The Half-Circle Protractor

(1)

www.everydaymathonline.com eToolkit ePresentations Interactive Teacher’s Lesson Guide Algorithms Practice EM Facts Workshop Game™ Assessment Management Family Letters Curriculum Focal Points Common Core State Standards Lesson 67 437

Advance Preparation

For the optional Enrichment activity in Part 3, obtain the book Sir Cumference and the Great Knight of Angleland by Cindy Neuschwander (Charlesbridge Publishing, 2001).

Teacher’s Reference Manual, Grades 4–6 pp. 44–46, 178–180, 234–236 Key Concepts and Skills

• Add and subtract to find unknown angle measures.

[Operations and Computation Goal 2] • Use reference points to estimate the

measures of angles; use a half-circle protractor to measure and draw angles. [Measurement and Reference Frames Goal 1] • Classify angles according to their measure.

[Geometry Goal 1] Key Activities

Students identify types of angles. They measure and draw angles with a half-circle protractor. They identify an angle as obtuse or acute to help them determine which protractor scale to use. They add and subtract to find unknown angle measures.

Ongoing Assessment:

Recognizing Student Achievement Use Mental Math and Reflexes. [Number and Numeration Goal 1]

Ongoing Assessment:

Informing Instruction See page 440. Key Vocabulary

acute angle obtuse angle reflex angle straight angle half-circle protractor base line

Materials

Math Journal 1, pp. 157 and 158 Study Link 66 half-circle protractor

protractor for demonstration purposes straightedge chart paper

World Tour Option: Visiting Europe Math Journal 1, p. 171 (optional); pp. 172, 173, 180, and 181 Student Reference Book, pp. 276, 277, 280, 284, 285, 297, and 302–305 Math Masters, pp. 419–421 (optional) Students resume the World Tour in Europe.

Math Boxes 67

Math Journal 1, p. 159

Students practice and maintain skills through Math Box problems. Study Link 67

Math Masters, p. 192 protractor

Students practice and maintain skills through Study Link activities.

READINESS Modeling Angles

Student Reference Book, p. 93 rope or string

Students model angles with a rope.

ENRICHMENT

Measuring Angles in Triangles Math Masters, p. 193

straightedge protractor glue or tape Students determine that the sum of the measures of the angles of any triangle is 180°.

ENRICHMENT

Exploring Angles in Literature Math Masters, p. 388 or 389

Students read Sir Cumference and the Great Knight of Angleland.

ELL SUPPORT Building Background for Mathematics Words

Students create a graphic organizer for the word angle.

Teaching the Lesson

Ongoing Learning & Practice

Differentiation Options

The Half-Circle

Protractor

Objectives

To guide students as they classify angles as acute,

right, obtuse, straight, and reflex; and to provide practice using a half-circle protractor to measure and draw angles.

437_EMCS_T_TLG1_G4_U06_L07_576817.indd 437

(2)

Links to the Future

Identifying and describing acute, obtuse, straight, and reflex angles is a Grade 5 Goal.

Ongoing Assessment:

Mental Math and Reflexes

Recognizing Student Achievement

Use Mental Math and Reflexes to assess students’ ability to identify places in decimals and the values of the digits in those places. Students are making adequate progress if they can correctly identify and express the values of digits through thousandths. Some students may correctly identify and express the values of digits through ten thousandths.

[Number and Numeration Goal 1]

1 Teaching the Lesson

Math Message Follow-Up

WHOLE-CLASS

ACTIVITY

(Math Journal 1, p. 157)

In discussing the answers, talk about the angles in terms of rotations: ∠ A is less than a _ 1

4 turn, ∠ B is more than a 1

_

4 turn but

less than a _ 1

2 turn, and ∠C is between a 1

_

2 turn and 1 full turn.

Call attention to the names for the various types of angles. To support English language learners, write the names on chart paper next to an example of each. Display the chart paper throughout the unit.

An acute angle measures between 0° and 90°.

An obtuse angle measures between 90° and 180°.

A reflex angle measures between 180° and 360°. Remind students that a 90° angle is called a right angle.

Complete the list of angle names by mentioning that a 180° angle is called a straight angle. Ask: Why is this a good name for this

angle?In a 180° angle, the two sides meet to form a straight line.

ELL

Getting Started

Math Message

Complete the Math Message problems on journal page 157.

Study Link 6

6 Follow-Up

Students compare answers. Make sure they understand that the directional arc shows the path and direction of the rotation.

Mental Math and Reflexes

Write decimals on the board and have volunteers read them aloud. Suggestions:

3.45 12.358 10.005

0.27 60.893 2.6074

6.89 83.591 26.0801

Ask questions such as:

• What digit is in the hundredths place? • What is the value of the digit x?

(3)

Drawing and Measuring Angles

LESSON

67

Date Time

Math Message

Use a straightedge to draw the following angles. Do not use a protractor.

∠A: any angle that ∠B: any angle that measures ∠C: any angle that measures less more than 90° and less measures more than 90° than 180° than 180°

∠A is called an acute angle. ∠B is called an obtuse angle. ∠C is called a reflex angle. Measuring Angles with a Protractor

Write whether the angle is acute or obtuse. Then measure it as accurately as you can.

∠SDE is acute. ∠COR is acute. ∠RTV is obtuse.

∠SDE measures 55 °. ∠COR measures 40 °. ∠RTV measures 140 °.

Answers may vary by 3 degrees either way.

D S E O R C T R V 92 93 141 Sample answers: A B C EM3MJ1_G4_U06_137-169.indd 157 1/13/11 3:28 PM Math Journal 1, p. 157

Student Page

Lesson 67 439

The arc indicates which angle to consider.

90 80 70 60 50 40 30 20 10 100110 120 130 140 150 160 170 180 0 100 110 120 130 140 150 160 170 8070 60 50 40 30 20 10 0 180 center baseline a 40° angle 90 80 70 60 50 40 30 20 10 100110 120 130 140 150 160 170 180 0 100 110 120 130 140 150 160 170 8070 60 50 40 30 20 10 0 180 a 115° angle

Introducing the Half-Circle

WHOLE-CLASS

ACTIVITY

Protractor

Have students examine their half-circle protractors. Students can use either the protractor on the Geometry Template or any other protractor. If some students’ protractors do not have labels for the 0° and 180° marks, have them write the labels. The marks may smear and disappear later, but they are helpful for this introduction to the half-circle protractor.

Ask partnerships to decide how the half-circle protractor is different from the full-circle protractor they used in the previous lesson. Review observations. Have students indicate “thumbs-up” if they had a similar answer.

The curved edge of the protractor is a half circle. The edge of the full-circle protractor is a full circle.

There are two scales on the half-circle protractor: one scale goes from 0° to 180° in a clockwise direction and the other from 0° to 180° in a counterclockwise direction. To support English language learners, discuss the meaning of scale in this context.

The 0° mark on one side of the half-circle protractor is connected with the 180° mark on the other side by a line segment. This segment is the base line of the protractor.

The midpoint of the base line is the center of the half-circle protractor. There is often a hole at the center.

Measuring Angles with a

WHOLE-CLASS

ACTIVITY

Half-Circle Protractor

Draw an angle on the board or overhead projector, but omit the arc that indicates the direction of the rotation. Mention that without the directional arc, this angle could represent a rotation less than 180° or a rotation greater than 180°. Draw directional arcs with arrowheads to show the two possible angles. Explain that if no arc is shown, the smaller of the two angles is intended. (See margin.)

Draw another angle on the board or overhead. Use your demonstration protractor to show how to measure the angle.

1. Students should first estimate whether the angle measures more or less than 90°. Ask: Is the angle acute or obtuse?

They can also use 45° and 180° as reference angles to refine their estimate. If students develop this good habit, they will seldom read the wrong scale.

2. Put the center of the protractor over the vertex of the angle.

3. Move the protractor so that one side of the angle is on the base line, as shown in the margin. Make sure the center of the protractor remains over the vertex.

ELL

EM3cuG4TLG1_438-442_U06L07.indd 439

(4)

Drawing Angles

LESSON

67

Date Time

1. Draw a 35° angle, using line segment GH as one of its sides.

2. Draw a 150° angle, using ray CD as one of its sides.

3. Draw a 60° angle, using ray EF as one of its sides.

4. Draw a 15° angle, using ray AB as one of its sides.

5. Draw a 330° angle, using ray IJ as one of its sides.

B A C D E F H G J I 143 Try This EM3MJ1_G4_U06_137-169.indd 158 1/13/11 2:28 PM Math Journal 1, p. 158

Student Page

Step 4 90 80 70 60 50 40 30 20 10 100 110 120 130 140 150 160 170 180 0 100 110 120 130 140 150 160 170 80 70 60 50 40 30 20 10 0 180 Steps 2 and 3 Step 1

Ongoing Assessment:

Informing Instruction

Watch that students are measuring carefully. A difference of 3 degrees either way should be enough leeway.

mark on the edge of the protractor.

5. Decide which of the two 0°-to-180° scales to use to determine the degree measure of the angle. If it is acute, use the smaller number; if it is obtuse, use the larger number.

Ask students to measure angles SDE,COR, and RTV at the bottom of journal page 157 and compare their measurements.

Drawing Angles with a

INDEPENDENT

ACTIVITY

Half-Circle Protractor

Have students complete Problem 1 on journal page 158 with a partner or on their own. Ask students to describe a procedure for using a half-circle protractor to draw angles while you or a student demonstrates at the board or overhead. One method:

1. Draw a ray.

2. Place the center of the protractor at the endpoint of the ray so that the base line is along the ray.

3. Use the scale that shows 0° where the ray crosses the edge of the protractor. Make a dot where the other ray should cross the edge of the protractor.

4. Draw a ray from the vertex through the dot.

Have students complete journal page 158 on their own. Remind students that they can check whether they have chosen the appropriate scale by noting if the angle is acute or obtuse.

Share strategies for Problem 5, which requires students to draw a reflex angle. Sample strategy: Subtract 330° from 360° (= 30°). Draw a 30° angle; then draw an arc on the “outside” of the angle.

Adding and Subtracting to

WHOLE-CLASS

ACTIVITY

Find Missing Angle Measures

Show students that you can add and subtract angle measures to find unknown angle measures. Draw the diagram below on the board or overhead.

30°

a

Point out that a straight angle measures 180° and the smaller angle measures 30°. Let a stand for the measure of the missing angle. Explain to students that the angle measure of the whole is the sum of the angle measures of the parts. Therefore, 30° +a= 180°, so a = 150°.

Pose additional problems on the board using either a straight angle or a 90° angle as the main angle. Have students use addition or subtraction to find the missing angle measures.

PROBLEMOOOOOOOOOBOOBOOOOOOOOOOBLBLEBLBLBLBLEBLEBLBLBLBLBLBLEBLEBLEBBLEBBBBBBBBBBLEBBBBBBBBBBBBBBBBBBBBBBBBBBBBBLLLLLLLLLEEMMMMMMMMMMMMMMMMMM PRO PRO PRO PRO PRO PRO PRO PRO PRO PRO PRO P PR P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P PROROROROROROROO P P P P P P PRRRRRRRRRRRRRRR LLLLLELELLLLELLLLLLLLLLLLLLLEEEEEEEEEEEEEEEEEEEEEEMMMMMMMMMMMMMMMMMMMM PROBLEM SOLVINGOBOOOOOOOOBBLBLBLBLBLBLBBBBBBBBBBBBBBBBBBBBBBLEEEEMMMMMMM ROOOO RO RO RO RO RO RO RO RO ROO LLLLLLLLLLLLLLLLLLLLLEMLLELELELELEEEEEEEMMMMMMMMMMMM R R R R R R R R R R R G G G L L L L L L L L L L L L LVINVINVVINVINVINVINVINVINVINVINVINVINNNNGGGGGGGGGGG OL O O OL O O OL OL O

OLLVINLLLLLLLVINVINVINVINVINVINVINVINVINVINVINVINVINVVVINVINNGGGGGGGGGGG O O OL OL OL OL OL OL O O OLLLLLLLLLLVVVVVVVVVVVV O SO SOO SO SO SO SO SO SO SOO SO SO SO SO SOOOOO SO SO SO SO S SOO SO SO SO SO SO SO SO SO SO S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVV S S S S S S S S S S S S S S S S S S S S S SOOOOOOOOOOOOOOOOOOO IIIIIIIIIIIIIIIIIIIIINNNNNNNNNNNNNNNNNNNNNNNNNNNGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG SOLVING

(5)

Math Boxes LESSON 67 Date Time 129 315 44 150 128 1 1 4 1 1 2 1 1 4 3 4 22 23 18 19 ( ) ( ) ( ) ( )

1. Insert parentheses to make each number sentence true.

a. 12 = 15 - 2 + 1

b. 66 - 16 ∗ 4 = 200

c. 49 = 4 + 3∗ 42 6

2. Draw a line segment that is 2 inches long. Mark and label the following inch measurements on the line segment:

_{_}1 4 ,

3

_ 4 , 1, 1 1_ 4 and 1 _ 12

3. Six classrooms collected newspapers for one week. If they collected a total of 582 newspapers by the end of the week, on average about how many newspapers did each class collect?

Number model with unknown: 582 6 = n Answer: 97 newspapers Summary number model: 582 6 = 97

4. Multiply with a paper-and-pencil algorithm. 67 ∗ 34 = 2,278

5. How many centimeters are in 9.7 meters? Circle the best answer.

A. 907

B. 900.7

C. 970

D. 9,700

6. Circle the square that has _{_}1 3 shaded. A. B. EM3MJ1_G4_U06_137-169.indd 159 1/13/11 3:28 PM Math Journal 1, p. 159

Student Page

STUDY LINK

67 Measuring Angles with a Protractor

Name Date Time

First estimate whether the angles measure more or less than 90°. Then use a half-circle protractor to measure them.

B C S R Q P N O M L K A 1.∠A: ° 2. ∠B: ° 3. ∠C: ° 84 150 60 4.∠QRS: ° 5. ∠NOP: 32 ° 6. ∠KLM: 300 ° 105 Practice 7. 93 ∗ 6 = 8. = 547 ∗ 7 9. 1,872 = 48 ∗ 39 10. 51 ∗ 64 = 3,264 3,829 558 Try This 143 EM3MM_G4_U06_177-202.indd 192 1/13/11 2:16 PM Math Masters, p. 192

Study Link Master

Lesson 67 441

Three students, acting as points, use rope to represent line segments and form an acute angle.

2 Ongoing Learning & Practice

World Tour Option:

SMALL-GROUP

ACTIVITY

Visiting Europe

(Math Journal 1, pp. 171–173, 180, and 181; Student Reference Book,

pp. 276, 277, 280, 284, 285, 297, and 302–305; Math Masters, pp. 419–421)

Social Studies LinkIf you have chosen to extend the scope of the World Tour for your class, divide students into groups of 4 or 5. Each group visits one of the remaining countries in Europe and records their country data on journal pages 180 and 181, or on Math Masters, pages 419 and 420.

Math Boxes 6

7

INDEPENDENT

ACTIVITY

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 6-5. The skill in Problem 6 previews Unit 7 content.

Writing/Reasoning Have students write a response to the

following: Wei said that both squares in Problem 6 have _ 1

3 shaded. Do you agree or disagree? Explain your answer.

Sample answer: I disagree. Both of the squares are divided into three parts and both of the squares have one part shaded. However, only A shows _ 1

3 because A is divided into equal parts. The three

parts in B are not equal.

Study Link 6

7

INDEPENDENT

ACTIVITY

(Math Masters, p. 192)

Home Connection Students measure angles using a

half-circle protractor. Some students may prefer to use a full-circle protractor from Lesson 6-6.

3 Differentiation Options

READINESS SMALL-GROUP _ACTIVITY

Modeling Angles

5–15 Min

(Student Reference Book, p. 93)

To explore estimating angle measures using a concrete model, have students use rope to model angles. (See margin.) Ask the “vertex” to tell the two “points” the type of angle or the measure of the angle that they should form. Have students use Student

Reference Book, page 93 as a guide.

EM3cuG4TLG1_438-442_U06L07.indd 441

(6)

LESSON 67

Name Date Time

Exploring Triangle Measures

py

g

p

You need 2 sheets of paper, a straightedge, and a protractor.

1. Draw a large triangle on each sheet of paper. The 2 triangles should not look the same.

2. Label the vertices of one triangle A,B, and C. Label the vertices of the other triangle D,E, and F. Be sure to write the labels inside the triangles.

3. Using your protractor, measure each angle as accurately as you can. Record the degree measures in the tables below.

4. Find the sum of the degree measures in triangle ABC and in triangle DEF.

5. Write a true statement about the sum of the measures of the 3 angles of a triangle.

The sum of the angle measures of a triangle is 180°.

Angle Degree Measure

∠A About ° ∠B About ° ∠C About ° Sum About 180° 90 45 45 143 Sample answers:

Angle Degree Measure

∠D About ° ∠E About ° ∠F About ° Sum About 180° 60 60 60 EM3MM_G4_U06_177-202.indd 193 1/13/11 2:13 PM Math Masters, p. 193

ENRICHMENT PARTNER _ACTIVITY

Measuring Angles in Triangles

15–30 Min

(Math Masters, p. 193)

To apply students’ understanding of measuring angles, have them draw two triangles, measure the angles, and find the sum of the angle measures for each triangle. The sum of the students’ measures of the angles of a triangle should range from 170° to 190°, with most sums close to 180°.

Guide students as they “prove” that the sum of the measures of the angles of any triangle is 180°. Have them cut off the three corners of one of their triangles and arrange them so that the three angles touch each other but do not overlap. From this, it should be clear that the angles form a straight angle, and so the sum of their measures is 180°.

Cut Cut C _A B Cut C A B

Encourage students to extend this exploration to “prove” that the sum of the measures of the angles of any quadrilateral is 360°. Students may reason that since any quadrilateral can be divided into two triangles, the sum of the angles is twice 180°, or 360°.

ENRICHMENT SMALL-GROUP _ACTIVITY

Exploring Angles in Literature

15–30 Min

(Math Masters, p. 388 or 389)

Literature Link To further explore the half-circle protractor and angles, have students read and discuss

Sir Cumference and the Great Knight of Anglelandby

Cindy Neuschwander (Charlesbridge Publishing, 2001). In a Math Log or on an Exit Slip, ask students to summarize what Radius learned on his quest.

ELL SUPPORT PARTNER _ACTIVITY

Building Background for

5–15 Min

Mathematics Words

To provide language support for angles, have students create a graphic organizer for the word angle. They may list words and draw pictures that are connected to the word angle. See