383A Chapter 6

About the MathProfessional Development

LESSON AT A GLANCE

Learning ObjectiveAdd and subtract mixed numbers with unlike denominators.

Language ObjectiveStudents write in their Math Journal the meaning of the term unlike denominators and whisper to a partner how you can add and subtract mixed numbers with unlike denominators.

MaterialsMathBoard

F C R Focus:Common Core State Standards5.NF.A.1 Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent

fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.

Also 5.NF.A.2

MATHEMATICAL PRACTICESMP1 Make sense of problems and persevere in solving them. MP2 Reason abstractly and quantitatively. MP6 Attend to precision.

F C R Coherence:Standards Across the GradesBefore4.NF.A.3c4.NF.A.3d

Grade 55.NF.A.1

After6.NS.A.1

F C R Rigor:Level 1: Understand Concepts....................Share and Show ( Checked Items)Level 2: Procedural Skills and Fluency.......On Your OwnLevel 3: Applications..................................Think Smarter and Go Deeper

F C R For more about how GO Math! fosters Coherence within the Content Standards and Mathematical Progressions for this chapter, see page 349J

FOCUS COHERENCE RIGOR

Professional Development Videos

If Students AskStudents who do not recognize the importance of using an estimate to check an answer may ask about the value of estimating. Remind these students that errors can be made when adding and subtracting mixed numbers with unlike denominators (as well as when solving other kinds of problems). Point out, for example, that students may add or subtract the fractions, but forget to add or subtract the whole numbers. They may also make mistakes writing equivalent fractions or simplifying an answer. Inadvertently performing the wrong operation is another possible error.

Give volunteers an opportunity to suggest errors that they themselves sometimes make.

Add and Subtract Mixed Numbers

LESSON 6.6

Interactive Student Edition

Personal Math Trainer

Math on the Spot

Animated Math Models

HMH Mega Math

Multimedia and Technology

ENGAGE1

Lesson 6.6 383B

Daily RoutinesCommon Core

Daily RoutinesCommon Core

How can you add and subtract mixed numbers

with unlike denominators?

with the Interactive Student Edition

Essential QuestionHow can you add and subtract mixed numbers with unlike denominators?

Making ConnectionsInvite students to tell you what they know about mixed numbers and fractions.

What is a mixed number? a number that includes both a whole number and a fraction What are some examples of mixed numbers? Possible answers: 5 1 _ 2 , 1 1 _ 3 , 2 1 _ 4 , etc. How do you subtract fractions with equal denominators? Subtract the lesser numerator from the greater numerator. The denominator remains the same. Why does the denominator remain the same when you subtract fractions with equal denominators? because the denominator represents the whole, which does not change

Learning Activity• How many hours of darkness does the town have at night?

10 3 _ 4 hours

• How many hours are the streetlights on when it is dark at night? 6 1 _ 2 hours

• What operation will you use to find the total hours when the streetlights are off when it is dark at night? subtraction

• What do you need to do to fractions with unlike denominators in order to subtract? write equivalent fractions that have the same denominators

Literacy and Mathematics• Have students write an explanation as to how solving the problem

would be different if the streetlights were on 6 1 _ 4 hours instead of 6 1 _ 2 hours.

Table Soccer, Anyone?

Literature ConnectionFrom the Grab-and-Go™ Differentiated Centers Kit

Students read about a carpentry project that involves adding and subtracting mixed numbers.

Problem of the Day 6.6Sarah placed 3 5 _ 8 quarts of strawberries and 2 1 _ 8 quarts of blueberries in a bowl. How many quarts of fruit did Sarah place in the bowl? 5 6 _ 8 quarts

Vocabulary

• Interactive Student Edition• Multimedia Glossary e

Fluency BuilderUnknown Numbers Review adding and subtracting fractions with unlike denominators. Have students find the unknown numbers in exercises such as the following.

4 __ 5 = ___

15 12

+ 1 __ 3 = + 5 __ 15

17 ___ = 2 __ 15; 1; 15

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EXPLORE2

1 1 _ 5 1 2 __ 10

Unlock the ProblemUnlock the Problem

MATHEMATICAL PRACTICES 2MathTalk

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Chapter 6 383

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Lesson 6.6

• What operation should you use to solve the problem?

• Do the fractions have the same denominator?

Add and Subtract Mixed NumbersEssential Question How can you add and subtract mixed numbers with unlike denominators?

Denise mixed 1 4 _ 5 ounces of blue paint with

2 1 __ 10

ounces of yellow paint. How many

ounces of paint did Denise mix?

Add. 1 4 __ 5 + 2 1 ___

10

To find the sum of mixed numbers with unlike denominators, you can use a common denominator.

STEP 1 Estimate the sum. ___

STEP 2 Find a common denominator. Use the common denominator to write equivalent fractions with like denominators.

So, Denise mixed _ ounces of paint.

STEP 3 Add the fractions. Then add the whole numbers. Write the answer in simplest form.

1 4 __ 5

=

+ 2 1 ___ 10

= +

1. MATHEMATICALPRACTICE 1 Evaluate Reasonableness Explain how you know whether your answer is reasonable.

2. What other common denominator could you have used?

Number and Operations—Fractions—5.NF.A.1 Also 5.NF.A.2

MATHEMATICAL PRACTICESMP1, MP2, MP6

Connect Symbols and Words Did you use the least common denominator? Explain.

addition

no

2 + 2 = 4

3 9 __ 10

Math Talk: Possible explanation: Yes, because I know that 10 is the least common multiple of 5 and 10, so 10 is the least common denominator.

3 9 ___ 10

1 8 ___ 10

2 1 ___ 10

Possible explanation: I can compare my answer to the estimate. Since 3 9 __ 10

is close to the

Answers will vary depending on the

common denominator used. Possible answer: 50

estimate, 4, the answer is reasonable.

See below.

Name

Add and Subtract Mixed Numbers

When you add or subtract mixed numbers, you may need to rename the fractions as fractions with a common denominator.

Find the sum. Write the answer in simplest form. 5 3 _ 4 1 2 1 _

3

Step 1 Model 5 3 _ 4 and 2 1 _

3 .

Step 2 A common denominator for 3 _ 4 and 1 _

3 is 12,

so rename 5 3 _ 4 as 5 9 __

12 and 2 1 _

3 as 2 4 __

12 .

Step 3 Add the fractions.

9 __ 12

1 4 __ 12

5 13 __ 12

Step 4 Add the whole numbers

5 1 2 5 7

Add the sums. Write the answer in simplest form.

13 __ 12

1 7 5 7 13 __ 12

, or 8 1 __ 12

So, 5 3 _ 4 1 2 1 _

3 5 8 1 __

12 .

Find the sum or difference. Write your answer in simplest form.

1. 2 2 __ 9

1 4 1 __ 6

2. 10 5

__ 6

1 5 3

__ 4

3. 11 7 __ 8

2 9 5

__ 6

4. 18 3

__ 5

2 14 1 __ 2

Lesson 6.6

Reteach

4 1 __ 10

2 1 __ 24

16 7 __ 12

6 7 __ 18

6-15 ReteachChapter Resources© Houghton Mifflin Harcourt Publishing Company

Name Lesson 6.6

Enrich

What do you call a scared dinosaur?

3 7 ___ 12

1 1 __ 2

2 1 __ 2

3 3 __ 4

1 3 ___ 10

3 5 __ 8

2 7 ___ 18

2 11 ___ 24

3 1 __ 6

1 1 __ 8

3 1 ___ 18

A. 1 1 __ 3

1 2 1 __ 4

5 ________________

E. 5 9 ___ 10

2 3 2 __ 5

5 ________________

E. 8 5 __ 8

2 7 1 __ 2

5 ________________

N. 4 3 __ 4

2 3 1 __ 4

5 ________________

O. 1 3 __ 4

1 1 7 __ 8

5 ________________

R. 2 1 __ 6

1 1 7 ___ 12

5 ________________

R. 1 9 ___ 10

1 1 4 ___ 15

5 ________________

S. 6 5 __ 6

2 4 3 __ 8

5 ________________

U. 8 5 __ 6

2 6 4 __ 9

5 ________________

V. 7 1 __ 2

2 6 1 __ 5

5 ________________

X. 1 1 __ 6

1 1 8 __ 9

5 ________________

Mixed Number Sums and Differences

Write equivalent fractions and then find the sum or difference. Write

the answer in simplest form. Write the letter of the exercise above its

sum or difference at the bottom of the page to answer the riddle. Two

have been done for you.

XERSUOVRENA

3 1 __ 18

1 3 __ 10

2 7 __ 18

2 11

__ 24

3 1 _ 6

3 3 _ 4

3 5 _ 8

1 1 _ 2

1 1 _ 8

2 1 _ 2

3 7 __ 12

6-16 EnrichChapter Resources© Houghton Mifflin Harcourt Publishing Company

1

2

3 DifferentiatedInstruction

383 Chapter 6

Unlock the ProblemMATHEMATICAL PRACTICES

Read and discuss the problem.• Why is addition used to solve this problem?

Possible answer: Addition is used because I am being asked to find a total amount.

Discuss the estimate shown in Step 1.• Why is it important to estimate the

sum? The estimate is used to check the answer for reasonableness.

MP6 Attend to precision.• How were benchmarks used to round the

mixed numbers? The mixed numbers were rounded to the nearest benchmark. The benchmarks can be whole numbers or can be mixed numbers that have 1 __ 2 as the part that is a fraction.

MP3 Construct viable arguments and critique the reasoning of others.• How do you know your answer is in

simplest form? I know because the numerator and the denominator do not have a common factor.

MathTalk Use Math Talk to focus students’

understanding of common denominators.

• Could you have used 20 as a common denominator? How would it have changed your answer? Yes, but my answer would not be in simplest form.

ELL Strategy: Model Language

Write these problems on the board:

+ 2 7 ___ 10 ______

+ 2 7 ___ 10 ______

• Use these sentence frames to help students practice using the terms unlike denominators and common denominators. The fractions in the first problem have ______ denominators, ___ and ___. The fractions in the second problem have a __________ __________, ___.

• Pairs use the sentence frames to describe other sets of fractions.

LESSON 6.6

unlike

common denominator

5

10

10

5.NF.A.1 Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.

Enrich 6.6Reteach 6.6

Meeting Individual Needs

DifferentiatedInstruction

COMMON ERRORS

342 1

6� 1 23� 1 7

125

Advanced Learners

384

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ExampleSubtract. 4 5 __

6 − 2 3 __

4

You can also use a common denominator to find the difference of mixed numbers with unlike denominators.

STEP 1 Estimate the difference. __

STEP 2 Find a common denominator. Use the common denominator to write equivalent fractions with like denominators.

STEP 3 Subtract the fractions. Subtract the whole numbers. Write the answer in simplest form.

3. MATHEMATICALPRACTICE 1 Evaluate Reasonableness Explain how you know whether

your answer is reasonable.

Share and ShShare and ShShare and Show MATHBOARDMATHBOARD

1. Use a common denominator to write equivalent fractions with like

denominators and then find the sum. Write your answer in simplest form.

2. 2 3 _ 4

+ 3 3 __ 10

3. 5 3 _ 4

+ 1 1 _ 3

4. 3 4 _ 5

+ 2 3 __ 10

Find the sum. Write your answer in simplest form.

4 5 __ 6

=

– 2 3 __ 4

= –

7 2 __ 5

=

+ 4 3 __ 4

= +

5 − 3 = 2

Possible explanation: I can compare my answer to the estimate. Since 2 1 __ 12

is

close to the estimate, 2, the answer is reasonable.

7 1 __ 12

6 1 __ 20

6 1 __ 10

2 2 ___ 24 = 2 1 ___ 12

4 20 ___ 24

2 18 ___ 24

7 8 ___ 20

4 15 ___ 20

11 23 ____ 20 = 12 3 ____

20

Lesson 6.6 384

Error Students forget to subtract the whole numbers because they do not include the whole number parts of the mixed numbers when they write equivalent fractions with like denominators.

Example

5 1 _ 2

− 3 2 _ 5

= 5 __ 10

− 4 __ 10

= 1 __ 10

Springboard to Learning Remind students of the importance of using estimates to check mixed number sums and differences.

ExampleDiscuss the various steps in the example.• Explain how the estimate in Step 1 was

made. Include the term benchmark in your answer. Possible explanation: The mixed numbers were rounded to whole-number benchmarks. To the nearest whole number, 4 5 _ 6 rounds to 5 and 2 3 _ 4 rounds to 3.

• Explain how the fraction parts of the mixed numbers can help you round those numbers. Possible explanation: Compare the numerator of each fraction to the denominator. If the numerator is more than half of the denominator, the mixed number rounds to the next whole number.

• In Step 2, describe how you chose a common denominator. Possible explanation: I found a common denominator by multiplying the denominators.

• Could 12 have been used as a common denominator? Explain why or why not. Yes. Possible explanation: 12 is a common multiple of both 6 and 4.

Students may choose different common denominators to solve the problem. Have them share which common denominators they used.

EXPLAIN3

Share and Show MATHBOARDMATHBOARDMBMMMBBBMATHABOARDMMMAAATHATHTHHAAAAAAAAATTAAAABOARDBOARDBOARD

The first problem connects to the learning model. Have students use the MathBoard to explain their thinking.

Materials index cards

Each student should write five mixed number problems and solutions. Each problem should be on its own index card, and each solution should be on its own index card.• Problems can have more than two numbers. For

example, a problem could be a + b + c.• Problems can mix addition and subtraction.• Problems can use both fractions

and mixed numbers.When students finish writing the problems and solutions on cards, each student may choose to trade card sets with another student to match the problems with the solutions.

VisualIndividual / Partners

DifferentiatedInstruction

Quick Check

If

Rt I 1

2

3

Quick Check

If

Rt I 1

2

3

On Your OwnOn Your Own

MATHEMATICAL PRACTICES 6MathTalk

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Chapter 6 • Lesson 6 385

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Find the difference. Write your answer in simplest form.

5. 9 5 _ 6

− 2 1 _ 3

6. 10 5 _ 9

− 9 1 _ 6

7. 7 2 _ 3

− 3 1 _ 6

8. 1 3 __ 10

+ 2 2 _ 5

9. 8 1 _ 6

+ 7 3 _ 8

10. 2 1 _ 2

+ 2 1 _ 3

Find the sum or difference. Write your answer in simplest form.

11. 12 3 _ 4 − 6 1 _ 6

12. 2 5 _ 8

− 1 1 _ 4 13. 14 7 __ 12

− 5 1 _ 4

14. 1 5 __ 12

+ 4 1 _ 6

15. 8 1 _ 2

+ 6 3 _ 5

16. 2 1 _ 6

+ 4 5 _ 9

17. 3 5 _ 8

+ 5 __ 12

Practice: Copy and Solve Find the sum or difference. Write youranswer in simplest form.

18. 3 2 _ 3

− 1 1 _ 6

19. 5 6 _ 7 − 1 2 __ 3

20. 2 7 _ 8

− 1 _ 2 21. 4 7 __ 12

− 1 2 _ 9

22. DEEPER Dakota makes a salad dressing by combining 6 1 _ 3 fluid ounces of oil and 2 3 _ 8 fluid ounces of vinegar in a jar. She then pours 2 1 _ 4 fluid ounces of the dressing onto her salad. How much dressing remains in the jar?

23. DEEPER This week, Maddie worked 2 1 _ 2 hours on Monday, 2 2 _ 3 hours on Tuesday, and 3 1 _ 4 hours on Wednesday. How many more hours will Maddie need to work this week to make her goal of 10 1 _ 2 hours a week?

Explain why you need to write equivalent fractions with common denominators to add 4 5 _ 6 and 1 1 _ 8 .

1 7 __ 18

7 1 _ 2 4 1 _

2

3 7 __ 10

1 3 _ 8

4 5 _ 6

9 1 _ 3

Possible explanation: Without common denominators, the fractions would not be made up of equal-size pieces and could not be easily added.

6 11 __ 24 � uid ounces

2 1 __ 12 hours

5 7 __ 12

15 1 __ 10

6 13 __ 18

4 1 __ 24

2 1 _ 2 4 4 __

21 2 3 _

8 3 13 __

36

15 13 __ 24

6 7 __ 12

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385  Chapter 6

On Your OwnIf students complete the checked exercises correctly, they may continue with the remaining exercises.Remind students to use an estimate to check the reasonableness of each sum or difference.

Practice: Copy and SolveFor Exercises 14–21, remind students to write their answers in simplest form.

a student misses the checked exercises

Differentiate Instruction with • Reteach6.6

• PersonalMathTrainer5.NF.A.1

• RtITier1Activity(online)

Then

MathTalk Use Math Talk to focus students’

understanding of the lesson’s concepts.

• Why can’t you just add the numerators of   5 _ 6   and   1 _ 8  ? Becausethe5referstolargerpiecesthanthe1.Ifyouaddedthem,itwouldn’tbeclearwhetherthatnumberrepresentedsixthsoreighths.

Use the checked exercises on page 384 and page 385 for Quick Check. Students should show their answers for the Quick Check on the MathBoard.

DO NOT EDIT--Changes must be made through “File info” CorrectionKey=B

ELABORATE4

Differentiated Centers Kit

DIFFERENTIATED INSTRUCTION INDEPENDENT ACTIVITIES EVALUATE5 Formative Assessment

ELABORATE4

Differentiated Centers Kit

DIFFERENTIATED INSTRUCTION INDEPENDENT ACTIVITIESD

Math on the Spot videos are in the Interactive Student Edition and at www.thinkcentral.com.

Paint Gavin Uses(in ounces)

Red Yellow

SunriseOrange

Tangerine

Mango

258

39

10

556

314

238

556

Shade

MATHEMATICAL PRACTICES M

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Use the table to solve 24–25.

26. SMARTER Martin won first place in the 100-meter dash with a time of

14 23 ___

100 seconds. Samuel came in second place with a time of 15 7 __

10 seconds.

For 26a–26d, select True or False for each statement.

26a. A common denominator of the mixed True False

numbers is 100.

26b. To find the difference between the True False

runners’ times, Samuel’s time needs

to be rewritten.

26c. Samuel’s time written with a denominator True False

of 100 is 15 70 ___

100 .

26d. Martin beat Samuel by 21 __ 25

second. True False

24. MATHEMATICALPRACTICE 2 Reason Quantitatively Gavin plans to mix a batch

of Tangerine paint. He expects to have a total of 5 3 __ 10

ounces of paint

after he mixes the amounts of red and yellow. Explain how you can

tell if Gavin’s expectation is reasonable.

25. SMARTER Gavin mixes the amount of red

from one shade of paint with the amount of

yellow from a different shade of paint. He mixes

the batch so he will have the greatest possible

amount of paint. What amounts of red and yellow

from which shades are used in the mixture?

Explain your answer.

Possible explanation: No; Gavin’s expectation is not

reasonable. If you estimate the amount of Tangerine paint

made, 4 + 2 1 _ 2 = 6 1 _

2 , the estimate is not close to Gavin’s

answer.

3 9 __ 10

oz of red from Tangerine and 5 5 _ 6 oz of yellow from Mango; I looked for the greatest

number in each column for colors. Since the numbers in the two columns for Mango are

the same, I looked for the next greatest number.

Lesson 6.6 386

Students match pictorial models to subtract fractions and mixed numbers with unlike denominators.

Students read about a carpentry project that involves adding and subtracting mixed numbers.

Students complete blue Activity Card 8 by using rulers to model adding mixed numbers.

GamesPicture Problems

LiteratureTable Soccer, Anyone?

ActivitiesMixed Measures

Essential QuestionUsing the Language ObjectiveReflect Have students write in their Math Journal the meaning of the term unlike denominators and whisper to a partner to answer the essential question. How can you add and subtract mixed numbers with unlike denominators? Possible answer: I find a common denominator and use it to write equivalent fractions with like denominators. Then I add or subtract the fractions, and I add or subtract the whole numbers. Finally, I make sure that the sum or difference is in simplest form.

Math Journal WRITE MathWrite your own story problem using mixed numbers. Show the solution.

MATHEMATICAL PRACTICES

MP2 Reason abstractly and quantitatively. For Problem 24, students use benchmark numbers to estimate a sum. They use this estimate to determine whether Gavin’s sum is reasonable.

SMARTER

Problem 25 involves higher order thinking skills because the solution is dependent upon an inference.

SMARTER

Exercise 26 assesses a student’s ability to subtract mixed numbers by rewriting them with a common denominator. Be sure students understand how to subtract the mixed numbers by asking them how many seconds Martin finished the race ahead of Samuel (1 47

___ 100 seconds).

Math on the Spot Video TutorUse this video to help students model and solve this type of Think Smarter problem.

Meeting Individual Needs

Problem Solving • Thinking

nt

Cross-Curricular

Problem SolvingProblem Solving

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Name

Add and Subtract Mixed Numbers

Find the sum or difference. Write your answer in simplest form.

1. 3 1 _ 2

− 1 1 _ 5

__

2. 2 1 _ 3

+ 1 3 _ 4

__

3. 4 1 _ 8

+ 2 1 _ 3

__

4. 5 1 _ 3

+ 6 1 _ 6

__

9. Jacobi bought 7 1 _ 2 pounds of meatballs. He

decided to cook 1 1 _ 4 pounds and freeze the rest.

How many pounds did he freeze?

_______

10. Jill walked 8 1 _ 8 miles to a park and then

7 2 _ 5 miles home. How many miles did

she walk?

_______

Chapter 6 387

5. 2 1 _ 4

+ 1 2 _ 5

__

6. 5 17 __

18 − 2 2 _

3

__

7. 6 3 _ 4

− 1 5 _ 8

__

8. 5 3 _ 7

− 2 1 _ 5

__

−1 1 _ 5

3 1 _

2 3 5 __

10

−1 2 __ 10

2 3 __ 10

COMMON CORE STANDARD—5.NF.A.1 Use equivalent fractions as a strategy to add and subtract fractions.

11. WRITE Math Write your own story problem using mixed numbers.

Show the solution.

Check students’ problems and solutions.

Lesson 6.6Practice and Homework

6 11

__ 24

11 1 _

2

6 1 _ 4 pounds 15 21

__ 40

miles

3 13

__ 20

4 1 __

12

5 1 _

8 3

8 __

35 3

5 __

18

Practice and HomeworkUse the Practice and Homework pages to provide students with more practice of the concepts and skills presented in this lesson. Students master their understanding as they complete practice items and then challenge their critical thinking skills with Problem Solving. Use the Write Math section to determine student’s understanding of content for this lesson. Encourage students to use their Math Journals to record their answers.

387 Chapter 6

• The water cycle is the process by which water circulates through the atmosphere. As the sun shines down on a lake, the heat from the sun causes some water to evaporate. The water vapor in the atmosphere then changes to a liquid when the temperature in the clouds becomes too cold. When the clouds can no longer hold any more condensation, precipitation begins. The water vapor now changes to rain, hail, sleet, or snow.

• Jeremy has a rain gauge outside to measure the amount of rain. After a two-day period, there were 6 3 _ 4 inches of rain. After two sunny days, there were 4 3 _ 8 inches of rain left in the rain gauge. How much of the rainwater had evaporated? 2 3 __ 8 inches of rainwater

• During the Revolutionary War, the colonists gained their independence from England in 8 5 __ 12 years. The Civil War was a battle between the Northern and Southern states.

• The Civil War lasted 3 3 _ 4 years. How long did both wars last altogether? 12 1 _ 6 years

SCIENCE SOCIAL STUDIES

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Personal Math Trainer

FOR MORE PRACTICE GO TO THE

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388

Lesson Check (5.NF.A.1)

Spiral Review (5.NBT.A.2, 5.NBT.B.6, 5.NBT.B.7)

1. Ming has a goal to jog 4 1 _ 2 miles each day. On Monday she jogged 5 9 __ 16 miles. By how much did she exceed her goal for that day?

2. At the deli, Ricardo ordered 3 1 _ 5 pounds of cheddar cheese and 2 3 _ 4 pounds of mozzarella cheese. How many pounds of cheese did he order all together?

5. What number is 100 times as great as 0.3? 6. Mark said that the product of 0.02 and 0.7 is 14. Mark is wrong. What is the product?

3. The theater has 175 seats. There are 7 seats in each row. How many rows are there?

4. During the first 14 days, 2,744 people visited a new store. The same number of people visited the store each day. About how many people visited the store each day?

1 1 __ 16

miles

25 rows

30

about 200 people

0.014

5 19 __ 20

pounds

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Continue concepts and skills practice with Lesson Check. Use Spiral Review to engage students in previously taught concepts and to promote content retention. Common Core standards are correlated to each section.

Lesson 6.6 388

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