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EM07TLG1_G4_U01_LOP08.qxd 1/29/06 11:33 AM Page 57

Objective To guide students in the construction of figures with a compass and straightedge.

1 Teaching the Lesson materials Key Activities Math Journal 1, pp. 21–24 Students continue their work with a compass and straightedge. They copy line segments, Student Reference Book, pp. 114, construct regular inscribed in circles, and divide a into six equilateral 117, and 118 (optional)

. Study Link 17 Key Concepts and Skills +, – Fact Triangles; compass; straightedge; Geometry Template; • Use a compass as a tool to measure distance. [Measurement and Reference Frames Goal 1] tape; scissors • Copy a line segment with a compass and straightedge. [Geometry Goal 1] board compass for demonstration • Use a compass to draw circles; construct a regular hexagon inscribed in a circle. purposes [Geometry Goal 2] • Verify that the sides of regular are the same length. [Geometry Goal 2] See Advance Preparation Ongoing Assessment: Informing Instruction See page 59. 2 Ongoing Learning & Practice materials

Students match descriptions of geometric figures with their names. Math Journal 1, pp. 25 and 26 Students practice and maintain skills through Math Boxes and Study Link activities. Study Link Master (Math Masters, p. 30) Ongoing Assessment: Recognizing Student Achievement Use journal page 26. [Operations and Computation Goal 1] 3 Differentiation Options materials READINESS ENRICHMENT EXTRA PRACTICE Teaching Masters (Math Masters, pp. 31–33) Students identify a regular Students create 6-point Students inscribe an hexagon in a design. designs. equilateral compass; straightedge; crayons or markers; paper inside a circle. See Advance Preparation

Additional Information Technology Advance Preparation For Parts 1 and 3, students should have plenty of paper for constructions. Assessment Management System Math Boxes, Problem 1 See the iTLG.

Lesson 18 57 EM07TLG1_G4_U01_L08.qxd 1/29/06 11:37 AM Page 58

Getting Started

Mental Math and Math Message Study Link 17 Reflexes Suppose your partner draws Follow-Up a line segment on a piece of paper. You Students take out their , Fact want to make an exact copy of the line Ask students to measure the radii of a Triangles and practice the facts in their segment without using a copying partner’s circles. They should compare Try Again piles. When time is up, students machine. How would you do it? Record the measurements and remeasure if transfer appropriate triangles to the OK your ideas on a half-sheet of paper. they disagree. pile, fasten their new piles with paper clips, and store them.

NOTE Some students may benefit from doing the Readiness activity before you begin 1 Teaching the Lesson Part 1 of the lesson. See the Readiness activity in Part 3 for details. PARTNER Math Message Follow-Up ACTIVITY

Before discussing the Math Message problem as a class, have partners discuss how they would copy the line segment. Strategies might include the following: Use tracing paper and trace the line segment. Measure the line segment with a ruler, and then draw another line segment that is the same length. If no student mentions the use of a compass and straightedge, ask if anyone can think of a way that these tools could be used to copy the line segment. Allow several minutes for partners to discuss the question. Then tell students that in this lesson they will learn how to use a compass and straightedge to copy a line segment. Student Page

Date Time INDEPENDENT Making Constructions with ACTIVITY LESSON 1᭜ 8 Copying a Line Segment Steps 1–4 below show you how to copy a line segment. a Compass and Straightedge Step 1 You are given line segment AB AB to copy. (Math Journal 1, pp. 21–24)

Step 2 Draw a line segment that is longer than line segment AB. Label one of its C endpoints C. The constructions in the previous lessons consisted mainly of

Step 3 Open your compass so that the drawing circles with a compass. The compass-and-straightedge anchor is on one endpoint of line segment AB and the pencil point constructions in this lesson involve marking equal distances is on the other endpoint. A B between points on a circle. Step 4 Without changing the compass opening, place the anchor on point C on your second line segment. Make a mark that At first, you probably will need to demonstrate the constructions crosses this line segment. Label the point where the mark crosses the line segment on the board or overhead projector. The directions also appear in with the letter D. C D the journal. Line segment CD should be about the same length as line segment AB. Line segments CD and AB are congruent.

Use a compass and straightedge to copy the line segments shown below. For each Copying a Line Segment problem, begin by drawing a line segment that is longer than the one given.

1. Go over the steps for copying a line segment shown on journal E F page 21. Students should practice this construction several times. Then have students copy line segments EF and MN onto the

2. M N journal page. Remind them that their straightedges are for drawing straight lines, not for measuring. 21 Math Journal 1, p. 21

58 Unit 1 Naming and Constructing Geometric Figures EM07TLG1_G4_U01_L08.qxd 1/29/06 11:37 AM Page 59

Student Page

Date Time

Constructing an Inscribed Regular Hexagon LESSON 1᭜ 8 Constructing an Inscribed, Regular Hexagon Discuss the examples of regular hexagons shown on journal Follow each step below. Draw on a separate sheet of paper. Repeat these steps several times. Cut out your best work, and tape it onto the bottom of this page.

page 22. Then demonstrate how to construct an inscribed regular Step 1 Draw a circle. (Keep the same compass opening for Steps 2 and 3.) Draw a dot on hexagon while students follow the directions on page 23. the circle. Place the anchor of your compass on the dot and make a mark on the circle. Language Arts Link Explain that the root word scribe Step 2 Place the anchor of your compass on comes from the Latin word that means “to write.” In the mark you just made and make another mark on the circle. mathematics, the word inscribe means “to write or draw a figure

inside another figure such that every vertex of the inside figure Step 3 Do this four more times to divide the circle into 6 equal parts. The 6th mark touches the outside figure.” should be on the dot you started with or very close to it. After completing Step 3, there should be 6 arcs. The points where Step 4 With your straightedge, connect the these 6 arcs cross the circle will become the 6 vertices of the 6 marks on the circle to form a regular hexagon. Use your compass to check that the sides of the hexagon are all about the hexagon. Before going on to Step 4, ask the following questions: same length.

● The hexagon is inscribed in the circle because What do you notice about the 6 marks? They divide the circle each vertex of the hexagon is on the circle. into 6 equal parts. ● Without measuring, how do you know that the marks are the same distance apart? The compass opening did not change.

● What do you think the next step should be? Connect the 6 marks. 23 Math Journal 1, p. 23 After Step 4, ask the following question: ● How do we know that all the vertices of the hexagon are the same distance from the center of the circle? All the vertices of the hexagon are on the circle. Because all points on a circle are the same distance from the center, all the vertices must be the same distance from the center. After students have practiced this construction several times, they should tape their best work onto journal page 23.

Ongoing Assessment: Informing Instruction

Watch for students who have difficulty connecting consecutive marks in Step 4 to form the regular hexagon. If students completed the Readiness activity in Part 3, Student Page refer them to the regular hexagon outlined in red. This will provide them with a Date Time LESSON visual image of the design they are trying to create. 1᭜ 8 More Constructions

Construct a regular hexagon on a separate sheet of paper. Then divide the hexagon into 6 equilateral triangles. Use your compass to check that the sides of the 6 equilateral triangles are all about the same length. Try this several times until you are satisfied with your work. Then cut out your best Dividing a Regular Hexagon work and tape it in the space below. Have students construct another inscribed hexagon and then divide it into six equilateral triangles by drawing appropriate diagonals. Encourage them to keep trying until they have one they can tape onto journal page 24.

Adjusting the Activity

Encourage students to do additional constructions from pages 114, 117, and 118 of the Student Reference Book.

AUDITORY KINESTHETIC TACTILE VISUAL

24 Math Journal 1, p. 24

Lesson 18 59 EM07TLG1_G4_U01_L08.qxd 1/29/06 11:37 AM Page 60

Student Page

Date Time

LESSON 1᭜ 8 Definition Match Match each description of a geometric figure in Column I with its name 2 Ongoing Learning & Practice in Column II. Some of the items in Column II do not have a match. 94–100

I II INDEPENDENT a. a with 4 right angles and f 4 sides of the same length Defining Geometric Figures ACTIVITY e (Math Journal 1, p. 25) b. a polygon with 4 sides, none of which are the same length h right angle Students match descriptions of geometric figures with names. c. a with exactly 1 pair c of opposite sides that is parallel

hexagon d. lines that never intersect INDEPENDENT a Math Boxes 1 8 ACTIVITY e. a with all sides the same length, but not a i (Math Journal 1, p. 26)

f. a polygon with 8 sides perpendicular lines

d parallel lines Mixed Practice Math Boxes in this lesson are paired g. a polygon with 5 sides with Math Boxes in Lesson 1-6. The skill in Problem 6 g h. an angle that measures 90° previews Unit 2 content. i. a triangle with all sides the same length b quadrangle Ongoing Assessment: Math Boxes 25 Problem 1 Math Journal 1, p. 25 Recognizing Student Achievement

Use Math Boxes, Problem 1 to assess students’ automaticity with subtraction facts. Students are making adequate progress if they compute the differences in Problems 1a–1f correctly. Some students may be able to explain how to use addition to check their answers. [Operations and Computation Goal 1]

INDEPENDENT Study Link 1 8 ACTIVITY (Math Masters, p. 30)

Home Connection Students inscribe polygons in circles.

Student Page Study Link Master

Date Time Name Date Time

LESSON STUDY LINK 1᭜ 8 Math Boxes 1᭜ 8 Inscribed Polygons

1. Subtract mentally. 2. Which of the shape(s) below are 1. Use a straightedge to inscribe a different Example: 96 97 A and C polygon in each of the circles below. 115 ଙa. 14 Ϫ 9 ϭ 5 polygons? Write the name of each polygon. b. 13 Ϫ 8 ϭ 5

ϭ Ϫ c. 9 18 9 Sample answers: A B C d. 17 Ϫ 8 ϭ 9

e. 4 ϭ 11 Ϫ 7

f. 9 ϭ 15 Ϫ 6 96

3. Draw a quadrangle that has 2 pairs of 4. Circle the concave (nonconvex) polygon(s). square Equilateral triangle parallel sides and no right angles. a. b. Sample answer:

What kind of quadrangle is this?

parallelogram c.hexagon d. Rectangle and triangle

99 100 97 2. Are any of the polygons that you drew regular polygons? Explain how you know. 5. Draw and label ray CA. 6. In the numeral 30,516, what does the Sample answer: Sides are all the same length, Draw point R on it. 3 stand for? Circle the best answer. and interior angles are all the same measure. Sample answer: A. 3,000 R B. 30 A C C. 30,000 Practice

What is another name for ray CA? D. 300,000 3. 41 ϩ 27 ϭ 68 4. 322 ϭ 263 ϩ 59 5. 461 ϩ 398 ϭ 859 CR៮៬ 6. 36 ϭ 72 Ϫ 36 7. 158 Ϫ 71 ϭ 87 8. 742 Ϫ 349 ϭ 393 91 4

26 Math Journal 1, p. 26 Math Masters, p. 30

60 Unit 1 Naming and Constructing Geometric Figures Name Date Time

LESSON 1᭜8 A Hexagon Design

1. Outline the regular hexagon in the design to the right using a red crayon or pencil. Use your crayons 3 Differentiation Options or pencils to color the design in an interesting way.

INDEPENDENT READINESS ACTIVITY 2. How do you know the polygon you outlined is a regular hexagon? Sample answer: All the sides are the same ᭤ Identifying a Regular Hexagon 5–15 Min length, and all the angles have the same measure. (Math Masters, p. 31)

To explore the concept of regular polygons, have students identify Math Masters, page 31 and describe a regular hexagon inscribed in a circle. Ask them to outline the regular hexagon in a design and then color the design in an interesting way. Have students share their definitions of a .

INDEPENDENT ENRICHMENT ACTIVITY

᭤ Creating 6-Point Designs 15–30 Min (Math Masters, p. 32)

Art Link To apply students’ ability to inscribe hexagons in circles, have them create hexagram designs. Suggestions are on Math Masters, page 32. Students can work on these designs first and then create some of their own as an ongoing project for the next few weeks. Their finished work could be displayed in a Geometry Art Exhibit.

Teaching Master

Name Date Time Sample hexagram design from Math Masters, page 32 LESSON 1 ᭜ 8 An Inscribed Equilateral Triangle

Follow each step below. Draw on a separate sheet of paper. Repeat these steps several times. Then cut out your best work and tape it to the bottom of this page.

INDEPENDENT Step 1 Draw a circle. (Keep the same EXTRA PRACTICE ACTIVITY compass opening for Steps 2 and 3.) Draw a dot on the circle. Place the anchor of your compass on the dot and draw a mark on the circle. ᭤ Inscribing an Equilateral 5–15 Min Step 2 Place the anchor of your compass on the mark you just made and draw Triangle in a Circle another mark on the circle. (Math Masters, p. 33)

Step 3 Do this 4 more times to divide the circle into 6 equal parts. The 6th mark should be on the dot you To provide practice inscribing polygons in circles, students use a started with or very close to it. compass to divide a circle into six equal parts. Then they inscribe an equilateral triangle by connecting three alternating points on Step 4 With your straightedge, connect 3 alternating marks (every other mark) on the circle to form an equilateral the circle. triangle. Use your compass to check that the sides of the equilateral triangle are about the same length.

Math Masters, p. 33

Lesson 1᭜8 61