Clock Fractions and Common Denominators · a clock face to add and subtract fractions. Remind...

$: Clock Fractions and Common Denominators · a clock face to add and subtract fractions. Remind students that elapsed time is the time that passes between a given starting and ending$
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Lesson 6�9 423

Advance PreparationFor Part 1, display the Multiplication Table Poster on the board or elsewhere for student use. When you

use the Everyday Mathematics posters with English language learners, display either the English version

only or both English and Spanish versions simultaneously; do not display the Spanish version only.

Teacher’s Reference Manual, Grades 4–6 pp. 141, 142, 242, 243

Clock Fractions andCommon Denominators

Objective To provide additional references for fraction concepts.

Key Concepts and Skills• Find common denominators.

[Number and Numeration Goal 5]

• Use clock models and pencil-and-paper

algorithms to add and subtract fractions.

[Operations and Computation Goal 4]

• Use benchmarks to estimate sums and

differences.


Key ActivitiesStudents use a clock face to find equivalent

fractions and to model addition and

subtraction of fractions. They use the

multiplication rule, a multiplication table, and

reference lists to find common denominators.

Ongoing Assessment: Recognizing Student Achievement Use journal page 194. [Operations and Computation Goal 4]

Key Vocabularycommon denominator � unlike denominators

MaterialsMath Journal 1, pp. 194–197

Student Reference Book, p. 401

Study Link 6�8

Math Masters, p. 177

Multiplication Table Poster � demonstration

clock

Playing Fraction CaptureMath Journal 1, p. 198


per partnership: 2 six-sided dice

Students practice comparing fractions

and finding equivalent fractions.

Math Boxes 6�9Math Journal 1, p. 199

Students practice and maintain skills

through Math Box problems.

Study Link 6�9Math Masters, p. 178

Students practice and maintain skills

through Study Link activities.

ENRICHMENTModeling Fractions with a Military ClockMath Masters, p. 179

Students apply the clock model to a different

unit by using military time.

EXTRA PRACTICE

Writing Elapsed Time Number StoriesMath Masters, p. 180

Students write a number story using fractions

to represent elapsed time.

Teaching the Lesson Ongoing Learning & Practice

132

4

Differentiation Options

��

eToolkitePresentations Interactive Teacher’s

Lesson Guide

Algorithms Practice

EM FactsWorkshop Game™

AssessmentManagement

Family Letters

CurriculumFocal Points

Common Core State Standards

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12 12

3

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8

9

1011

Clock FractionsLESSON

6�9

Date Time

Part 1: Math Message

The numbers on a clock face divide one hour into

twelfths. Each �112� of an hour is 5 minutes.

How many minutes does each of the following fractions and mixed

numbers represent? The first one has been done for you.

1. �112� hr � min 2. �

152� hr � min 3. �

12

� hr � min

4. �13

� hr � min 5. �14

� hr � min 6. �16

� hr � min

Part 2

Using the clock face, fill in the missing numbers. The first one has been done for you.

7. �14

� hr � hr 8. �182� hr � hr 9. �

13

� hr � hr

10. hr � �56

� hr 11. hr � �192� hr 12. �

122� hr � hr

13. 1�12

� hr � hr 14. �53

� hr � hr 15. �142� hr � hr

Part 3

Use clock fractions, if helpful, to solve these problems. Write each answer as a fraction.

Example: �34

� � �13

� � ?

Think: 45 minutes � 20 minutes � 25 minutes

So �34

� � �13

� � �152�

16. �152� � �

132� � 17. �

34

� � �24

� � 18. �1112� � �

132� �

19. 1 � �23

� � 20. �54

� � �24

� � 21. �23

� � �16

� �

22. �14

� � �13

� � 23. �13

� � �14

� � 24. �56

� � �34

� �

101520

30255

312

1012

64

2012

16

32

43

31

62

�1

8

2�, or �

2

3� �

5

4�, or 1�

1

4� �

1

8

2�, or �

2

3�

�1

3�

�1

7

2� �

1

1

2� �

1

1

2�

�3

4� �

5

6�

Rule

hour

Whole

�

Math Journal 1, p. 194

Student Page

424 Unit 6 Using Data; Addition and Subtraction of Fractions

Getting Started

1 Teaching the Lesson

▶ Math Message Follow-Up WHOLE-CLASSDISCUSSION

(Math Journal 1, p. 194)

Review the answers to Part 1 with the class. You might want to pose a few additional easy problems that have mixed numbers or fractions greater than 1. Suggestions:

● How many minutes are in 2 1 _ 2 hours? 150 min

● In 5 _ 2 hours? 150 min

● In 5 _ 4 hours? 75 min

● 3 _ 2 hours is equivalent to how many minutes? 90 min

▶ Using a Clock to Add and PARTNER ACTIVITY

Subtract Fractions(Math Journal 1, p. 194)

Make sure students understand that they may use the clock model to help them answer the problems on the journal page. At times, students might want to “think minutes,” as in the example for Part 3. At other times, students might want to look at the clock face divided into twelfths. Using a demonstration clock, work several of the problems in Part 2 with the class before students work in partnerships. Assign the remainder of the journal page.

Ongoing Assessment: Journal

Page 194 �Problems 16–24Recognizing Student Achievement

Use journal page 194, Problems 16–24 to assess students’ ability to use a

visual model to add and subtract fractions with unlike denominators. Students

are making adequate progress if they correctly solve Problems 16–24.


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Math MessageComplete Part 1 on journal page 194.

Study Link 6�8 Follow-UpHave partners compare answers and resolve differences.

Mental Math and Reflexes Fraction addition and subtraction: Write the problems on the board, and have students estimate using benchmarks and then solve. Students use the estimate to assess the reasonableness of the answers.

7

_ 8 + 2

1

_ 4 = ? 3

1

_ 8

3 5

_ 8 - 1

3

_ 4 = ? 1

7

_ 8

2 1

_ 4 + 2

1

_ 2 + 1

1

_ 4 = ? 6

1 5

_ 8 + 1

1

_ 16

+ 3 3

_ 4 = ? 6

7

_ 16

6 1

_ 2 - 4 + 2

1

_ 4 = ? 4

3

_ 4

4 3

_ 8 + 2 - 1

3

_ 4 + 1

1

_ 2 = ? 6

1

_ 8

NOTE Working with elapsed time can

provide students with more practice using

a clock face to add and subtract fractions.

Remind students that elapsed time is the time

that passes between a given starting and

ending time. Give them various problems that

involve finding elapsed time and then adding

or subtracting the amounts. Ask students to

give the elapsed time in minutes and then in

fractions of an hour. They can then use the

clock face to add or subtract.

EM3cuG5TLG1_424-428_U06L09.indd 424EM3cuG5TLG1_424-428_U06L09.indd 424 12/9/10 9:29 AM12/9/10 9:29 AM

Number Strip FractionsLESSON

6�9

Date Time

Name the strips that you used for the numerator and denominator. Then list the

fractions formed by the two strips. Sample answers:

Problem 1

Strip Fractions List

Name

Numerator:

Denominator:

Problem 2


Name

Numerator:

Denominator:

Problem 3


Name

Numerator:

Denominator:

Problem 4

Explain how you can use a multiplication table to find equivalent fractions for �2

9

7�.

the numbers in the 3 row.

other fractions made by the numbers in the 1 row that correspond to

and the 3 row has 27 for the denominator. I can then list the

the rows for those numbers. The 1 row has 9 for the numerator,

I can look for a column that has both 9 and 27. Then I can find

3

410

97

1

�7

9�, �

1

1

4

8�, �

2

2

1

7�, �

2

3

8

6�, �

3

4

5

5�, �

4

5

2

4�, �

4

6

9

3�, �

5

7

6

2�, �

6

8

3

1�, �

7

9

0

0�

�1

4

0�, �

2

8

0�, �

3

1

0

2�, �

4

1

0

6�, �

5

2

0

0�, �

6

2

0

4�, �

7

2

0

8�, �

8

3

0

2�, �

9

3

0

6�, �

1

4

0

0

0�

�3

1�, �

6

2�, �

9

3�, �1

42�, �

1

5

5�, �

1

6

8�, �

2

7

1�, �

2

8

4�, �

2

9

7�, �

3

1

0

0�


Student Page

Adjusting the Activity

Lesson 6�9 425

▶ Discussing Strategies for WHOLE-CLASSDISCUSSION

Adding and Subtracting Fractions(Math Journal 1, p. 194)

Discuss students’ solutions to Part 3. Expect that some students converted most fractions to minutes, did the operation, and then converted the answer in minutes back to a fraction. Others may have converted all fractions to twelfths and found the answer without any reference to the clock or time.

Pose fraction problems with denominators of 30 and 60.

Suggestions follow:

• 3

_ 4 +

5

_ 60

50

_ 60

, or 5

_ 6 •

18

_ 30

+ 1

_ 3 28

_ 30

, or 14

_ 15

• 5

_ 12

+ 25

_ 60

50

_ 60

, or 5

_ 6

A U D I T O R Y � K I N E S T H E T I C � T A C T I L E � V I S U A L

NOTE A clock face is a convenient model for fraction operations involving

halves, thirds, fourths, fifths, sixths, twelfths, and even thirtieths and sixtieths. The

link between fractions and their equivalents in minutes allows students to add and

subtract fractions with unlike denominators without rewriting the fractions with a

common denominator.

▶ Using a Multiplication Table

WHOLE-CLASS ACTIVITY

to Explore Equivalent Fractions(Math Journal 1, p. 195; Math Masters, p. 177)

Any two rows of a multiplication table can be used to form equivalent fractions. Display the Multiplication Table Poster on the board. Ask partners to cut the strips from Math Masters, page 177 and place them in the middle of their workspace.

� Each student takes one strip. Ask students to make true statements about the numbers on their strip. The numbers are multiples of the first number on the strip; the numbers have a common factor. Tell them that a strip can be named by its smallest number, for example, the “4 strip.”

� One partner is “numerators” and the other is “denominators.” Partners then match their strips, laying the numerator strip above the denominator strip. Tell students that the columns form fractions.

� For Problem 1, on journal page 195, ask students to write down the strip names and then list all of the fractions formed by the matches.

� Partners then take two different strips and repeat this process for Problems 2 and 3. Each strip should be used only once.

Circulate and assist.

Ask students what they notice about their lists of fractions. The numerators and denominators are multiples of the numbers in the first column; the fractions are equivalent.


Using a Common DenominatorLESSON

6�9

Date Time

Study the examples. Then work the problems below in the same way.

1. �23� � �

29� � ?

Unlike CommonDenominators Denominators

�23�

� �29�

3. �13� � �

25� � ?


�13�

� �25�

�34� � �1

92�


�23� �

46�

� �16� � �

16�


�56� �

1102�

� �34� � �1

92�

Example 1: �23� � �

16� � ? Example 2: �

56� � �

34� � ?

�56� �1

12�

�23� � �

46� �

56� � �

1102�

�56� � �

1158�

�69�

�29�

�89�

�23� � �

69�

�13� � �1

55�

�25� � �1

65�

�155�

�165�

�1115�

�

�

2. �1136� � �

34� � ?


�1136�

� �34�

4. �56� � �

49� � ?


�56�

� �49�

�49� � �1

88�

�34� � �

1126�

�1136�

�1126�

�116�

�

�1158�

�188�

�178�

�


Student Page


Ask volunteers to match two of their remaining strips and write the fraction from the first column on the board. Use these fractions to demonstrate the multiplication rule. Example:

4 _ 9 = 4 ∗ 2 _ 9 ∗ 2 = 8 _ 18

Ask students to use the appropriate strips to give another equivalent fraction for 4 _ 9 . Then ask a volunteer to write the number model for this change using the multiplication rule. Sample response: 32 _ 72 = 4 ∗ 8 _ 9 ∗ 8

Refer students to the Multiplication Table Poster. Explain that for any two rows, the equivalent fractions are the result of multiplying the fraction in the first column by another name for 1, such as 2 _ 2 or 8 _ 8 , depending on the column. So the second column is the result of multiplying by 2 _ 2 , the third column is the result of multiplying by 3 _ 3 , and so on.

▶ Using a Common Denominator PARTNER ACTIVITY

(Math Journal 1, pp. 196 and 197; Student Reference

Book, p. 401)

Algebraic Thinking Introduce the next activity by discussing the following points:

● It is easy to add or subtract fractions if they have the same denominator, usually called a common denominator. To support English language learners, discuss the meaning of common in this mathematical context.

● One way to add or subtract fractions with different denominators, usually called unlike denominators, is to rewrite the fractions with a common denominator.

● One way to find common denominators is to use the multiplication rule (or the division rule) for finding equivalent fractions. Ask volunteers to express the rules with variables. a _ b = a ∗ n _ b ∗ n ; a _ b = a ÷ n _ b ÷ n

Have students look at Example 1 and Example 2 at the top of Math Journal 1, page 196. Ask: How could you use benchmarks to estimate the solution to each problem? Sample answer: For Problem 1, I know that 2 _ 3 is greater than 1 _ 2 and 1 _ 6 is less than 1 _ 2 , so theanswer will be close to 1. For Problem 2, I know that 5 _ 6 and 3 _ 4 are both close to 1, so my answer will be close to zero.

NOTE Some students may realize that 1 _ 6 is less than 1

_ 3 and conclude that the

answer will be less than 1.

Then work through the examples to illustrate the use of the multiplication rule to find common denominators. Pose one or two similar problems as needed.

ELL

Adjusting the ActivityRefer students to the Equivalent

Fractions, Decimals, and Percents table on

Student Reference Book, page 401. Ask

students to describe the similarities and

differences between the structure of this

table and their work with multiplication table

number strips. 1 row of the table is similar

to two number strips; the table shows the

equivalent decimals and percents for the

fractions.

A U D I T O R Y � K I N E S T H E T I C � T A C T I L E � V I S U A L

424-428_EMCS_T_TLG1_G5_U06_L09_576825.indd 426424-428_EMCS_T_TLG1_G5_U06_L09_576825.indd 426 2/18/11 1:16 PM2/18/11 1:16 PM

Math Boxes LESSON

6�9

Date Time

5. Measure each line segment to the nearest�18� in.

a.

in.

b.

in.

1. In the figure below, write the correctfraction in each of the smaller regions.Check to see that the fractional parts addup to 1.

3. Esther did 5 standing jumps. Her longestjump was 50 in. Could her average jumpbe 53 in.?

Create a data set for Esther’s jumps thatcould have this average.51, 52, 53, 54, 55

No

4. Measure the length and width of each of thefollowing objects to the nearest centimeter.

a. pinkie finger

length: ______ cm

width: ______ cm

c. notebook

length: ______ cm

width: ______ cm

6. Rename each fraction as a mixed numberor a whole number.

a. �559� �

b. �8181� �

c. �1270

� �

d. �944� �

e. �1062

� � 17

8

2.

Estimate the measure of �M:

The measure of �M is about .37°

14

12

116

316

M

b. pencil

length: ______ cm

width: ______ cm

2�28�, or 2�

14�

1�18�

11�45�

17�17�

23 �24�, or 23 �

12�

Answers vary.

Answers vary.

75

121

183

204

183

62 63


Student Page

Using a Common Denominator continuedLESSON

6 �9

Date Time

5. �142� � �

32� � ?


�142�

7. A piece of ribbon is 7�12� in. long. If a piece

2�136� in. long is cut off, how long is the remaining piece? in.

Write a number sentence to show how you solved the problem.

8. Three boards are glued together. The diagram below shows the thickness of each board. What is the total thickness of the three boards? in.

Write a number sentence to show how you solved the problem.

5�156�

7�186� � 2�1

36� � 5�1

56�

3�58� � �

48� � 2�

68� � 6�

78�

3 85

21 2 4

3" " "

6�78�

6. 1�116� � �

38� � ?


1�116�

� �38�

�142�

�64��

�1176�

�166�

�32� � �

64�

1�116� � �

1176�

�38� � �1

66�

�148�, or 4�

12�

� �32�

�1116�

�


Student Page

Lesson 6�9 427

In addition to using the multiplication rule to find equivalent fractions, students can also refer to the Table of Equivalent Fractions, Decimals, and Percents on page 401 of the Student Reference Book.

Assign journal pages 196 and 197. Remind students of the importance of using benchmarks to estimate the solution and then assess the reasonableness of their answers. Students may choose to solve Problems 7 and 8 by finding a common denominator.

NOTE In Problems 1, 2, 5, 6, 7, and 8, on journal pages 196 and 197, the

common denominator is the same as one of the original denominators. In

Problems 3 and 4, the common denominator is different from both of the

original denominators.

2 Ongoing Learning & Practice

▶ Playing Fraction Capture PARTNER ACTIVITY

(Math Journal 1, p. 198; Math Masters, p. 460)

Players roll dice, form fractions, and claim corresponding sections of squares. The rules are on Math Journal 1, page 198, and the gameboard is on Math Masters, page 460. Remind students of the importance of using the benchmark fraction of 1 _ 2 when playing this game.

▶ Math Boxes 6�9

INDEPENDENT ACTIVITY

(Math Journal 1, p. 199)

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

Writing/Reasoning Have students write a response to the following: Explain your answer to the question in Problem 3 and how you chose the values for the data set.

Because the average cannot be greater than the maximum in the data set, 53 inches cannot be Esther’s average since 50 is the maximum number. I chose 5 numbers for the data set that could be added together and divided by 5 so that the average would equal 53.

▶ Study Link 6�9


(Math Masters, p. 178)

Home Connection Students solve problems similar to those on journal pages 196 and 197. This page reinforces the idea that a common denominator can be determined by finding fractions equivalent to the given fractions.

EM3cuG5TLG1_424-428_U06L09.indd 427EM3cuG5TLG1_424-428_U06L09.indd 427 1/19/11 3:00 PM1/19/11 3:00 PM

Multiplication Rule

To find a fraction equivalent to a given fraction,

multiply the numerator and the denominator

of the fraction by the same number.

STUDY LINK

6�9 Adding and Subtracting Fractions

Name Date Time

�ab� � �

ab

º

º

nn�

Example 1: �4

9� � �

1

3� � ?

�1

3� � �

2

6� � �

3

9� � �

1

4

2� � �

1

5

5� � �

1

6

8� � ...

9 is a common denominator.

�4

9� � �

1

3� � �

4

9� � �

3

9� � �

1

9�

Find a common denominator. Then add or subtract.

1. �2

3� � �

4

5� � 2. �

8

9� � �

5

6� � 3. �

3

4� � 1�

1

2� �

4. Lisa was 4 feet 10�1

2� inches tall at the end of fifth grade. During the year, she

had grown 2�3

4� inches. How tall was Lisa at the start of fifth grade?

feet in.

5. Bill was baking two different kinds of bread. One recipe called for 3�1

2� cups of

flour. The other called for 2�1

3� cups of flour. How much flour did Bill need in all?

cups

4

2�14

��118�1�

175�

Example 2: �5

8� � �

2

5� � ?

�5

8� � �

1

1

0

6� � �

1

2

5

4� � �

2

3

0

2� � �

2

4

5

0� � �

3

4

0

8� � ...

�2

5� � �

1

4

0� � �

1

6

5� � �

2

8

0� � �

1

2

0

5� � �

1

3

2

0� � �

1

3

4

5� � �

1

4

6

0� � �

1

4

8

5� � ...

Both fractions can be rewritten with the

common denominator 40.

�5

8� � �

2

5� � �

2

4

5

0� � �

1

4

6

0� � �

4

4

1

0�, or 1�

4

1

0�

�23

� � �1105� �

8

9� � �

1168� 1�

12

� � �3

2� � �

64

�

�45

� � �1125� �

56

� � �1158� �

34

� � �64

� � �94

�, or 2�14

�

�1105� � �

1125� � �

2125�, �

1168� � �

1158� � �

118�

or 1�175�

5�56

�

6568–71

7�34

�


Study Link Master

LESSON

6�9

Name Date Time

Fractions in Military Time

On a military clock, the whole is 1 day or 24 hours. �2

1

4� is one hour. The time

shown on this clock face is 08:14:42 (8 hours, 14 minutes, and 42 seconds).

Using the clock face, write the fractions as days, hours, and minutes. The first

one has been done for you.

1. �2

2

4� �

1of a day � 2 hours � minutes

2. �1

2

8

4� � of a day � hours � minutes

3. �1

2

0

4� � of a day � hours � minutes

4. �1

2� hour � of a day

5. Explain how you found your answer for Problem 4.

Sample answer: �12

� hour is equal to 30 minutes.

48

1

6001012

5

1,080184

3

12012

00

20

16

12

08

04

2322

21

19

18

17

1514

13 1110

09

07

06

05

0302

01

055

50

40

3530

25

20

15

10

5

45

Rule

day

Whole

�1,

34040� � �

1344� � �

418�.

In one day, there are 1,440 minutes, so


Teaching Master Teaching Master

LESSON

6�9

Name Date Time

Writing Elapsed-Time Number Stories

The numbers on a clock face divide one hour into twelfths. Each �1

1

2� of an hour is

5 minutes.

Use fractions to represent amounts of elapsed time and write a number story for a

partner to solve.

Example:

Maria started her piano practice at 3:15. She practiced for �1

8

2� of an hour. At what time did

she finish practicing?

Think: �1

1

2� hour � 5 minutes; �

1

8

2� hour is 8 � 5, or 40 minutes; 40 minutes more than 3:15 is 3:55.

Maria finished practicing at 3:55.

Your Elapsed-Time Number Story:

Answers vary.

Your Partner’s Solution:

Answers vary.

Explain your answer.

Answers vary.

12 12

3

4567

8

9

1011

Rule

hour

Whole



3 Differentiation Options

ENRICHMENT PARTNER ACTIVITY

▶ Modeling Fractions with a 15–30 Min

Military Clock(Math Masters, p. 179)

To apply students’ understanding of the fractional units on a 12-hour clock face, have students use a 24-hour military clock face model to add, subtract, and find

equivalent fractions. When they have finished the page, have students describe similarities and differences between using the 12-hour clock and the 24-hour clock.

EXTRA PRACTICE


▶ Writing Elapsed Time 15–30 Min

Number Stories(Math Masters, p. 180)

Students write a number story using fractions to represent elapsed time. Ask students to exchange and solve each other’s problems and then share their solution strategies.


Clock Fractions and Common Denominators · a clock face to add and subtract fractions. Remind...

Documents

Transcript of Clock Fractions and Common Denominators · a clock face to add and subtract fractions. Remind...