ar chapter 6 student text

8/3/2019 AR Chapter 6 Student Text

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CHAPTER 6 FractionsFractions | 247

Chapter

6

Fractions

Chapter Objectives

Define fractions using the number line. Describe quantities and verbal information using fractions and units. Multiply and divide fractions. Understand the concept of equivalent fractions. Order and compare fractions.

Lessons

Lesson 26

Numbers in Between

Whole Numbers

Lesson 27

Fractions of Units

Lesson 28

Commutative Propertyof Multiplication

Lesson 29

Fractions and Area

Lesson 30

Equivalent Fractions

Lesson 31

Dividing Fractions

Lesson 32

Equivalent Rates

ARE YOU READY?

Evaluate each expression.

1. 15 5

__

7

2.3 jumps_______

4 seconds 108 seconds

3. 40 boxes 5 boxes_______4 statues

Solve ford.

4. 724 + d = 903

5. You earned $5 in the first 2 hours, then $4 in the next 3 hours. If you

continue earning money at this average rate, how long will it take to earn

$45?


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248 | CHAPTER 6 Fractions

LESSON 26 Numbers in BetweenWhole Numbers

Objectives Use the concept of a whole multiplied by a ratio to define fractions on the number

line.

Read and write fractions.

Express whole numbers as fractions.

Compare fractions on a number line using symbols >, )

inequality

less-than sign (


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LESSON 26 Numbers in Between Whole NumbersNumbers in Between Whole Numbers | 249

LESSON 26 Numbers in betweenWhole Numbers

Concepts and Skills: RN.1, RN.4,

RN.10

Up until this lesson we have only discussed whole numbers. These are the points on

the number line that are multiples of 1. But what about the points on the number line

between whole numbers? We will begin discussing those numbers in this lesson.

In lesson 2 we defined a whole number as a multiple of a whole. In other words, all

whole numbers can be expressed as the repeated addition of 1. For example, 3 is a

whole number because we can express it as 1+1+1:

3 = 1 + 1 + 1

Using multiplication, this equation becomes:

3 = 3 1

So far, we have named points on the number line by multiplying 1 by a whole

number. In this lesson we will start to name more points by multiplying 1 by a

ratio:

Fractionson the

Number Line

The above equation means that we will use the ratioa_b to name the point specified

bya_b 1. Here the variables a and b represent whole numbers (except b cant be

zero). The numbera_b belongs to what are called rational numbers. The word

rational comes from the word ratio.

Lets look at an actual example by finding the value of the following expression:

To find the value, we first express 1 using 4 equal jumps:

After multiplying by the3_4 ratio, we get 3 of those jumps, and we name the resulting

point3_4:

In other words, to find the point 3_4, we break a whole into 4 equal parts then wekeep 3 of those parts. You might recall that another word for break is fracture

(such as a fractured bone). For this reason, rational numbers written this way are

often called fractions. The fraction3_4 is typically pronounced three-fourths

because when you break a whole into 4 equal parts, each part is called a fourth.


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250 | CHAPTER 6 FractionsFractions

Here are the numbers one-fourth, two-fourths, three-fourths and a whole on

the number line:

Notice how we get to3_4using 3 jumps of + 1_4. This leads to the following equation:

This is repeated addition of1_4. We can simplify this by expressing the repeated

addition as multiplication:

3

__

4 = 3

1

__

4

Lets look at another example.

Where is5_3on the number line?

We find the location5_3by evaluating the following expression:

To find the value, we first express 1 using 3 equal jumps:

Here we have expressed 1 as three jumps of + 1_3. After multiplying by a 5_3ratio, weturn the 3 jumps into 5 jumps:

We call this number five-thirds, and we can express it as repeated addition of1_3:

Instead of repeated addition, we can simplify the above using multiplication:


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Check forUnderstanding

1. The jumps on each line are equal. Express each variable as a fraction:The jumps on each line are equal. Express each variable as a fraction:

a.

b.

c.

2. Express the variable as a fraction:Express the variable as a fraction:

a.1

__

5+1

__

5+1

__

5 +1

__

5 = f

b. 3

1

__

9

= g

3. Express7

__

3 using repeated addition of1_3

4. Express5

__

6 as repeated addition, then express the repeated addition as

multiplication.

Vocabulary There is a lot of technical vocabulary associated with fractions. Lets talk about acouple of useful ones. We have been expressing fractions as a ratio between two

whole numbers. The two whole numbers have different roles, and we use different

names to refer to them:

The number on the bottom is called the denominator. It tells us how many equalparts we are using to express a whole. The top number is called the numerator. It

tells us how many of the equal parts we have.

For example, in the fraction5_3, the 3 is the denominator and the 5 is the numerator.

The 3 tells us that we broke a whole into 3 equal parts. The 5 tells us that we have

5 of those equal parts.


5. In the fraction2_7, what is the 2 called? What does it mean? What is the 7

called? What does the 7 mean?

6. Both number lines show equal jumps. Which shows n =8_5?

a. b.


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One is aFraction Too

Its important to point out that rational numbers include whole numbers. To see

why this is, lets start with the number 1, which we call a whole. We can represent

1 as a fraction where the numerator equals the denominator. In other words, the

following is an identity:

Here the denominator ofb means that it takes b parts to make a whole. Since we

have that many parts (as indicated by the numerator), we have a whole. Lets

look at3_3as an example. To find where 3_3is located, we first express a whole using

3 equal jumps:

We can express a whole as 3 jumps of + 1_3

:

To make3_3, we also use 3 jumps of 1_3:

Therefore 3_3 is the same number as 1:is the same number as 1:is the same number as 1:

Whole Numbersare Fractions

Too

All the other whole numbers can be expressed as fractions as well. For example,lets find the value of the following expression:

The denominator of 1 means we first express a whole using a single jump:

After multiplying by 4_1 we get 4 of those jumps:we get 4 of those jumps:we get 4 of those jumps:


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This is 1+1+1+1, which is the definition of 4. In other words, the fraction 4_1is thesame as the whole number 4. These are two names for the same number:

All whole numbers can be represented as fractions in this way:

The above equation is an identity. This means that all whole numbers are rational

numbers.


7. Which of these fractions are whole numbers? Explain why.

a.8

__

1 b.5

__

5 c.7

__

3 d.1

__

4 e.d__d f. 0__5

8. Write each fraction as a whole number. (Assume the variables are wholenumbers):

a.5

__

1 b.6

__

6 c.n__1 d. h__h e. 0__k

ComparingRational

Numbers

All rational numbers are points on the number line. For example, here are rational

numbers 0, 1_2and 1:

Because 1 is to the right of 1_2, we say that 1 is greater than 1_2. To indicate this usingsymbols, we write:

1 >1

__

2

We call > the greater-than sign.

We can also say that 1_2 is less than 1, and we indicate this using


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To find if one number is greater than, less than, or equal to another number, we

simply find the location of both numbers on the number line. If both numbers are

located at exactly the same point, then they are equal. Otherwise, the number

farther to the right (in the positive direction) is greater than the other number. Letspractice this.

Which number is greater,2__3 or 5__

3?

To answer this question, lets locate 2_3 andandand 5__3 on the number line:on the number line:on the number line:

As we can see,5_3 is greater than 2_3. This makes sense because 5 jumps of + 1_3 is

obviously greater than 2 jumps of +1_3. We can represent this relationship as

follows:

We cant call the above an equation because the two sides are not equal. Instead we

call it an inequality (which means not equal). Lets look at another example:

Which is greater,1__2 or 1__

3?

Lets plot both numbers on the number line. We find 1_2by breaking a whole into 2by breaking a whole into 2by breaking a whole into 2equal parts:

Now we find 1_3by breaking a whole into 3 equal parts:by breaking a whole into 3 equal parts:by breaking a whole into 3 equal parts:

As we can see, 1_2 is greater thanis greater thanis greater than 1_3. In symbols, we indicate this with the followinginequality:

Since each 1_2jump is bigger than a 1_3jump, we can also conclude that the followingjump, we can also conclude that the followingjump, we can also conclude that the followingis true in general (as long as n > 0):


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In other words, n jumps of 1_2 is greater thanis greater thanis greater than n jumps of 1_3. This is true as long as nis greater than 0. Ifn equals zero, we have an equation:

Zero jumps of any amount is zero.


9. Earlier we showed that5_3is greater than 2_3. Generalize this for comparing

any fractions with the same denominator.

10. Use the symbols , or = to make the statement true.

a. b. c. d.

11. Rewrite each inequality using the < sign.

a.1

__

5 >>>1

__

7 b.2

__

2 >>>1

__

9 c.9

__

8 >>>2

__

2 d.3

__16 >>> 2__16

Problem Set 1. Copy the sentences below. Then fill in the blanks with the correct words tomake the statement true.

A fraction represents a _____________ on the number line. The __________

____ of the fraction tells you how many __________ jumps express a whole.

The _____________ of a fraction tells you how many of those jumps you use

to get to the ____________ on the number line.

Each number line below shows equal jumps. Find the fractions represented by

points a, b and c on each number line.

2.

3.

4.

5.


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What is the fraction represented byp on each number line?

6. 7.

What fraction does the expression equal?

8.1

__

2+

1

__

2+

1

__

2+

1

__

2+

1

__

2 9.

1

__

3+

1

__

3

10.1

__

4+

1

__

4+

1

__

4+

1

__

4+

1

__

4+

1

__

4+

1

__

4 11.1

__

c+ 1__c+ 1__c+ 1__c

A fraction that has a numerator of 1 is called a unit fraction (see the

chapter review for an explanation). Express each of the fractions below as

repeated addition of a unit fraction. Then express the repeated addition as

multiplication.

12.4

__

3 13.3

__

5 14.5

__

2

Find the value of the variable in each equation.

15. 16. 17.

18. 19. 20.

Use the symbols , or = to make each statement true.

21. 22. 23. 24.

25. 26. 27. 28.

29. 30.


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Challenge Problems

1. Find the value ofh in the following equation:

1

__

h + 1__

h + 1__

h = 5__5

2. Estimate which point has the value7

__

6:

Multiple Choice Practice

1. All fractions are:

whole numbers less than a whole

greater than a whole rational numbers

2. What fraction is the best estimate of the pointp on the number line below?

1

__

2

1

__

54

__

56

__

5

3. What fraction is the best estimate of the pointp on the number line below?

1

__

21

__

54

__

56

__

5

4. Which fraction is greater than a whole?

15__16 3__4 8__7 6__6

5. Which fraction is less than a whole?

20__21 20__20 20__19 6__5

6. Which fraction is equal to a whole?

20__21 20__20 20__19 6__5

7. Which fraction is greater than3

__

5?

3

__

72

__

53

__

41

__

2


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Math Journal Questions

1. Explain how to find where8_5 is located on the number line. Draw pictures

to support your explanation. Now explain how to find where the generalized

fractionk__w is located on the number line.

2. Explain why a fraction cant have a denominator of zero. Use the meaning of

the denominator of a fraction to explain.

3. The following two questions involve comparing fractions.

a. What is the general rule for the values m andgthat will make the following

statement is true?

m__7 > g__7

b. What is the general rule for the whole numbers m andgthat will make this

next statement true?

7

__

m > 7__g

Find the Errors

A student made 3 mistakes below. Find and correct each mistake.

1.

2. 3. 4.

LOOKING BACK

Vocabulary: denominator, equal (=), inequality, greater than (>),

fractions, less than ( mean? How do I know which one

to use?


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LESSON 27 Fractions of Units | 259

LESSON 27 Fractions of Units

Objectives

Use symbols to express verbal and graphical information about fractional

quantities.

Understand and apply the concept of a whole and its parts.

Interpret fractions of a group as a quantity multiplied by a fraction.

Interpret fractions of a whole as a whole multiplied by a fraction.

Explain equivalent fractions in the context of units (such as units of area).

Concepts and Skills

RN.2 Understand fractions as parts of a set and as parts of a whole. Write the

fraction represented by a drawing of parts of a figure; represent a given

fraction with a drawing.RN.3 Know that taking a fraction of a unit or a quantity is the same as

multiplying the fraction by the unit (e.g., 3_4of 2 pizzas is the same as

3_4 2 pizzas).SN.2 Translate verbal descriptions into mathematical expressions.

Remember from Before

What is the definition of a fraction on the number line?

How do you multiply a whole number by a fraction?

Get Your Brain in Gear

1. Use mental math to find the value of each expression.

a.5

__

4 8 b.

5

__

4 24

c.3

__

5 30 d.

1

__

2 18

2. Estimate the fractionn__5 that best represents the point p on the number line

below.

Vocabulary

areaequal parts

fractions


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LESSON 27 Fractions of Units


SN.2

In the previous lesson we introduced rational numbers and represented them as

fractions. Earlier, in lesson 6, we used the concept of a unit to describe real-world

quantities with whole numbers. Here we will extend the use of units to describe

quantities with fractions.

Lets first review the concept of a unit. Units are a way of using the number line

to describe quantities in the real world. A unit is a quantity that corresponds to a

whole. In other words, a unit is represented by a jump of +1 on the number line. For

example, lets define the following whole pizza as a unit:

With this unit we can now use numbers to represent any quantity of pizza. What

would it mean to have 3 of the above pizzas?To have 3 of something means to have 3 that thing. In this case, 3 of onepizza means 31 pizzas:

Instead of saying 3 of a pizza, we usually simply say 3 pizzas. These are two

ways of expressing the same quantity.

Along this same line of reasoning, what would the following mean?

5_4of a pizza

Just as we said before, this means:

5

__

4 1 pizzas

We find the value of this expression by first breaking a whole pizza into 4 equal

parts:

After multiplying by5_4 we end up with 5 of those parts:we end up with 5 of those parts:we end up with 5 of those parts:

We could call this 5_4pizzas, but when fractions are involved, we usually use the

word of and say it the longer way: 5_4of a pizza, or "five-fourths of a pizza".


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LESSON 27 Fractions of UnitsFractions of Units | 261


1. Below we broke a whole pizza into 2 equal parts and then made 7 of those

parts. What fraction of a pizza is this?

2. Below we broke a whole pizza into 8 equal parts and kept 5 of those parts.What fraction of a pizza is this?

3. What fraction of a pizza best describes each amount of pizza below?

Lets look at another example, this time using units of people. What does 2_3of 12

people mean?

Translating this into symbols, 2_3of 12 people becomes:2

__

3 12 people

To find the value of the above expression, we first break the 12 people into 3 equal

groups:

Here each group has 4 people in it. After multiplying by 2_3, we end up with 2 ofthese groups:

This is a total of 8 people. From this we conclude that2_3of 12 people is 8 people:

2

__

3 12 people = 8 people

Lets look at a different example. Here is a box containing 6 circles:


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Lets color 1_3 of the circles in the box blue. Since there are 6 circles total, 1_3ofthem means:

1

__

3 6 circles6 circles

To find this amount, we separate the circles into 3 equal groups. By equal, we

mean each group has the same number of circles. Here is one of many ways to form

these 3 equal groups:

Now there are 3 equal groups, and each group has 2 circles. Because we are

multiplying by 1_3, it means we color 1 of the groups blue:

Now 1_3 of the circles in the box are blue. We colored 2 circles because:1

__

3 6 = 2


4. Which picture shows3_4 of the circles in the box colored blue?

5. Draw a diagram for each expression:

a. 4_3of 6 peopleb.

5_2of a circlec.

3_4of 8 pencilsd. 2_3of the letters in the word fractions

More Pizza Lets go back to pizzas again. What would 3_4 of 2 pizzas mean?Translating this into symbols,

3_4of 2 pizzas becomes:3

__

4 2 pizzas


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To find the value of this expression, we first take 2 pizzas and break them into 4

equal parts:

Notice that each of these parts is equal to 1_2 of a pizza.After multiplying by

3_4, we end up with 3 of these parts:

Here we have 3 halves of a pizza, which we can write as:

3

__

2of a pizza

From this we conclude that3_4of 2 pizzas is equal to 3_2 of a pizza:

3

__

4

2 pizzas =3

__

2pizza


6. Draw a diagram for each expression:

a.7

__

8of 2 chocolate bars

b.5

__

2of 3 cakes

c.2_3of 6 circles

Units of Area As we have discussed before, area is described using unit squares. These are squareswhere the width and height are each 1 unit in length:

With unit squares we can find the area of any shape. Lets take the following shape

as an example:


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To find the area, we fill the shape with non-overlapping unit squares:

From this we say that the above shape has an area of 3. Extending this reasoning

beyond whole numbers, how would we describe the area of the following

triangle?

In order to cover the above triangle, we break a unit square into 2 equal parts, along

the diagonal:

Now we use one of these parts to fill our triangle:

As a result, we say that the triangle has an area of1_2of a unit square. We can also

simply say an area of1_2 because its understood that we measure area using unit

squares.

The above isnt the only way we can split a square into 2 equal areas. For example, 1_2of each of the following squares is shaded gray:

Assuming the following unit square is broken into equal areas, how much of itis shaded gray?


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We have 9 equal parts and 2 are shaded gray. This means that 2_9of the area is shadedgray.

What would it mean to shade7_4 of a unit square gray?

In symbols, we express7_4 of a unit square as follows:of a unit square as follows:of a unit square as follows:

7

__

4 1 unit square

To create such an area, we first break a unit square into 4 equal parts. Heres one

way to do this:

After multiplying by7_4, we end up with 7 of those parts, and we shade them gray:

Notice that7_4of a unit square is more than 1 unit square.


7. Assuming the following unit squares are broken into equal parts, what

fraction of each unit square is shaded gray?

8. Draw a picture showing5_4 of a unit square.

Fractionsare about

Equal Parts

Lets look at another example using circles in a box.Lets look at another example using circles in a box.

What fraction of the circles in the following box are blue?

There are a total of 6 circles, so we can break the circles up into 6 equal parts, each

part will be 1 circle:


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Since 3 of the groups are blue, we can say 3_6of the circles in the box are blue. In

symbols this corresponds to:

3

__

6 6 circles = 3 circles

Alternatively, we could have broken the circles up into 2 equal groups where each

group has 3 circles:

Since 1 of the two groups is blue, we can say 1_2of the circles in the box are blue.In symbols, we write:

1

__

2 6 circles = 3 circles

Both ways give us 3 circles as the answer. This means1_2 is equal to 3_6:

1

__

2 =

3

__

6

As we've seen before, we can describe the same situation in different butequivalent ways. Sometimes one way is more convenient than another, and well

talk more about this in lesson 30.


9. Fill in the blank for each description:


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10. Which figures appear to show 1_2 of the area shaded gray? For the onesthat dont, explain why they dont show 1_2 of the area shaded.

Money Before we finish this lesson, lets discuss fractions of money. Well use units ofdollars.

What is3_5 of 10 dollars?

To answer this, we find the value of the following expression:

3

__

5 10 dollars

We first break 10 dollars into 5 equal parts. To do this, we dont tear a ten-dollar bill

into pieces. Instead, lets use 10 one-dollar bills to form 5 equal groups:

We have expressed $10 as 5 groups of $2. After multiplying by3_5, we end up with

3 groups of $2:

From this we conclude that 3_5 of $10 is $6. Lets state this result as an equation:of $10 is $6. Lets state this result as an equation:of $10 is $6. Lets state this result as an equation:3

__

5 10 dollars = 6 dollars


How much is8_4 of a dollar?

To find out, we first express 1 dollar using 4 equal parts. Lets do this on the

number line:

Since 4 quarters equals a dollar, each of the above jumps represents a quarter. After

multiplying by8_4, we get 8 quarters:


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From this we see that 8_4 of a dollar equals 2 dollars:of a dollar equals 2 dollars:of a dollar equals 2 dollars:8

__

4 1 dollar = 2 dollars1 dollar = 2 dollars= 2 dollars= 2 dollars2 dollars2 dollars

Check for

Understanding

11. Solve.

a. What is2

__

3 of $30? b. What is

7

__

2 of $1?

c. What is5

__

4of $20?

Problem Set Use symbols to express each description.

Example:3

__

4of a cup Answer:3

__

4 1 cup

1.

5

__

6of a candy bar 2.

5

__12

of 36 people

3.9

__

8 of 21 dollars 4.

2

__

7 of a week

Write an equation for each sentence.

Example:2

__

3 of 6 apples is 4 apples Answer:

2

__

3 6 apples = 4 apples

5.5

__

6of 18 cars is 15 cars 6.5

__

3 of 6 boxes is 10 boxes

7.3

__10 of 100 dollars is 30 dollars 8. 8__2 of 30 dollars is 120 dollars

Find the value of the variable to make each sentence true.

9. Herea__3of the circle is colored blue:

10. Here2

__

a of the people are small, and 5__b of the people are big:

11. Here1

__

aof the rectangles are green and 6__bof the rectangles are black:

12. Herea

__

bof the triangles are red:


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13. Herea__3 of the squares are blue:

14. Here9

__

a of a circle is colored green:

15. Here isa__2of 10 dollars:

16. Here isa__2of 10 dollars:

17. Here is5

__

2ofn dollars:

18. Here is where7

__

a of a whole is located on the number line:

19. We can say that6

__

a of the following unit square is colored green, or we can saythat

3

__

b of the unit square is colored green:

20. We can say thata__6 of the happy faces are yellow, or we can say that b__3 of the

happy faces are yellow, or we can say that4

__

c of the happy faces are yellow:

21. Here the length of each hanger is1

__

a of the length ofb pants:


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22. In the word algebra2

__

a of the letters are the letter a.

23. Here is1

__

2ofn pizzas:

Answer each question. Use an equation to represent the result.

24. In this problem set,a__30of the problems include pictures or diagrams.

25. After you finish this problem (and all the previous problems) you will be5

__

6 of

the way though the 30 problems in this set. How many is5

__

6of 30 problems?

26. If only4

__

5 of the students in the class did their homework, and there are a total

of 35 students in the class, how many did their homework? How many didnt

do their homework?

27. Mariana got to keep 3

__

4of the $10,000 prize money. How much money did she

get?

28. If the class spent9

__

7 of the $280 they had budgeted, how much money did they

spend?

29. In a box of 144 books,1

__

3 of the books are new. How many are new?

30. A sale advertised2

__

3off every price in the store.

a. If you buy a coat that was originally $147, how much will you save?

b. Since the store removed

2

__

3of the price, you only need to pay for

1

__

3of the

price. How much do you have to pay for the coat (before taxes)?

Challenge Problems

1. The owner of a small software company owns1

__

5 of the total shares of the

company. If he owns 1,500 shares, how many total shares are there?


1. Estimate which fraction best describes how much of the circle is colored blue:

1

__

63

__

6

5

__

67

__

6


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1. The following three questions deal with the importance of equal parts.

a. If someone tells you that 3 out of the 4 groups of people below are going tothe party, do you know how many people they are talking about? Explain

why it isnt clear.

b. In this next situation, if someone tells you that 3 out of the 4 groups of people

below are going to the party, is it now clear how many people they are talking

about? Explain.

c. Describe how equal parts allow you to specify quantities in a clear way

using fractions.

2. In the lesson we drew7

__

4of a unit square (page 265). Was that the only way to

show7

__

4of a unit square? What are some other ways we could have illustrated

this quantity?

Find the Errors What is inaccurate about the way the student drew the picture below? How

would you improve the drawing?

LOOKING BACK

Vocabulary: area, equal parts, fractions

Student Self Assessment: Do I get it?

1. How are fractions related to the real world? What examples can Igive?

2. How do I break something into equal parts?

3. How do I know what fraction is represented in a partially shaded

diagram?

4. How do I nd a fraction of something?


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LESSON 28 Commutative Propertyof Multiplication

Objectives Understand and use the commutative property of multiplication.

Explain and apply the interpretation of fractions as division of whole numbers

by whole numbers.

Determine if a given fraction is equal to a whole number.

Concepts and Skills

WO.7 Represent the area of a rectangle using multiplication.

RN.5 Understand that the division of a whole number by another whole

number can be represented as a fraction.

PR.4 Understand and identify the commutative property of multiplication.

PR.9 Simplify expressions, generate equivalent expressions and equations and

solve equations using the following properties of rational numbers: the

commutative and associative properties of addition and multiplication,

the distributive property, and the special properties of 0 and 1.

PR.10 Understand that multiplication and division are inverse operations.

Use the inverse relationship of multiplication and division to generateequivalent expressions, evaluate expressions, verify the results of

computations, and solve equations.


What is the commutative property of addition?

What operation is the inverse of multiplication? Explain.

How do you multiply a whole number by a fraction?

How do you divide a whole number by a ratio?

How is long division used to test if a number is divisible by another number?



a. 8__3 15 b. 7__4

24

c.9

__

7 56 d.

11__2 82. Use mental math to divide.

a. 60,248 2 b. 38 2

c. 94 2 d. 76 2

Vocabulary

commutative

property of

multiplication


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LESSON 28 Commutative Property of MultiplicationCommutative Property of Multiplication | 273

LESSON 28 Commutative Propertyof Multiplication

Concepts and Skills: WO.7, RN.5,

PR.4, PR.9, PR.10

Before we go on discussing fractions, we need to improve our understanding of

multiplication. We will spend this lesson studying a powerful mathematical property

called the commutative property of multiplication.

The commutative property of multiplication states that the order in which we

multiply numbers doesnt change the value of the product. In symbols, we describe

this property with the following identity:

ab = ba

This identity holds true for all rational numbers.

Lets look at some of the implications of this property in situations that use

multiplication, such as the area of a rectangle. For example, consider the followingrectangle:

This rectangle has a width of 4 units and a height of 2 units. As usual, we describe

this area as 42, which equals 8 unit squares:

4 2 = 8

The commutative property of multiplication tells us that if we swap the widthfor the height, the resulting rectangle will maintain the same area. In other words,rotating a rectangle doesnt change its area:

We can describe this rotated rectangle as 2 stacks of 4 unit squares, or 24. This

also equals 8, giving us the following equation.

4 2 = 2 4

This result is true in general. A rectangle with width w and height h has the same

area as a rectangle with width h and height w:

wh = hw

This is true for all rational numbers.


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1. Find the pairs of rectangles that have the same area according to the

commutative property of multiplication:

The commutative property of multiplication provides us with two ways of

looking at the same multiplication problem. To see how this is useful, lets examine

the following situation:

We have $5, and we want to share it equally among 4 people. How much

does each person get?

The most straightforward way to express this situation is using division, 5 dollars

divided by 4:

5 4

Lets rewrite this expression as multiplication by the inverted ratio:

1

__

4 5

To find the value of this, we first break $5 into 4 equal parts. Lets represent this on

the number line:

After multiplying by 1_4, we end up with 1 of those parts:

From this we see that each person gets a little more than 1 dollar but how muchmore? What is the value of that point on the number line? To get a better description

of that points location, lets look at the problem in a different way.

The commutative property of multiplication gives us 2 ways to look at the same

product:

1__4 5 = 5 1__

4


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We already looked at 1_4 5. Lets now examine 5 1_4. On the number line, this isrepresented as 5 jumps of + 1_4:

From this we see that 5 1_4 equals the fraction 5_4:

5 1

__

4 ===5

__

4

In the context of money,5_4means 5 quarters. As a result we see that to share $5

equally among 4 people, we can give each person 5 quarters. We can represent this

result in symbols as follows:

5 4 =5

__

4

This is a powerful result! It shows us that we can express division as a fraction. This

result is true in general, providing us with the following identity:

a b =a__b

In upcoming lessons we will limit our use of the symbol. Instead, we will express

division using fraction notation.


2. Write the division as a fraction.

a. 6 3 33 b. 2 7 77 c. 7 8 d. 9 4

3. Find the value of the variable.

a. m1__5 =

7

__

5 b.1

__

8n =

9

__

8 c. 4

1

__

d = 4__6 d. 2 k__3 = 2__3

Division is NotCommutative

In an earlier lesson, we studied the commutative property of addition. Here we

just learned that multiplication is also commutative. What about division? Is abequal to ba? Lets find out by finding the values of 12 and 21.

As we saw earlier, 1 2 equals1_2. Here is the location of 1_2on the number line:


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On the other hand, 21 simply equals 2. Here is the location of 2 on the

number line:

Obviously, 2 does not equal 1_2. This means that division is not commutative. Forthis reason, it is often beneficial to rewrite division as multiplication by the inverse.

Well discuss this more in future lessons.


4. Find each set of equivalent expressions.

Which Fractionsare WholeNumbers?

A couple of lessons ago, we learned that all whole numbers can be expressed as

fractions. We also learned that the same number can be expressed as a fraction indifferent ways. This leads to the question: Which fractions are whole numbers?

Lets investigate this question.

Here is a generic fraction:

a__b

As we discussed earlier, this fraction is the answer to the division problem ab:

a__b = a b

This means that a_bis a whole number whenis a whole number when ab equals a whole number. We can testthis using the long division algorithm. If the final remainder is zero, then the result

is a whole number. Lets try this out.


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Is the fraction174___3 a whole number?

We can use long division to find out:

The result is 58 with a remainder of 0. This means174___3 is equal to the whole

number 58:

174___3 = 58= 58= 585858

Since we can write 58 as58__1, we now know of at least two ways to represent 58 as

a fraction:

174

___

3 =58

__

1

There are actually endless ways to represent any whole number as a fraction, and

well discuss this more in the next lesson.


5. Which of the following fractions are whole numbers?

a)16__7 b) 486___3 c) 765___6 d) 9___360

CommutingMultiplication

Lets finish this lesson by returning our discussion to the commutative property of

multiplication. One simple place this becomes useful is in calculating problems

such as the following:

We havent discussed yet how to deal with multipliers larger than 10 (well discuss

this in lesson 52). But for this case, we can use the commutative property of

multiplication to rearrange the problem so we have a single-digit multiplier:

Well discuss more powerful uses of the commutative property of multiplication in

future lessons.


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6. Use the commutative property of multiplication to simplify the

calculation.

a. b.

Problem Set Here are the mathematical properties we have learned so far:

a. Commutative property of addition

b. Distributive property of multiplication over addition

c. Multiplicative property of 1

d. Multiplicative property of 0

e. Additive property of 0

f. Associative property of addition

g. Commutative property of multiplication

Name the property represented by each of the equations below.

1. (1 + k) + 7 = 1 + (k+ 7) 2. 7 (k+ 1) = 7 k+ 7 1

3. k7

__

7 = k 4.0

__

7k = 0

5.1

__

7k = k

1

__

7 6. k+ 7 = 7 + k

7. k+0

__

7 = k

Express the division as a fraction.

8. 9 2 9. 7810. 134 11. 4 38Is the fraction equal to a whole number? If so, what whole number?

12.592___593 13. 7___371 14. 256___2

15.205___6 16. 558___9 17. 371___7

Each number line shows the division of a whole number by another whole

number. Find the fraction represented by point p on each number line.

18.

19.


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20.

21.

Apply the commutative property of multiplication and then find the value of

the expression.

22. 23.

Find the value of the expression. Write the result as an equation. Dont forget

the units!

24. 25.

26.

Solve the following problems. Write the result as an equation.

27. Sound travels at a speed of 761 miles per hour. It would take sound about 4

hours to travel from the west coast to the east coast of the United States. Using

this information, about how many miles is it across the United States?

28. In geometry you learned that the diameter is the distance across a circle, and

the circumference is the distance around a circle:

You can find the circumference of a circle by multiplying the diameter by the

number called pi (pronounced pie). A common approximation for pi is 22__7. Ifa circular dining table has a diameter of 56 inches, what is the circumference?


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29. There are 86,400 seconds in a day. How many seconds are in a week?

30. There are about 52 weeks in a year. About how many weeks old is a 5-year-old

kindergartner?

Challenge Problems

1. What is the only situation where ab will equal ba? Explain your reasoning.

2. Here is a multi-step problem:

A quarter coin weighs 6 grams. My tennis racquet weighs 336 grams. If a pile

of quarters weigh the same amount as my tennis racquet, is the value of the

quarters a whole number of dollars?

Here are the steps we can use to solve this problem:

Step 1: Use the rate of 1 quarter per 6 grams to convert 336 grams to

quarters.

Step 2: Now use the rate of 1 dollar for every 4 quarters to convert the quarters

to dollars.

Is the value a whole number? How many dollars is it? Thats how much I paid

for my racquet. I got it used at a garage sale. Its worth its weight in quarters!


1. Which is NOT a whole number?

832 million 015__5 8__3

2. What fraction is equal to the following division?

7

__

2

2

__

772__1 1__72

3. Estimate where5

__

2is located on the number line below.

Math Journal Questions1. What does 1_4 5 = 5 1_4 mean when the units are dollars? Explain.2. How does the following picture illustrate that 1_2 3 is equal to 3 1_2?


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3. A student drew the following picture to help her remember how long division is

related to a fraction. Explain what the student was trying to describe with this

picture and why it might have been useful to her.

4. Explain why division is not commutative but multiplication is.

Find the Errors

A student made 3 mistakes below. Find and correct each mistake.

1. 2.

3. 4.

LOOKING BACK

Vocabulary: commutative property of multiplication


1. What is the commutative property of multiplication? How is it

useful?

2. What are four dierent ways to write division?

3. How does a fraction represent division?

4. How do I determine if a fraction is a whole number?

5. Why is division not commutative but multiplication is commutative?

Explain.


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LESSON 29 Fractions and Area

Objectives

Understand the meaning of multiplication of fractions.

Explain the meaning of multiplication of fractions using area of rectangles and

jumps on the number line.

Define multiplication of fractions symbolically and apply this definition to

solving problems.

Concepts and Skills

RN.2 Understand fractions as parts of a set and as parts of a whole. Write thefraction represented by a drawing of parts of a figure; represent a given

fraction with a drawing.

RO.1 Multiply positive fractions.

RO.5 Interpret fraction multiplication in terms of the area of part of a unit

square. For example, if a rectangles width is 1_2of the width of the squareand its length is

3_5of the length of the square, then the rectangles areais

3__10.Remember from Before

How is multiplication used to find the area of a rectangle?

How do you find a fraction of a unit square?

What is the definition of multiplication by a fraction?


1. Use mental math to apply the commutative property of multiplication and thenevaluate the expression.

a. 42 5

__

6

b. 81 2

__

9

c. 14 3

__

7

d. 100 19__10

Vocabulary

unit fraction


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LESSON 29 Fractions and Area | 283

LESSON 29 Fractions and Area

Concepts and Skills: RN.2, RO.1,

RO.5

In the previous lesson we deepened our knowledge of multiplication. In this lesson

well use this knowledge to discuss the area of rectangles where the width and

height are fractions.

Fractional Area Here is the unit square we will use in this lesson:

Lets use the above unit square to create a rectangle with a width of2_3and a height

of 1. To find a width of2_3, we first break 1 unit length into 3 equal parts:

Then we keep 2 of those parts:

Now we can use this as the width for our rectangle:

The above rectangle shaded gray has a width of 2_3 and a height of 1. We left theoutline of the whole unit square for comparison. The above gray rectangle has an

area of 2_3, which we describe as follows:2

__

3 1 =

2

__

3


Assuming the following unit square is broken into equal parts, what is the area of

the gray region?

Since the unit square is broken into 5 equal parts, and 2 of the parts are shaded gray,

we say that the area is 2_5. This time the width is 1 and the height is 2_5:1

2

__

5 =2

__

5


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1. For each unit square, find the fraction of the shaded area. Write your answer

as an equation in the form width height = area.

FractionalWidth and

Height

Lets build a rectangle of width 2_3 and height 1_2. In other words, build a rectangulararea described by the following expression:

2_31_2

Earlier in this lesson we already determined what a width of 2_3 looks like:looks like:looks like:

To determine what a height of 1_2 looks like, we first divide a 1 unit length into 2looks like, we first divide a 1 unit length into 2looks like, we first divide a 1 unit length into 2equal parts, then we keep one of those parts:

Now we use the above width and height to form our rectangle:

We can describe the area of this rectangle as width height:

2

__

3

1

__

2

We can also describe the area as a single fraction. The unit square is broken into 6

equal parts:

And 2 of the parts are shaded gray:


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This means that the gray part is 2_6 of the unit square. This leads us to the followingof the unit square. This leads us to the followingof the unit square. This leads us to the followingequation:

2

__

3

1

__

2 =2

__

6


Assuming the following unit square is broken into equalparts, lets find the width,

height and area of the gray region:

We see that the width is broken into 4 equal parts and 3 are used. This makes a

width of3_4.

The height is broken into 3 equal parts and 2 are used. This makes a height of2_3.

From this we can express the area as width height:

3

__

4

2

__

3

We can also express the area as a single fraction. The unit square is broken into 12

equal parts, and 6 of the parts are gray. This makes an area of6__12. Therefore we have

the following equation:

3

__

4

2

__

3 =6

__12

Check for

Understanding

2. Use fractions to express the width, height and area of the gray regions for

each of the following unit squares. Write the answer as an equation.

3. Draw a diagram to show each product as an area. Express the area as a

single fraction.

a.1

__

2

1

__

4 b.1

__

3

1

__

3 c.1

__

3

1

__

2

MultiplyingFractions

on theNumber Line

What is1_3of a half? In other words, what is the value of the following

expression?

1

__

3

1

__

2


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Let's use our knowledge of the number line to find the value of this expression. The

fraction 1_2 is the following point on the number line:is the following point on the number line:is the following point on the number line:

To multiply this number by 1_3, we first express the number using 3 equal jumps:

After multiplying by 1_3, we end up with 1 of those jumps:

We now know where1_31_2 is located on the number line, but how do we

express this point as a single fraction?

Lets create the variablep to represent the product1_31_2:

1

__

3

1

__

2 = p

We know that it takes 3 jumps of +p to equal1_2:

This means it takes 6 jumps of +p to equal a whole:

The above expression shows a whole broken into 6 equal parts. By definition,

one of the above parts equals 1_6. This means thatp equals 1_6. We can now express ourproduct as a single fraction:

1

__

3

1

__

2 =

1

__

6

When we break a whole into 2 parts, and then break each part into 3 subparts, weend up with 2 3 parts. We can easily visualize this using area:


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The gray rectangle above has a width of 1_3, a height of 1_2and an area of 1_6, whichgives us the same equation as before:

1

__

3

1

__

2 ===1

__

6

We can generalize the above result with the following identity:

1

__

j1

__

k ==1

___

j kCheck for

Understanding 4. Find the product.

a.1

__

7

1

__

5 b.1

__

3

1

__

8 c.

1

__

9

1

__

6 d.

1

__

4

1

__

4

Writing aFraction as a

Product

Before we generalize how to find the product of any two fractions, lets review what

the numerator and denominator of a fraction mean. Heres an example fraction:

5

__

3

The denominator is 3. This means to break a whole into 3 equal parts. On the

number line, we can express a whole as 3 jumps of +1_3:

The numerator is 5, which means that we have 5 of these + 1_3jumps:

As we can see, the numerator acts as a multiplier. It tells us how many of the equal

parts we have. The fraction5_3 means 5 jumps of + 1_3. We can describe this with the

following equation:

5

__

3 = 5= 5= 555

1

__

3

In general, any fraction a_b can be rewritten with the numerator as a multiplier:a__b== a1__b

Here 1_b is called ais called a unit fraction because the numerator is 1.


5. Rewrite each fraction as multiplication of a unit fraction.

a.8

__

5 b.1

__

2 c.

3

__

9


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MultiplyingFractions

Lets now conclude this lesson by determining how to find the product of any two

fractions. Well use the following as an example:

4

__

5

2

__

3

As we discussed earlier, we can rewrite4_5as 4 1_5, and rewrite 2_3as 2 1_3:

4 1

__

5

22 1

__

3

Now, using the commutative property of multiplication, we can reorder the

expression like this:

4 2 1

__

5

1

__

3

Since 53 equals 15, we can rewrite 1_51_3simply as 1__15:4 22 1__

15

Now we can rewrite 4 2 as 8:

8 1__15

This is 8 jumps of + 1__15, which equals 8__15. Therefore, 8__15 is the value of 4_52_3:4

__

5

2

__

3 ===8

__15We can easily visualize this result using area:

Here 8 of the 15 equal parts are shaded gray.

GeneralizingFraction

Multiplication

Lets now generalize this result with the following identity:

This shows us that when multiplying fractions we multiply the numerators and

multiply the denominators. This is a very powerful result. It will save us a lot ofwork when multiplying fractions. For example, we can now easily find the value of

the following product:

5

__

6

4

__

3

Using the identity we know that:


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Since 54 equals 20, and 63 equals 18, we end up with the following:

5

__

6

4

__

3 ===20__18

Check for

Understanding

6. Find the value of each product.

a.2

__

7

3

__

4 b.

5

__

4

3

__

5 c.4

__

3

6

__

5 d.2

__

5

7

__

4

Problem Set Each unit square below is broken into equal parts. What fraction of each unitsquare is shaded gray? Write the result as an equation where width height

equals the area.

1. 2. 3.

4. 5. 6.

Find the value ofp on each number line.

7.

8.

9.

Find the value of each expression.

10.1

__

7

1

__

8 11.4

__

9

7

__

6 12. 6

8

__

5

13.3

__372__5 14. 7__419__7 15. 63 5__9


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Find the value when a= 7.

16. a 1

__

6 17.5

__

a

2

__

6 18.2

__

3 a

Find the value when h=4 and d=6.

19. h__37__d 20. d2__h 21. 9__d5__h

Solve for the variable in each equation.

22. 6 1

__

3=1

__

3

6

__

m 23. 7__55__g= 35__40 24. w__97__6= 28__54

25. b2

__12= 18__12 26. 7 = 7 5__p 27. 2__5n__8= 0

28. In the picture below, the unit square was broken into 9 equal parts. Then the

top left part was broken into 4 equal parts, and one of the 4 parts was colored

green:

This means that 1_4 of 1_9 of the unit square is colored green. Calculate 1_41_9 tofind what fraction of the unit square is green.

29. Here is a circle divided into 3 equal parts:

If you divide the top part into 5 equal parts, and color 2 of the parts red, you

get:

What fraction of this circle is colored red?

30. The dog, cat, bird and turtle equally shared a pizza. The bird only ate 2_5of itsshare. How much of the whole pizza did the bird eat?


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Challenge Problems

1. What fraction of a dollar is 2_5of a quarter? What coin is worth 2_5of a quarter?2. The jumps below are all equal. What is the value ofp?

3. The farmer picked 168 apples from his apple trees. He gave away3

__

4of the apples

and kept 1_4for himself. He then ate 2_3of his apples but had to throw the rest away.How many apples did he eat? How many did he throw away?


1. If the jumps are equal, what is the value ofp on the number line below?

1

__

2

1

__

8

1

__16 1__44

2. Which point on the number line represents the value of the product 4_35_3?


1. Is the following equation an identity? Explain why it is or why it isnt.

a__bb__a = 1

2. Explain how you multiply fractions on the number line. Give examples.

3. Does multiplication by a fraction always produce a smaller number? Explainwhy or why not, and give examples to support your reasoning.


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Find the Errors

A student made a mistake multiplying. Identify and correct the mistake.

What is inaccurate about the way the student drew this picture? How would

you improve it?

LOOKING BACK

Vocabulary: area, denominator, height, multiplication, numerator,

unit fraction, width


1. How would I use a number line to explain how to multiply fractions?

2. How would I use a unit square to explain how to multiply fractions?

3. How do I describe fraction multiplication with an identity?


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LESSON 30 Equivalent Fractions | 293

LESSON 30 Equivalent Fractions

Objectives

Understand and apply the concept of equivalent fractions.

Explain how equivalent fractions represent the same point on the number line.

Understand the concept of a simplified fraction.

Concepts and Skills

RN.6 Know that equivalent fractions represent the same point on the number

line.

RN.7 Determine if two fractions are equivalent.

RN.8 Understand that if the numerator of a fraction is equal to the denominator,

then the fraction is equal to 1.

RO.2 Multiply a fraction by a fraction of the formn

__

n to generate an equivalentfraction.Remember from Before

What is the definition of fraction multiplication?

In general, what fractions are equal to a whole?

What does it mean when two expressions are equal?

What does it mean to simplify an expression?

Get Your Brain in Gear1. Use mental math to find the value of each expression.

a.5

__

5

3

__

5 b.1

__

2

4

__

3

c.7

__

8

9

__

4 d.

2

__

3

7

__

7

Vocabulary

equivalentfractions

simplied fraction


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LESSON 30 Equivalent Fractions


RN.8, RO.2

In the previous lessons we learned that3_3and 5_5both equal 1. This is an example of

how there are many ways to represent the same number using fractions. Here we

will study this issue in more detail.A fraction is a way of naming a point on the number line. Two fractions are equal

when they name the same point. When two fractions are equal, they are said to be

equivalent fractions. Lets look at some examples.

A few lessons ago we learned that all fractions of the formn__n are equal to 1. Inare equal to 1. Inare equal to 1. In

other words, if the numerator equals the denominator, then the fraction is equal

to a whole. For example, 2_2 equals 1, andequals 1, andequals 1, and 4_4 equals 1. This means thatequals 1. This means thatequals 1. This means that 2_2 andandand 4_4 areareareequivalent fractions:

2

__

2 =4

__

4

It is easy to see why these two fractions are equal by illustrating this equation onthe number line:

Both expressions arrive at a whole. We can see that the top jumps and bottom jumps

also meet at the same point halfway between 0 and 1. This means1_2 andandand 2_4 are alsoare alsoare also

equivalent fractions:

1

__

2 =

2

__

4

What other fractions are equal to 1_2?We can find all equivalent fractions by going back to lesson 26 and the definition

of how ratios name points on the number line. The ratio 1_2 names the point on thenames the point on thenames the point on thenumber line we arrive at by multiplying

1_2 times a whole:times a whole:times a whole:1

__

2 =

1

__

2 1

As we just discussed, we can rewrite 1 asn__n:

1

__

2 =1

__

2n__n

This now gives us a way to find fractions equivalent to1_2. If we give n a value of 2,

then we get:

1

__

2 ===

1

__

2

2

__

2


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LESSON 30 Equivalent FractionsEquivalent Fractions | 295

In the previous lesson, we learned how to multiply two fractions. The value of1_22_2 equals 2_4:

1

__

2 ===2

__

4

This produces the equivalent fractions we found earlier. Using n=3, we can create

another fraction equivalent to

1_

2:

1

__

21

__

2

3

__

31 33____2 33 3__6

This shows that3_6 is also equivalent to 1_2:

1

__

2 ===

3

__

6

By allowing n to be any whole number (except 0), we can find infinitely many

fractions equal to 1_2. We can do this with any fraction. The following identity gives us a way of findingfractions equivalent to

a__b:

a__b === a__

bn__n


1. Multiply:Multiply:

a.2

__

3

4

__

4 b.4

__

9

3

__

3 c.5

__

6

8

__

8

2. Make an equivalent fraction for each:

a. 1_8 b. 2__5 c. 3__9

Equivalent Area Lets now examine equivalent fractions in terms of area. Well start with a whole.Here is a unit square:

This square has a width of 1 and a height of 1, and we can express this area using

multiplication:

1 = 1 1


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Since 2_2equals 1, we can use 2_2as the height without changing the area. To illustratethis, we break the height into 2 parts, and we keep both parts:

This obviously still has the same area as the previous unit square. We are just

representing the height in a different (but equivalent) way:

1 = 1= 1= 111 2

__

2

The fraction 4_4 is also equal to 1. Lets use 4_4 as the height of our unit square:

There are endless ways to represent a whole, and therefore endless ways to representa unit square. The above are just a couple of examples.

Lets next build an area of1_2.

The gray shaded region of the following unit square has a width of1_2and a height

of 1:

We find the area by multiplying the width times the height:

1

__

2 1 =1 ===

1

__

2

This means the rectangle has an area of 1_2.Now lets create an equivalent fraction by using a height of

2_2:

Since 2 of the 4 equal areas are shaded, we can express this shaded area as 2_4whichis equivalent to 1_2.


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Here are other areas equal to 1_2 along with their equivalent fractions:


3. Here are 4 ways to represent an area of3_4. Name each equivalent fraction.

SimpliedFractions

If there are endless ways to represent the same fraction, which way should

we use?

The answer to this question depends on the situation. If a pizza is sliced into 4

equal parts, and someone eats two of the parts, it might be informative to say that

the person ate2

__

4 of the pizza. Then again, its simpler to say that they ate1_2 of

the pizza.

Usually people prefer to use the equivalent fraction with the smallest possible

denominator. This is called the simplified fraction. For example

1_

2is a simplifiedis a simplifiedis a simplified

fraction, and2_4 is not. A simplified fraction is simple because it uses the fewestis not. A simplified fraction is simple because it uses the fewestis not. A simplified fraction is simple because it uses the fewest

number of equal parts to represent the number.

In a later lesson well develop useful methods to find simplified fractions, but lets

first try to visualize them. To visualize simplified fractions, well use the following

long strip of blue paper as our unit:

By defining the above strip as a unit, we are saying its a whole with a value equalto 1. Lets now break the whole into 2 equal parts:

The above is still equal to 1, but now we call it2_2.


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5. 6.

The fraction of the top strip that is filled blue is equivalent to the fraction of the

bottom strip that is filled blue. Name the equivalent fractions.

7.

8.

9. The valuep on the number lines below is the same value. Each number line

shows equal jumps. What two equivalent fractions are represented?

10. The two circles below have equal fractions colored green. What two

equivalent fractions best describe how much of the circles are colored

green?

Find the value ofkin each equation.

11.1

__

3 =

8

__

k 12. 4__5 = k__30 13. k__7 = 18__14

14.2

__

k = 6__30 15. k__8 = 0 16. 6__k = 42__7

Find the fractionn_d that forms a solution to the equation. Use the result to

determine if the fractions are equivalent. Place = or in the circle.

17.2

__

7n__d = 4__14 18. 4__5n__d = 8__15 19. 3__2n__d = 9__8

2

__

74

__14 4__5 8__15 3__2 9__8


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20.1

__

5n__d = 2__10 21. 7__6n__d = 21__24 22. 8__3n__d = 48__18

1

__

52

__10 7__6 21__24 8__3 48__1823.

8

__

3n__d = 32__9 24. 5__9n__d = 20__36

8

__

3 32__9 5_9 20__36Find the value ofkin each equation. Write the value ofkas a fraction.

25. 35 7

__

5 = k 35 26.

1

__

4

2

__

3 = k 27. 7 3 =1

__

3k

28.2

__

9k =

6

__27 29. 8__8 = k1__8 30. 5 6 = 5 k

Challenge Problems

1. An extra large pizza is first cut in half. Then each half is cut in half. Then each

of the resulting pieces is then cut into three slices. Whatn__nbest represents how

the whole pizza is cut? What fraction of the whole pizza is each slice? Draw apicture if you need help visualizing this.

2. What is the only value ofw that forms a solution to the equation below?

w__4 = w__8

3. The pointp has the same value on both number lines below. What is the value

ofm?


1. If two fractions are equivalent, then which statement is always true?

The numerators are equal.

The denominators are equal.

The fractions represent the same point on the number line.

When you multiply the fractions together you get a whole.


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2. Which of the fractions below is NOT equivalent to the others?

6

__15 3__5 12__20 21__35

3. All of the fractions below are equivalent. Which is the simplified fraction?

75___100 9__12 15__20 3__4


1. What mathematical property is represented by the following identity?

gn__n = g

Explain how this property allows us to generate equivalent fractions.

2. When and why is it useful to show fractions in their simplified form? When

might it be useful to show a fraction that isnt simplified? Give examples to

explain your reasoning.

Find the Errors A student made 2 mistakes below. Identify and correct each mistake.

1. 2. 3.

LOOKING BACK

Vocabulary: equivalent fractions, simplify, simplied fractions


1. What does it mean when fractions are equivalent?

2. How do I make equivalent fractions?

3. What is a simplied fraction?


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LESSON 31 Dividing Fractions | 303

LESSON 31 Dividing Fractions

Objectives

Define and understand division of fractions.

Apply fraction division to the testing of fraction equivalence.

Apply fraction division to comparing and ordering fractions.

Concepts and Skills

RO.3 Divide positive fractions by first rewriting the division as multiplication

by the inverted fraction.

RO.4 Use division to compare fractions.


What does it mean to invert a ratio?

How do you divide a whole number by a ratio?

How are the multiplication and division operations related?

What is the value ofn n when n is any number other than zero?

When is k b greater than a whole? When is k b less than a whole?



a. 15 3

__

5 b. 20 4

__

7

c. 35 5

__

2 d.3

__

8

2

__

7

e.2

__

9

3

__

5 f.8

__

5

1

__

4

Vocabulary

equivalentfractions


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LESSON 31 Dividing Fractions

Concepts and Skills: RO.3, RO.4 In the previous lesson we discussed equivalent fractions. One way to determine if

two fractions are equivalent is to divide them. Well discuss how this works in thislesson.

We learned how to divide by a ratio in lesson 20. As a review, lets divide 3 by 1_2:3

1

__

2

As we learned in lesson 20, dividing by a ratio is the same as multiplying by

the inverted ratio. Lets rewrite the above expression using multiplication by the

inverted fraction:

3 2__1

The fraction2_1 is equivalent to the whole number 2, giving us:

3 2

Since 32 equals 6, we conclude that 3 divided by1

__

2 equals 6:

3 1

__

2 = 6

Lets look at a different problem. Below we are dividing2_3by 4_5:

2

__

3

4

__

5

As we did before, we can rewrite the division as multiplication by the inverted

fraction:

2

__

3

4

__

52

__

3

5

__

4

Now we multiply the fractions as we learned a couple of lessons ago:

2

__

3

5

__

42 5____3 4 10__12

From this we conclude that 2_3divided by 4_5 equals 10__12:2

__

34

__

5 =10__12


1. Rewrite each division as multiplication by the inverted fraction, then

evaluate the expression.

a.1

__

3

1

__

4 b.

1

__

3

2

__

4 c.

1

__

2

2

__

4 d.

3

__

5 5

__

3


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LESSON 31 Dividing FractionsDividing Fractions | 307


3. Place the symbol , or = in each circle to make the statement true.

a.3

__

8

4

__

9 b.

5

__

8

6

__

9 c.

4

__

716__28

d.5

__11 6__12 e. 7__10 2__3

Problem Set Rewrite the division as multiplication by the inverted fraction.

1.3

__

5 2 2.6

__

5 17__19 3. 8 3__4 4. 99__4 3

Find the value of the expression.

5.5

__

3

8

__

5 6.5

__

4 3 7. 5

1

__

6

8. 2

__

3 4

__

5

9. 11__7 5__3 10. 1__6 12__7Given the equation compare the fractions. Write , or = in the circle to

make the statement true.

Example:2

__

5 4

__

9 =18__20 Solution: 18 < 20

2

__

54

__

9

11.1

__

2

1

__

4 =

4

__

2 12.

2

__

3

4

__

6 =

12__12 13. 8__10 10__12 = 96___1001

__

21

__

42

__

34

__

68

__10 10__12

14.3

__37 2__37 = 108___102 15. 5__75 2__30 = 150___150 16. 15__19 7__8 = 120___1333

__37 2__37 5__75 2__30 15__19 7__8

Use division to compare the fractions. Place , or = in the circle to make the

statement true.

17.5

__

3

8_5 18. 3__4 7__6 19. 4__5 2__3

20.20__6 30__9 21. 7__13 17__8 22. 15__9 14__8

23.5

__

6

13__16 24. 6__15 14__35 25. 8__5 5__3


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LESSON 32 Equivalent RatesEquivalent Rates | 311

LESSON 32 Equivalent Rates

Concepts and Skills: RR.10,

RR.12, RR.13, SN.8

We spent the past couple of lessons discussing equivalent fractions. We can apply

the same concepts to equivalent ratios and equivalent rates. This is the topic of the

current lesson.Lets start by answering the following rate problem:

It takes 3 potatoes to feed 5 people:

How many potatoes does it take to feed 15 people?

This is an easy problem. We are going from people to potatoes, so we use the

following rate:

To see how many potatoes are needed to feed 15 people, we multiply 15 people by

the above rate:

Since 15 5 equals 3, and 33 equals 9, the above expression equals 9 potatoes:

This means 9 potatoes are needed to feed 15 people:

EquivalentRates

Its important to realize that 9 potatoes for 15 people is an equivalent rate to 3

potatoes for 5 people. In other words, the two rates are equal:


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In the previous lesson we used division to test if two fractions are equal. Lets use

that method to show that the above two rates are equal. If they are equal, dividing3_5

by9__15 should equal 1. Lets test this:

Dividing by a fraction is the same a multiplying by the inverted fraction:

We know that 5 9 equals 45:

Now lets calculate 315:

This is also equal to 45, which tells us that 3_5 9__15 equals 45__45:

Since45__45 equals a whole, we conclude that 3_5 is equivalent to 9__15:

This verifies that 9 potatoes for 15 people is an equivalent rate to 3 potatoes for 5

people:

3 potatoes________5 people

=9 potatoes________15 people


1. Answer each question. Then use the result to write two equivalent rates.

Divide the rates to verify that they are equivalent.

a. If the pay is 7 dollars an hour, how much is made in 8 hours?

b. It took 2 cars to drive 3 families. If there were 12 families, how many

cars were needed?

c. Two quarters has the same value as 10 nickels. How many quarters is

60 nickels worth?


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LESSON 32 Equivalent RatesEquivalent Rates | 317

27. On average Kelly makes 6 out of 10 attempted free throw shots. If she made 54

free throw points in the season, how many attempts did she make?

28. The t-shirts are on sale, 4 shirts for $19. How much will 24 shirts cost?

29. If a student makes $8 an hour, how much will the student make in a week if he

works 40 hours a week? Write the result as an equivalent weekly rate.

30. Continuing from the previous problem, how much will the student earn in a year

if the student works all 52 weeks in the year? Write the result as an equivalentyearly rate (dollars per year). This rate is the students yearly salary.

Challenge Problems

Here are some multi-step problems.

1. You are filling a 108 gallon jacuzzi with water. You have two faucets going into

the jacuzzi. One faucet releases water at the rate of 5 gallons per minute. The

other faucet goes at the rate of 4 gallons per minute. Using both faucets at the

same time, how long will it take to fill the 108 gallon jacuzzi?

2. Glamour Plus offers 5 tubes of lip gloss for $10, where each tube is 3 ounces.A-Plus Beauty Supply offers 2 tubes of lip gloss for $5, but each tube is 4 ounces.

Which store has the better deal?

If you need help, try this: If you spend the same amount at each store, which

store will give you more ounces of lip gloss?


1. Find the rate equivalent to4 errors________6 problems?

9 errors_________

15 problems

8 errors_________18 problems

6 errors________9 problems 12 errors_________24 problemsMath Journal Questions

1. In the previous lesson we learned how to compare fractions using division.

You can use the same technique to compare rates. For example, compare the

following two brands of yogurt:

Yummy Yogurt gives you 10 ounces for 78 cents.

Healthy Plus Yogurt gives you 8 ounces for 60 cents.

Which brand gives you more yogurt for your money?

2. This question continues from the previous math journal question. See if you

compared the rates correctly by finding how much it would cost to buy 80

ounces of each brand of yogurt. Which one costs less for the 80 ounces? Explain

the results you get and how they compare to what you found in the previous

problem.


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Summary and Review

Chapter 6: Fractions

ChapterAccomplishments

We understand the definition of a fraction on the number line. We know how to describe quantities and verbal information using fractions and

units.

We know how to multiply and divide fractions and apply the commutativeproperty of multiplication.

We understand the concept of equivalent fractions and we know how to applythis concept to solving problems involving rates.

We know how to compare fractions.

Vocabulary fromthe Chapter

area

commutative property

of multiplication

denominator

equal parts

equivalent fractions

equivalent rates

fractions

greater-than sign (>)

inequality

less-than sign (

ar chapter 6 student text

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