5
www.everydaymathonline.com Lesson 5 7 369 Advance Preparation Use a board compass and meterstick (or another long straightedge) to demonstrate Part 1 constructions on the board. Teacher’s Reference Manual, Grades 4–6 pp. 42–44, 192–199, 209–211 Key Concepts and Skills • Use appropriate strategies to measure angles to the nearest degree. [Measurement and Reference Frames Goal 1] • Label the points and vertices of a preimage and its image. [Measurement and Reference Frames Goal 3] • Use a straightedge and compass to construct a figure that is congruent to the original. [Geometry Goal 2] Key Activities Students review two basic compass-and- straightedge constructions—copying a line segment and copying a triangle. Ongoing Assessment: Informing Instruction See page 371. Key Vocabulary compass-and-straightedge construction anchor (of a compass) concentric circles Materials Math Journal 1, pp. 188 and 189 Student Reference Book, pp. 188–190 Study Link 5 6 compass straightedge ruler Geometry Template/protractor tape scissors board compass and meterstick (for demonstration purposes) Playing 2-4-8 or 3-6-9 Frac-Tac-Toe (Decimal Versions) Student Reference Book, pp. 314–316 and 372 Math Masters, pp. 439–441 per partnership: 4 each of number cards 0–10 (from the Everything Math Deck, if available) 2 different-colored counters or pennies calculator Students practice renaming fractions as decimals. Ongoing Assessment: Recognizing Student Achievement Use an Exit Slip (Math Masters, page 404).  [Number and Numeration Goal 5] Math Boxes 5 7 Math Journal 1, p. 190 ruler Students practice and maintain skills through Math Box problems. Study Link 5 7 Math Masters, p. 166 Students practice and maintain skills through Study Link activities. READINESS Practicing Circle Constructions Math Masters, p. 167 compass scissors tape or glue Students make circle constructions to practice anchoring and rotating a compass. ENRICHMENT Constructing an Octagon Math Masters, p. 168 compass straightedge Students construct an octagon using a compass and a straightedge. EXTRA PRACTICE Making Congruent Shapes on Geoboards Math Masters, p. 419 (optional) per partnership: 2 geoboards with rubber bands Students practice making and identifying congruent shapes. Teaching the Lesson Ongoing Learning & Practice 1 3 2 4 Differentiation Options Compass-and-Straightedge Constructions Part 1 Objective To construct figures with a compass and a straightedge. eToolkit ePresentations Interactive Teacher’s Lesson Guide Algorithms Practice EM Facts Workshop Game™ Assessment Management Family Letters Curriculum Focal Points Common Core State Standards

Compass-and-Straightedge Constructions Part 1 circlesare circles that have the same center. Use a compass to draw 3 concentric circles. Compass-and-Straightedge Constructions Many

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Page 1: Compass-and-Straightedge Constructions Part 1 circlesare circles that have the same center. Use a compass to draw 3 concentric circles. Compass-and-Straightedge Constructions Many

www.everydaymathonline.com

Lesson 5�7 369

Advance PreparationUse a board compass and meterstick (or another long straightedge) to demonstrate Part 1 constructions on the board.

Teacher’s Reference Manual, Grades 4–6 pp. 42–44, 192–199, 209–211

Key Concepts and Skills• Use appropriate strategies to measure

angles to the nearest degree. [Measurement and Reference Frames Goal 1]

• Label the points and vertices of a preimage and its image. [Measurement and Reference Frames Goal 3]

• Use a straightedge and compass to construct a figure that is congruent to the original. [Geometry Goal 2]

Key ActivitiesStudents review two basic compass-and-straightedge constructions—copying a line segment and copying a triangle.

Ongoing Assessment: Informing Instruction See page 371.

Key Vocabularycompass-and-straightedge construction �

anchor (of a compass) � concentric circles

MaterialsMath Journal 1, pp. 188 and 189Student Reference Book, pp. 188–190Study Link 5�6compass � straightedge � ruler � Geometry Template/protractor � tape � scissors � board compass and meterstick (for demonstration purposes)

Playing 2-4-8 or 3-6-9 Frac-Tac-Toe (Decimal Versions)Student Reference Book, pp. 314–316 and 372Math Masters, pp. 439– 441 per partnership: 4 each of number cards 0–10 (from the Everything Math Deck, if available) � 2 different-colored counters or pennies � calculatorStudents practice renaming fractions as decimals.

Ongoing Assessment: Recognizing Student Achievement Use an Exit Slip (Math Masters, page 404).  [Number and Numeration Goal 5]

Math Boxes 5�7Math Journal 1, p. 190ruler Students practice and maintain skillsthrough Math Box problems.

Study Link 5�7Math Masters, p. 166 Students practice and maintain skillsthrough Study Link activities.

READINESS

Practicing Circle ConstructionsMath Masters, p. 167compass � scissors � tape or glueStudents make circle constructions to practice anchoring and rotating a compass.

ENRICHMENTConstructing an OctagonMath Masters, p. 168compass � straightedge Students construct an octagon using a compass and a straightedge.

EXTRA PRACTICE Making Congruent Shapes on GeoboardsMath Masters, p. 419 (optional)per partnership: 2 geoboards with rubber bandsStudents practice making and identifying congruent shapes.

Teaching the Lesson Ongoing Learning & Practice

132

4

Differentiation Options

Compass-and-StraightedgeConstructions Part 1

Objective To construct figures with a compass and a straightedge.

������

eToolkitePresentations Interactive Teacher’s

Lesson Guide

Algorithms Practice

EM FactsWorkshop Game™

AssessmentManagement

Family Letters

CurriculumFocal Points

Common Core State Standards

369_EMCS_T_TLG1_G6_U05_L07_576833.indd 369 2/9/11 1:28 PM

Page 2: Compass-and-Straightedge Constructions Part 1 circlesare circles that have the same center. Use a compass to draw 3 concentric circles. Compass-and-Straightedge Constructions Many

Concentric circles are circles that have the same center.

Use a compass to draw 3 concentric circles.

Compass-and-Straightedge Constructions

Many geometric figures can be drawn using only a compass and straightedge. The compass is used to draw circles and mark off lengths. The straightedge is used to draw straight line segments.

Compass-and-straightedge constructions serve many purposes:

♦ Mathematicians use them to study properties of geometric figures.

♦ Architects use them to make blueprints and drawings.♦ Engineers use them to develop their designs.♦ Graphic artists use them to create illustrations on a computer.

In addition to a compass and a straightedge, the only materialsyou need are a drawing tool (a pencil with a sharp point is thebest) and some paper. For these constructions, you may notmeasure the lengths of line segments with a ruler or the sizes ofangles with a protractor.

Draw on a surface that will hold the point of the compass (alsocalled the anchor) so that it does not slip. You can draw on astack of several sheets of paper.

The directions below describe two ways to draw circles. For eachmethod, begin in the same way:

♦ Draw a small point that will be the center of the circle.♦ Press the compass anchor firmly on the center of the circle.

Method 1 Hold the compass at the top and rotate the pencilaround the anchor. The pencil must go all the way around tomake a circle. Some people find it easier to rotate the pencil as far as possible in one direction, and then rotate it in theother direction to complete the circle.

Method 2 This method works best with partners. One partner holds the compass in place while the other partnercarefully turns the paper under the compass to form the circle.

Geometry and Constructions

Architect’s drawing of a house plan

Method 1

Method 2

concentric circles

Student Reference Book, p. 188

Student Page

370 Unit 5 Geometry: Congruence, Constructions, and Parallel Lines

Getting Started

Math MessageRead the directions on page 188 of the Student Reference Book. Use a compass to draw three concentric circles on a separate sheet of paper.

Mental Math and Reflexes Students write numbers in scientific notation. Suggestions:

702,000,000 7.02 ∗ 108

0.0000513 5.13 ∗ 10–5

393.4 3.934 ∗ 102

865 _ 10,000 8.65 ∗ 10–2

1 Teaching the Lesson

▶ Math Message Follow-Up WHOLE-CLASSDISCUSSION

(Student Reference Book, p. 188)

Discuss page 188 in the Student Reference Book. Make sure students understand that in a compass-and-straightedge construction, they are allowed to use only a compass, a straightedge, and a pencil. They may not use rulers and protractors. Tracing is not allowed.

NOTE A straightedge is any tool used for drawing straight lines. A ruler is a straightedge and a measuring tool.

Remind students to work on top of a notebook or several sheets of paper when using a compass with a sharp point. Doing so will keep the anchor (the sharp point) of the compass from slipping as the pencil (or paper) is rotated.

Concentric circles are circles that have the same center but radiuses of different lengths. When drawing concentric circles, students should maintain the anchor’s position at the center of the circle and change only the compass opening.

▶ Reviewing Compass-and-

WHOLE-CLASS ACTIVITY

Straightedge Constructions(Student Reference Book, pp. 189 and 190)

Review pages 189 and 190 in the Student Reference Book with the class. The pages give step-by-step directions for copying line segments and triangles. When copying figures, it is helpful to name points in the copied figure after points in the original. The symbol (') can be added after the letter to distinguish between the original figure and its copy. Consider modeling several constructions on the board with a board compass and meterstick.

Study Link 5�6 Follow-UpPoint out that the copy of the path in Problem 2 is the mirror image of the original path. The copy of the path in Problem 3 is obtained by rotating the original path 180° in either direction—clockwise or counterclockwise.

Mathematical PracticesSMP1, SMP3, SMP5, SMP6Content Standards6.NS.6b

370-373_EMCS_T_TLG1_G6_U05_L07_576833.indd 370 3/19/12 9:17 AM

Page 3: Compass-and-Straightedge Constructions Part 1 circlesare circles that have the same center. Use a compass to draw 3 concentric circles. Compass-and-Straightedge Constructions Many

Constructing Line SegmentsLESSON

5�7

Date Time

Use only a compass, a straightedge, and a sharp pencil for the constructions below. Use rulers and protractors only to check your work. Do not trace.

1. Copy this line segment. Label the endpoints of your copy A� and B�. (These symbolsare read A prime and B prime.)

2. Construct a line segment twice as long as CD�. Label the endpoints of your segmentC� and D�.

3. Construct a line segment as long as EF� and GH� together. Label the endpoints ofyour segment E � and H �.

4. Construct a segment with a length equal to the length of IJ� minus the length of KL�. Label the endpoints of your segment I � and K�.

K L

I J

B'A'A B

C D C' D'

E F G H

E' H'

Try This

I' K'

189

Math Journal 1, p. 188

Student Page

Constructing TrianglesLESSON

5�7

Date Time

Try This

Use only a compass, a straightedge, and a sharp pencil for the constructions below. Use rulers and protractors only to check your work. Do not trace.

Make your constructions on another sheet of paper. If your compass has a sharp point,work on top of a piece of cardboard or a stack of several sheets of paper. When you are satisfied with a construction, cut it out and tape it onto this page.

1. Copy triangle ABC. Label the vertices of your copy A�, B�, and C�.

2. Construct a triangle with each side the same length as DE�.

4. Is it possible to draw a triangle with sides 3 inches, 3 inches, and 7 inches long?

Explain.

3. Use a ruler to draw a line segment 2 inches long and another line segment3 inches long. Then use a compass and astraightedge to construct a triangle with sides2 inches, 2 inches, and 3 inches long.

A

C

B A'B'

C'

D E

2 in. 2 in.

3 in.

With a base of 7 inches, two 3-inch sides would nevermeet to form the third angle.

no

Math Journal 1, p. 189

Student Page

Lesson 5�7 371

Ongoing Assessment: Informing InstructionRemind students that while doing a construction, they should not measure line segments with a ruler or measure angles with a protractor. They can, however, use a ruler and/or protractor to check their work when they have completed the construction.

▶ Constructing Line Segments PARTNER ACTIVITY

and Triangles(Math Journal 1, pp. 188 and 189)

After students have learned how to copy line segments and triangles, have them use these constructions to complete the problems on journal pages 188 and 189.

After students have completed Problem 3, have them compare their triangles with a partner and decide if the two are congruent. Give students time to work with a partner to investigate whether it is possible to construct a triangle with the same three side lengths as their triangles, but is NOT congruent to them. For an additional case, have them try to construct two non-congruent triangles with side lengths measuring 2 inches, 3 inches, and 4 inches. Discuss students’ triangles. Help them conclude that only one unique triangle can be created from three given side lengths.

Using students’ responses to Problem 4, lead the class to understand that a triangle cannot have the given side lengths. Challenge students to start with side lengths 7 inches and 3 inches and experiment with other measures for the third side until they successfully construct a triangle. On the board, record and organize side lengths students used that did and did not lead to a triangle. Make sure students notice that a triangle cannot be created if one side is larger than or equal to the combined lengths of the other two sides.

2 Ongoing Learning & Practice

▶ Playing 2-4-8 or 3-6-9 PARTNER ACTIVITY

Frac-Tac-Toe (Decimal Versions)(Student Reference Book, pp. 314–316 and 372; Math Masters, pp. 439–441)

Students use Math Masters, page 439 to organize the cards when playing the game. Depending on students’ needs and abilities, use Math Masters, page 441 to work with denominators of 3, 6, and 9 or Math Masters, page 440 to work with denominators of 2, 4, and 8. Players may use a calculator or the Table of Decimal Equivalents for Fractions (Student Reference Book, p. 372) to convert their fractions to decimals.

SOLVING

NOTE You may wish to have students use tools in the eToolkit, such as the line and compass tools, to construct figures.

370-373_EMCS_T_TLG1_G6_U05_L07_576833.indd 371 3/19/12 12:28 PM

Page 4: Compass-and-Straightedge Constructions Part 1 circlesare circles that have the same center. Use a compass to draw 3 concentric circles. Compass-and-Straightedge Constructions Many

Math Boxes LESSON

5�7

Date Time

2. Solve mentally.

a. 40% of 55 �

b. �56� of 72 �

c. � 50% of 2�12�

d. � �45� of 100 80

1.2560

22

3. a. Draw a line segment that is 8.6 cm long.

b. By how many centimeters would you need to extend the line segment you drew to make it 10 centimeters long?

1.4 centimeters

180 234 49 50 88

209

241–24388

5. Find the value that makes the numbersentence true.

a. �1x2� � 12 x �

b. 3 � (3 � b) � 0 b �

c. 7 � (8 � n) � 21 n � 51

1444. Multiply or divide.

a. 63 � �19� �

b. 81 � 9 �

c. � 140 � �17�

d. � 180 � 6 3020

97

1. If point A (0,5) is reflected over the x-axis,what are the coordinates of A�?

Fill in the circle next to the best answer.

A (5,0)

B (�5,0)

C (5,5)

D (0,�5)

Math Journal 1, p. 190

Student Page

STUDY LINK

5 �7 Angle Relationships

163 233

Name Date Time

Write the measures of the angles indicated in Problems 1–6. Do not use a protractor.

1.

m∠r =

m∠s =

m∠t =

3. Figure CBDA is a parallelogram. ∠DAE is a straight angle.

m∠A =

m∠B =

m∠C =

5. Angles x and y have the same measure.

m∠x =

m∠y =

m∠z =

x

y z

60°A

E

D

C

B

2. ∠JKL is a straight angle.

m∠NKO =

4. Angles a and t have the same measure.

m∠a =

m∠c =

m∠t =

6.

m∠p =

a

tc

21°

135°

MJ

K

L O

N

15°

93°

62°

p

36°

133°rs t

Practice

7. 0.09 º 0.03 = 8. 0.15 º 0.8 =

9. 0.07 º 0.07 = 10. 0.75 º 0.3 =

47°133°47°

57°114°57°

120°120°60°

10°

45°45°135°

0.2250.00490.120.0027

54°

EM3cuG6MM_U05_145-179.indd 166 3/29/10 5:16 PM

Math Masters, p. 166

Study Link Master

372 Unit 5 Geometry: Congruence, Constructions, and Parallel Lines

Ongoing Assessment: Exit Slip �Recognizing Student Achievement

Have students use an Exit Slip (Math Masters, p. 404) to record fraction and decimal equivalents they cover on the gameboard. Use students’ recorded equivalencies to assess their ability to convert between fractions and decimals. [Number and Numeration Goal 5]

▶ Math Boxes 5�7

INDEPENDENT ACTIVITY

(Math Journal 1, p. 190)

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 5-5. The skills in Problems 4 and 5 preview Unit 6 content.

Writing/Reasoning Have students write a response to the following: Sally said, “To solve Problem 1, I don’t have to graph the points. I only have to find an opposite.” Do you agree with Sally? Explain your answer. Sample answer: I agree. I know that when a point is reflected across the x-axis, the x-coordinate stays the same and the y-coordinate changes sign. So I only have to find the opposite of 5 to know that the answer is (0,–5).

▶ Study Link 5�7

INDEPENDENT ACTIVITY

(Math Masters, p. 166)

Home Connection Students solve problems involving angle relationships.

NOTE Study Links 5-7 and 5-8 do not reinforce the activities in these two lessons. Few students have the tools necessary to perform constructions at home. Instead, Study Links 5-7 and 5-8 provide practice with other skills and concepts presented in Unit 5.

3 Differentiation Options

READINESS

INDEPENDENT ACTIVITY

▶ Practicing Circle 15–30 Min

Constructions(Math Masters, p. 167)

To provide practice anchoring and rotating their compasses, have students complete Math Masters, page 167.

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Page 5: Compass-and-Straightedge Constructions Part 1 circlesare circles that have the same center. Use a compass to draw 3 concentric circles. Compass-and-Straightedge Constructions Many

LESSON

5�7

Name Date Time

Circle Constructions

Use the directions and the pictures below to make one or both of the constructions.You may need to make several versions of the construction before you are satisfiedwith your work. Cut out your best constructions and tape or glue them to anothersheet of paper.

Construction #1

Step 1 Draw a small point that will be the center of thecircle. Press the compass anchor firmly onthe center of the circle.

Step 2 Hold the compass at the top and rotate thepencil around the anchor. The pencil must go allthe way around to make a circle. You may find iteasier to firmly hold the compass in place andcarefully turn the paper under the compass.

Step 3 Without changing the opening of thecompass, draw another circle that passesthrough the center of the first circle.Mark the center of the second circle.

Step 4 Repeat Step 3 to draw a third circle thatpasses through the center of each of thefirst two circles.

Construction #2

Follow the steps to make the three circles fromConstruction #1. Then draw more circles to createthe construction shown at the right.

Math Masters, p. 167

Teaching Master

LESSON

5�7

Name Date Time

Octagon Construction

Use the directions and the pictures to construct an octagon.

Step 1 Draw a circle with a compass. Label thecenter of the circle A. Then draw adiameter through A. Label the two pointswhere the diameter intersects the circleS and T.

Step 2 Use the length of ST� to set the compass opening.Then place the anchor of your compass on S anddraw an arc below the circle and another arc above the circle. Without changing the compassopening, place the compass anchor on T and draw another set of arcs above and below thecircle. Label the points where the arcs intersectas D and E. Draw a line through D and E.Label the points where DE��� intersects the circleas M and N.

Step 3 Set the compass opening so that it is equal tothe length of SM�. Then place the compass anchoron S and draw an arc; reposition the anchoron M and draw another arc. Label the point where the arcs intersect as G. Draw a linethrough G and A. Label the points where GA���

intersects the circle as H and I.

Step 4 Repeat Step 3, using the length of MT� to setthe compass opening. Label the points ofintersection as X, Y, and Z.

Step 5 Connect the points on the circle to form an octagon.

AS T

A

G

H

D

M

N

E

S

I

T

A

G

H

D

M X Y

N

E

Z

S

I

T

Math Masters, p. 168

Teaching MasterTeaching Aid MasterName Date Time

Dot Paper (1 cm)

Math Masters, p. 419

Lesson 5�7 373

ENRICHMENT

INDEPENDENT ACTIVITY

▶ Constructing an Octagon 15–30 Min

(Math Masters, p. 168)

To extend students’ knowledge of geometric constructions, they complete Math Masters, page 168. In this activity, students draw diameters and chords in circles, using a compass and a straightedge. Then they use these parts of the circle to construct an octagon.

EXTRA PRACTICE PARTNER ACTIVITY

▶ Making Congruent Shapes 5–15 Min

on Geoboards(Math Masters, p. 419)

To provide practice making and identifying congruent shapes, have students work in partnerships. One partner uses one or more rubber bands to make a shape on a geoboard. The other partner makes an exact copy of the shape on a different geoboard and explains why the two shapes are congruent. Then partners switch roles. If geoboards are not available, have students draw and copy shapes on dot paper (Math Masters, page 419).

EM3cuG6TLG1_370-373_U05L07.indd 373 12/13/10 8:58 AM