UNIT 6 OPERATIONS WITH FRACTIONS - … 6 – Media Lesson 3 Problem 2 MEDIA EXAMPLE – Adding Fractions with Unlike Denominators Use the diagrams given to represent the values in

Unit 6 – Media Lesson

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UNIT 6 – OPERATIONS WITH FRACTIONS

INTRODUCTION

In this Unit, we will use the multiple meanings and representations of fractions that we studied in the previous

unit to develop understanding and processes for performing operations with fractions. You are likely familiar

with algorithms for computing fraction operations. Please accept the challenge of really thinking through the

meaning of operations and the different contexts of fractions so you understand why we perform operations as

we do. If you accept this challenge, this unit will help you see the meaning of these processes not just the steps.

The table below shows the learning objectives that are the achievement goal for this unit. Read through them

carefully now to gain initial exposure to the terms and concept names for the lesson. Refer back to the list at the

end of the lesson to see if you can perform each objective.

Learning Objective Media Examples You Try

Add fractions with like denominators 1 4

Add fractions with unlike denominators 2 4

Add improper fractions and mixed numbers 3 4

Subtract fractions with like denominators 5 8

Subtract fractions with unlike denominators 6 8

Subtract improper fractions and mixed numbers 7 8

Multiply a unit fraction times a whole number 9 11

Multiply a composite fraction times a whole number 10 11

Multiply a whole number times a fraction 12 15

Multiply two fractions 13 15

Multiply mixed numbers 14 15

Divide a whole number by a fraction 16 19

Divide fraction with common denominators 17 19

Divide fraction with uncommon denominators 18 19

Divide mixed numbers 20 21

Perform +, −, ×, ÷ on signed fraction 22 24

Use the order of operations with signed fractions 23 24


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UNIT 6 – MEDIA LESSON

SECTION 6.1: ADDING FRACTIONS In this section, we will learn to visualize the addition of fractions using an area model and number lines. Recall

that the operation of addition is combining two amounts or adding one amount on to another. We will achieve

this with fractions by ensuring we have a common unit fraction (common denominator) for the numbers we are

combining. We can then add the number of copies of each fraction while retaining the common denominator.

Problem 1 MEDIA EXAMPLE – Adding Fractions with Like Denominators

Use the diagrams given to represent the values in the addition problem and find the sum. Then perform the

operation and represent the sum using the symbolic representation of the algorithm.

1. Jon had 1

5 of a pepperoni pizza and

2

5of a mushroom pizza. How much of one whole pizza did Jon

have?

1

5 is _____ copies of _______

2

5 is _____ copies of _______

Combined, we have a total of _____ copies of _______ or the fraction__________.

Symbolic Representation of Algorithm:

2. Christianne sprinted 3

8 of a mile and then jogged another

7

8 of a mile. How far did she run in total?

3

8 is _____ copies of _______

7

8 is _____ copies of _______


Simplified result as an improper fraction:________ Simplified result as a mixed number:__________



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Problem 2 MEDIA EXAMPLE – Adding Fractions with Unlike Denominators



a) 1 1

3 2

Step 1: Rewrite the given fractions with a common denominator.

1

3 is equivalent to ________

1

2 is equivalent to__________

Step 2: Rewrite the equivalent fractions with the common denominator using copies of language using a

common unit fraction.

1

3: _____ is _____ copies of _______

1

2:_______ is _____ copies of _______

Step 3: Combine (add).


Step 4: Simplify if necessary.


b) 1 3

4 8



4

c) 5 4

6 9


Problem 3 MEDIA EXAMPLE – Adding Improper Fractions and Mixed Numbers



a) 7 5

4 4



5

b) 2 1

4 33 2

Step 1: Rewrite the fractional parts of the mixed numbers with a common denominator



2

3: _____ is _____ copies of _______

1

2:_______ is _____ copies of _______

Step 3: Combine the fractional parts and the whole number parts.

Fractional Parts: Combined, we have a total of _____ copies of _______ or the fraction__________.

Whole Numbers plus fractional parts: Combined we have a total of ___________________________



c) 3 9

25 5



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Problem 4 YOU TRY - Adding Numbers with Fractions



a) 2 1

3 4

Step 1: Rewrite the given fractions with a common denominator.

2

3 is equivalent to ________

1

4 is equivalent to__________



2

3: _____ is _____ copies of _______

1

4:_______ is _____ copies of _______

Step 3: Combine (add).




b) 1 3

14 8



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SECTION 6.2: SUBTRACTING FRACTIONS In this section, we will learn to visualize subtracting fractions using an area model and number lines. We will

investigate this idea using our two models of subtraction; subtraction as taking away a part of a whole, and

subtraction as comparing two quantities.

Problem 5 MEDIA EXAMPLE – Subtracting Fractions with Like Denominators

Use the diagrams given to represent the values in the subtraction problem and find the difference. Then

perform the operation and represent the difference using the symbolic representation of the algorithm.

a) The day after Thanksgiving, there was 3

5 of a pumpkin pie remaining. Lara ate

1

5 of a pie for

breakfast. How much pie is leftover now?


b) Billy ran 5

8 of a mile. Roberta ran

11

8 of a mile. How much further did Roberta run?



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Problem 6 MEDIA EXAMPLE – Subtracting Fractions with Unlike Denominators



a) 1 1

2 3


b) 5 2

6 3


c) Perform the operation and represent the difference using the symbolic representation of the algorithm.

7 3

12 8



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Problem 7 MEDIA EXAMPLE – Subtracting Improper Fractions and Mixed Numbers



a) 8 6

5 5


b) 1 1

4 33 2


c) 3 11

25 10



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Problem 8 YOU TRY - Subtracting Numbers with Fractions



a) Michael lives 5

18

of a mile from school. Rachel lives 7

8 of a mile from school. How much closer to

school does Rachel live?


b) 1 9

38 8


c) Barney had 7

8 of his weekly salary left after paying his bills. He then spent

1

4 of his weekly salary on

a weekend trip. What fraction of his weekly salary remains?



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SECTION 6.3: MULTIPLYING FRACTIONS In this section, we will examine multiplying fractions using the idea that a b or a b is equivalent to a copies

of b.

Problem 9 MEDIA EXAMPLE – Multiplying a Unit Fraction and a Whole Number

Use the diagrams given to represent the multiplication problem and find the product. Then perform the

operation and represent the product using the symbolic representation of the algorithm.

a) Ebony ate 1

3 of a pizza. The pizza had 12 slices. How many slices did she eat?


b) Kimber was 20 miles from home. She travelled 1

5 of this distance

while listening to her favorite song. How many miles did she travel while listening to her favorite song?


c) Logan has a 5 gallon bucket. He fills it 1

4 of the way to the top. How much water is in the bucket?



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Problem 10 MEDIA EXAMPLE – Multiplying a Composite Fraction and a Whole Number



a) Ebony ate 2

3 of a pizza. The pizza had 12 slices. How many slices did she eat?


b) Kimber was 20 miles from home. She travelled 3

5 of this distance while listening to her favorite song.

How many miles did she travel while listening to her favorite song?


c) Logan has a 5 gallon bucket. He fills it 3

4 of the way to the top. How much water is in the bucket?



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Problem 11 YOU TRY – Multiplying Unit or Composite Fractions with Whole Numbers



a) Carese was running a 10 kilometer race. She ran 1

5 of the race in 12 minutes. How many kilometers

did she run in 12 minutes?


b) Casey’s gas tank holds 12 gallons. The gas gage says it is 2

3 full. How many gallons of gas are in her

tank?



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Problem 12 MEDIA EXAMPLE – Multiplying Whole Numbers and Fractions



a) Ashley has 3 packs of cupcakes. There are 4 cupcakes per pack. How many cupcakes does Ashley

have?

Picture:

Copies of Language:


b) Anderson jogged around a lake 5 times this week. The distance around the lake is 1

3 of a mile. How

far did Anderson jog in total over the week?

Picture:

Copies of Language:


c) Knia is having a dinner party with a total of 7 people. She bought 2

5 of a pound of cold cuts per person.

How many pounds of cold cuts did she buy in total?

Picture:

Copies of Language:



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Problem 13 MEDIA EXAMPLE – Multiplying Two Fractions



a) Yesterday, Cameron walked 1

3 of a mile to school. Today, Cameron’s friend picked him up after he

had walked 1

2 of the way to school. How far to Cameron walk today?

Picture:

Copies of Language:


b) Jassey bought a rectangular piece of land to grow vegetables. The land is 2

3 of a mile long and

3

4 of a

mile wide. How many square miles of land did Jassey buy?

Picture:

Copies of Language:


c) Ray made a tray of brownies. He ate 1

5 of the tray after they had cooled. The next day, he ate

3

4 of

what was left over in the tray. How much of the whole tray did Ray eat the next day?

Picture:

Copies of Language:



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Problem 14 MEDIA EXAMPLE – Multiplying Mixed Numbers

Create a diagram to represent the multiplication problem and find the product. Then perform the operation and

represent the product using the symbolic representation of the algorithm.

a) According to the Bureau of Labor Statistics, the buying power of the dollar is 4

15

times larger in 2016

when compared to 1991. Determine the comparable buying power in 2016 of $10 in 1991.

Picture:

Copies of Language:


b) J’Von bought a rectangular piece of land in Alaska. The land was 3

24

miles wide and 5

38

miles long.

How many square miles of land did he buy?

Picture:

Copies of Language:



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Problem 15 YOU TRY – Multiplying Mixed Numbers, Fractions, and Whole Numbers

Create a diagram to represent the multiplication problem and find the product. Then perform the operation and

represent the product using the symbolic representation of the algorithm.

a) Rick walked around a track 6 times this month. The distance around the track is 3

4 of a mile. How far

did Rick walk in total over the month?

Picture:

Copies of Language:


b) Kate has a rectangular piece a land to raise horses. The land is 2

5 of a mile long and

5

8 of a mile wide.

How many square miles of land does Kate own?

Picture:

Copies of Language:


c) Javier makes 1

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times as much an hour as when he first started his job. If he made $9 an hour when

he first started, how much does Javier make now?

Picture:

Copies of Language:



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SECTION 6.4: DIVIDING FRACTIONS In this section, we will learn to visualize dividing fractions using an area model and number lines. We will

investigate this idea using two models of division; dividing as partitioning the dividend into a known number of

copies, and dividing as determining how many copies of a given size are in the dividend.

In general, we do not usually find a common denominator to divide fractions. However, we will begin with

examples by finding common denominators to illustrate a general process for dividing fractions.

Problem 16 MEDIA EXAMPLE – Dividing Whole Numbers by Fractions

Use the diagrams given to represent the division problem and find the quotient.

a) Tia made 3 cakes for her guests. If each guest receives 1

5 of a cake, how many guests can Tia serve?

Picture:

Copies of Language:

Symbolic Representation:

b) Elaine ran 6 miles this month. If she ran 3

5 of a mile every day she ran, how many days did Elaine run?

Picture:

Copies of Language:



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Problem 17 MEDIA EXAMPLE – Dividing Fractions Using Common Denominators

Use the diagrams given to represent the division problem and find the quotient.

a) 9 1

2 2

Copies of Language:


b) 8 2

5 5

Copies of Language:


c) 18 3

4 2

Copies of Language:


d) What patterns do you observe in problems a through c?


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Results: Based on the previous examples, we can divide two fractions by dividing their numerators (left to

right) and their denominators (left to right) just like we multiply the numerators and denominators of fractions

to multiply them.

The last two problems illustrate this fact below.

8 2 8 2 4 18 3 18 3 6

4 35 5 5 5 1 4 2 4 2 2

or

Problem: The previous two problems worked out nicely because dividing the numerators and denominators

resulted in an integer in both the resulting numerator and denominator.

Consider this problem that doesn’t work out as nicely.

77 2 7 2 2

55 3 5 33

Using the division method actually made the problem worse! We want the result as a fraction of one integer

over another integer, not a fraction with fractions as numerators and denominators.

Solution: We will find a common denominator for the two fractions we are dividing and perform the division

like we did for the first problems. Then we will check for patterns to simplify the process.

1. Original Problem 7 2

5 3

2. Rewrite the two fractions with a common denominator of 15.

7 3 2 5 21 10

5 3 3 5 15 15

3. Divide the numerators and the denominators.

21 10 21 10

21 1015 15 1

4. Rewrite the division in the numerator as a fraction. 21

21 1010

Pattern: Let’s look at the original problem and the final answer.

Final Solution: 7 2 21

5 3 10 Comparable Method:

7 2 7 3 21

5 3 5 2 10

Rule: To divide two fractions, multiply the first fraction by the reciprocal of the second fraction.

(Note: The reciprocal of a fraction a

b is

b

a. In this example, the reciprocal of

2

3 is

3

2.


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Problem 18 MEDIA EXAMPLE – Dividing Fractions with Unlike Denominators

Divide the fractions by multiplying the first fraction by the reciprocal of the second fraction. Simplify your

result if necessary. Write any answers greater than 1 as both an improper fraction and a mixed number.

a) 12 3

13 5 b)

18 3

5 10 c)

2 12

3 5

Problem 19 YOU TRY – Dividing Fractions

a) A snail crawled 5 meters this week. If he crawled 5

3 of a meter every day he crawled, how many days

did he crawl?

Picture:

Copies of Language:


b) Divide the fractions by dividing the numerators and the denominators. Simplify your result if necessary.

i. 18 9

11 11 ii.

12 3

14 7 iii.

15 17

8 8

c) Divide the fractions by multiplying the first fraction by the reciprocal of the second fraction. Simplify

your result if necessary.

i. 19 5

3 2 ii.

16 4

3 6 iii.

4 8

7 3


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Problem 20 MEDIA EXAMPLE – Dividing Mixed Numbers

Rewrite any mixed numbers as improper fractions. Then perform the division by multiplying the first fraction

by the reciprocal of the second fraction. Simplify your result if necessary and rewrite your result as a mixed

number when possible.

a) 1 1

22 2 b)

4 33 2

5 8 c)

2 31 3

5 5

Problem 21 YOU TRY – Dividing Mixed Numbers

Rewrite any mixed numbers as improper fractions. Then perform the division by multiplying the first fraction

by the reciprocal of the second fraction. Simplify your result if necessary and rewrite your result as a mixed

number when possible.

a) 2 2

53 3 b)

3 16 3

5 6 c)

2 32 4

5 5


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SECTION 6.5: SIGNED FRACTIONS AND THE ORDER OF OPERATIONS In this section, we will use our knowledge of operations on integers and the order of operations to perform

operations with signed fractions and the order of operations with fractions.

Problem 22 MEDIA EXAMPLE – Operations on Signed Fractions

Perform the indicated operations on the fractions and/or mixed numbers using your knowledge of signed

numbers.

a) 3 2

7 3 b)

2 7

5 5 c)

2 32 4

5 5

d) 3 1

5 2

e) 13 5

8 8

f) 4 2

4 85 3

Problem 23 MEDIA EXAMPLE – The Order of Operations and Signed Fractions


numbers and the order of operations.

a) 1 7 8

2 9 9

b) 9 3 7

5 5 2 c)

25 1 3

3 2 4


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Problem 24 YOU TRY – The Order of Operations and Signed Fractions


numbers and the order of operations.

a) 5 1 1

4 3 4

b) 9 1 3

4 3 2

c)

22 2

43 3

Documents

UNIT 6 OPERATIONS WITH FRACTIONS - … 6 – Media Lesson 3 Problem 2 MEDIA EXAMPLE – Adding Fractions with Unlike Denominators Use the diagrams given to represent the values in