# Lesson 1.6 Drawing Circles with a Compass

## Transcription

Lesson 1.6 Drawing Circles with a Compass

EM07TLG1_G4_U01_LOP06.qxd 1/29/06 11:32 AM Page 47 Objective 1 To provide practice using a compass. materials Teaching the Lesson Key Activities Students draw circles with a compass and construct a square inscribed in a circle by folding paper. Key Concepts and Skills • • • • Use a compass to measure distance. [Measurement and Reference Frames Goal 1] Use a compass to draw circles. [Geometry Goal 2] Construct an inscribed square. [Geometry Goal 2] Verify that the sides of a square are the same length. [Geometry Goal 2] Math Journal 1, pp. 14 and 15 Study Link 1 5 slate; compass; paper (colored, if available); straightedge; cardboard; scissors (optional) Key Vocabulary compass • circle • center (of a circle) • inscribed square 2 Ongoing Learning & Practice Students play Polygon Pair-Up to practice identifying properties of polygons. Students practice and maintain skills through Math Boxes and Study Link activities. Ongoing Assessment: Recognizing Student Achievement Use journal page 16. [Geometry Goal 1] materials Math Journal 1, p. 16 Student Reference Book, p. 258 Study Link Master (Math Masters, p. 23) Game Masters (Math Masters, pp. 496 and 497) scissors See Advance Preparation 3 Differentiation Options ENRICHMENT Students solve an inscribed square puzzle. materials EXTRA PRACTICE Students use a compass to create circle designs. Teaching Master (Math Masters, p. 24) straightedge; compass; scissors; glue; colored paper (optional) Ed Emberley’s Picture Pie: A Circle Drawing Book See Advance Preparation Additional Information Advance Preparation In Part 2, students cut apart the Polygon Deck and Property Deck from Math Masters, pages 496 and 497. Consider copying the cards on cardstock for students and making overhead transparency cards for demonstrations. For the optional Extra Practice activity in Part 3, obtain the book Ed Emberley’s Picture Pie: A Circle Drawing Book by Ed Emberly (Little Brown, 1984). Technology Assessment Management System Math Boxes, Problem 3 See the iTLG. Lesson 1 6 47 EM07TLG1_G4_U01_L06.qxd 1/29/06 11:44 AM Page 48 Getting Started Mental Math and Reflexes Math Message Write numbers such as those listed below on the board. Your job is to draw a large circle on the playground. How will you do it? Discuss the problem with a partner. Record your ideas on a half-sheet of paper. 2,510 9,246 3,082 7,682,041 4,502,639 67,314,851 32,756 172,908 530,175 Study Link 1 5 Follow-Up For each number, ask questions such as the following: Have small groups of students compare answers to Problems 1–3. Then ask students to pose their own riddles to the group. • What is the value of the digit x? • Which digit is in the thousands place? 1 Teaching the Lesson Math Message Follow-Up WHOLE-CLASS DISCUSSION Encourage students to share ideas. One method is to anchor one end of a rope to the ground, attach a piece of chalk to the other end, pull the rope taut, and then rotate it around the anchor to draw the circle. Let students judge for themselves the relative effectiveness of the methods they suggest. You might have them try out their methods on the playground. Student Page Date LESSON 16 䉬 One way of drawing a circle without a compass Time An Inscribed Square Follow the directions below to make a square that you will tape on the next page. Step 1 Use your compass to draw a circle on a sheet of colored paper. The circle should be small enough to fit on the next page. Cut out the circle. Step 2 With your pencil, make a dot in the center of the circle, where the hole is, on both the front and the back. Step 3 Fold the circle in half. Make sure that the edges match and that the fold line passes through the center. Be sure to make sharp creases. Step 4 Fold the folded circle in half again so that the edges match. Step 5 Unfold your circle. The folds should pass through the center of the circle and form 4 right angles. Step 6 Using a straightedge, connect the endpoints of the folds at the edge of the circle to make a square. Cut out the square. 116 Tell students that in this lesson they will learn how to use a compass to draw circles. Students who have never used a compass will need plenty of practice with the most basic construction—that of a circle—before attempting more difficult constructions. Drawing Circles with a Compass Demonstrate two methods for drawing circles with a compass. (These directions are for right-handed students.) Students should draw on top of cardboard or several sheets of paper to prevent damage to the desk or tabletop and to keep the anchor from slipping as the pencil is rotated. For either method, the compass should be held at the top, not by its arms. (See margin on page 49.) 14 Math Journal 1, p. 14 48 WHOLE-CLASS ACTIVITY Unit 1 Naming and Constructing Geometric Figures EM07TLG1_G4_U01_L06.qxd 1/29/06 11:45 AM Page 49 Method 1: Students should experiment with both methods and use the one with which they feel more comfortable. Have them draw circles of various sizes. Remind students of the following: As with polygons, the interior of a circle is not part of the circle. The point located at the anchor of the compass is called the center of the circle; the center is not part of the circle. All points on a circle are the same distance from the center of the circle. Constructing an Inscribed Square INDEPENDENT ACTIVITY (Math Journal 1, pp. 14 and 15) Construction of an inscribed square (a square whose vertices all lie on a circle) relies on paper folding. Demonstrate the construction while students follow your directions, or have students follow the directions on their own. 1. Press the anchor of the compass firmly onto the paper. (Some teachers find it helpful to tape the paper to the work surface.) 2. Rotate the pencil point of the compass around the anchor, keeping the paper fixed in place. 3. If the pencil is rotated clockwise, start with the pencil close to where the 5 would be located on a clock face. (If rotating counterclockwise, start near the 7.) Adjusting the Activity Have students construct an inscribed regular octagon. One way to do this is to draw and cut out a circle and fold it into eighths. The fold lines meet the circle in eight equally spaced points. These points become the vertices of the octagon. If students need a hint, you might tell them to start as though they were making a square using the method on journal page 14. A U D I T O R Y K I N E S T H E T I C T A C T I L E V I S U A L Method 1 Method 2: Rotate a single sheet of paper, keeping the anchor and pencil point fixed in place. This method is especially useful when drawing smaller circles. Student Page Date LESSON 16 䉬 Time An Inscribed Square continued Now use your compass to find out whether the 4 sides of your square are about the same length. Place the anchor on one endpoint of a side and the pencil point on the other endpoint of the side. Then, without changing the compass opening, try to place the anchor and pencil point on the endpoints of each of the other sides. Method 2 If the sides of your square are about the same length, tape the square in the space below. If not, follow the directions on page 14 again. Tape your best square in the space below. NOTE These directions are for a traditional, two-arm compass. Other compasses are available, including ruler-type compasses. These methods work with both types of compasses. 15 Math Journal 1, p. 15 Lesson 1 6 49 EM07TLG1_G4_U01_L06.qxd 1/29/06 11:45 AM Page 50 2 Ongoing Learning & Practice Playing Polygon Pair-Up PARTNER ACTIVITY (Student Reference Book, p. 258; Math Masters, pp. 496 and 497) Students play Polygon Pair-Up to practice identifying properties of polygons. Consider playing a game or two against the class on the overhead projector to help students learn the rules. Adjusting the Activity Polygon Cards from Math Masters, page 496 Use these game variations as appropriate: Variation 1: Use only the Property Cards. Players take turns drawing cards and tracing a shape from their Geometry Template that matches the property described on the card. The first time a shape from the Geometry Template is used, a player earns 3 points for a correct match. If the same shape is traced again, players earn only 1 point. All sides are the same length. All angles are right angles. When time runs out, the player with the highest score wins. Variation 2: Place all the Polygon Cards faceup. Place the Property Cards facedown between the players. Players take turns drawing a Property Card and searching the Polygon Cards to find a match. Property Cards from Math Masters, page 497 The player with the most pairs wins. A U D I T O R Y 1 6 䉬 a. 9 ⫺ 5 ⫽ b. 11 ⫺ 2 ⫽ c. d. 7 8 3 B and C ⫽ 14 ⫺ 7 A ⫽ 12 ⫺ 4 e. 13 ⫺ 6 ⫽ f. polygons? ⫽ 12 ⫺ 9 96 夹 V I S U A L INDEPENDENT ACTIVITY Writing/Reasoning Have students write a response to the following: Explain why the shapes you chose in Problem 2 are not polygons. Sample answer: B has curved sides, and A does not have sides that connect end to end. C B 7 3. Draw a quadrangle with only 1 right angle. Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 1-8. The skill in Problem 6 previews Unit 2 content. 2. Which of the shape(s) below are NOT 4 9 T A C T I L E (Math Journal 1, p. 16) Math Boxes 1. Subtract mentally. Time LESSON K I N E S T H E T I C Math Boxes 1 6 Student Page Date 4. Circle the convex polygon(s). Draw in the right angle symbol. Sample answers: Ongoing Assessment: Recognizing Student Achievement How do you know it is a right angle? It is a square corner. Use Math Boxes, Problem 3 to assess students’ understanding of right angles. Students are making adequate progress if they are able to draw an appropriate quadrangle. Some students may be able to explain how they know the angle is a right angle. 6. In the numeral 42,318, the 2 stands Draw point T on it. for 2,000. Sample answer: HA T a. The 1 stands for ? What is another name for HA b. The 8 stands for HT 91 c. The 4 stands for d. The 3 stands for 10 . 8 . 40,000 . 300 . [Geometry Goal 1] 4 16 Math Journal 1, p. 16 50 97 93 99 5. Draw and label ray HA. Math Boxes Problem 3 Unit 1 Naming and Constructing Geometric Figures EM07TLG1_G4_U01_L06.qxd 1/29/06 11:45 AM Page 51 Study Link Master Name Study Link 1 6 INDEPENDENT ACTIVITY Date STUDY LINK Time Properties of Geometric Figures 1 6 䉬 (Math Masters, p. 23) 96–100 Home Connection Students match geometric figures and properties. A B C D E F G H I Write the letter or letters that match each statement. 3 Differentiation Options ENRICHMENT Solving an Inscribed-Square INDEPENDENT ACTIVITY 1. These are polygons. 2. These are regular polygons. 3. These are quadrangles. 4. These are concave. 5. These are NOT parallelograms. A, B, C, E, F, G, and I B and C C, E, F, and I A A, B, D, F, G, H, and I 6. These do NOT have any right angles or angles whose measures are larger than a right angle. D, G, and H 15–30 Min Puzzle Try This Take a paper clip and two pencils. Create a homemade compass. You may not bend or break the paper clip. How many different size circles can you make with it? 7. 2 Practice (Math Masters, p. 24) 30 50 8. 60 11. To apply students’ understanding of inscribing polygons within circles, have them solve an inscribed-square puzzle. Polygon 1 is said to be inscribed in Polygon 2 if all of the vertices of Polygon 1 are on Polygon 2. 80 9. 80 20 12. 100 10. 250 140 120 70 13. 460 230 40 60 50 390 230 Math Masters, p. 23 Have students describe the squares they drew. Encourage words like center of circle and inscribed. EXTRA PRACTICE Creating Circle Designs INDEPENDENT ACTIVITY 30+ Min Art Link To provide practice using a compass to draw circles, have students create circle designs based on the ones in Ed Emberley’s Picture Pie: A Circle Drawing Book by Ed Emberley (Little Brown, 1984). Circles that have been constructed with a compass and cut into halves, fourths, or eighths are the basis for the artwork in the book. The circles can be constructed on paper of various colors and can be used to form elaborate designs. Teaching Master Name Date LESSON 1 6 䉬 Time A Crowded-Points Puzzle Nine points are crowded together in a large, square room. The points do not like crowds. 1. 2. Use a straightedge to draw 2 squares so that each point will have a room of its own. Explain what you did to solve this puzzle. 䉬 Describe the squares you drew using vocabulary words you learned in class. 䉬 Tell how you know that the 2 polygons you drew are squares. Sample answer: I drew 2 squares, one inside the other. They are inscribed squares because their vertices touch the other squares. Both shapes are squares because they have equal sides and right angles. Math Masters, p. 24 Lesson 1 6 51