rotations and angles - everyday math · pdf filethe other half of the straw clockwise to the...

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eToolkitePresentations Interactive Teacher’s

Lesson Guide

Algorithms Practice

EM FactsWorkshop Game™

AssessmentManagement

Family Letters

CurriculumFocal Points

Common Core State Standards

Lesson 6�5 425

Advance Preparation

Teacher’s Reference Manual, Grades 4–6 pp. 178–180

Key Concepts and Skills• Use multiples of 30.

[Number and Numeration Goal 3]

• Form angles of a given measure.

[Measurement and Reference Frames Goal 1]

• Describe right angles.

[Geometry Goal 1]

• Rotate objects a given number of degrees.

[Geometry Goal 3]

• Investigate the relationship between

rotations and degrees.

[Geometry Goal 3]

Key ActivitiesStudents review clockwise rotations. They

make a full-circle protractor by measuring

rotations in degrees; then they use the

protractor to form angles of given measures.

Students solve problems that involve

measuring elapsed time in degrees.

Ongoing Assessment: Informing Instruction See page 427.

Key Vocabularyrotation � turn � clockwise � degree �

right angle

MaterialsMath Journal 1, pp. 152 and 153

Study Link 6�4

drinking straws � demonstration clock

Making a Bar GraphMath Journal 1, p. 154

Student Reference Book, p. 301

Students make a bar graph showing

percent of population (ages 0–14) for

Region 2 countries.

Ongoing Assessment: Recognizing Student Achievement Use journal page 154. [Data and Chance Goal 1]

Solving Elapsed-Time ProblemsMath Journal 1, pp. 154A and 154B

demonstration clock (optional) �

calculator (optional)

Students solve elapsed-time

problems.

Math Boxes 6�5Math Journal 1, p. 151

Students practice and maintain skills

through Math Box problems.

Study Link 6�5Math Masters, p. 185

Students practice and maintain skills

through Study Link activities.

READINESS

Matching Alternate Time DisplaysMath Masters, pp. 186–188

scissors

Students match alternate ways of

naming time.

ENRICHMENTMeasuring Elapsed Time in DegreesMath Journal 1, pp. 152 and 153

Math Masters, p. 189

Students determine elapsed time for 1° increments on a clock face.

EXTRA PRACTICE

Playing RobotStudents practice making rotations of a

given size.

ELL SUPPORT

Building a Math Word BankDifferentiation Handbook, p. 140

Students add the term degree to their Math

Word Banks.

Teaching the Lesson Ongoing Learning & Practice Differentiation Options

�� Rotations and AnglesObjectives To review rotations; and to guide students as they

make and use a full-circle protractor.

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426 Unit 6 Division; Map Reference Frames; Measures of Angles

Making a Full-Circle ProtractorLESSON

6�5

Date Time

There are 360 marks around the circle. They divide the edge of the circle into

360 small spaces. Twelve of the marks are longer than the rest. They are in the same

positions as the 12 numbers around a clock face. Your teacher will tell you how to label

the 12 large marks on the circle.

12

6

11

5

10

4

1

7

2

8

39

330° 30°

300° 60°

270° 90°

240° 120°

210° 150°180°

360°/0°

92

Math Journal 1, p. 152

Student Page

1 Teaching the Lesson

� Math Message Follow-Up WHOLE-CLASS ACTIVITY

The Math Message reminds students who used Third Grade Everyday Mathematics of previous experiences with rotations. Review answers using a clock with an hour and minute hand or a demonstration clock to model movements of the minute hand.

Pose additional problems: A 3 _ 4 turn? 45 min A 1 _ 6 turn? 10 min A 1 _ 3 turn? 20 min A 2 _ 3 turn? 40 min

� For a 3 _ 4 turn—Since there are 3 five-minute intervals in 1 _ 4 of a turn (3 = 1 _ 4 of 12), there are three times as many in 3 _ 4 of a turn, or 9 five-minute intervals. Therefore, it takes the minute hand 45 minutes to move through 3 _ 4 of a turn.

� For a 1 _ 6 turn—Since there are 12 five-minute intervals in 1 full turn of the minute hand, there are 2 five-minute intervals in 1 _ 6 of a turn. Therefore, it takes the minute hand 10 minutes to move through a 1 _ 6 turn.

� For a 2 _ 3 turn—Since there are 4 five-minute intervals in 1 _ 3 of a turn (4 = 1 _ 3 of 12), there are twice as many in 2 _ 3 of a turn, or 8 five-minute intervals. Therefore, it takes the minute hand 40 minutes to move through 2 _ 3 of a turn.

� Investigating Rotations WHOLE-CLASS ACTIVITY

and Degree Measures(Math Journal 1, p. 152)

Tell the class that in this lesson they will investigate the markings on a full-circle protractor and compare them to familiar markings on an analog clock.

ELL

Getting Started

Mental Math and Reflexes Have students imagine standing in the center of a clock with their right hand extended as the minute hand. Ask them to rotate their bodies to make turns such as the following:

1 _ 2 turn clockwise

1 _ 4 turn clockwise

full turn clockwise

90° turn clockwise

180° turn clockwise

360° turn clockwise

3

_ 4 turn counterclockwise

3

_ 4 turn clockwise

1

_ 2 turn counterclockwise

Math MessageHow many minutes does it take the minute hand to move through a full turn on the face of a clock? 60 min A

1

_ 2 turn? 30 min A

1

_ 4 turn? 15 min

Study Link 6�4 Follow-Up Have students discuss how they handled the remainders in Problems 1 and 2.

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Lesson 6�5 427

30°

0°

90°

30°

NOTE Think of an angle as “in motion”

opening from 0° to the desired angle. For

example, to measure a 15° angle, start with a

0° angle and open the angle to 1°, 2°, and so

on until 15° is reached. Thinking this way can

help students realize that counting the spaces

in between the marks of the protractor is

more accurate than counting the marks.

Links to the Future

Discuss the marked circle on journal page 152.

� There are three different lengths of marks.

� The shortest marks divide the circle into 360 small spaces. 360 ∗ 1° = 360°

� The longest marks are in the same positions as the 12 numbers around a clock face. These 12 long marks divide the circle into 12 spaces. 12 ∗ 30° = 360°

� The middle-size marks divide the circle into 72 spaces. 72 ∗ 5° = 360°

Ask students to write 0° beneath the large mark at the 12 o’clock position on the circle.

Ask students to fold a straw in half. Show them how to place it on the circle on journal page 152. The bend of the straw should touch the center of the circle, and both halves of the straw should point to the 0-degree mark.

Keeping one part of the straw pointing to the 0-degree mark, move the other half of the straw clockwise to the first large mark, or 1 _ 12 of a turn.

Students will discuss counterclockwise rotations in Lesson 6-6.

The straw-halves form an angle. Remind students that angles are measured in degrees and that the degree symbol (◦) is often used in place of the word degree. To support English language learners, write degree on the board and explain that this word has different meanings when it is used to measure angles and temperature.

Now show students how to measure the straw angle they just made: To measure the angle, count the number of small spaces created by the shortest marks. (See note in margin.) 30 spaces, so the angle measures 30 degrees, or 30◦ Ask students to write 30◦ at the first large mark on the circle.

Tell students to move the straw-half back to its original position and then repeat the routine for a 1 _ 4 turn. (See margin.)

● What is the measure of the angle? 90◦ Ask students to write 90◦ at the 1 _ 4 -turn mark on the circle. (See margin.)

Ongoing Assessment: Informing Instruction

Watch for the different strategies that students use to determine that the result of

a 1

_ 4 turn of the straw is an angle that measures 90°.

� Count the 90 spaces along the circle between the sides of the straw.

� Recognize that the angle is 3 times as large as the first angle, and multiply

by 3 to get 90°.

� Recognize that the angle is a right angle, and right angles measure 90°.



154

Date Time

76 301

The table below shows the percent of

the population (number of people out

of 100) who are 14 years old or

younger in the Region 2 countries.

Percent of Population Ages 0–14

Franc

e

Gre

ece

Hun

gary

Icelan

dIta

ly

Net

herla

nds

Nor

way

Polan

d

Spain

Unite

d Kin

gdom

0

5

10

15

20

25

Perc

ent

Country in Region 2

1. Make a bar graph to display the information given in the table above.

2. Why might it be important to know what percent of the population of a country

is 0 through 14 years of age?

Sample answer: A government will know how much money

Country Percent of Population Ages 0–14

France 19

Greece 15

Hungary 16

Iceland 23

Italy 14

Netherlands 18

Norway 20

Poland 18

Spain 15

United Kingdom 19

to give schools for grades 8 and below.

Population Bar GraphLESSON

6�5

�

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Student Page

Clock AnglesLESSON

6�5

Date Time

92 141

12 12

3

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8

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Use the clock below and the full-circle protractor on journal page 152 to help you

answer the questions.

1. How many minutes and how many degrees does the minute hand move

a. from 3:00 to 4:00? 60 minutes 360 °

b. from 7:00 to 7:45? 45 minutes 270 °

c. from 8:15 to 8:45? 30 minutes 180 °

d. from 6:30 to 6:50? 20 minutes 120 °

e. from 5:15 to 5:30? 15 minutes 90 °

f. from 1:00 to 1:10? 10 minutes 60 °

g. from 12:00 to 12:05? 5 minutes 30 °

h. from 5:00 to 5:01? 1 minutes 6 °

2. How many degrees does the hour hand move

a. in 1 hour? 30 °

b. in 1 _ 2 hour? 15 °

c. in 10 minutes? 5 °

3. Explain how you solved Problem 2c.

Sample answer: I know that the hour hand

moves 30° in 1 hour. There are 6 groups of

are 6 groups of 5° in 30°, or 30°/ 6 = 5°.

10 minutes in 1 hour (60 minutes). So there

Try This

EM3MJ1_G4_U06_137-169.indd 153 1/13/11 2:24 PM


Student Page● What is another name for a 90-degree angle? A right angle

To support English language learners, review the different meanings of right discussed in Unit 1—right answer, right hand, right angle.

Now tell students to move the straw-half back to its original position, and then repeat the procedure for a full turn.

● What is the measure of the angle? 360°

Ask students to write 360° at the full-turn mark on the circle, right next to the 0° mark. Have students label the rest of the large tick marks on the circle until all 12 marks have been labeled.

� Forming Angles of WHOLE-CLASS ACTIVITY

Given Measures(Math Journal 1, p. 152)

Direct students to use their bent straws to form angles of various degree measures. For example, Show me a 120° angle; a 45° angle; a 77° angle.

� Measuring Elapsed Time PARTNER ACTIVITY

in Degrees(Math Journal 1, pp. 152 and 153)

Students solve problems and share solution strategies about the number of degrees the minute and hour hands of a clock move in a given amount of time.

2 Ongoing Learning & Practice

� Making a Bar Graph INDEPENDENTACTIVITY

(Math Journal 1, p. 154; Student Reference Book, p. 301)

Social Studies Link Students make a bar graph to show percent of population (ages 0–14 years) for Region 2

countries. Direct students to page 301 of the Student Reference Book for additional data.

Ongoing Assessment: Journal

page 154 �Recognizing Student Achievement

Use journal page 154 to assess students’ ability to create a bar graph. Students

are making adequate progress if they can draw the bars at the appropriate

height on the graph. Some students may be able to provide a title and label

each axis.

[Data and Chance Goal 1]

PROBLEMBBBBBBBBBBOOOOOOOOOOBBBBBBBBBBBBBBBBBBBBBBBBBB MMMMMEEEEEMMMMMLEBLELEBLEBLELLLBLEBLEBLEBLEBLEBLEBLEBLEEEMMMMMMMMMMMMMMOOOOOOOOOOOOBBBBBBBBLBLBLBLBLBLBLLLLLPROPROPROPROPROPROPROPROPROPROPROPRPRPROPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPROROROROROROROOPPPPPPP MMMMMMMMMMMMMMMMMMMEEEEEEEEEEEELLELEEEEEEEEEELLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRRRRRRRPROBLEMSOLVING

BBBBBBBBBBBBBBBBBBBBBLELEELEMMMMMMMMMOOOOOOOOOOBLBBLBBLBLBBBBLBLROOORORORORORORORORORROO LELELELEEEEEELEMMMMMMMMMMMMLEMLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRGGGGGLLLLLLLLLLLLLVVINVINVINVINNNNVINVINNVINVINVINVINVINV GGGGGGGGGGGOLOOOOOOLOLOLOO VVINVINLLLLLLLLLVINVINVINVINVINVINVINVINVINVINVINVINVINVINNGGGGGGGGGGOOOLOLOLOLOLLOLOO VVVLLLLLLLLLLLVVVVVVVVVVVSOSOSOOSOSOSOSOSOSOOSOSOSOOOSOOSOOSOSOSOSOSOSOSOSOOSOSOSOSOSOSOSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS VVVVVVVVVVVVVVVVVVVVVVLLLLLLVVVVVVVVVLLLVVVVVVVVLLLLLLLLVVVVVLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLSSSSSSSSSSSSSSSSSSSSSSSS GGGGGGGGGGGGGGGGGOOOOOOOOOOOOOOOOOOO GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGNNNNNNNNNNNNNNNNNNNNNNNNNNIIIIIIIIIIIIIIIIIIIIISOLVING

Adjusting the Activity Have students describe a strategy

for labeling the remaining large tick marks without using the folded straw. Sample

answer: Count by 30s; each large tick mark corresponds to a multiple of 30.

AUDITORY � KINESTHETIC � TACTILE � VISUAL


� Solving Elapsed-Time INDEPENDENTACTIVITY

Problems(Math Journal 1, pp. 154A and 154B)

Students solve elapsed-time problems. Students may use the demonstration clock, calculators, or anything else that may help. Some students may find it helpful to use an open number line to illustrate the strategy of counting up in hours and minutes. See Lesson 3-6 for more information.

� Math Boxes 6�5 INDEPENDENTACTIVITY

(Math Journal 1, p. 151)

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

� Study Link 6�5 INDEPENDENTACTIVITY

(Math Masters, p. 185)

Home Connection Students follow directions, given as fractions of turns and distances, to trace a path on a coordinate grid. When reviewing answers, point out that

the length of each horizontal line segment equals the difference of the x-coordinates and that the length of each vertical line segment equals the difference of the y-coordinates.

Lesson 6�5 429

Elapsed Time continuedLESSON

6�5

Date Time

Read the time on each clock. What time will it be in 50 minutes?

8. 9. 10.

8:50

4:15

8:42

For each time, record what time it will be in 1 hour and 20 minutes.

11. 11:00 A.M. 12. 6:45 P.M. 13. 9:53 P.M.

12:20 P.M. 8:05 P.M. 11:13 P.M.

Read the time on each clock. What time was it 30 minutes ago?

14. 15. 16.

7:15

10:50

4:39

For each time, record what time it was 2 hours and 15 minutes ago.

11. 10:15 A.M. 12. 2:05 P.M. 13. 1:12 A.M.

8:00 A.M. 11:50 A.M. 10:57 P.M.

12 12

3

4567

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12 12

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12 12

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12 12

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12 12

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12 12

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EM3MJ1_G4_U06_137-169.indd 154B 1/14/11 10:32 AM

Math Journal 1, p. 154B

Student Page

Elapsed TimeLESSON

6�5

Date Time

Record the times on each pair of clocks. Then record the elapsed time.

1. Start End 2. Start End

3. Start End 4. Start End

Record how much time has passed between the start time and the end time.

5. Start 11:00 A.M. Elapsed time: 5 hours, 30 minutes End 4:30 P.M.

6. Start 2:20 P.M. Elapsed time: 4 hours, 15 minutes End 6:35 P.M.

7. Start 9:12 A.M. Elapsed time: 2 hours, 31 minutes End 11:43 P.M.

12 12

3

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9

101112 1

2

3

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12 12

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2

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2

3

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2

3

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2 hours, 57 minutes1:05 4:02 11:32 6:28

6 hours, 56 minutes

12 12

3

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101112 1

2

3

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8

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7 hours1:00 8:00

40 minutes2:35 3:15

EM3MJ1_G4_U06_137-169.indd 154A 1/14/11 10:32 AM

Math Journal 1, p. 154A

Student Page

Math Boxes LESSON

6�5

Date Time

128150

22 23 18 19

129 44

21 1211

234

( )

( )

( ) ( )

1. Insert parentheses to make each number

sentence true.

a. 15 + 5 ∗ 6 = 120

b. 7 + 9 ∗ 2 = 25

c. 77 = 1 + 6 ∗ 6 + 5

2. Draw a line segment that is 2 inches

long. Mark and label the following inch

measurements on the line segment:

1 _ 2 ,

3

_ 4 , 1, 1 1 _ 2 , and 2

3. The Sports Boosters raised $908 at their

annual chili supper. Four athletic teams

will share the money equally.

How much money will each team receive?

Number model with unknown:

908 � 4 = d

Answer: $227

Summary number model:

908 � 4 = 227

4. Multiply with a paper-and-pencil algorithm.

66 ∗ 62 = 4,092

5. Complete.

a. 9 m = 900 cm

b. 1,500 cm = 15 m

c. 350 cm = 3.5 m

d. 458 cm = 4 m 58 cm

e. 3.2 m = 320 cm

Sampleanswers:

6. a. Shade 1 _ 2 of

the square.

b. Shade 2 _ 3 of

the square.

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Student Page

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3 Differentiation Options

READINESS PARTNER ACTIVITY

� Matching Alternate Time Displays 5–15 Min

(Math Masters, pp. 186–188)

To explore alternate ways of naming time, have students match cards that indicate the same time in analog, digital, and word forms.

ENRICHMENT INDEPENDENTACTIVITY

� Measuring Elapsed Time 5–15 Min

in Degrees(Math Journal 1, pp. 152 and 153; Math Masters, p. 189)

To further investigate the relationship between elapsed time and angle measures, have students use the full-circle protractor on journal page 152 to determine how long it takes the minute and the hour hands to move 1 degree.

EXTRA PRACTICE PARTNER ACTIVITY

� Playing Robot 5–15 Min

To practice rotations expressed as both fractions of turns and degree measures, have students play Robot. One partner is the “Controller” and the other is the “Robot.” The Controller picks a destination. The Controller gives the Robot directions for the amount of each turn and the number of steps to take until the Robot reaches the destination. The amount of each turn may be given as a fraction of a full turn or as a degree measure.

Example: “Make a half-turn clockwise and go forward 5 steps. Now turn clockwise a quarter-turn (90 degrees), and go back 3 steps.”

ELL SUPPORT PARTNER ACTIVITY

� Building a Math Word Bank 5–15 Min

(Differentiation Handbook, p. 140)

To provide language support for angle rotations, have students use the Word Bank template found on Differentiation Handbook, page 140. Ask students to write the term degree, draw a picture or give an example to represent the term, and write other related words. See the Differentiation Handbook for more information.

12 12

3

4567

8

9

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past5 o’clock

Alternate ways of naming time from Math Masters,

pages 186–188

LESSON

6�5

Name Date Time

Clock Angle Challenge

Use the full-circle protractor and the clock from journal pages 152

and 153 to help you solve the problems below.

1. How long does it take the hour hand to move 1°?

Explain.

2. How long does it take the minute hand to move 1°?

Explain.

60 by 6, I get that it moves 1� every 10 seconds.

so it moves 6� every minute (or 60 seconds). Dividing

It takes the minute hand 60 minutes to move 360�,

10 seconds

takes 10 minutes to move 5� and 2 minutes to move 1�.

The hour hand takes 60 minutes to move 30�, so it

2 minutesSample explanations:

141 142

Math Masters, page 189

STUDY LINK

6�5 Treasure Hunt

Name Date Time

Marge and her friends are playing Treasure Hunt. Help them find the treasure.

Follow the directions. Draw the path from the oak tree to the treasure. Mark the

spot where the treasure is buried.

1. Start at the dot under the oak tree; face north. Walk 4 steps.

2. Make a quarter turn, clockwise. Walk 5 steps.

3. Face south. Walk 2 steps.

4. Face east. Walk 2 1

_ 2 steps.

5. Make a 3

_ 4 turn, clockwise. Walk 5 steps.

6. Make a 3

_ 4 turn, clockwise. Walk 6

1

_ 2 steps.

7. Make an X to mark the spot where you end.

1 step

Oak Tree

N

W E

S

10

9

8

7

6

5

4

3

2

1

109876543210

0

Practice

8. 88 ÷ 3 = 9. = 71 ÷ 6

10. = 603 � 7 11. 934 � 5 = 186 R4 86 R1

11 R5 29 R1

107

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Math Masters, p. 185

Study Link Master


rotations and angles - everyday math · pdf filethe other half of the straw clockwise to the...

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