In Lesson 29, you learned how to use a protractor to measure and … · 2019-08-07 · 278 L30: Add and Subtract ith Angles Ad n ubtrac it ngles Lesso 30 Part 1 Introduction In Lesson

©Curriculum Associates, LLC Copying is not permitted.L30: Add and Subtract With Angles278

Add and Subtract With AnglesLesson 30 Part 1: Introduction

In Lesson 29, you learned how to use a protractor to measure and draw angles. Now you will learn about adding and subtracting angle measures. Take a look at this problem.

Flora cuts a rectangular sheet of paper into two pieces on the dotted line.

908 ba

What is the sum of angle a and angle b?

Explore It

Use the math you already know to solve the problem.

Angle a is than 90°.

Angle b is than 90°.

Describe where the 90° angle starts and stops as it turns through the bottom left corner of the uncut sheet of paper.

Describe where angle a starts and stops as it turns through the bottom left corner of the cut sheet of paper.

Describe where angle b starts and stops as it turns through the bottom left corner of the cut sheet of paper.

Compare where the turning starts and stops for the 90° angle and for turning through angle a and going on through angle b.

Develop Skills and Strategies

CCLS4.MD.C.7

©Curriculum Associates, LLC Copying is not permitted.279L30: Add and Subtract With Angles

Lesson 30Part 1: Introduction

Find Out More

When Flora cut the sheet of paper, she split up a 90° angle into two smaller angles. You can decompose, or split up, any angle into smaller angles.

Remember that an angle’s measure is the number of one-degree angles it turns through. You can think of any angle as a group of one-degree angles with a common vertex.

Picture a 10° angle.

1°

You can decompose a 10° angle in many different ways. Here are two possible ways.

10°6°

4° 4 1 6 5 10

10°7°

3° 3 1 7 5 10

Reflect

1 Is it possible to decompose a 100° angle into two 60° angles? Explain.

Lesson 30

©Curriculum Associates, LLC Copying is not permitted.

L30: Add and Subtract With Angles280

Part 2: Modeled Instruction

Read the problem below. Then explore different ways to understand combining smaller angles to form a larger angle.

Waylon and Andres play a game where Piece#2Piece

#1Piece

#350°

game piece tray

?

the goal is to fill a tray with three

same-size triangular game pieces.

There are no gaps and no overlaps.

What is the measure of the bottom

angle of the tray?

Picture It

You can use a sketch to help understand the problem.

Imagine putting the three pieces together in the tray. The vertices of the 50° angles become the common endpoint of a greater angle. This is the purple angle at the bottom of the tray.

50°50°50°

The three 50° angles compose, or combine to form, the greater angle.

Model It

You can also use a protractor to help understand the problem.

Look at a protractor. Start at 0°. Count three jumps of 50° each.

9080

100

70110

60120

50130

40

140

30150

20160

10170

0180 0

1017016020

15030

14040

50120

60110

7010080

180

130

Lesson 30


Part 2: Guided Instruction

Connect It

Now you will solve the problem from the previous page using an equation.

2 How many 50° angles compose the bottom angle of the tray?

3 Does addition or subtraction best express putting two or more angles together to make a greater angle?

4 You can write an equation to combine the 50° angles to compose the bottom angle of the tray.

bottom angleof tray

1 1 5

1 1 550°angle

50°angle

50°angle

The bottom angle of the tray measures degrees.

5 Could you compose an angle measuring 210° from three angles with unequal measures? If so, give an example.

Try It

Use what you just learned to solve these problems.

6 The angle between each spoke on a wheel of Sophia’s bicycle is 15°. Sophia puts reflectors on two spokes as shown to the right. What is the angle between the spokes with the reflectors?

7 Gina sets two floor tiles as shown. What is the measure of the purple angle?

15°

reflectorreflector

108°53°?

Lesson 30



Read the problem below. Then explore different ways to understand using addition and subtraction to find unknown angle measures.

A door swings open 85° and then gets stuck. Randy pushes on the door, and it

opens some more. Altogether, the door opens 100°. How many degrees does the

door open after Randy pushes it?

Picture It

You can use a sketch to help you understand the problem.

door opens an unknown amountafter push

total dooropening is 100°

door opens 85°before push

The 100° angle is composed of two smaller angles. One angle measures 85°, and the other measure is unknown.

Model It

You can use a protractor to help you understand the problem.

Look at a protractor. Start at 0°. Count on 85°. How many more degrees do you need to count on to get to 100?

9080

100

70110

60120

50130

40

140

30150

20160

10170

0180 0

1017016020

15030

14040

50120

60110

7010080

180

130

Part 3: Modeled Instruction

Lesson 30


Connect It

Now you will solve the problem from the previous page using an equation.

8 Write a sentence that describes how the unknown angle measure is related to the 85° and 100° angles.

9 Does addition or subtraction best express this relationship?

10 Write an equation to describe how the unknown angle measure is related to the 85° and 100° angles.

5

11 How would the measure of the unknown angle change if the door opened a total of 120°?

12 Imagine an angle that is composed of three smaller angles. If you know the measure of the composed angle and the measure of two of the smaller angles, explain how you could find the measure of the third small angle.

Try It

Use what you just learned to solve these problems.

13 A game includes an 8-second timer. The timer’s pointer turns through 135° as it counts down from 8 seconds to 5 seconds, as shown to the right. How many more degrees does the pointer have to turn through to count down to 0?

14 A snake’s jaws open to form a 180° angle. The snake closes its jaws until they form a 60° angle. By how many degrees did the snake’s jaws close?

Part 3: Guided Instruction

? 5

1808 608

135°0

4

6 2

1

5

7

3

Student Model

Part 4: Guided Practice Lesson 30



Once you know the angle from 4 to 6, how could you find the angle from 4 to 5?

Pair/Share

I know the hands make a 180° angle when they are on 12 and 6.

What operation did you use to solve the equation?

Pair/Share

One full turn is equal to 360°. So the sum of 60, 215, and the unknown angle is equal to 360.

Study the model below. Then solve problems 15–17.

Halah turns a jar lid 60° and then 215° more. How many more

degrees does Halah need to turn the lid to make one full turn?

Look at how you could show your work using a drawing and

an equation.

215°60°

?8

firstturn

secondturn

60 1 215 1 ? 5 360 275 1 ? 5 360 ? 5 85

Solution:

15 When the hands of a clock are on 12 and 4, they form a 120° angle. What angle is formed if the hands are moved to 4 and 6?

Show your work.

Solution:

85°

Part 4: Guided Practice Lesson 30


A circle has 360 degrees, so a half-circle has 180 degrees.

How could you check your answer?

Pair/Share

The 140° angle is composed of 3 angles: 40, 75, and ? degrees. The sum of these three angles must be 140.

Does Ellen’s answer make sense?

Pair/Share

16 Tyra’s front door has a half-circle window. What is the angle of the center piece of glass?

15° 15°30° 30°

?

Show your work.

Solution:

17 A windshield wiper turns through 140°. The window cleaner sprays across 75°. If the wiper goes 40° before it gets to the sprayed area, how far past the sprayed area does the wiper go? Circle the letter of the correct answer.

40°140°

75°sprayed area

wiperstops here

wiperstarts here

?

A 25°

B 35°

C 115°

D 255°

Ellen chose D as the correct answer. How did she get that answer?

Part 5: Common Core Practice Lesson 30



Solve the problems.

1 Keith uses a can opener. Every time he twists the knob on the opener, the opener moves 36° around the can’s lid. Which best describes the can after 5 twists?

A one-tenth open

B one-fifth open

C half open

D completely open

2 A rope swing hangs straight down. Then a child gets on, swings forward 50°, and swings back 95°. How many degrees forward must the swing go to return to its starting position?

50°

startingposition

swingforwardswing

backward

95°

?

A 5°

B 45°

C 50°

D 95°

Part 5: Common Core Practice Lesson 30


3 Choose either Yes or No to tell whether there is an angle of the given measure in the diagram.

a. 225° Yes No

b. 265° Yes No

c. 70° Yes No

d. 320° Yes No

e. 90° Yes No

160°

40°65°

25°

4 In the figure below, angle ABC measures 110°.

D

A B

C

Use a protractor to measure angle ABD. Use this measure and the measure of angle ABC to find the measure of DBC.

5 Lilit opens a pair of pliers 65°. Then she closes the pliers by 60° to grasp a bolt. Then she reopens the pliers by 100°. How many degrees open are the pliers now?

Show your work.

Answer The pliers are open degrees.

Go back and see what you can check off on the Self Check on page 207.Self Check

Add and Subtract With AnglesLesson 30

L30: Add and Subtract With Angles 301©Curriculum Associates, LLC Copying is not permitted.

Develop Skills and Strategies

(Student Book pages 278–287)

Lesson objectives

• Explain how one angle can be broken down into several smaller angles.

• Explain how several smaller angles can be put together to form one large angle.

• Solve addition and subtraction problems to find unknown angles.

Prerequisite skiLLs

In order to be proficient with the concepts/skills in this lesson, students should:

• Recognize an angle as a geometric shape.

• Recognize the relationship between an angle and a circle.

• Measure an angle correctly.

• Draw an angle of a specific degree.

vocabuLary

decompose: to split up into parts

the Learning Progression

Earlier in Grade 4, students learned that angles are geometric shapes. They explored how angles are measured in reference to a circle. They learned how to measure and draw angles using protractors. In this lesson, students use this knowledge to explore the idea of breaking apart (decomposing) an angle into smaller parts and putting together (composing) smaller angles to form one larger angle. Additionally, students learn to find unknown angles using addition and subtraction.

Teacher Toolbox Teacher-Toolbox.com

✓

✓

✓

✓

✓

Prerequisite Skills 4.MD.C.7

Ready Lessons

Tools for Instruction

Interactive Tutorials ✓

ccLs Focus

4.MD.C.7 Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.

aDDitionaL stanDarDs: 4.OA.A.3, 4.OA.B.4, 4.MD.C.5, 4.MD.C.6, 4.G.A.1, 4.G.A.2 (See page A44 for full text.)

stanDarDs For MatheMaticaL Practice: SMP 1–6 (See page A11 for full text.)

L30: Add and Subtract With Angles302©Curriculum Associates, LLC Copying is not permitted.

Part 1: Introduction Lesson 30

at a gLance

Students answer a series of questions designed to help them find the sum of two angle measures.

steP by steP

• Tell students that this page models how to find the sum of two angle measures.

• Have students read the problem at the top of the page.

• Work through Explore It as a class.

• Ask students to estimate the measures of angles a and b. [Possible answer: 45°; 45°]

• Ask students to find the sum of their estimates. Ask, How does the sum compare to a 908 angle? [Possible answer: they are the same.]

sMP tip: Students are guided to use given information to construct arguments related to the addition of angles. They learn to build upon and follow a logical progression of statements in order to make conjectures about angle addition. (SMP 3)

• What are some other combinations of angles you can add together to get a 90° angle? a 180° angle?

Students should list angle pairs that add up to 90° and pairs that add to 180°. Students may also list 3 or 4 angles that can be added together to get 90° or 180°.

• What type or types of angles are you adding together to get a 90° angle? What type or types of angles can you add to get 180°?

Students should recognize that only acute angles can be added together to get a 90° angle. Students should further recognize that to get a 180° angle they can add 2 right angles, 1 acute angle and 1 obtuse angle, or multiple acute angles. Be sure that students understand that you can only have one obtuse angle when adding angles to get a sum of 180°.

Mathematical Discourse

©Curriculum Associates, LLC Copying is not permitted.L30: Add and Subtract With Angles278

add and subtract With anglesLesson 30 Part 1: introduction

in Lesson 29, you learned how to use a protractor to measure and draw angles. now you will learn about adding and subtracting angle measures. take a look at this problem.

Flora cuts a rectangular sheet of paper into two pieces on the dotted line.

908 ba

What is the sum of angle a and angle b?

explore it

use the math you already know to solve the problem.

Angle a is than 90°.

Angle b is than 90°.

Describe where the 90° angle starts and stops as it turns through the bottom left corner of the uncut sheet of paper.

Describe where angle a starts and stops as it turns through the bottom left corner of the cut sheet of paper.

Describe where angle b starts and stops as it turns through the bottom left corner of the cut sheet of paper.

Compare where the turning starts and stops for the 90° angle and for turning through angle a and going on through angle b.

Develop skills and strategies

ccLs4.MD.c.7

less

less

it turns from bottom edge of paper to left side.

it turns from bottom edge to dotted line.

it turns from the dotted line to the left side.

Possible answer: if you turn through a, then through b, you start and end

at same places as the 90° angle—from bottom edge to left side. going “a,

then b” turns through same number of degrees as the 90° angle.


Part 1: Introduction Lesson 30

at a gLance

Students learn how to find the sum of two angle measures.

steP by steP

• Read Find Out More as a class.

• Make sure students understand that when an angle is broken apart into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts.

• Ask a volunteer to explain how adding with angle measures is similar to adding with whole numbers. [You can add angle measures in the same way you add whole numbers.]

Emphasize the combinations of angles that can and cannot be made when splitting an angle into two smaller angles.

• Give students an angle measure and then have them list as many possible combinations of angle pairs that add to the measure of the original angle. Have students justify their answers by showing the correct sum. For example, if the original angle measures 45°, it can be split into a 308 and 158 angle because 308 1 158 5 458.

• Next, have students list angle combinations that do not add up to the original angle measure. Guide the students so that they have examples of angles that have sums that are greater than and less than the original angle. For example, a 45° angle cannot be split into a 30° and 40° angle because 30° 1 40° 5 70°, and that is greater than the original angle. Another example is using a 10° and 30° angle because 10° 1 30° 5 40°, which is less than the original angle.

concept extension Ask students to brainstorm examples of everyday situations where they might see angles being divided into smaller angles. Displaying images may help spark ideas and help students better understand the concept.

Examples: cutting slices of a round pizza or pie; a clock with hour, minute, and second hands; edges of desks or tables when pushed together in groups; panes of glass in a window

real-World connection


Lesson 30Part 1: introduction

Find out More

When Flora cut the sheet of paper, she split up a 90° angle into two smaller angles. You can decompose, or split up, any angle into smaller angles.

Remember that an angle’s measure is the number of one-degree angles it turns through. You can think of any angle as a group of one-degree angles with a common vertex.

Picture a 10° angle.

1°

You can decompose a 10° angle in many different ways. Here are two possible ways.

10°6°

4° 4 1 6 5 10

10°7°

3° 3 1 7 5 10

reflect

1 Is it possible to decompose a 100° angle into two 60° angles? Explain.

no, because 60° 1 60° is greater than 100°.

304 L30: Add and Subtract With Angles


Lesson 30Part 2: Modeled instruction

at a gLance

Students find the sum of three angle measures.

steP by steP

• Read the problem at the top of the page as a class.

• Read Picture It. Have a volunteer explain how to find the angle measure of the purple angle. [Add the three angle measures.]

• Ask students to determine another operation they could use to find the measure of the purple angle. [Because the three angle measures are the same, you can multiply one angle measure by 3.]

• Read Model It.

• Ask students to determine which scale they should use to count three jumps of 50° each. [outer scale]

use noodles to understand combining smaller angles to form a larger angle.

Materials: uncooked spaghetti noodles, modeling clay, white paper, protractors

• Distribute several pieces of uncooked spaghetti, a small piece of clay, a sheet of paper, and a protractor to each student. Instruct students to carefully break the noodles in half.

• Instruct students to make a set of angles using two pieces of noodles held together at the vertex with a small piece of clay. Each student should form angles that measure 30°, 45°, 60°, 90°, and 120°. Students should use a protractor to measure the angles as they create them.

• Have students work in pairs. Have them make angles that measure 75°, 90°, and 120° using one angle from each student. Students should record their angle combination and then verify the sum of the two angles using a protractor.

• Repeat with additional angles as time allows.

hands-on activity

• If you add the measures of angles that have been combined to find their sum and then measure the larger angle with a protractor and get different solutions, what are some of the possible errors that you might have made?

Some possible errors are that they used the protractor incorrectly to measure, their arithmetic was incorrect, or there was a gap between the angles when they measured the larger angle. Students should be looking to identify which answer doesn’t make sense and why.


Lesson 30



Part 2: Modeled instruction

read the problem below. then explore different ways to understand combining smaller angles to form a larger angle.

Waylon and Andres play a game where Piece#2Piece

#1Piece

#350°

game piece tray

?

the goal is to fill a tray with three

same-size triangular game pieces.

There are no gaps and no overlaps.

What is the measure of the bottom

angle of the tray?

Picture it

you can use a sketch to help understand the problem.

Imagine putting the three pieces together in the tray. The vertices of the 50° angles become the common endpoint of a greater angle. This is the purple angle at the bottom of the tray.

50°50°50°

The three 50° angles compose, or combine to form, the greater angle.

Model it

you can also use a protractor to help understand the problem.

Look at a protractor. Start at 0°. Count three jumps of 50° each.

9080

100

70110

60120

50130

40

140

30150

20160

10170

0180 0

1017016020

15030

14040

50120

60110

7010080

180

130


Lesson 30Part 2: guided instruction

at a gLance

Students revisit the problem on page 280.

steP by steP

• Read Connect It as a class. Be sure to point out that the questions refer to the problem on page 280.

• Ask students to compose an angle measuring 1508 from three angles with different measures. One possible solution is angles that measure 208, 408, and 908.

• Students can verify that their answer to problem 5 is correct by using a protractor to draw an angle measuring 2108 and drawing the three angles they listed. The three angles they draw should fit inside the 2108 angle without overlapping.

try it soLutions

6 Solution: 1058; There are 7 angles, each measuring 158. One way to find the measure is to find the sum: 158 1 158 1 158 1 158 1 158 1 158 1 158 5 1058.

7 Solution: 161°; The measure of the purple angle is the sum of the measures of the two black angles: 108° 1 53° 5 161°.

sMP tip: Encourage students to draw and use diagrams to help conceptualize and solve problems. This will help the students make sense of problems as they become more complex. (SMP 1)

ERROR ALERT: Students who wrote 90° only added six 15° angles. Students who wrote 45° only added three 15° angles.

Lesson 30


Part 2: guided instruction

connect it

now you will solve the problem from the previous page using an equation.

2 How many 50° angles compose the bottom angle of the tray?

3 Does addition or subtraction best express putting two or more angles together to make a greater angle?

4 You can write an equation to combine the 50° angles to compose the bottom angle of the tray.

bottom angleof tray

1 1 5

1 1 550°angle

50°angle

50°angle

The bottom angle of the tray measures degrees.

5 Could you compose an angle measuring 210° from three angles with unequal measures? If so, give an example.

try it

use what you just learned to solve these problems.

6 The angle between each spoke on a wheel of Sophia’s bicycle is 15°. Sophia puts refl ectors on two spokes as shown to the right. What is the angle between the spokes with the refl ectors?

7 Gina sets two fl oor tiles as shown. What is the measure of the purple angle?

15°

reflectorreflector

108°53°?

3

addition

150

161°

105°

yes; Possible example: 40°, 80°, 90°.



Lesson 30Part 3: Modeled instruction

at a gLance

Students use pictures and benchmark angles to solve a problem about an angle.

steP by steP

• Read the problem at the top of the page as a class.

• Read Picture It. Have a volunteer explain how they would solve this problem.

• Ask students to explain why they cannot add the angle measures given to solve this problem. [The measure of the composed angle is given. So you need to subtract the measure of the angle the door opens before the push from the measure of the composed angle to solve this problem.]

• Read Model It. Ask students to determine which operation they would use to count on. [addition]

• Have a volunteer write an addition equation where x represents the unknown amount the door opens after Randy pushes it. The equation should be 85° 1 x° 5 100°.

sMP tip: Students should be familiar with using a protractor to measure angles and solve problems involving angles. They will be able to identify when it is most appropriate to use a protractor to help them solve problems. (SMP 5)

• Why would an answer of 185° not make sense for this situation?

Students should reason that it does not make sense to open a door more than 1808. Most doors have frames that do not allow them to be opened past a certain point.


Cut out triangles of various sizes to demonstrate adding and subtracting angles. Make triangles with related angle measures, such as one triangle that has a 35° angle, another that has a 45° angle, and a third that has an 80° angle.

• Using a protractor, have students measure and label the angles in each triangle.

• Have students put together the sides of two triangles and then separate them to visually see how two angles compose or decompose to form a larger angle.

• Students can trace the angles formed by the triangles and record the sum of the angle measures.

• Repeat the steps above where one triangle is inside another triangle. In this case, students will have to find the difference of the angle measures.

visual Model

Lesson 30



read the problem below. then explore different ways to understand using addition and subtraction to find unknown angle measures.

A door swings open 85° and then gets stuck. Randy pushes on the door, and it

opens some more. Altogether, the door opens 100°. How many degrees does the

door open after Randy pushes it?

Picture it

you can use a sketch to help you understand the problem.

door opens an unknown amountafter push

total dooropening is 100°

door opens 85°before push

The 100° angle is composed of two smaller angles. One angle measures 85°, and the other measure is unknown.

Model it

you can use a protractor to help you understand the problem.

Look at a protractor. Start at 0°. Count on 85°. How many more degrees do you need to count on to get to 100?

9080

100

70110

60120

50130

40

140

30150

20160

10170

0180 0

1017016020

15030

14040

50120

60110

7010080

180

130

Part 3: Modeled instruction


Lesson 30Part 3: guided instruction

at a gLance

Students revisit the problem on page 282.

steP by steP

• Read Connect It as a class. Be sure to point out that the questions refer to the problem on page 282.

• Point out that students can check their answer to problem 10 by adding the measures of the two smaller angles: 858 1 158 5 1008.

• In problem 11, have students think about the difference between the door opening 120° and the door opening 100°. They should see that the door opens 20 more degrees, so the total amount the door opens after the push is 15° 1 20° 5 35°.

• Give students an additional example after they answer problem 12. Draw an angle that is composed of three smaller angles. The composed angle has a measure of 125°. Two of the smaller angles have measures of 34° and 42°. Ask students to find the measure of the third angle. [49°]

try it soLutions

13 Solution: 2258; There are 3608 in a circle. Subtract the measure of the angle of the elapsed time from the degrees in a circle: 3608 2 1358 5 2258.

14 Solution: 1208; Subtract the measure of the angle when the snake’s jaw is partially closed from the measure of the angle when the snake’s jaw is completely open: 1808 2 608 5 1208.

use straws and pipe cleaners to understand how to find unknown angle measures.

Materials: drinking straws (cut in half), pipe cleaners, protractors

• Instruct students to make adjustable angles by placing two cut straws on a pipe cleaner. Push the straws next to each other and bend the pipe cleaner to make the vertex. Show students how to move the angle into different sizes.

• Give students an angle measure to make with the straws. They should use a protractor to measure their angle.

• Tell students how many degrees to add to or subtract from the initial angle. Adjust the angle along the protractor to show the specified change.

• Students should record their initial angle measure, the number of degrees added or subtracted, and the measure of the resulting angle on their paper.

• Repeat with additional angle measures.

hands-on activity

ERROR ALERT: Students who answered 1358 moved clockwise from 5 to get to 0. Students who answered 458 thought there were 1808 in a circle instead of 3608.

Lesson 30


connect it

now you will solve the problem from the previous page using an equation.

8 Write a sentence that describes how the unknown angle measure is related to the 85° and 100° angles.

9 Does addition or subtraction best express this relationship?

10 Write an equation to describe how the unknown angle measure is related to the 85° and 100° angles.

5

11 How would the measure of the unknown angle change if the door opened a total of 120°?

12 Imagine an angle that is composed of three smaller angles. If you know the measure of the composed angle and the measure of two of the smaller angles, explain how you could fi nd the measure of the third small angle.

try it

use what you just learned to solve these problems.

13 A game includes an 8-second timer. The timer’s pointer turns through 135° as it counts down from 8 seconds to 5 seconds, as shown to the right. How many more degrees does the pointer have to turn through to count down to 0?

14 A snake’s jaws open to form a 180° angle. The snake closes its jaws until they form a 60° angle. By how many degrees did the snake’s jaws close?

Part 3: guided instruction

? 5

1808 608

135°0

4

6 2

1

5

7

3

120 is 20° greater than 100. the unknown angle would be

15 1 20 5 35°.

Possible answer: you could subtract the sum of the two known smaller

angles from the measure of the composed angle.

the unknown angle measure is the difference

between 85° and 100°.

subtraction

225°

120°

100 2 85 15



Lesson 30Part 4: guided Practice

at a gLance

Students add and subtract angle measures.

steP by steP

• Ask students to solve the problems individually and label units in their calculations.

• When students have completed each problem, have them Pair/Share to discuss their solutions with a partner or in a group.

soLutions

Ex There are 3608 in a full turn. Another possible solution is 360 2 60 2 215 5 85.

15 Solution: 608; The hands on a clock make a 1808 angle when they are on 12 and 6. Subtract the measure of the angle when the hands are on 12 and 4 from the measure of the angle when the hands are on 12 and 6: 1808 2 1208 5 608. (DOK 2)

16 Solution: 908; There are 3608 4 2 5 1808 in a half-circle. Subtract the measures of the given angles from the measure of the degrees in a half-circle: 1808 2 158 2 308 2 308 2 158 5 908. (DOK 1)

17 Solution: A; 1408 2 758 2 408 5 258.

Explain to students why the other two answer choices are not correct:

B is not correct because 140 2 (40 1 75) is not 35.

C is not correct because you need to subtract 40 and 75 from 140, not add. (DOK 3)

Part 4: guided Practice Lesson 30


A circle has 360 degrees, so a half-circle has 180 degrees.

How could you check your answer?

Pair/share

The 140° angle is composed of 3 angles: 40, 75, and ? degrees. The sum of these three angles must be 140.

Does Ellen’s answer make sense?

Pair/share

16 Tyra’s front door has a half-circle window. What is the angle of the center piece of glass?

15° 15°30° 30°

?

Show your work.

Solution:

17 A windshield wiper turns through 140°. The window cleaner sprays across 75°. If the wiper goes 40° before it gets to the sprayed area, how far past the sprayed area does the wiper go? Circle the letter of the correct answer.

40°140°

75°sprayed area

wiperstops here

wiperstarts here

?

a 25°

b 35°

c 115°

D 255°

Ellen chose D as the correct answer. How did she get that answer?

she added the sum of 75 1 40 to 140 instead of subtracting

the sum from 140.

90°

Student Model

Part 4: guided Practice Lesson 30



Once you know the angle from 4 to 6, how could you find the angle from 4 to 5?

Pair/share

I know the hands make a 180° angle when they are on 12 and 6.

What operation did you use to solve the equation?

Pair/share

One full turn is equal to 360°. So the sum of 60, 215, and the unknown angle is equal to 360.

study the model below. then solve problems 15–17.

Halah turns a jar lid 60° and then 215° more. How many more

degrees does Halah need to turn the lid to make one full turn?

Look at how you could show your work using a drawing and

an equation.

215°60°

?8

firstturn

secondturn

60 1 215 1 ? 5 360 275 1 ? 5 360 ? 5 85

Solution:

15 When the hands of a clock are on 12 and 4, they form a 120° angle. What angle is formed if the hands are moved to 4 and 6?

Show your work.

Solution:

85°

60°


Lesson 30Part 5: common core Practice

at a gLance

Students add and subtract angle measures to solve problems that might appear on a mathematics test.

soLutions

1 Solution: C; There are 5 angles, each measuring 368. Find the sum: 368 1 368 1 368 1 368 1 368 5 1808 or multiply 5 3 36°. 1808 is half of a circle, so the can is half open. (DOK 2)

2 Solution: B; Subtract the angle measure that the child swings forward from the angle measure that the child swings back: 958 2 508 5 458. (DOK 1)

3 Solution: a. No; b. Yes; c. Yes; d. Yes; e. Yes (DOK 2)

4 Solution: 70°; Angle ABD measures 40°. Since angle ABC is made up of angles ABD and DBC, you can subtract the measure of angle ABD from ABC to find DBC. 110 2 40 5 70. (DOK 2)

5 Solution: 1058; Subtract the angle measure that Lilit closes the pliers from the angle measure that the pliers were open: 658 2 608 5 58. Add the measure of the angle the pliers are open to the measure of the angle the pliers are reopened: 58 1 1008 5 1058. (DOK 2)

Part 5: common core Practice Lesson 30



Solve the problems.

1 Keith uses a can opener. Every time he twists the knob on the opener, the opener moves 36° around the can’s lid. Which best describes the can after 5 twists?

A one-tenth open

B one-fi fth open

C half open

D completely open

2 A rope swing hangs straight down. Then a child gets on, swings forward 50°, and swings back 95°. How many degrees forward must the swing go to return to its starting position?

50°

startingposition

swingforwardswing

backward

95°

?

A 5°

B 45°

C 50°

D 95°

Part 5: common core Practice Lesson 30


3 Choose either Yes or No to tell whether there is an angle of the given measure in the diagram.

a. 225° Yes No

b. 265° Yes No

c. 70° Yes No

d. 320° Yes No

e. 90° Yes No

160°

40°65°

25°

4 In the fi gure below, angle ABC measures 110°.

D

A B

C

Use a protractor to measure angle ABD. Use this measure and the measure of angle ABC to fi nd the measure of DBC.

5 Lilit opens a pair of pliers 65°. Then she closes the pliers by 60° to grasp a bolt. Then she reopens the pliers by 100°. How many degrees open are the pliers now?

Show your work.

Answer The pliers are open degrees.

Go back and see what you can check off on the Self Check on page 207.self check

105

70°

33333

Differentiated Instruction Lesson 30

L30: Add and Subtract With Angles310©Curriculum Associates, LLC Copying is not permitted.

assessment and remediation

hands-on activity challenge activityMeasure angles formed by diagonals of polygons.

Give students a worksheet with several regular polygons printed on it. Students should draw two diagonals on the polygon starting from the same vertex but ending at different vertices. Have students calculate the measure of each of the three angles formed by the diagonals at the initial vertex and also find the sum of all three angles. Have students compare their answers for each type of polygon.

use a clock to understand adding and subtracting with angles.

Materials: practice analog clocks, protractors

• Have students position the hands of a clock to show a time that you give them. Then have them measure the angle formed between the minute and hour hands. Students should record the time and the measure of the angle formed.

• Give students an amount of time to advance their clocks. Students should find the new angle formed between the minute and hour hand and then record the new time and angle measure.

• Using the information gathered, students should calculate the change in the angle measures of the clock hands between the original time and the final time on the clock.

• An angle has a measure of 948. Two of the three smaller angles that make up the larger angle have measures of 188 and 338. Ask students to find the measure of the third angle. [438]

• For students who are still struggling, use the chart below to guide remediation.

• After providing remediation, check students’ understanding. An angle has a measure of 1128. Two of the three

smaller angles that make up the larger angle have measures of 528 and 298. Ask students to explain their

thinking while finding the measure of the third angle. [318]

• If a student is still having difficulty, use Ready Instruction, Level 4, Lesson 29.

if the error is . . . students may . . . to remediate . . .

518 have added the angle measures of the two smaller angles.

Remind students they can count on from 518 to get to the measure of the composed angle, 948. The measure of the third angle is the amount they add to 518 to get 948.

618 or 768 have subtracted only one of the smaller angles from the larger angle.

Demonstrate by drawing and labeling the three angles that make the composed angle. Show students that they must subtract both angle measures to find the measure of the third angle.

1458 have added the three angle measures given.

Demonstrate using a picture that students need to subtract the measures of the two smaller angles from the measure of the composed angle.

Documents

In Lesson 29, you learned how to use a protractor to measure and … · 2019-08-07 · 278 L30: Add and Subtract ith Angles Ad n ubtrac it ngles Lesso 30 Part 1 Introduction In Lesson