0cog reteaching estimating sums and differences of … 10... · reteaching 10-1 estimating sums and...

Name

Copyright © Pearson Education, Inc., or its affiliates. All Rights Reserved. 5

Reteaching

10-1Estimating Sums and Differences of Mixed NumbersYou can use rounding to estimate sums and differences of fractions and mixed numbers.

How to round fractions:

If the fractional part is greater than or equal to 1 _ 2 , round up to the next whole number.

Example: Round 3 5 _ 7 to the nearest whole number.

5 _ 7 is greater than 1 _ 2 , so 3 5 _ 7 rounds up to 4.

If the fractional part is less than 1 _ 2 , drop the fraction and use the whole number you already have.

Example: Round 6 1 _ 3 to the nearest whole number.

1 _ 3 is less than 1 _ 2 , so drop 1 _ 3 and round down to 6.

How to estimate sums and differences of fractions and mixed numbers:

Round both numbers to the nearest whole number. Then add or subtract.

Example: Estimate 4 1 _ 8 ! 7 2 _ 3 .

4 1 _ 8 rounds down to 4.

7 2 _ 3 rounds up to 8.

4 ! 8 " 12

So, 4 1 _ 8 ! 7 2 _ 3 is about 12.

Round to the nearest whole number.

1. 8 5 _ 6 2. 13 8 _ 9 3. 43 1 _ 3

4. 7 40 __ 81 5. 29 4 _ 5 6. 88 2 _ 4

7. 19 3 __ 34 8. 63 41 __ 49

Estimate each sum or difference.

9. 7 1 _ 9 ! 8 2 _ 5 10. 14 5 _ 8 # 3 7 __ 10

11. 2 1 _ 4 ! 5 1 _ 2 ! 10 3 _ 4 12. 11 3 _ 5 # 4 1 __ 12

13. 9 ! 3 11 __ 14 ! 5 1 _ 9 14. 15 6 _ 7 # 12 2 __ 10

R 10•1

Name


Practice

10-1Estimating Sums and Differences of Mixed NumbersRound to the nearest whole number.

1. 3 3 _ 8 2. 6 5 __ 11 3. 1 11 __ 20 4. 12 6 __ 13

Estimate each sum or difference.

5. 3 1 _ 4 + 2 5 _ 6 6. 5 6 _ 9 − 1 3 _ 4

7. 5 5 __ 13 + 8 3 _ 5 8. 11 − 6 3 _ 7 + 2 2 _ 5

Robert and May are competing in a track meet. The table at the right shows the results of their events.

9. Robert says his better jump was about 1 ft longer than May’s better jump. Is he correct?

10. Use the table above. If the school record for the softball throw is 78 ft, about how much farther must Robert throw the ball to match the record?

A 15 ft B 16 ft C 18 ft D 20 ft

11. Consider the sum of 3 _ 5 + 3 _ 4 . Round each fraction and estimate the sum. Add the two fractions using a common denominator and then round the result. Which estimate is closer to the actual answer?

Participant

Robert

May

Event Results/Distance

Long jump

Softball throw

Long jump

Softball throw

1. 6 5 __ 12 ft 2. 5 2 _ 3 ft

62 1 _ 5 ft

1. 4 2 _ 3 ft 2. 4 3 _ 4 ft

71 7 _ 8 ft

P 10•1

Name


Enrichment

10-1Bake SaleJill and her mother are going to prepare some foods for the school bake sale. They can choose to bake any of the foods in the chart. They can bake as many different things as they like. Use the chart to answer the questions.

1. Estimate to the nearest cup the amount of flour needed to make 4 pumpkin pies.

2. Jill wants to bake at least two different items. If Jill has 2 c of sugar and 3 c of flour, name a combination of baked goods that she could prepare.

3. Would it take more sugar and flour combined to make the cookies or the pumpkin pie?

4. Jill is thinking about baking 36 oatmeal cookies. How much flour would she and her mom need for this?

5. Estimate the amount of flour needed to bake everything listed in the chart.

Item Sugar Flour ServingsOatmeal cookies 1 1 _ 3 c 2 1 _ 4 c 18 cookies

Carrot cake 1 _ 2 c 2 _ 3 c 12 slices

Pumpkin pie 1 1 _ 2 c 1 3 _ 4 c 1 pie

Peach cobbler 1 _ 3 c 1 1 _ 2 c 8 pieces

E 10•1

1

1

1

1

18

18

1

1

1

18

18

18

18

18

18

18

18

18

18

118

18

18

18

18

18

18

18

18

18

18

18

88 1

28 left

1

1

15

1

15

15

15

15

45

15

15

Name


Reteaching

10-2Modeling Addition and Subtraction of Mixed NumbersExample 1: Draw a model to add 1 7 __ 8 + 2 3 __ 8 .

Step 1 Model each mixed number using fraction strips.

Step 2 Add the fractions. Regroup if you can.

7 __ 8

+ 3 __ 8

10 ___ 8 =1 2 __ 8

Step 3 Add the whole numbers to the regrouped fractions. Write the sum. Simplify, if possible.

So, 1 7 __ 8 + 2 3 __ 8 = 4 1 __ 4 .

Example 2: Draw a model to subtract 2 1 __ 5 – 1 2 __ 5 .

Step 1 Model the number you are

subtracting from, 2 1 __ 5 .

Express the part of the model that is not crossed out as a fraction or mixed number. So, 2 1 __ 5 –1 2 __ 5 = 4 __ 5 .

Use fraction strips to find each sum or difference. Simplify, if possible.

1. 3 1 __ 2 + 1 1 __ 2 2. 2 5 __ 8 + 4 3 __ 8 3. 5 2 __ 6 + 3 5 __ 6 4. 2 2 __ 4 + 6 3 __ 4

5. 6 1 __ 8 – 3 5 __ 8 6. 8 3 ___ 12 – 2 5 ___ 12 7. 12 1 __ 3 – 5 2 __ 3 8. 9 7 ___ 10 – 6 9 ___ 10

Step 2 Rename 2 1 __ 5 as 1 6 __ 5 . Cross out one

whole and 2 __ 5 to show subtracting 1 2 __ 5 .

R 10•2

1

18

18

18

18

18

18

18

1

18

18

18 1

1

15

15

15

15

15

15

Name


Practice

10-2Modeling Addition and Subtraction of Mixed NumbersFor 1 and 2, use each model to find each sum or difference.

1. 1 3 __ 8 + 1 7 __ 8 2. 3 1 __ 5 – 1 4 __ 5

Use fraction strips to find each sum or difference. Simplify, if possible.

3. 2 1 __ 3 + 1 2 __ 3 4. 3 5 __ 6 + 4 3 __ 6 5. 5 1 __ 4 – 1 2 __ 4 6. 12 3 __ 8 – 2 5 __ 8

7. 8 1 __ 6 – 3 5 __ 6 8. 4 6 ___ 10 + 5 7 ___ 10 9. 7 1 __ 9 – 4 2 __ 9 10. 6 2 __ 5 + 3 4 __ 5

11. 1 1 __ 6 + 3 5 __ 6 12. 2 4 __ 9 + 6 7 __ 9 13. 6 3 __ 5 – 4 3 __ 5 14. 5 1 __ 3 – 4 2 __ 3

15. Jerome’s rain gauge showed 13 9 __ 10 centimeters (cm) at the end of last month. At the end of this month, the rain gauge showed 15 3 __ 10 centimeters. How many more centimeters of rain fell this month?

A 29 2 ___ 10 cm B 15 3 ___ 10 cm C 2 4 ___ 10 cm D 1 4 ___ 10 cm

16. You are adding 3 2 _ 3 + 2 2 _ 3 using fraction strips. Explain how you rename the fraction part of the problem.

P 10•2

Name


Enrichment

10-2Prime TimeRemember that a prime number is a whole number greater than 1 that has exactly two factors, itself and 1.

Every number greater than 5 can be written as the sum of three prime numbers. For example, 13 can be written as the sum of 3 + 3 + 7.

Express the numbers below as the sum of three prime numbers. Write the primes in the squares.

1. ! ! ! 17

2. ! ! ! 10

3. ! ! ! 15

4. ! ! ! 21

5. ! ! ! 32

6. ! ! ! 30

7. ! ! ! 26

8. ! ! ! 50

E 10•2

34 ! 3 9

12 3

56 ! 2 10

122

1 1

1 1

1

Tirzah’s garden 23

4 yards

346 yards

Name


Reteaching

10-3Adding Mixed NumbersRandy talks on the telephone for 2 5 _ 6 hours, and then surfs the Internet for 3 3 _ 4 hours. How many hours does he spend on the two activities?

Step 1. Write equivalent fractions with the least common denominator. You can use fraction strips to show the equivalent fractions.

Step 2. Add the fraction part of the mixed number first. Then add the whole numbers.

9 ___ 12 + 10 ___ 12 = 19 ___ 12 3 + 2 = 5

19 ___ 12 + 5 = 5 19 ___ 12

Step 3. Simplify the sum if possible.

5 19 ___ 12 = 6 7 ___ 12 hours

So, 2 5 __ 6 + 3 3 __ 4 = 6 7 ___ 12 .

In 1 through 6, find each sum. Simplify if possible.

1. 2 5 __ 6 2. 1 3 __ 8 3. 5 2 __ 5

+ 3 1 __ 4 + 6 3 __ 4 + 4 1 __ 2

4. 10 1 __ 3 + 7 __ 9 = 5. 3 1 __ 4 + 6 2 __ 3 = 6. 1 5 __ 7 + 3 1 __ 2 =

7. Tirzah wants to put a fencearound her garden. She has 22 yards of fence material. Does she have enough to go all the way around the garden?

R 10•3

Name


Practice

10-3Adding Mixed NumbersIn 1 through 6, find each sum. Simplify, if possible. Estimate for reasonableness.

1. 7 2 __ 3 + 8 5 __ 6 2. 4 3 __ 4 + 2 2 __ 5

3. 11 9 ___ 10 + 3 1 ___ 20 4. 7 6 __ 7 + 5 2 __ 7

5. 5 8 __ 9 + 3 1 __ 2 6. 21 11 ___ 12 + 17 2 __ 3

7. Write two mixed numbers that have a sum of 3.

8. What is the total measure of an average man’s brain and heart in kilograms (kg)?

9. What is the total weight of an average woman’s brain and heart in pounds (lb)?

10. What is the sum of the measures of an average man’s brain and an average woman’s brain in kilograms?

11. Which is a good comparison of the estimated sum and the actual sum of 7 7 _ 8 + 2 11 __ 12 ?

A Estimated < actual C Actual > estimated

B Actual = estimated D Estimated > actual

12. Can the sum of two mixed numbers be equal to 2? Explain why or why not.

Vital Organ Measures

Average woman’s brain 1 3 ___ 10 kg 2 4 __ 5 lb

Average man’s brain 1 2 __ 5 kg 3 lb

Average human heart 3 ___ 10 kg 7 ___ 10 lb

P 10•3

Name


Enrichment

10-3Mixed Sums 1. Annie walked her dog 2 1 _ 4 miles from her house to the

dog park and 3 7 _ 8 miles around the park. Then she walked the same distance home. How many miles did she walk in all?

2. John’s full backpack weighs 5 1 _ 2 pounds, and Tyrone’s full backpack weighs 6 3 _ 5 pounds. What is the total weight of both boys’ backpacks?

3. Julie has two extension cords with lengths of 22 1 _ 6 feet and 26 3 _ 4 feet. How long a cord can she make by attaching them together?

4. Terry weighed his two cats at the veterinarian’s office. Boots weighed 12 9 __ 10 pounds, and Tiger weighed 13 1 _ 4 pounds. What is the total weight of both cats?

5. David ran 9 3 _ 4 miles on Saturday and 7 1 __ 10 miles on Sunday. How many miles did he run on the weekend?

6. Amanda’s library books weigh 4 3 _ 8 pounds, and her water bottle weighs 1 7 __ 16 pounds. What is the total weight of her books and water bottle?

E 10•3

13

13

16

161616

161616

111111111111

2312 4

612

125 3

65

11111

12

Name


Reteaching

10-4Subtracting Mixed NumbersThe Plainville Zoo has had elephants for 12 2 _ 3 years. The zoo has had zebras for 5 1 _ 2 years. How many years longer has the zoo had elephants?

For 1 through 4, find each difference. Simplify, if possible.Remember: You may have to rename a fraction in order to subtract.

1. 4 3 __ 5

− 2 1 __ 3

2. 5 6 __ 7

− 1 1 __ 2

3. 30

− 1 3 __ 4

4. 6 5 __ 6

− 5 1 __ 2

5. To find the difference of 7 − 3 5 __ 12 , how do you rename the 7?

6. Robyn ran 5 3 _ 4 miles last week. She ran 4 1 __ 10 miles this week. How many more miles did she run last week?

Step 1: Write equivalent fractions with the least common denominator. You can use fraction strips.

Step 2: Find the difference of 12 4 _ 6 − 5 3 _ 6 . Subtract the fractions. Then subtract the whole numbers. Simplify the difference if possible.

4 __ 6 − 3 __ 6 = 1 __ 6 12 − 5 = 7

So, 12 2 __ 3 − 5 1 __ 2 = 7 1 __ 6 years.

Example: Sometimes you may have to rename a fraction so you can subtract.Find the difference of 6 − 2 3 _ 8 .

6 rename 5 8 __ 8

− 2 3 __ 8 − 2 3 __ 8 3 5 __ 8

R 10•4

Name


Practice

10-4Subtracting Mixed NumbersFor 1 through 10, find each difference. Simplify, if possible.

1. 10 3 __ 4

− 7 1 __ 4

2. 7 3 __ 7

− 2 8 ___ 21

3. 3

− 2 2 __ 3

4. 17 7 __ 8

− 12 3 ___ 12

5. 9 5 __ 9 − 6 5 __ 6 6. 4 3 __ 4 − 2 2 __ 3

7. 6 1 __ 4 − 3 1 __ 3 8. 5 1 __ 5 − 3 7 __ 8

9. 8 2 __ 7 − 7 1 __ 3 10. 2 9 ___ 10 − 2 1 __ 3

The table shows the length and width of several kinds of bird eggs.

11. How much longer is the Canada goose egg than the raven egg?

12. How much wider is the turtledove egg than the robin egg?

13. Which is the difference of 21 15 __ 16 − 18 3 _ 4 ?

A 2 7 ___ 16 B 2 9 ___ 16 C 3 3 ___ 16 D 3 9 ___ 16

14. Explain why it is necessary to rename 4 1 _ 4 if you subtract 3 _ 4 from it.

Egg Sizes in Inches (in.)

Bird Length Width

Canada goose 3 2 __ 5 2 3 ___ 10

Robin 3 __ 4 3 __ 5

Turtledove 1 1 __ 5 9 ___ 10

Raven 1 9 ___ 10 1 3 ___ 10

P 10•4

Name


Enrichment

10-4Mixed DifferencesAnswer each question. Write your answers in simplest form.

1. Jenna bought a spool of ribbon for her craft projects. It contained 5 3 _ 4 meters of ribbon. She used 3 2 _ 5 meters of ribbon for a project. How many meters of ribbon does Jenna have left?

2. Brandon has a male puppy with a mass of 3 1 _ 2 kilograms and a female puppy with a mass of 2 1 _ 5 kilograms. How much greater is the male puppy’s mass than the female puppy’s?

3. Jeff’s sister drives 14 miles to her college, but his brother only drives 5 7 __ 10 miles to his college. How much farther does Jeff’s sister drive than his brother?

4. Montel bought a spool of string for making kites. It contained 10 3 __ 20 meters of string. He used 6 9 __ 10 meters of string for kites. How many meters of string does Montel have left?

5. Janet grew a pumpkin that weighs 13 1 _ 2 pounds and a melon that weighs 8 1 _ 4 pounds. How much heavier is the pumpkin than the melon?

6. Aidan roller-skated 3 1 __ 10 miles around the lake. Josh roller-skated 2 1 _ 2 miles around the park. How much farther did Aidan roller-skate than Josh?

E 10•4

Name


Reteaching

10-5More Adding and Subtracting Mixed NumbersYou can use what you know about adding and subtracting with mixed numbers when you simplify expressions with mixed numbers.

Simplify ( 4 1 __ 8 + 6 1 __ 4 ) – 2 1 __ 2 .

Step 1 Add the mixed numbers in parentheses first. Find a common denominator.

Step 2 Subtract 2 1 __ 2 from the sum you found. Find a common denominator.

You can’t subtract 4 __ 8 from 3 __ 8 .

Step 3 Rename if possible.

In 1 through 9, simplify each expression. Remember to rename mixed numbers if possible.

1. ( 12 4 __ 7 + 2 3 ___ 14 ) – 2 6 ___ 14 2. ( 5 1 __ 2 + 2 3 __ 4 ) – 3 1 __ 2 3. 10 5 ___ 16 – ( 5 1 __ 4 + 2 9 ___ 16 )

4. 6 __ 9 + 5 ___ 18 + 1 3 __ 6 5. 1 4 ___ 10 + 1 3 ___ 20 + 1 1 __ 5 6. ( 3 3 __ 8 – 1 1 __ 5 ) + 1 7 __ 8

7. 1 2 ___ 12 + 1 __ 6 + 7 3 __ 4 8. ( 1 5 __ 8 + 3 1 __ 4 ) – 1 20 ___ 24 9. 5 1 __ 4 + 7 3 ___ 20 + 1 3 __ 4

10. Suzy spent 6 7 __ 8 days working on her English paper, 3 1 __ 6 days doing her science project, and 1 1 __ 2 days studying for her math test. How many more days did Suzy spend on her English paper and math test than on her science project?

4 1 __ 8 + 6 1 __ 4

4 1 __ 8 + 6 2 __ 8 = 10 3 __ 8

10 3 __ 8 – 2 1 __ 2

10 3 __ 8 – 2 4 __ 8

9 11 ___ 8 – 2 4 __ 8 = 7 7 __ 8

R 10•5

Name


Practice

10-5More Adding and Subtracting Mixed NumbersIn 1 through 13, simplify each expression.

1. 2 1 __ 8 + 4 2 __ 3 2. 9 1 __ 6 – 3 2 __ 9 3. 8 1 __ 4 – 6 5 __ 8 4. 6 4 __ 5 + 5 3 ___ 10

5. ( 3 1 __ 2 + 8 2 __ 3 ) – 3 5 __ 6 6. ( 14 1 __ 6 – 4 5 __ 9 ) + 1 7 ___ 18 7. 7 7 ___ 12 + ( 3 1 __ 2 + 1 7 __ 8 ) 8. ( 10 1 __ 4 – 5 5 __ 8 ) – 1 3 __ 8 9. 100 3 ___ 10 – 60 2 __ 3 – 5 2 ___ 15 10. 25 3 __ 8 – ( 10 4 __ 5 + 5 7 __ 8 ) 11. ( 2 2 __ 3 + 4 5 __ 6 ) + 3 3 __ 8 12. ( 30 1 __ 9 + 4 1 __ 3 ) – 19 5 __ 6 13. 7 2 __ 3 + ( 5 1 __ 6 – 1 5 __ 9 ) 14. Which shows three mixed numbers that have sum of 10?

A 1 2 __ 3 + 3 5 ___ 12 + 4 3 __ 4 C 2 3 __ 8 + 5 1 __ 2 + 1 1 __ 4

B 3 1 __ 3 + 3 1 __ 4 + 3 5 ___ 12 D 5 1 __ 4 + 1 7 __ 8 + 3 7 __ 8

15. What is a reasonable estimate for the sum of 4 1 __ 8 + 3 2 __ 3 + 5 1 __ 2 ?

16. Veronica is buying cubed cheese from Mr. Sand’s deli. She asks for 1 3 __ 4 pounds. When Mr. Sand places some cheese in a container and weighs it, the scale shows 1 1 __ 4 pounds. The container weighs 1 ___ 16 pound. How many more pounds of cheese would Mr. Sand need to add to the scale to get the amount that Veronica asked for? Explain how you solved the problem.

P 10•5

Name


Enrichment

10-5Table TimeMake tables to solve. Write each answer in a complete sentence.

1. A train has 3 engines, 52 boxcars, and 1 caboose. At every stop, it picks up 8 more boxcars. How many total cars (engines, cars, and cabooses) will the train have after 5 stops?

2. Eileen likes to keep scrapbooks. She already has 4 scrapbooks filled with 40 pages each. If she fills 5 pages every month, how many months will it take her to fill up 2 more 40-page scrapbooks?

3. Phil’s Garage charges $50 for towing and $40 per hour to fix a car. Cliff’s Cars charges $60 for towing and $38 per hour to fix a car. After how many hours of working on a car will the cost of towing and fixing a car be the same at the two repair shops?

4. Dominic got a new video game. The first time he played the game he scored 80 points. After that, each time he played he increased his score by 60 points. How many times will he have to play before he scores 500 points?

5. A scientist is studying certain germs. She places 3 germs in a special solution that will help the germs grow. The number of germs doubles every hour. How many germs will there be after 8 hours?

6. Ed saved $50 one week. For the next 6 weeks, he saved $25 more than he saved the week before. How much did he save in all? One student solved this problem using the expression $50 ! 6($25) " $200. What error was made?

E 10•5

381 in.

34 in.

Name


Reteaching

10-6Problem Solving: Draw a Picture and Write an EquationA jeweler has a strand of gold wire that is 1 3 _ 8 inches. He cuts 3 _ 4 of an inch of wire to make a loop. How long is the remaining piece of wire?

From his house, Jason rode his bike 1 1 _ 3 miles to the post office. He then rode in the same direction to the park, which is 1 _ 4 of a mile from the post office. How far did Jason ride?

1. To the right, draw a picture to represent the problem to be solved. Let x = the distance Jason rode from his house to the park.

2. Write an equation that represents this distance. Then solve for x.

Read and Understand

What do you know? The length of the wire is 1 3 _ 8 inches.

The length he cuts off is 3 _ 4 of an inch.

What are you trying to find? The length of the wire that is left over.

Plan and Solve

Draw a picture for what you know.

Write an equation. Let x = the length of wire left over.

1 3 __ 8 – 3 __ 4 = x

Solve the problem. 1 3 __ 8 – 6 __ 8 = 11 ___ 8 – 6 __ 8 = 5 __ 8

Write the answer in a sentence. The remaining wire is 5 _ 8 inch long.

Look Back and Check

Is your answer correct? Yes, 3 __ 4 + 5 __ 8 = 6 __ 8 + 5 __ 8 = 11 ___ 8 = 1 3 __ 8

R 10•6

Name


Practice

10-6Problem Solving: Draw a Picture and Write an EquationDraw a picture and write an equation. Then solve.

3. Stewart draws a triangle, and each side is 2 1 _ 6 inches long. Judith draws a square, and each side is 1 5 _ 8 inches long. Which figure has the greater perimeter, the triangle or the square?

4. Cristoff practices playing his guitar for 1 1 _ 2 hours each weekday. He practices this amount of time plus an additional 1 1 _ 2 hours on Sundays. Let x = the number of hours Cristoff practices on Sundays. Draw a picture and write an equation and solve to find the number of hours he practices on Sundays.

5. Which of these fractions, when added to 2 1 _ 3 , will give you a sum greater than six?

A 3 1 __ 2 B 3 5 ___ 12 C 3 7 ___ 12 D 3 3 __ 4

6. Dennis says that 1 1 _ 2 , 1 2 _ 4 , and 1 3 _ 6 are all equivalent. Is he correct? Draw a picture and explain your answer.

1. Mr. Flanders drives 1 2 _ 3 miles to school and 1 2 _ 3 miles home each day. He also drives an extra 2 2 _ 7 miles to go to the gym. How many miles does he drive in one day?

2. Alison is making a 16-inch necklace. The first 4 1 _ 4 inches are filled with red beads and 8 3 _ 8 inches are filled with blue beads. The rest has white beads. How many inches are filled with white beads?

P 10•6

Name


Enrichment

10-6Snack Mix Max and his friends are comparingrecipes for their favorite outdoor snacks. Use the table to answer the questions.

1. Estimate to the nearest cup the amount of fruit needed to make 16 servings of Pam’s hiking medley.

2. Max is making 12 servings of his snack. How many total cups of fruit and nuts will he make?

3. How many total cups of fruit and nuts are in Pam’s and Sue’s recipes?

4. Which recipe uses the most ingredients?

5. Tom combines Max’s and Pam’s recipes. How many cups of nuts will he use?

6. Estimate the amount of nuts needed to make all 4 snacks listed in the table.

Item Nuts Fruit ServingsMax’s trail mix 2 2 _ 3 c 1 c 6

Tom’s trek snack 3 _ 4 c 2 _ 3 c 3

Pam’s hiking medley 3 _ 4 c 2 2 _ 3 c 8

Sue’s energy munchies 2 1 _ 3 c 1 1 _ 2 c 9

E 10•6

0cog reteaching estimating sums and differences of … 10... · reteaching 10-1 estimating sums and...

Documents