UNIT. K Explore. Investigation 1. Vocabulary. polygon vertex

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1 UNIT K Explore Patterns in Geometry Suggested Grouping: Groups of 3 or 4 Prepare Display a 4 4 square and ask students to write down the number of squares they see. If some students can find only 16 squares, point out one larger square. UNIT K Inv 1 Polygons 172 Inv 2 Angles 176 Inv 3 Classify Polygons 180 Inv 4 Triangle Sides 184 Patterns in Geometry In this lesson, you will work with two-dimensional geometric figures. You will classify polygons and find angle measures. Explore How many squares are in this design? (Hint: The answer is more than 16.) 30 (sixteen 1 1 squares, nine 2 2 squares, four 3 3 squares, one 4 4 square) Play Ask students to find all of the possible squares. Encourage students to use the draw a picture or diagram problem-solving strategy. Use transparencies to outline a 2 2 square. Then move the transparency over the 4 4 square and match it up with as many 2 2 squares as possible. Do the same with 3 3 and 4 4 squares. Report Have a reporter from each group share their findings to the class. Discuss how to determine which groups are correct. Score Give each group a point for each square they find. Investigation 1 Vocabulary polygon vertex 1 The first two shapes are not made from line segments. In the third shape, two of the segments touch only one other segment. In the fourth shape, some of the segments touch more than two other segments. Polygons Polygons are flat, two-dimensional geometric figures that have these characteristics. They are made of line segments. Each segment touches exactly two other segments, one at each of its endpoints. These shapes are polygons. These shapes are not polygons. Think & Discuss Look at the shapes above that are not polygons. Explain why each of these shapes does not fit the definition of a polygon. See 1. Investigation 1 On Your Own Exercises Pages Exercises 1 3, 16, 17 Polygons It is assumed in this investigation that students are familiar with the terms segment, endpoint, and line. Review how lines and segments are different. Think & Discuss Discuss the examples of nonpolygons with the class. 172 Unit K Patterns in Geometry Reaching All Learners ELL English Language Learners English language learners may have difficulty remembering how to classify each polygon because the vocabulary is new. Encourage them to use note cards to write the names of polygons in English and in their native languages. Have them include examples on the note cards. 172 UNIT K Patterns in Geometry

2 Objectives U N I T K To name polygons and classify them by regularity, concavity, and line symmetry To use Venn diagrams to classify objects To find the measures of identical angles summing to 90, 180, or 360 To estimate angle measures To determine whether three segments or angles could form a triangle Patterns in Geometry This lesson focuses on patterns in geometric shapes, specifically polygons and angles. Students are introduced to terminology of and relationships among geometric shapes and to well-established mathematical patterns and rules. The first two investigations introduce students to polygons and angles. Students develop intuitions about angle measure as they use deduction and comparison to estimate the measures of angles. Students will use protractors in the next lesson. Investigation 3 introduces students to new characteristics of polygons and the use of Venn diagrams. In the inquiry investigation, students discover the triangle inequality, although the formal name of the former is not used in this text. The inequality is discovered in a concrete manner by using linkage strips and physical triangles. Summary Materials On Your Own Exercises (pp ) Assessment Opportunities Investigation 1 (p. 172) Pacing: 2 days This investigation introduces students to polygons and the terminology used with them. Students identify polygons and name polygons using vertices. blank transparencies (optional) Practice & Apply: 1 3 Connect & Extend: 16, 17 Share & Summarize (p. 176) Troubleshooting (p. 175) Investigation 2 (p. 176) Pacing: 1 day Students develop their sense of angle as they estimate angle measures by comparing angles to benchmarks: 90, 180, and 360. Unit K Master 1, scissors (optional) Practice & Apply: 4 11 Connect & Extend: On the Spot Assessment (p. 179) Share & Summarize (p. 179) Investigation 3 (p. 180) Pacing: 1 day Students classify polygons using a variety of criteria. Venn diagrams are introduced as an organizational tool. Unit K Masters 2 and 3, scissors (optional), string (optional) Practice & Apply: Connect & Extend: On the Spot Assessment (p. 182) Share & Summarize (p. 183) Inquiry Investigation 4 (p. 184) Pacing: 1 day Students focus on patterns in and special properties of triangles in this investigation, concluding with the triangle inequality. They also make conjectures about the sum of the angles in a triangle. *linkage strips or Unit K Master 4, scissors (optional), fasteners, rulers On the Spot Assessment (p. 185) 172A UNIT K Patterns in Geometry

3 Differentiated Instruction Reaching All Learners Below are suggestions on differentiating the materials presented in this unit. Additional modifications should be considered. Approaching Level AL Beyond Level BL Flash Cards Students having difficulty identifying polygons and/or angles may need extra practice. Use index cards to create flash cards showing a shape or angle on one side, and the correct term on the other side. Have students work in pairs or small groups. Tell them to take turns holding up a card and having the others identify the shape or angle shown. As students progress, have them show the side with the term while the other student draws an illustration of the term. If the flash cards are helpful to students, allow individuals to make a set of flashcards for themselves. Encourage them to take the cards home and practice with a family member. Draw It Some students will quickly grasp the concept of measuring around a polygon. Divide these students into pairs and tell them to take turns drawing a polygon for the other partner to measure around. Have each student draw a figure and determine whether the distance around it would be called the perimeter or the circumference. Then have them trade and find the specified measurements. Remind students to use formulas whenever possible, such as the formula for finding circumference. Once each student has found the measurement for the figure, have students trade back and check one another s work. Monitor for correct use of formulas and measurement strategies. KEY AL Approaching Level OL On Level BL Beyond Level ELL English Language Learners Unit Overview 172B

4 Real-World Link In Greek, poly means many and gon means angle. Polygon names refer to the number of angles. For example, pentagon means five angles. Polygons can be classified according to the number of sides they have. You have probably heard many of these names. Name Sides Examples Triangle 3 Quadrilateral 4 Pentagon 5 Hexagon 6 Heptagon 7 Prefixes Refer to the Real-World Link and discuss the names for different polygons. Point out the prefixes used in these names, and ask students what other words they can think of that use these prefixes. Some prefixes are used less frequently and students may not be able to provide examples. In those cases, give them the examples below and have them look up the words in the dictionary as homework. Prefix triquadpentahexaheptaoctanonadeca- Examples triangle, tripod quadruple, quadrilateral pentathlon, pentagram hexose, hexapod heptarchy, heptameter octopus, octet nonagenarian decathalon, decapod Octagon 8 Nonagon 9 Decagon 10 Investigation Also, ask students why there is not a name for a two-sided polygon. Point out that if two segments meet only at the endpoints, they must make a V or they must make a straight line segment, neither of which is a polygon because not all of the endpoints meet. If students argue that the line segments could be placed on top of one another so the endpoints meet, point out that the result is actually one segment, not two, so there is no such thing as a two-sided polygon. Real-World Link Emphasize that polygons are named only by their number of sides and not by their angle measures. Investigation 1 173

5 Teacher Tips Discuss the naming convention for polygons with more than 10 sides. You may want to let students know that some of those polygons do have special names; for example, a polygon with 12 sides is called a dodecagon. Introduce the term vertex and its plural, vertices, and demonstrate how to label the vertices of a polygon with capital letters. Math Link Students may not understand that the diagonals of any quadrilateral divide the quadrilateral into two triangles. Show several examples. Example Illustrate how to name triangles and quadrilaterals using their vertices by going through the example. Make sure students understand that it is important to list the vertices in order when naming figures. You might ask students why they think this was mentioned for quadrilaterals but not for triangles. Possible answer: Any way the vertices are listed will be in order for a triangle, but this is not true for figures with more than three sides. Math Link A diagonal is a segment that connects two vertices of a polygon but is not a side of the polygon. In quadrilateral ABCD, the diagonal is AC. Most polygons with more than ten sides have no special name. A polygon with 11 sides is described as an 11-gon, a polygon with 12 sides is a 12-gon, and so on. Each of the polygons below is a 17-gon. Each corner of a polygon, where two sides meet, is called a vertex. The plural of vertex is vertices. Labeling vertices with capital letters makes it easy to refer to a polygon by name. Example This figure can be seen as two triangles and one quadrilateral. To name one of the polygons in the figure, list its vertices in order as you move around it in either direction. One name for the white triangle is ABC. Other names are possible, including BCA and ACB. One name for the green triangle is ADC. The quadrilateral in the figure could be named quadrilateral ABCD, or BCDA, or DCBA, or DABC. All of these names list the vertices in order as you move around the quadrilateral. The name ACBD is not correct. 4. Polygon Names Score Triangle RQT, RST, RUV, WVT 12 Quadrilateral QRST, QRUW, TSUW, QRVW, TSUV 20 Pentagon RQTVU, RSTWV 10 Hexagon RQTSUV, RQWVTS 12 Total Score Possible answer: Polygon Names Score Triangle ABC, ADF, DEF, BED, ECF 15 Quadrilateral BDFE, ADEF, CEDF, ADEC, BDFC, ABEF 24 Pentagon Hexagon Total Score Unit K Patterns in Geometry Reaching All Learners OL On Level Some students may not think it is important to name the vertices in order for the quadrilateral in the example. Point out that if the vertices are not put in order, ACBD instead of ABCD for example, a diagonal of the quadrilateral may be included in the name. 174 UNIT K Patterns in Geometry

6 Real-World Link Polygons, such as triangles and octagons, are used for traffic signs. Develop & Understand: A You will now search for polygons in given figures. Each figure has a total score that is calculated by adding the following. 3 points for each triangle 4 points for each quadrilateral 5 points for each pentagon 6 points for each hexagon As you work, try to discover a systematic way to find and list all the polygons in a figure. Be careful to give only one name for each polygon. Record your work in a table like this one, which has already been started for Exercise 1. Polygon Names Score Triangle ABC, ADC 6 Quadrilateral ABCD 4 Pentagon Hexagon Total Score See table above. See margin. 5. Now create your own figure that is worth at least 30 points. Label the vertices. List each of the triangles, quadrilaterals, pentagons, and hexagons in your figure. See margin. 2. Polygon Names Score Triangle XYZ, XWZ, XYV, XWV, ZWV, ZVY, XYW, WYZ 24 Quadrilateral WXYZ 4 Pentagon YXWZV, XWZYV, WZYXV, ZYXWV 20 Hexagon Total Score 48 Investigation See below. See margin. Additional Answer 3. Polygon Names Score Triangle Quadrilateral LMNO, LMPR, NORP, SMNT, LSTO, MPQS, NTQP, OTQR, LRQS 36 Pentagon Hexagon PMLOTQ, PNOLSQ, RONMSQ, RLMNTQ 24 Additional Answers for Develop & Understand: A Exercises 4 and 5 are on page 174. C01-032A C01-034A C01-033A C01-035A Develop & Understand: A Suggested Grouping: Individuals Exercise 1 Make sure students understand that the table shown on this page is for this exercise only. Watch for students who do not name the quadrilateral correctly. Exercises 2 4 Some students may need strategies to help them count the triangles and quadrilaterals and then look for their complements. For example, in Exercise 2, pentagon YXWZV includes everything in the figure except ΔVYZ. Others may choose one vertex (say, X) and count all the triangles that contain vertex X and then all the quadrilaterals that contain vertex X, and so on. Then they move on to vertex Y and count all the triangles that contain vertex Y but not vertex X, and so on. Point out that it is helpful to record all the names using the same direction, either clockwise or counterclockwise so they may notice if a polygon is listed more than once. Exercise 3 Some students may think SQ, QT, OR, and RL are sides of quadrilateral LSQTORL. Point out that LSQTORL is not a quadrilateral. A quadrilateral has exactly four vertices. Exercise 5 Have students share their results for Exercises 1 4 before doing this exercise and have them check how they named the polygons. This will help ensure that they understand the concepts. Use a transparency to highlight the polygons when there is a question because it may be difficult matching the names to the polygons without a visual connection. Troubleshooting One of the difficulties some students encounter when studying geometric shapes is vocabulary. To help students keep the terminology clear, you may want to make a sheet of terms, with examples of each for reference. Encourage students to add any new terms from future investigations to their vocabulary lists. Investigation 1 175

7 Share & Summarize For Exercise 1, have students share their examples and explanations and make sure everyone agrees that they are correct. For Exercise 2, the strategies students list are important. Students sometimes have difficulty explaining their strategies, so pair them up and have them explain the strategies to each other. Have them apply their strategies to example polygons they present. Investigation 2 On Your Own Exercises Pages Exercises 4 11, Directions This investigation is designed to develop students sense of angle as well as estimation techniques for angle measurement. Introduce angles informally by using the following activity. Show students the path below, and ask them to write directions so that someone could follow this path exactly from A to B (without actually seeing the path). Investigation 2 Vocabulary angle Materials paper polygons or pattern blocks 1 Possible answer: I counted all the polygons containing a certain vertex first, then I moved on to other vertices. I counted a polygon and then counted the rest of the figure that did not contain that polygon. I always listed the vertices in counterclockwise order, so it was easier to spot repeats. Share & Summarize 1. Draw two polygons. Also draw two shapes that are not polygons. Explain why the shapes that are not polygons do not fit the definition of a polygon. Answers will vary. 2. In Exercises 1 5, you had to find ways to list all the polygons in a figure without repeating any. Describe one strategy you used. See 1. Angles You probably already have a good idea about what an angle is. You may think about an angle as a rotation, or a turn, about a point. Examples include an arm bending at the elbow or hinged boards snapping shut at the start of a movie scene. You may also think about an angle as two sides that meet at a point, like the hands of a clock or the vanes of a windmill. Or you may think of an angle as a wedge, like a piece of cheese or a slice of pizza. Possible directions: Walk two steps forward. Turn 90 to the left. Walk one step forward. Turn 45 to the left. Walk one step forward. Turn 45 to the left. Walk one step forward. Turn 90 to the right. Walk two steps forward. 176 Unit K Patterns in Geometry Reaching All Learners AL Approaching Level Ask students who still have difficulty identifying angles to give examples of real-life angles such as a partially opened book, the corner of a table or desk, or an open door. 176 UNIT K Patterns in Geometry

8 In mathematics, an angle is defined as two rays with the same endpoint. A ray is straight, like a line. It has an endpoint where it starts, and it goes forever in the other direction. Ray 2 Ray 1 Angles can be measured in degrees. Below are some angles with which you may be familiar. The angle at the vertex of a square measures 90. You can think of a 90 angle as a rotation 1_ 4 Ray 2 90 Ray 1 of the way around a circle. Two rays pointing in opposite directions form a 180 angle. A 180 angle is a rotation 1_ 2 of the way around a circle. Angles Introduce the formal definition of angle using the concept of ray. You may need to discuss how a ray is different from a line or from a line segment. Review the degree symbol ( ) and discuss the benchmark angles 0, 90, 180, and 360. These angles provide a point of reference for measuring other angles. If there is a student in your class who likes to skateboard or snowboard, you might ask him or her to describe how those angles are used in the sport and demonstrate for the class what they mean. Illustrate how to use benchmark angles to estimate the measures of other angles. Real-World Link In snowboarding, skateboarding, and other sports, the term 360 is used to mean a full turn. The term 180 is used to refer to a half turn. 180 Ray 2 Ray 1 A 360 angle is a rotation around a complete circle. In a 360 angle, the rays point in the same direction. 360 Ray 1, Ray 2 Real-World Link Point out that the tip of the skateboard can be thought of as making a complete circle when there is a full turn (360 ) and of as making a half-circle when there is a half turn (180 ). You can use 90, 180, and 360 angles to help estimate the measures of other angles. For example, the angle below is about a third of a 90 angle, so it has a measure of about 30. Investigation Reaching All Learners OL On Level Students may notice that for each angle of 30 formed by a ray rotating counterclockwise, there is also an associated angle of 330 formed by the same ray rotating clockwise. Investigation 2 177

9 Think & Discuss Ask students to offer as many ways as they can to figure out the angle measure that is marked in the star. In addition to the sample answer, some students may say that eight angles make 360, so they divide 360 by 8. Others may say that two angles make 90, so they divide 90 by 2. You may find it helpful to have an overhead transparency of the design shown in the Think & Discuss. As you discuss the measures of the angles, fill in the angles that are adding to the benchmark in question so that students can visually see which angles have been included. For example, the figure below shows three angle measures that have been counted ; Possible explanation: Since there are eight identical angles that fill 360, they must each measure 360 8, or 45. Think & Discuss Copies of the polygon at right can be arranged to form a star. What is the measure of the angle that is marked in the star? How do you know? See 1. Develop & Understand: A 1. You will be given several copies of each polygon below. Your job is to determine the angle measures for each polygon. To find the measures of the angles, you can use 90, 180, and 360 angles as guides, and you can compare the angles of the polygons with one another. Teacher Tips In this exercise set, students develop their angle estimation skills. They will need pattern blocks or polygons cut from Unit K Master 1. Develop & Understand: A Suggested Grouping: Individuals Exercise 1 Students may use one of these strategies to estimate the angle measures for these polygons. Make designs like those in Think & Discuss by placing equal angles around a vertex. For example, in Δ FGE, six equal angles measure 360, so one angle measures 60. Compare benchmark angles. For example, the angles in square ABCD are 90. Students can use a corner of a piece of paper or of an index card to confirm the measure. Compare the angles to other angles whose sizes have already been determined. For example, KHI is the same size as FGE, so it is 60. Similarly, LON has the same measure as two of FGE, so it is Unit K Patterns in Geometry Additional Answer 1. Angles Measure P, R 30 E, F, G, H, I, L, N 60 A, B, C, D 90 J, K, M, O, T, U, V, W, X, Y 120 S, Q 150 Your answers should be a record of each vertex, A Y, and the measure of the angle at that vertex. For many of the polygons, two or more of the angles are identical. So, you only have to find the measure of one of them. See margin. You will now use the angles you found in Exercise 1 to help estimate the measures of other angles. 178 UNIT K Patterns in Geometry

10 Real-World Link Think about the corners of index cards when estimating angle measures. They form approximate 90 angles ; Possible explanation: It is about half the size of each angle of the square ; Possible answer: A circle can be completed with angle S, which is ; Possible explanation: It is about half of angle R ; Possible explanation: A circle can be completed with the 60 angle of the triangle ; Possible explanation: It is the same as two of the 120 angles of the trapezoid, angles M and O ; Possible explanation: A circle can be completed with the 90 angle of the square ; Possible answer: The circle can be completed with one of the 30 angles of the rhombus, angle P or R. On the Spot Assessment Develop & Understand: B Estimate the measure of each angle. To help make your estimates, you can compare the angles to 90, 180, and 360 angles and to the angles of the polygons in Exercise 1. For each angle, explain how you made your estimate Share & Summarize 1. Describe how you can estimate the measure of an angle. 2. Moria said the angles below have the same measure. Stella said Angle 2 is larger than Angle 1. Who is correct? Explain. Angle 1 Angle 2 1_ -rotation and a 4 2 -rotation. 3. Explain the difference between a 1_ 1. Possible answers: You can compare the angle to 90, 180, and 360 angles and to other angles whose measures you already know. 2. Moria; Possible explanation: An angle s size does not depend on the length of its rays, only the amount of the angle is opened. The rays forming both angles actually go on forever. A picture shows only part of the rays. 3. Possible answer: A 1_ 4 -rotation is a 90 angle. A 1_ -rotation is 2 a 180 angle. Investigation Students may have trouble with the angles that are larger than 180. Suggest that they extend one of the rays through the endpoint to indicate a 180 angle and estimate the measure of the angle formed by the new line and the ray that was not extended. Then they can add the two measures. For example, to estimate the measure of the angle in Exercise 3, extend the ray and estimate the measure of the small angle, about 30. Add that measure to the measure of the straight angle, , or 210. Develop & Understand: B Suggested Grouping: Pairs Exercise 2 Emphasize that students should compare their angles to the angles of the polygons on page 178. Tell students they should look for fractions of benchmark angles. For example, this angle is about half the size of a 90 angle, so it is about 45. Exercises 3, 5, 6 8 When measuring angles greater than 180, estimate the other smaller angle and then subtract the result from 360. For example, for Exercise 7, the other angle is about 90, so the angle to be measured is about , or 270. Exercise 4 Compare the angle to ones whose measures are known. This angle is about half of the angle in Exercise 2, so it is about 22. Real-World Link Point out that folding down a corner of the index card to make a triangle can show the approximate size of a 45 angle. Share & Summarize For Exercise 1, have students list at least two different ways to estimate angles. If students do not mention the sample strategies listed in the notes for Develop & Understand, you may want to bring them up yourself. Exercise 2 emphasizes the fact that the length of the rays does not affect the size of the angle. Students may argue that the Angle 2 picture is obviously larger; if so, clarify that when discussing whether an angle is larger, the measure of the angle is what is considered. Some students may better understand that the angles are the same if you ask them to draw an angle, estimate its measure, and then extend its rays. Ask students to estimate its measure again and determine whether the angle measure has changed. Investigation 2 179

11 Investigation 3 On Your Own Exercises Pages Exercises 12 15, Classify Polygons In this investigation, students classify polygons according to type of shape as well as the following properties: concave/not concave, regular/not regular, line symmetry/no line symmetry. Students use circle diagrams as an organizational tool for classifying shapes. Before students open their books, give them copies of Unit K Master 2. This master shows the polygons on page 180. Ask students to quickly cut out the polygons and sort them into groups. Many students will sort based on number of sides. Ask them to do it again, this time not using that characteristic. Explain there are other characteristics of polygons that can be used to group them together. Teacher Tips Discuss the differences between concave and convex polygons. Point out the angles in the first set of figures that measure more than 180. Discuss the differences between regular and irregular polygons. Have students identify the characteristics of the irregular polygons on page 180. Demonstrate line symmetry with a paper rectangle. Show the two lines of symmetry by folding the rectangle lengthwise and widthwise. Ask students whether there are any other lines of symmetry. If students do not suggest folding along the diagonal, ask them why the diagonal is not a line of symmetry. To determine if a polygon is concave or convex, stretch a rubber band around the polygon. The rubber band can touch all vertices of a convex polygon. A concave polygon will have a gap. Example: Investigation 3 Vocabulary concave polygon line symmetry regular polygon Materials set of polygons and category labels large Venn diagram 180 Unit K Patterns in Geometry Reaching All Learners AL Approaching Level Have students who are having difficulty understanding line symmetry cut out the shapes in Unit K Master 2 and work with folding them so the two halves match exactly. Monitor them so they make the folds accurately. Point out that the folding line is called the line of symmetry. Have them draw the line(s) of symmetry on the shapes. Classify Polygons Polygons can be divided into groups according to certain properties. Concave polygons look like they are collapsed or have a dent on one or more sides. Any polygon with an angle measuring more than 180 is concave. The polygons below are concave. The polygons below are not concave. Such polygons are sometimes called convex polygons. Regular polygons have sides that are all the same length and angles that are all the same size. The polygons below are regular. The polygons below are not regular. Such polygons are sometimes referred to as irregular. A polygon has line symmetry, or reflection symmetry, when you can fold it in half along a line and the two halves match exactly. The folding line is called the line of symmetry. 180 UNIT K Patterns in Geometry

12 Real-World Link The United Nations building located in New York City is an example of line symmetry in modern-day architecture. The polygons below have line symmetry. The lines of symmetry are shown as dashed lines. Notice that three of the polygons have more than one line of symmetry. These polygons do not have line symmetry. Think & Discuss Consider the polygons below. This diagram shows how these four polygons can be grouped into the categories concave and not concave. Concave Not Concave Now make a diagram to show how the four polygons can be grouped into the categories line symmetry and not concave. Use a circle to represent each category. See margin. Investigation Line Symmetry If students cannot find a line of symmetry, ask them to trace the polygon onto a sheet of paper, cut it out, and then try to find the line of symmetry by folding. Nonrectangular parallelograms may be especially tricky for some students, so ask students to experiment finding a line of symmetry with a paper parallelogram. If students do not understand why a regular hexagon has line symmetry, explain that the angles in the polygon have as much to do with line symmetry as the sides, and ask them to experiment with folding paper hexagons. Think & Discuss Ask students how to group the polygons as those with line symmetry and those that are not concave. If necessary, suggest that polygon Y must be placed outside both regions because it does not have either characteristic. Students should see that polygon Z fits both criteria. Some students may suggest adding an extra circle labeled both; others may suggest putting one copy of the shape in each region. Give your class time to suggest overlapping the circles, or discuss how using overlapping circles is an option. Real-World Link Point out that if a figure has line symmetry, it can be folded along the line of symmetry so that each side of the resulting half figure is identical. Additional Answer for the Think & Discuss Possible answer (students may draw the circles with no overlap and draw polygon Z twice, once in each circle): Investigation 3 181

13 Develop & Understand: A Suggested Grouping: Groups of 3 or 4 Exercise 1 For this game, each group will need a copy of Unit K Master 3, which reproduces the set of polygons and labels, and scissors. Students can also use attribute or pattern blocks instead of the paper polygons on Unit K Master 3. Emphasize that students can write the names of the polygons, rather than drawing the shapes, to record their work. Math Link For this game, the region where all three circles intersect represents the polygons that have the attributes common to all three circles. Additional Examples See the polygons on page 182 or Unit K Master 3 to answer the questions below. 1. What do polygons G, H, and Q have in common? They are all concave. 2. Complete a two-circle Venn diagram with the labels quadrilateral and hexagon. 3. There should be no polygons in the overlap of the circles for Exercise 2. Why? A polygon cannot be a quadrilateral and a hexagon at the same time. Math Link A Venn diagram uses circles to represent relationships among sets of objects. 1. E, F, J, K, O, P G, H, L, M, Q, R Concave Triangle B, C A You will now play a polygon-classification game with your group. Your group will need a set of polygons, category labels, and a large Venn diagram. Here are the polygons used for the game. Regular Develop & Understand: A Here are the category labels. Regular Concave Triangle Not Regular Not Concave Not Triangle Quadrilateral Pentagon Hexagon Not Quadrilateral Not Pentagon Not Hexagon Line Symmetry No Line Symmetry 1. As a warm-up for the game, put one of the labels Regular, Concave, and Triangle next to each of the circles on the diagram. Work with your group to place each of the polygons in the correct region of the diagram. Record your work. Sketch the three-circle diagram, label each circle, and record the polygons you placed in each region of the diagram. Record just the letters. You do not need to draw the polygons. D, I, N 182 Unit K Patterns in Geometry On the Spot Assessment If a group has difficulty guessing the labels, you can ask them if they think there are any misplaced polygons. Encourage students to ask the leader to check the placement of any polygons that seem wrong. If necessary, look at the labels to see if there are any misplaced shapes and then help the leader reposition them. If the diagram is correct, but the students guessing do not see the patterns, have them focus on one circle at a time, and ask which labels could or could not fit that circle. 182 UNIT K Patterns in Geometry

14 3. Possible answer: diagram with labels Pentagon, Triangle, and Hexagon. 4. Possible answer: diagram with labels Not Pentagon, Not Triangle, and Not Hexagon 2. Now you are ready to play the game. Choose one member of your group to be the leader. Use the following rules. Answers will vary. The leader selects three category cards and looks at them without showing them to the other group members. The leader uses the cards to label the regions, placing one card face down next to each circle. The other group members take turns selecting a polygon. The leader places the polygon in the correct region of the diagram. After a player s shape has been placed in the diagram, he or she may guess what the labels are. The first player to guess all three labels correctly wins. At the end of each game, work with your group to place the remaining shapes. Then copy the final diagram. Take turns being the leader until each member of the group has had a chance. 3. Work with your group to create a diagram in which no polygons are placed in an overlapping region, that is, no polygon belongs to more than one category. 4. Work with your group to create a diagram in which all of the polygons are placed either in the overlapping regions or outside the circles, that is, no polygon belongs to just one category. Exercise 2 Students play the game. You may want to decide on a set amount of time for the game depending on the length of your class period. Be sure to leave time for students to complete Exercises 3 and 4 as well as a Share & Summarize discussion. Exercises 3 and 4 Students are told to create Venn diagrams using different categories than in Exercise 2. Ask students to explain how they chose their categories. They should understand that using Not to form a category (for example, Not Triangle) gives them a category that is the complement of a category (Triangle in this case). Target Market Real-World Link Venn diagrams are named after John Venn ( ) of England, who made them popular. Venn diagrams are used in business to create visual models. 2. A polygon is either regular or not regular; it cannot be both. Share & Summarize 1. Determine what the labels on this diagram must be. Use the category labels on page 182. Label 3 Label 2 Label 1 Label 1: Not Regular Label 2: Regular Label 3: Pentagon 2. Explain why there are no polygons in the overlap of the Label 1 circle and the Label 2 circle. 3. Explain why there are no polygons in the Label 3 circle that are not also in one of the other circles. Every pentagon must be either regular or not regular. Investigation Real-World Link Point out that visual models like Venn diagrams are used because they can convey information quickly and concisely. Share & Summarize Use these questions to see whether students understand the vocabulary or how to use Vann diagrams. For Exercise 1, have students share answers and discuss any differences. For Exercises 2 and 3 point out the key features of a Venn diagram. Students answers will show how well they understand Venn diagrams. Reaching All Learners ELL English Language Learners The directions for the game described in Exercise 2 may be difficult for some students to understand. Play the game once as a class, or condense the rules as follows: Leader picks three cards. Do not show them to the group. Leader places one card face down by each circle. Player picks a polygon and asks the leader where to place it. Leader answers. Player guesses what the cards say. Investigation 3 183

15 Inquiry Inquiry Investigation 4 Suggested Grouping: Individuals Inquiry Inquiry Investigation 4 Materials linkage strips and fasteners Triangle Sides In many ways, triangles are the simplest polygons. They are the polygons with the fewest sides. Any polygon can be split into triangles. For this reason, learning about triangles can help you understand other polygons as well. Materials and Preparation Students will need linkage strips and fasteners. Triangle Sides In this investigation, students look at the relationship among the sides of a triangle. In this investigation, you will build triangles from linkage strips. The triangles will look like the one below. The sides of this triangle are 2, 3, and 4 units long. Notice that a unit is the space between two holes. Begin the investigation by showing students how to split various polygons into triangles. Challenge students to draw polygons on the board that cannot be split, and then let other students show how to split them. Demonstrate how to make a triangle by using linkage strips to make a 2-unit-by-3- unit-by-4-unit triangle, perhaps projecting it on the overhead so all students can see. Explain how to measure the length of the sides using the linkage strips. (The distance from one hole to the next is one unit.) Ask students to name characteristics of the triangle such as convex, irregular, and has no lines of symmetry. Do you think any three segments can be joined to make a triangle? You will investigate this question. Build the Triangles 1. Copy the table on the next page. Do the following steps for each row. Try to build a triangle with the given side lengths. In the Triangle? column, enter yes if you could make a triangle and no if you could not. If you could make a triangle, try to make a different triangle from the same side lengths. (Hint: For two triangles to be different, they must have different shapes.) In the Different Triangle? column, enter yes if you could make another triangle and no if you could not. Build the Triangles Exercise 1 Be sure students understand how to use linkage strips to make the triangles. Have students created their own linkage strips using copies of Unit K Master 4 on card stock. As students try to build a different triangle, ask: Is it possible to connect the sides in a different order? Can you press on the sides of the triangle to change its shape? 184 Unit K Patterns in Geometry Reaching All Learners AL Approaching Level The directions for Exercise 1 may be difficult. Reading them with the class and constructing the first triangle as an example can help. Make sure students understand that the column Different Triangle? is asking whether two or more different triangles can have the same side lengths. Two triangles are different if they cannot be turned or flipped so that they match up perfectly. 184 UNIT K Patterns in Geometry

16 2. No; Possible explanation: If you connect the 4-unit segments in a straight line, they will not reach the ends of the 10-unit segment, so there is no way to bend them to reach the ends of the 10-unit segment. 3. Yes; Possible explanation: The 10-unit and 15-unit segments connected together are longer than the 16-unit segment, so you can bend them where they connect to reach the ends of the 16-unit segment. Inquiry Side 1 Side 2 Side 3 Triangle? Different Triangle? 4 units 4 units 4 units Yes No 5 units 4 units 3 units Yes No 4 units 4 units 2 units Yes No 4 units 4 units 1 unit Yes No 4 units 3 units 1 unit No 4 units 2 units 2 units No 3 units 5 units 6 units Yes No 3 units 3 units 1 unit Yes No 3 units 2 units 2 units Yes No 3 units 2 units 1 unit No 3 units 1 unit 1 unit No Analyze the Results 2. Do you think you could make a triangle with segments 4, 4, and 10 units long? Explain your answer. 3. Do you think you could make a triangle with segments 10, 15, and 16 units long? Explain your answer. What Did You Learn? 4, 5. See margin. 4. Describe a rule you can use to determine whether three given segments will make a triangle. Test your rule on a few cases different from those in the table until you are convinced it is correct. 5. Do you think you can make more than one triangle with the same set of side lengths? Explain. Inquiry Analyze the Table Students should write no in the Triangle? column if the sum of each pair of two sides of the triangle is not greater than the third. For example, strips with length four units, three units, and one unit do not make a triangle because is not greater than 4. Analyze the Results Exercises 2 and 3 Have students who are unsure of the relationship use linkage strips to try to make a model of each triangle. Encourage them to determine how long they would need to make the third side in order to make a triangle. What Did You Learn? Exercise 5 Students should draw conclusions based on what they have tried so far. Teacher Tips After filling in the table for Build the Triangles, point out the Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Give several examples of how to apply the theorem to three side lengths. On the Spot Assessment For Exercise 4, watch for students who do not see the relationship between the sums of the side lengths. Encour age them to look back and think about why the triangle in Exercise 2 could not be made. Additional Answers 4. Possible answer: If the sum of the lengths of the two shorter segments is greater than the length of the longest segment, the segments can make a triangle. 5. No; Possible explanation: In each case we tried, at most, one triangle was possible. We could not press in the sides or vertices to create a different shape. Investigation 4 185

17 On Your Own Exercises Unit K On Your Own Exercises Unit K Investigation 1 Pages Practice & Apply: 1 3 Connect & Extend: 16, 17 Investigation 2 Pages Practice & Apply: 4 11 Connect & Extend: Investigation 3 Pages Practice & Apply: Connect & Extend: Exercises 1 and 3 You may want to make a transparency of each figure to facilitate discussion when reviewing the answers. Exercise 2 It will be difficult to share the names of the polygons in the figure as a class because students will be using different letters to name the vertices. For this reason, you may want to collect answers and check each student s work. Exercise 3 Ask students to name the different polygons they found. Make sure the class agrees that the naming is correct. You might also ask for other possible names for each polygon. Practice & Apply 2a. 1. How many triangles are in this figure? Do not count just the smallest triangles Look at the figure in Exercise 1. a. Copy the figure. Label each vertex with a capital letter. b. In your figure, find at least one of each of the following polygons. quadrilateral Possible answer: Quadrilateral BCHI, pentagon Pentagon CEFHA, hexagon Hexagon DBIJHF Use your vertex labels to name each shape. c. Find the polygon with the maximum number of sides in your figure. Use the vertex labels to name the shape. Possible answer: Heptagon IBCEFHJ 3. List all the polygons in the figure below. Compute the figure s score using the following point values. 3 points for each triangle 4 points for each quadrilateral 5 points for each pentagon 6 points for each hexagon Record your work in a table like the one below. Polygon Names Score Triangle EFC, BGF, BCF 9 Quadrilateral ABCD, AGED, BGEC, BCEF, BCFG 20 Pentagon ABFCD, ABFED, DCFGA 15 Hexagon FGADCB, FEDABC 12 Total Score Unit K Patterns in Geometry 186 UNIT K Patterns in Geometry

18 On Your Own Exercises On Your Own Exercises ; Since 5 of the same angle make 360, one has measure ; Since 10 of the same angle make 360, one has measure ; Since 2 of the same angle make 360, one has measure About 51.4 ; Since 7 of the same angle make 360, one has measure Real-World Link During a Ferris wheel ride, the wheel makes several complete rotations. 10. Angles will vary. Possible explanation: They are larger than the angle at the corner of a book. In Exercises 4 7, several identical angles have the same vertex. Find the measure of the marked angle. Explain how you found it A 180 angle is sometimes called a straight angle. Explain why that name makes sense. The two rays that make the angle form a straight line. 9. You know that a 360 rotation is one complete rotation around a circle. Find the degree measures for each of these rotations. a. half a rotation 180 b. two complete rotations 720 c. 1 1_ rotations d. three complete rotations 1,080 e. 2 1_ rotations f. five and one-half rotations 1, Draw two angles that each measure more than 90. Explain how you know they measure more than Draw two angles that each measure less than 90. Explain how you know they measure less than 90. Angles will vary. Possible explanation: They are smaller than the angle at the corner of a book. Exercises 4 7 Some students may just approximate the angle measures based on the activities completed on page 179. Although this is correct, explain that they can find the exact angle measure since they know all of the angles around the given vertex are the same size. Emphasize that the approximation is always a good first step because it helps to evaluate whether the calculation makes sense. This is especially useful when students divide incorrectly or do not remember where to place the decimal point. Students are most likely to make this mistake in Exercise 7. Real-World Link The Ferris wheel was introduced by George Ferris at the World s Columbian Exposition in A Ferris wheel typically carries 50 to 100 passengers. On Your Own Exercises 187 On Your Own Exercises 187

19 On Your Own Exercises On Your Own Exercises Exercise 12 Extension Ask students what kind of polygon would be placed in the three sections that do not have any figures. They should recognize that an equilateral triangle could be placed in the overlap of the three circles. No polygon can be placed in the overlap of the circles labeled triangle and regular since that figure would be an equilateral triangle with all three characteristics. No polygon can be placed in the circle labeled regular without also being placed in line symmetry because all regular polygons have line symmetry. Exercises If time permits, have students share their figures. There is only one possible shape for Exercise 13 (a square), although it can be different sizes. Exercise 16 When assigning this exercise, review the definition of diagonal. Have students look at the diagonal drawn for the polygon at the far right, and tell them that when a diagonal is outside of a polygon, the polygon is concave. 12a. Label 1: Regular Label 2: Line Symmetry Label 3: Triangle 12b. Shape A: in the overlap of all three circles Shape E: in the Line Symmetry (Label 2) circle Shape F: in the Line Symmetry (Label 2) circle Connect & Extend 14. Possible polygon: 12. The diagram shows the result of one round of the game of polygon classification described on page 182. Label 3 Label 2 Label 1 a. Figure out what the labels must be. Use the category labels from the polygon-classification game. b. Where would you place each of these shapes? In Exercises 13 15, draw a polygon that fits the given description, if possible. If it is not possible, say so. 13. a regular polygon with four sides 14. a concave polygon with a line of symmetry 15. a triangle with just one line of symmetry 16. A diagonal of a polygon is a segment that connects two vertices but is not a side of the polygon. In each polygon below, the dashed segment is one of the diagonals. 15. Any isosceles triangle that is not equilateral will work. The number of diagonals you can draw from a vertex of a polygon depends on the number of vertices the polygon has. 188 Unit K Patterns in Geometry 188 UNIT K Patterns in Geometry

20 On Your Own Exercises On Your Own Exercises Math Link Triangles are the only polygons that are rigid. If you use linkage strips to build a polygon with more than three sides, you can push on the sides or vertices to create an infinite number of different shapes. a. Copy each of these regular polygons. On each polygon, choose a vertex. Draw every possible diagonal from that vertex. Possible answers: Exercise 16b When discussing students answers, you may want to have the table for Part b already made to facilitate the discussion. Most students should be able to see the pattern for the number of diagonals from a vertex, but they may have difficulty finding the rule for the total number of diagonals in Part e. 16c. Number of diagonals = number of vertices d. Possible answer: If you pick a vertex, you cannot draw a diagonal from the vertex to itself. You cannot draw a diagonal from the vertex to the vertices on either side of it. That leaves (number of vertices - 3) vertices to which you can draw a diagonal. b. Copy and complete the table. Polygon Vertices Diagonals from a Vertex Triangle 3 0 Quadrilateral 4 1 Pentagon 5 2 Hexagon 6 3 Heptagon 7 4 Octagon 8 5 c. Describe a rule that connects a polygon s number of vertices to the number of diagonals that can be drawn from each vertex. d. Explain how you know your rule will work for polygons with any number of vertices. e. Challenge Describe a rule for predicting the total number of diagonals you can draw if you know the number of vertices in a polygon. Explain how you found your rule. Add a column to your table to help you organize your thinking. 17. Look for polygons in your home or school. Describe at least three different polygons that you find. Tell where you found them. Answers will vary. Total Diagonals The sequence rule (add 2, add 3, add 4, and so on) is the easiest for students to see, but the explicit rule, while easier to use, may require more explanation. Math Link To understand triangle rigidity, students may need to see that only one triangle can be constructed from three given side lengths. On Your Own Exercises 189 Additional Answer 16e. See completed column in Part e; Possible rules and explanations: total diagonals = number of vertices (number of vertices - 3) 2; There are (number of vertices - 3) diagonals from each vertex, and there are (number of vertices) vertices. That makes (number of vertices - 3) (number of vertices) diagonals. But this counts each diagonal twice (once for each vertex it connects), so divide by 2. The sequence in the second column is 0, 2, 5, 9, 14, 20,.... The pattern is add 2, add 3, add 4, and so on. This sequence can be continued to find the total number of diagonals for any polygon. On Your Own Exercises 189