangle that is possible from any three lengths. 2. Side Lengths of 3, 5, 10 a. Is it possible to build a triangle with these side lengths? YES or NO b. Explain why or why not. Include a sketch of the shape. Possible Answers: The sum of any two sides is not longer than the other side: 3 + 5 < 10 5 units 3 units 10 units c. If yes, is there more than one triangle? Explain. None possible. Checkup 1: Answers 53 © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. SAMPLE
3. Side Lengths of 2, 6, 7 a. Is it possible to build a triangle with these side lengths? YES or NO b. Explain why or why not. Include a sketch of the shape. Possible Answers: The sum of any two sides is longer than the other side. 7 units 6 units 2 units c. If yes, is there more than one triangle? Explain. There is only triangle that is possible from any three lengths. 4. Side Lengths of 6, 6, 9, 9 a. Is it possible to build a quadrilateral with these side lengths? YES or NO b. Explain why or why not. Include a sketch of the shape. Possible Answers: The sum of three sides is larger than the other side. Same order of side measures with different angle measures (parallelogram and rectangle) 9 9 6 6 9 9 6 6 Different order of side measures (kite) 6 6 9 9 c. If yes, is there more than one quadrilateral? Explain. Not unique; more than one quadrilateral possible © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. 54 Checkup 1: Answers SAMPLE
5. Side Lengths of 7, 7, 7, 7 a. Is it possible to build a quadrilateral with these side lengths? YES or NO b. Explain why or why not. Include a sketch of the shape. Possible Answers: The sum of three sides is larger than the other side. No variation in order of side measures, but different shapes because of angle measures 7 units 7 units 7 units 7 units 7 units 7 units 7 units 7 units 7 units 7 units c. If yes, is there more than one quadrilateral? Explain. Not unique; more than one quadrilateral possible (a square and many parallelograms) 6. Side Lengths of 3, 5, 6, 15 a. Is it possible to build a quadrilateral with these side lengths? YES or NO b. Explain why or why not. Include a sketch of the shape. Possible Answers: The sum of three sides is less than the other side. 3 + 5 + 6 < 15 6 units 3 units 15 units 5 units c. If yes, is there more than one quadrilateral? Explain. None possible © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. Checkup 1: Answers 55 SAMPLE
Implementation Key Terms Materials Resources Problem 2.1 Groups of 2 Pacing 2 days angle benchmark angle degree right angle angle measures For each pair of students • Learning Aid 2.1A: Four in a Row Game Boards For each student • Learning Aid 2.1B: Angles For the class • Teaching Aid 2.1: Introduction to Angle Measures Problem 2.2 Groups of 2 Pacing 1–2 days For each student • Learning Aid 2.2A: Bees and Angle Measurements • Learning Aid 2.2B: Amelia Earhart For the class • Teaching Aid 2.2: Using an Angle Ruler Problem 2.3 Think, Pair, Share Pacing 2 days parallel vertical angles supplementary complementary 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: Compliment Angles (optional) Mathematical Reflection Whole Class Individual Notes Pacing 1 day For the class • large poster paper (optional) word bank created by students and/or teacher (optional) Assessment Partner Quiz Partners Pacing __1 2 day For each pair of students • Partner Quiz INVESTIGATION 2 PLANNING CHART Designing with Angles INVESTIGATION 2 Materials for All Investigations: calculators; student notebooks; colored pens, pencils, or markers 56 SAMPLE
NOW WHAT DO YOU KNOW? How are benchmark angles useful in solving problems? Key Terms Materials angle degree right angle angle measures For each pair of students • Learning Aid 2.1A: Four in a Row Game Boards For each student • Learning Aid 2.1B: Angles For the class • Teaching Aid 2.1: Introduction to Angle Measures Pacing 1–2 days Groups 2 students A 1–6 C 20–22 E 28–29 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. *After students have played the game several times and worked on the Initial Challenge questions, stop them for a brief discussion before sending them off into the rest of the problem. Be sure that students have the right idea about rotation and angle measurement, especially for angles larger than 90°. Facilitating Discourse Teacher Moves LAUNCH CONNECTING TO PRIOR KNOWLEDGE Problem 2.1 extends students’ understanding of angles from their elementary experiences. To solve problems constructing polygons, it is important for students to have a sense of angles as rotations and to be able to estimate angle size. PRESENTING THE CHALLENGE Refer students to the two grids on Learning Aid 2.1A: Four in a Row Game Boards. Point out that these grids have concentric circles and lines forming angles. Suggested Questions • The name of the game is Four in a Row. How do you think you play the game? • What are the measures of some of the angles in this grid? • Using these grids, describe a point by giving two numbers. The first number tells how far to move from the center of the grid. The second number tells the amount of turn measured in degrees. How would I find the point (3, 90°)? You might find it helpful to play one game with the teacher against the whole class first in order to make the rules clear. Implementation Note* You might want to assign one or two angles from What If . . . ? Situation C to each group to save time. Problem 2.1 Four in a Row Game: Angles and Rotations 57 Four in a Row Game: Angles and Rotations At a Glance The goals of this problem are to develop student understanding of angles as rotations or turns, as a geometric shape, and expand their understanding of common benchmark angles that are multiples of 30°. In the Initial Challenge, students will play a game called Four in a Row, where the object is to get four in a row on a circular grid with 30° or 45° intervals. In the What If . . . ? situations, students will use this understanding to estimate given angles and draw angles. Arc of Learning Exploration Analysis Exploration PROBLEM 2.1 SAMPLE
Facilitating Discourse (continued) Teacher Moves (continued) EXPLORE PROVIDING FOR INDIVIDUAL NEEDS Suggested Questions • Is the angle greater than, less than, or equal to 90°? • Is it less than 180°? Is it greater than 180°? • Is it less than 270°? • For What If . . . ? Situation B, how might using 30°, 45°, 90°, 180°, 270°, and so on help you? For students that are ready for an extra challenge, sketch these angles: Angle 6: two-thirds of a straight angle Angle 7: one and two-thirds times a right angle Angle 8: one and __1 6 times a straight angle PLANNING FOR THE SUMMARY As students are working on the What If . . . ? questions, look for ways they are using benchmark angles to draw new angles. Have students share their strategies in the summary. Agency, Identity, Ownership SUMMARIZE DISCUSSING SOLUTIONS AND STRATEGIES Have students share their estimates for the angle measures in What If . . . ? Situation A, their sketches for What If . . . ? Situation C, and how they solved the equations in What If . . . ? Situation D. MAKING THE MATHEMATICS EXPLICIT Suggested Questions • What is the angle measure of the first figure in What If . . . ? Situation A? The second figure? • What benchmark angles were helpful to you? • What made these benchmark angles helpful to you? • How did you know how to draw the angles in What If . . . ? Situation C? • How did you solve for N in What If . . . ? Situation D? As you finish the mathematical discussions, have students reflect on the Now What Do You Know? question(s). Compare Thinking 2.1 58 Investigation 2 Designing with Angles SAMPLE
Problem Overview The goals of this problem are to develop student understanding of angles as rotations or turns, as a geometric shape, and expand their understanding of common benchmark angles that are multiples of 30°. To help students estimate angle measure, a degree is defined as ___1 90of a right angle. This problem will confront directionality and build estimation skills for angle measurement. The problem is also a review or introduction, depending on a student’s prior experience, of the conventions for labeling angles. Launch (Getting Started) Connecting to Prior Knowledge Problem 2.1 extends students’ understanding of angles from their elementary experiences. To solve problems constructing polygons, it is important for students to have a sense of angles as rotations and to be able to estimate angle size. This sense of angle size is needed for solving problems involving angles, too. Presenting the Challenge Explain to students that, in the same way we use benchmark fractions to help in reasoning with rational numbers, we use benchmark angles to help with reasoning about angles. (Language) Suggested Question • Can you jump and turn through angles of 90°, 180°, 270°, or even 360°? You could also have students demonstrate how they would jump and turn for each of these angles. Refer students to the two grids on Learning Aid 2.1A: Four in a Row Game Boards. Point out that these grids have concentric circles and lines forming angles. Have students look closely at the grid on the left. • What are the measures of some of the angles in this grid? Look at angles with a vertex at the center of the grid. (Point out the 45° angle, 90° angle, 135° angle, and so on. Help students to see that the angles are all multiples of 45°.) • The name of the game is Four in a Row. How do you think you play the game? (Answers will vary.) EXTENDED LAUNCH—EXPLORE—SUMMARIZE Extended Launch—Explore—Summarize 59 SAMPLE
Use two polystrips (or an angle ruler) to form an angle. Start with an angle of zero degrees, and then gradually open it up. Continue to rotate one of the polystrips (or opening the angle ruler), creating a larger and larger angle. Stop after a 45°, 90°, 135°, and 180° angle, and ask about the measure of each angle. Now have students look at the grid on the right, and ask the following: • What are the measures of some of the angles in this grid? (Point out the 30° angle, 60° angle, 90° angle, 120° angle, and so on. Help students to see that the angles are all multiples of 30°.) • Using these grids, describe a point by giving two numbers. The first number tells how far to move from the center of the grid. The second number tells the amount of turn measured in degrees. How would I find the point (3, 90°)? (Using either grid, help students to see that to locate this point, you move out 3 units in the positive direction along the 0° line (x-axis) and then rotate counterclockwise 90° along that circle.) The Four in a Row game will help students develop angle benchmarks. You might find it helpful to play one game with the teacher against the whole class first in order to make the rules clear. If students suggest a pair of numbers that don’t work, the teacher can dramatically show how it falls off the grid. Then, suggest that students play several more games in the Initial Challenge before moving on to the rest of the problem. Pass out Learning Aid 2.1A: Four in a Row Game Boards (one per partner group) and Learning Aid 2.1B: Angles. Explore (Digging In) Providing for Individual Needs As students play several games against each other, circulate to see that they are reading the coordinates correctly. Remind them to think about strategies for winning the game. Look for students’ ability to estimate and reason about angle size. Also, have them note any winning strategies. Suggested Questions • Is the angle greater than, less than, or equal to 90°? (Answers will depend on whether the angle is acute, right, or obtuse.) • Is it greater than 90°? Is it less than 180°? (Answers will depend on whether the angle is acute or obtuse.) • Is it greater than 180°? Is it less than 270°? (Answers will depend on whether the angle is obtuse or a fraction of a rotation greater than a straight angle.) 60 Investigation 2 Designing with Angles LES SAMPLE
• For What If . . . ? Situation A, how might using 30°, 45°, 90°, 180°, 270° (and so on) help you? For What If . . . ? Situations B and C, some students may find it helpful to have the grid from the Four in a Row game to help them visualize the angle measures. For students that are ready for an extra challenge, have them sketch the following angles: Angle 6: __ 2 3of a straight angle Angle 7: 1 __ 2 3 times a right angle Angle 8: 1 __1 6 times a straight angle (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? How are benchmark angles useful in solving problems? (As students are working on the What If . . . ? questions, look for ways they are using benchmark angles to draw new angles. Have students share their strategies in the summary.) Summarize (Orchestrating the Discussion) Discussing Solutions and Strategies Discuss any of the points A, B, C, D, or E that were troublesome for your students to place on the grid in the Initial Challenge. Stress that angles are shapes, and discuss how to label angles using ∠ABC with B being the vertex of the angle. Using the points they placed on their grids for A, B, C, D, and E, have them name each angle. Suggested Question • How could you use the Four in a Row game board to help you estimate the measures of the angles in the What If . . . ? (Answers will vary. Don’t press for perfect answers at this time. It is meant to be a way to spark ideas for students to get started.) Have students share their estimates for the angles measures in What If . . . ? Situation A. Have them discuss how they got their estimates. Have students discuss the student claims in What If . . . ? Situation B. Extended Launch—Explore—Summarize 61 LES SAMPLE
Have students share their sketches for What If . . . ? Situation C. Discuss how they drew the angle to match the given measure or criteria. Have students share how they solved the equations in What If . . . ? Situation D to find N. Making the Mathematics Explicit Suggested Questions • What is the angle measure of the first figure in What If . . . ? Situation A? (≈ 125°) • Repeat for the other angles. • What benchmark angles were helpful to you? (Answers will vary.) • What made these benchmark angles helpful to you? (Answers will vary.) For What If . . . ? Situation B, have students share their ideas about each claim. Focus on the claims that caused the most conversations during the Explore. • How did you know how to draw the angles in What If . . . ? Situation C? (Answers will vary. Look for students using benchmarks. Look for how students are using the information in the clues given to find the new angle measure.) • How did you solve for N in What If . . . ? Situation D? (Answers will vary. Students will not have a formal way to solve equations at this time. Look for students to use fact families or guess and check. This is fine for now. As they describe their process, you can script it to start leading them toward a more formal way to write their ideas, which will be developed in Moving Straight Ahead later in the year.) • For example: __1 3 N = 20° A student might say that they knew that __1 3of some number would be 20° and to them that meant that a number was being divided into three parts and resulting in 20°. Using fact families, they knew that 20 × 3 would get them to N. You might write: __1 3 N = 20° is the same as N ÷ 3 = 20°. 20 × 3 = N N = 60 Have students sketch the angle for N for each card and explain how it fits with the equation. 62 Investigation 2 Designing with Angles LES SAMPLE
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? Extended Launch—Explore—Summarize 63 LES SAMPLE
Equipment › Four in a Row Game Boards › pencil, pen, or marker Rules Teams can be partners or small groups. › Team 1 chooses a point where a circle and grid line meet. Team 1 says the coordinates of the point. › Team 2 makes sure that the coordinates are correct. If they are, Team 1 marks the point with an X. If they are not, Team 1 does not mark the point. › Team 2 chooses a point where a circle and grid line meet. Team 2 says the coordinates of the point. › Team 1 makes sure that the coordinates are correct. If they are, Team B marks the point with an O. If they are not, Team 2 does not mark the point. › Players repeat the steps. › The first team to get four marks in a “row,” either along a grid line or around a circle, wins the game. • Play Four in a Row several times. Play games with the 30° and the 45° grids. Describe any winning strategies you used. Student strategies will vary. • Use the circular grids to show angles A, B, C, D, and E described here. Will everyone in class have the same points marked to show the angles? Why or why not? Angles A and C have a range of answers, so there are many answers for those angles. (Shown below by the shaded answer key.) Angles B, D, and E are all the same-size angles. 64 Investigation 2 Designing with Angles INITIAL CHALLENGE Four in a Row Game Answers Embedded in Student Edition Problems Four in a Row Game: Angles and Rotations PROBLEM 2.1 Answers SAMPLE
Angle A The angle measure for angle A is greater than 120°. Angle B The angle measure for angle B is equal to 0. Angle C The angle measure for angle C is less than 90°and greater than 30°. Angle D Angle D is a rotation of 1.5 turns. Angle E Angle E is a rotation of 2 turns. Angle A ∠A > 120° can go anywhere in the shaded region on each drawing. D 45° 0° 0 1 2 3 0 60° E 30° 0° 1 2 3 Angle B ∠B = 0° is the ray on the horizontal axis to the right of the origin. (See red below.) 0 D 45° 0° 1 2 3 0 60° E 30° 0° 1 2 3 Angle C 30° < ∠C < 90° can go anywhere in the shaded region on this drawing. D 45° 0° 0 1 2 3 0 60° E 30° 0° 1 2 3 Problem 2.1 Four in a Row Game: Angles and Rotations 65 2.1 Answers SAMPLE
Angle D ∠D = 540° Students will need to show a turn of 1.5 • 360° = 540°. D 45° 0° 1 2 3 0 60° E 30° 0° 1 2 3 Angle E ∠E = 720° Students will need to show a turn of 2 • 360° = 720°. 0 D 45° 0° 1 2 3 0 60° E 30° 0° 1 2 3 WHAT IF . . . ? Situation A. Using Benchmark Angles to Estimate 1. On a circular grid, mark the benchmark angles 30°, 45°, 60º, 90°, 120°, 180°, and 270°. 2. Amit claims he used benchmark angles to estimate the measure of each angle. Is that possible? Explain. Angle W or ∠W Angle X or ∠X W X Angle Y or ∠Y Angle Z or ∠Z Y Z 66 Investigation 2 Designing with Angles 2.1 Answers SAMPLE
Angle W The measure of ∠W ≈ 120°. ∠W seems about a third of the whole 360° turn. It is about the size of angle A in the Initial Challenge. 0 60° 90° E About 30° 30° 0° 1 2 3 W Angle X ∠X ≈ 240° This almost looks like the first picture was flipped over. So the angle will be the whole rotation minus the measure of ∠W, or 360° − 120°. The angle is not the remainder of ∠W, but it is a good estimate without measuring or placing the angles together. X W Angle Y ∠Y ≈ 90° ∠Y looks like it is about __1 4 the way around a complete turn. So it would be about 270°. Also, it looks like you can match a “corner” of a piece of paper to the angle. Angle Z ∠Z ≈ 65° ∠Z is about halfway between 90° and 45°. 0 Z D 45° 0° 1 2 3 0 Z E 30° 0° 60° 1 2 3 Problem 2.1 Four in a Row Game: Angles and Rotations 67 2.1 Answers SAMPLE
Situation B. Students’ Estimation Claims Examine each claim. Do you agree with them? Explain why. Juan’s Claim In Situation A, angle W is about 30 degrees more than a __1 2 turn. Maria’s Claim In Situation A, angle W is about 45º less than a __1 2 turn. Disagree. He would be closer if he said, “Angle W is about 30° less than a __1 2 turn.” 30 degrees less than a __1 2 turn or __1 2(360) − 30 = 150 Also, the estimate of 30° less is too little to take away. It is closer to 45° less than a half turn. It looks more like the angles on the 45° interval Four in a Row game board. W ≈45° So it is closer to __1 2 turn − 45° ≈ 135°. Agree. The “amount missing” to create the straight angle looks more like the angles on the 45° interval Four in a Row game board. W ≈45° __1 2(360) − 45 = 135 Agree. Tabia: 30 degrees more than a __1 4 turn or 30 + __1 4(360) = 120 Just estimating visually, it might appear that the measure beyond the 90° may be 30°. The measure is larger than 30°. 60° 90° W 30° Turn 1 4 ≈35° 0° Various answers. Example answers might include: 220° About halfway between 180 (__1 2 turn) and 270 (__3 4 turn) 50° 45° is halfway between 0° and 90°. 50° is a few degrees more than 45°, which is the first interval in the 45° interval Four in a Row game board. 130° If we think about the 30° interval on the Four in a Row game board, the marks are at 30, 60, 90, 120. S, 130° is a little more than the 120° mark on the board. Tabia’s Claim Angle W is about 30 degrees more than a __1 4 turn. Amit’s Benchmark Claim I can use benchmark angles to sketch a rotation angle with approximately the following measures. 220° 50° 130° 300° 68 Investigation 2 Designing with Angles 2.1 Answers SAMPLE
Angle 1 One-third of a right angle of 90 = 30 1 3 30° Angle 2 One and a half times a right angle 1.5 • 90 = 135 135° Angle 3 Three times a right angle 270° 3 • 90 = 270 Angle 4 Three and a half times a right angle 315° 3.5 • 90 = 315 Or we can think of a right angle or a corner of a piece of paper (90°) added to a third of a right angle. 300° 30° more than a __3 4 turn. Situation C. Sketching Angles Kwun challenges his classmates to sketch an angle with the following directions. Sketch each angle, and label its measure in degrees. Angle 1 Angle 2 Angle 3 Angle 4 Angle 5 One-third of a right angle One and a half times a right angle Three times a right angle Three and a half times a right angle Twice a straight angle Answers Angle 5 Twice a straight angle 360° 2 • 180 = 360 Problem 2.1 Four in a Row Game: Angles and Rotations 69 2.1 Answers SAMPLE
Situation D. The Algebra Connection Hank was designing angles and remembers the Algebra Connections game he created for some of the grade 6 units. He added a few more cards to his Algebra Connections game. 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. Card 1 Card 2 Card 3 What is N? N + 120° + 360° N is a rational number. What is N? N is a rational number. 3 2 N = 45° What is N? N is a rational number. 1 3 N = 20° N = 60° N = 30° N = 240° NOW WHAT DO YOU KNOW? How are benchmark angles useful in solving problems? Benchmarks are helpful to estimate and sketch angle measures. Students should have a set of benchmark angles: 360° (1 turn), 270° (__3 4 turn), 180° ( __1 2 turn), 120° ( __1 3 turn), 90° ( __1 4 turn), 45° ( __1 8 turn), 30° ( __1 12 turn). 70 Investigation 2 Designing with Angles 2.1 Answers SAMPLE
Ways Angles Are Formed Wedge ray ray vertex Vertex and Two Rays vertex ray ray Rotation 90° 60° 45° 30° 15° 0°/360° 330° 315° 270° 285° 240° 220° 200° 180° 165° 135° 110° 60° angle of rotation Marking and Naming Angles Curved arcs indicate the counterclockwise rotation. Since there are two angles indicated by two rays, curved lines are also used to indicate which angle is being used. A P B A P B We can identify an angle with three letters. angle BPA or ∠BPA Or with just the vertex angle P or ∠P angle APB or ∠APB Or with just the vertex angle P or ∠P Note: As the first unit in grade 7, we will name angles by just the vertex in Shapes and Designs. © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. Problem 2.1 Four in a Row Game: Angles and Rotations 71 Introduction to Angle Measures 2.1 TEACHING AID SAMPLE
Angle Measures A one-quarter rotation is 90°. A right angle measures 90°. Right angles are commonly marked with a small square. 90° __1 4 Turn 90° 60° 45° 30° 15° 0°/360° 330° 315° 270° 285° 240° 220° 200° 180° 165° 135° 110° 90° 90° 90° 1/4 Turn 1/4 Turn 1/4 Turn 1/4 Turn 90° Divide a right angle into two angles of equal measure. Each angle would be a 45° angle. 45° __1 8 Turn 90° 60° 45° 30° 15° 0°/360° 330° 315° 270° 285° 240° 220° 200° 180° 165° 135° 110° 45° 45° 45° 45° 45° 45° 45° 45° 1/8 Turn Draw 89 rays to divide a right angle into 90 angles of equal measure. Each angle would have a measure of 1°. ____ 1 360 Turn 45° 45° 15° 15° 15° 5° 5° 5° 1° 1° 1° 1° 1° 1° A rotation of one-half turn defines a straight angle. It measures 180°. __1 2 Turn 180° 180° Angle Measure Names acute angles measure < 90° right angle = 90° 90° < obtuse angles measure < 180° straight angle = 180° 180° < reflex angles measure < 360° Note: In mathematics, we tend to measure counterclockwise. This matches the rotational spin of Earth from west to east as viewed from the North Pole star Polaris. © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. 72 Investigation 2 Designing with Angles SAMPLE
Name Date Class Four in a Row Game Boards LEARNING AID 2.1A © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. 45° 0° 10 2 3 30° 60° 0° 10 2 3 45° 0° 10 2 3 30° 60° 0° 10 2 3 Problem 2.1 Four in a Row Game: Angles and Rotations 73 Four in a Row 45º Game Board Four in a Row 30º Game Board Four in a Row 45º Game Board Four in a Row 30º Game Board 45° 0° 10 2 3 30° 60° 0° 10 2 3 45° 0° 10 2 3 30° 60° 0° 10 2 3 SAMPLE
angle A or ∠A A angle B or ∠B B angle C or ∠C C angle D or ∠D D © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. Name Date Class 74 Investigation 2 Designing with Angles Angles 2.1B LEARNING AID 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 The goal of this problem is to develop student understanding of two standard tools for angle measurement—the goniometer, or angle ruler, and the protractor. The contexts of the bee dance in the Initial Challenge and Amelia Earhart’s fateful journey in the What If . . . ? situations give a context for the purpose of accuracy in angle measurement. The Bee Dance and Amelia Earhart: Measuring Angles and Distance PROBLEM 2.2 Arc of Learning Exploration Analysis NOW WHAT DO YOU KNOW? What are the advantages and disadvantages of estimating angle measures? What are the advantages and disadvantages of using a protractor or angle ruler to measure an angle? Key Terms Materials For each student • Learning Aid 2.2A: Bees and Angle Measurements • Learning Aid 2.2B: Amelia Earhart Pacing 1–2 days Groups 2 students A 7–12 C 23–25 E 30–31 Facilitating Discourse Teacher Moves LAUNCH CONNECTING TO PRIOR KNOWLEDGE Draw an angle, and ask students to estimate its measure. Before students use a tool to measure an angle, they should have a reasonable estimate of the angle’s size so that they can tell if they have measured incorrectly. Tell students that in this problem, they will use tools to measure the size of angles. PRESENTING THE CHALLENGE Introduce the tools, angle ruler and protractor. Demonstrate measuring an angle using each tool. The angle ruler reinforces angles as rotations. Encourage students to use it as they are learning about angle measures being the opening between the two rays. Students are probably more familiar with protractors; however, to make sure students are using them correctly, you will need to review how to use them. Problem 2.2 The Bee Dance and Amelia Earhart: Measuring Angles and Distance 75 SAMPLE
Facilitating Discourse (continued) Teacher Moves (continued) EXPLORE PROVIDING FOR INDIVIDUAL NEEDS As students use an angle ruler, monitor their work to be sure that they are placing the angle ruler correctly and reading the measures accurately. Since the tool is easily flipped over and the arms interchanged, be aware of the importance of placing the arm with the scale on it down first. It needs to be placed over the initial side of the angle, with the rivet over the vertex, before rotating the other arm counterclockwise to the terminal side. If students use a protractor, ask them to check their answer with the goniometer/angle ruler and vice versa. This gives them practice with both tools. As students are working on the What If . . . ? Situation B table and equation about Amelia Earhart’s journey, refer them back to their work with equations in grade 6, Comparing Quantities or Variables and Patterns. PLANNING FOR THE SUMMARY As students are working on this problem, look for how students are using the tools to measure angles accurately. Have students demonstrate how they find the angle measures. SUMMARIZE DISCUSSING SOLUTIONS AND STRATEGIES Have students share the angle measures they found for the Initial Challenge and explain their strategies for using the tools. Have students share their angle measures and distance measures for the What If . . . ? on Amelia Earhart’s journey. MAKING THE MATHEMATICS EXPLICIT Have students report their findings in a class discussion. Take time to explore all the strategies they used to arrive at their answers. Have students record their measurements for some of the angles. Look for large discrepancies. If large discrepancies occur, have the group measure again. Discuss what might account for any differences in the measurements. In this discussion, students will begin to understand the issues involved in making precise measurements; relate this to the fateful journey of Amelia Earhart. Suggested Questions • How do you decide the number of degrees for angles between the 5° intervals on the angle ruler? • What things do you check to make sure you are making accurate measurements? • We didn’t always get the same measurements. Why did this happen? Have students complete this sentences “I used to think angles and angle measurement were . . .” and “Now, I think angles and angle measurement are . . .” to gain insight on their current understanding. As you finish the mathematical discussions, have students reflect on the Now What Do You Know? question(s). Compare Thinking I Used to Think . . . Now I Think . . . ProblemSolving Environment Language 2.2 76 Investigation 2 Designing with Angles SAMPLE
Problem Overview The goal of this problem is to develop student understanding of two standard tools for angle measurement—the goniometer, or angle ruler, and the protractor. You will need to provide students with each tool for the problem. The angle ruler is based on the goniometer, which is used to measure joint flexibility. It is accurate and easy for students to use. The more common protractor can be more difficult to use correctly. The contexts of the bee dance and Amelia Earhart’s fateful journey give a context for the purpose of accuracy in angle measurement. Launch (Getting Started) Connecting to Prior Knowledge In Problem 2.1, students developed a sense of angle size by estimating size using benchmark angles. In this problem, students will practice using tools to measure angles more accurately. Draw an angle, and ask students to estimate its measure. Before students use a tool to measure an angle, they should have a reasonable estimate of the angle’s size so that they can tell if they have measured incorrectly. Tell students that in this problem, they will use tools to measure the size of angles. Also tell them that to avoid using the tools incorrectly, they will estimate the angle’s size before measuring. Presenting the Challenge Use the description in the Did You Know? in the Student Edition to introduce students to the context of the bee dance that they will be using in the Initial Challenge. Introduce the tools, angle ruler and protractor. Demonstrate measuring an angle using each tool. Teaching Aid 2.2: Using an Angle Ruler can help you introduce this tool. Have students try to measure an angle they draw in their notes. Before students go to work measuring angles in this problem, demonstrate how to measure one of the angles as they follow along at their desks. Then have them measure two or three separate angles on a polygon together and record their angle measures. By recording the measures, you can show students that they will get slightly different measures. Talk to them about the fact that all measures are approximations; that is, no matter how precise the tool, there is always some error in measurement. EXTENDED LAUNCH—EXPLORE—SUMMARIZE Extended Lanuch—Explore—Summarize 77 SAMPLE
To get an accurate angle measurement using the angle ruler, make sure students place the rivet over the vertex. Then make sure they understand that the ruler arm remains stationary over the initial side while the other arm is rotated counterclockwise until it is over the terminal side. The angle ruler reinforces angles as rotations. Encourage students to use it as they are learning about angle measures being the opening between the two rays. Students are probably more familiar with protractors; however, to make sure students are using them correctly, you will need to review how to use them. Note: Students have been introduced to angles and whole number angle measures prior to grade 7. It may be helpful to spend more time exploring angles in this problem so students feel confident understanding angle, how to measure angles beyond whole numbers, and how to use the tools. The angle ruler is helpful for students to see and “feel” the rotation of the angles. Using regular polygons from Learning Aid Template: Shapes Set and having students measure the angles is a fun way to get comfortable measuring angles and will be helpful in later problems in this unit. The Smithsonian Channel offers a video online of the bee dance situation. It shows a worker bee using the bee dance to indicate to the colony how to find a source of pollen. (Language) Tell students that after they’ve worked on the Initial Challenge, which is about the bee dance, they will be investigating the mystery of Amelia Earhart’s disappearance in the What If . . . ? section. Pass out Learning Aid 2.2A: Bees and Angle Measurements and Learning Aid 2.2B: Amelia Earhart. Have students work in partners to complete this problem. Explore (Digging In) Providing for Individual Needs As students use an angle ruler, monitor their work to be sure that they are placing the angle ruler correctly and reading the measures accurately. Since the tool is easily flipped over and the arms interchanged, be aware of the importance of placing the arm with the scale on it down first. It needs to be placed over the initial side of the angle, with the rivet over the vertex, before rotating the other arm counterclockwise to the terminal side. If students use a protractor, ask them to check their answer with the goniometer/angle ruler and vice versa. This gives them practice with both tools. LES 78 Investigation 2 Designing with Angles SAMPLE
As you work with groups, look for interesting ways students reasoned for you to share in the summary. The question about which direction the bee uses to indicate the location may come up. In the diagram, it is counterclockwise, but it could be clockwise. As students are working on the What If . . . ? Situation B table and equation about Amelia Earhart’s journey, refer them back to their work with equations in grade 6, Comparing Quantities or Variables and Patterns. (Problem-Solving Environment) Suggested Questions • What are the variables in this situation? (time and distance) • How can you compare the variables in this situation? • Are there ratios in this situation that can help us see patterns? • Is there a unit rate in this situation? 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 are the advantages and disadvantages of estimating angle measures? What are the advantages and disadvantages of using a protractor or angle ruler to measure an angle? (As students are working on this problem, look for how students are using the tools to measure angles accurately. Have students demonstrate how they find the angle measures.) Summarize (Orchestrating the Discussion) Discussing Solutions and Strategies Have students share the angle measures they found for the Initial Challenge and explain their strategies for using the tools. (Language) Have students share their angle measures and distance measures for the What If . . . ? on Amelia Earhart’s journey. Making the Mathematics Explicit Have students report their findings in a class discussion. Take time to explore all the strategies they used to arrive at their answers. You may want to display benchmark strategies for students to refer to as they proceed through the unit. LES Extended Lanuch—Explore—Summarize 79 SAMPLE
Have students record their measurements for some of the angles. Look for large discrepancies. If large discrepancies occur, have the group measure again. (Portrayal) Discuss what might account for any differences in the measurements. Students do not need to agree on the degree measurements, but their measurements should be close. In this discussion, students will begin to understand the issues involved in making precise measurements; relate this to the fateful journey of Amelia Earhart. You might want to ask students for their suggestions for getting a more accurate measurement using the angle ruler. As you discuss the angle measures, you may ask which angles are acute, obtuse, or right, to review these ideas from elementary school. (Language) Encourage students to try both the angle ruler and protractor. Suggested Questions • How do you decide the number of degrees for angles between the 5° intervals on the angle ruler? (Answers will vary. One possibility: I use the degree marks on the arm of the ruler and line up the zero with a 5° mark.) • What things do you check to make sure you are making accurate measurements? (Answers will vary. One possibility: I check that the rivet is over the vertex and the centerline of the ruler arm is over the initial side. Then, I rotate the other arm counterclockwise until it lines up over the terminal side.) • We didn’t always get the same measurements. Why did this happen? (Measurements are not exact. Discuss possible sources of error. Stress that these tools may provide more accurate measures of angles than estimating but that they do not provide exact measures. No measuring tool is absolutely precise, so there is a little error in every measurement.) • Which tool do you prefer for measuring angles? Why? (Answers will vary.) An interesting activity to enhance students’ understanding of angle measures and developing their angle sense is to make a paper protractor. Find directions online for how to make one. There are also videos online about why we have 360° in a circle, which may interest some of your students. Have students complete the sentences “I used to think angles and angle measurement were . . .” and “Now, I think angles and angle measurement are . . .” to gain insight on their current understanding. LES 80 Investigation 2 Designing with Angles SAMPLE
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 Extended Lanuch—Explore—Summarize 81 SAMPLE
INITIAL CHALLENGE A worker bee has located flowers with nectar and is preparing to do her dance. The following picture shows the bee’s search for honey on several different days. The dotted lines represent the angle formed by the hive, the sun, and the flowers. Sun Monday Sun Tuesday Sun Wednesday Sun Thursday Flowers Flowers Flowers Flowers Hive Hive Hive Hive • Estimate the measure of each angle, and then use an angle ruler or protractor to check your estimates. Monday: 30 degrees Tuesday: 45 degrees Wednesday: 135 degrees Thursday: 90 degrees • Suppose on Friday, the angle formed by the two rays to the sun and flower is 180º. Make a sketch of the angle using the hive and sun. Friday Sun Flowers Hive Answers Embedded in Student Edition Problems The Bee Dance and Amelia Earhart: Measuring Angles and Distance PROBLEM 2.2 Answers 82 Investigation 2 Designing with Angles SAMPLE
• Repeat for an angle of 120º on Saturday. Saturday Sun Flowers Hive WHAT IF . . . ? Situation A. Looking for Amelia Earhart Bees know a lot about angles, but angles are also important to humans. In 1937, the famous aviator Amelia Earhart tried to become the first woman to fly around the world. She began her journey on June 1, when she took off from Miami, Florida. She reached Lae, New Guinea, and then headed east toward Howland Island in the Pacific Ocean. She never arrived at Howland Island. In 2012, investigators found evidence of the crash on the deserted island of Nikumaroro, far off her intended course. An error may have been made in plotting Earhart’s course. 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 Problem 2.2 The Bee Dance and Amelia Earhart: Measuring Angles and Distance 83 2.2 Answers SAMPLE
1. Estimate the number of degrees off course Earhart’s crash site was from her intended destination. Check your estimate with a measuring too. About 8 to 12 degrees 2. Amelia Earhart apparently flew several degrees south of her intended course. Suppose you start at New Guinea and are trying to reach Howland, but you fly 20° south of your intended course. On which island might you land? How can you use an angle ruler or protractor to find the island? If you fly 20° south of the intended course, you might end up in the Samoa Islands. Situation B. The Flight of Two Planes Suppose two planes fly the paths formed by the rays of the angle shown on the map. The planes leave at the same time. They fly at the same speed. The map shows their progress every four hours. Amit uses the scale on the map to estimate the distance between the planes. 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 About 80 miles About 160 miles About 240 miles 4 hours 8 hours 12 hours Lae 1. Record the data in a table like this one. Time (hours) 4 8 12 t Distance Apart (miles) 80 160 240 20 d 84 Investigation 2 Designing with Angles 2.2 Answers SAMPLE
2. Look for patterns in the table. Use words to describe the relationships in the pattern. Possible Answers: The angle doesn’t change. But the distance apart increases as you go farther out on the rays of the angle. The distance apart is about 80 miles for an increase of 4 hours. Every 4 hours, they are 80 more miles apart. They get about 20 miles farther apart every 1 hour. 3. Write an equation to represent the relationship between the time t in hours and the distance d apart. Distance is equal to the time times 20. d = 20 • 1t or d = 20t Note: In grade 6 Variables and Patterns, students learned to write equations by describing the pattern in words, then changing the words to numbers, variables, and operation signs. 4. Use your equation to find how far apart they are after 10 hours. D = 20 • (10) D = 200 miles NOW WHAT DO YOU KNOW? What are the advantages and disadvantages of estimating angle measures? What are the advantages and disadvantages of using a protractor or angle ruler to measure an angle? Estimating gives you an idea of the size of an angle. This will help you know the approximate size of an angle and if you are using a measuring tool correctly. Using an angle ruler or protractor gives you a more precise measurement. Even with a measuring tool, it is not easy to get a measurement. No measurement is exact, but we try to be as precise as possible. Problem 2.2 The Bee Dance and Amelia Earhart: Measuring Angles and Distance 85 2.2 Answers SAMPLE
© 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. 86 Investigation 2 Designing with Angles Measuring Angles One common tool to use for measuring angles is the angle ruler. An angle ruler has two arms linked by a rivet. The rivet allows the arms to spread apart to form angles of various sizes. One arm is marked with a circular ruler showing degree measures from 0° to 360°. 0 1 2 3 4 5 6 0 1 2 3 180 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 10 20 30 40 50 60 70 100 80 110 120 130 140 150 160 170 170 160 150 140 130 120 110 100 80 90 70 60 50 40 30 20 10 Angle Ruler Center Line Center Line Rivet To measure an angle with an angle ruler, do the following: 0 1 2 3 180 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 0 350 340 330 320 310 300 290 250 260 280 240 230 220 210 200 190 190 200 210 220 230 240 250 260 280 270 290 300 310 320 330 340 350 1500 • First place the rivet over the vertex. • Set the center line of the arm marked as a ruler on the first side of the angle. • Swing the other arm counterclockwise until its centerline lies on the second side of the angle. • Read the angle measure on the circular ruler. Using an Angle Ruler 2.2 TEACHING AID SAMPLE
© 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 87 When you use an angle ruler to measure a polygon in the Shapes Set or another object, place the object between the two arms of the angle ruler. 0 1 2 3 2 180 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 0 350 340 330 320 310 300 290 260 280 250 240 230 220 210 200 190 190 200 210 220 230 240 250 260 280 270 290 300 310 320 330 340 350 1200 2 3 1 4 R Then read the size of the angle. Angle 1 measures 120º in shape R. Another tool for measuring angles in degrees is the protractor. It is usually semicircular and has a scale in degrees. The protractor below shows how to measure ∠AVB. C A B Protractor V 90 120 150 180 60 30 0 What is the measure of ∠AVB in degrees? Notice in the diagram that ∠CVB and ∠AVB share a side. Both angles have VB⟶ as a side of the angle. Angles that share a side are called adjacent angles. SAMPLE
Name Date Class Sun Monday Sun Tuesday Sun Wednesday Sun Thursday Flowers Flowers Flowers Flowers Hive Hive Hive Hive Bees and Angle Measurements 2.2A LEARNING AID © 2025 Michigan State University. From Connected Mathematics® 4 published by Lab-Aids. All rights reserved. 88 Investigation 2 Designing with Angles SAMPLE