Flags and Fractions - Carnegie Learning...said, “Your flags don’t seem correct. Look at our flags.” Dante’s Flag Fourths Miko’s Flag Thirds Ashley looked at all four flags

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3.1 Modeling Parts of a Whole • 95

Key Terms fraction

numerator

denominator

Learning GoalsIn this lesson, you will:

Determine equal parts of a whole.

Draw different representations of equal parts.

Did you know that the first United States flag had 13 stars on it? You

might have seen some historic flags with the 13 stars in a circle within a field

of blue, or maybe you saw the 13 stars in rows. Because there were no

government guidelines about how the flag’s stars were to be organized in the

blue field in the early days of the United States, the placement of stars varied.

Since 1776, the United States has grown to include 50 states, so, the current

flag has 50 stars. Do you remember what the 13 red and white stripes represent?

FlagsandFractionsModeling Parts of a Whole

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Problem 1 Part-Whole

In this chapter you will be adding to your knowledge of fractions. As you learned in

elementary school, a fraction represents a part of a whole object, set, or unit. A fraction is

written using two whole numbers separated by a bar. The number above the bar is the

numerator, and the number below the bar is the denominator. The denominator (bottom

number) indicates how many parts make up the whole, while the numerator (top number)

indicates how many parts are counted.

96 • Chapter 3 Fractions

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There are 10 total bowling pins and 3 of the pins are knocked

down.

You can represent this situation as:

3 ___ 10

→ numerator ____________ denominator

→ the number of bowling pins knocked down

_______________________________________ the total number of bowling pins

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The Student Bowler Association (SBA) is an organization of student bowlers in Grades 3

through 8. Each of the SBA bowling teams consists of two student bowlers. Each team is

asked to design two flags to represent the two players on the team. Each team flag must

be labeled and evenly divided into thirds, fourths, fifths, sixths, eighths, or twelfths. Each

flag has the same dimensions as shown.

1. Describe the dimensions and total area of the flag.

Team members Yvonne and Matthew each designed a flag and labeled them “Fourths”

and “Thirds.”

Matthew’s Flag

Fourths

Yvonne’s Flag

Thirds

If labels aren't provided,

use the word "unit"” to describe the

dimensions.

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They showed their flags to team members Dante and Miko. Dante looked at the flags and

said, “Your flags don’t seem correct. Look at our flags.”

Dante’s Flag

Fourths

Miko’s Flag

Thirds

Ashley looked at all four flags and said, “You are all correct! Each flag shows equal parts

of a whole.”

2. For each question, explain why Ashley’s statement is correct.

a. How are both Matthew’s and Dante’s flags of fourths correct? Explain your answer

by describing how each flag shows equal parts of the whole.

b. How are both Yvonne’s and Miko’s flags of thirds correct? Explain by describing

how each flag shows equal parts of the whole.

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Let’s analyze another fractional representation.

3. A rectangular flag is divided into 24 equal parts, and 15 of those parts are shaded.

a. Represent the portion of the flag that is shaded as a fraction. Then describe what

each number of the fraction represents.

b. Does the shaded portion of the rectangular flag shown represent 15 ___ 24

?

Explain your reasoning.

c. Shade a different representation for 15 ___ 24

. Explain how you know you are correct.

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Fractions can be represented in different ways as long as they have the same equal

number of units dividing the whole. This means the way the model of the fraction looks

does not affect the value of the fraction.

4. Draw three different flags for each fraction. Show how you know you have equal parts

by writing how many square units are in each part. Then describe how you made your

flags. Extra grids are included for workspace.

a. Halves

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b. Thirds

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c. Fourths

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d. Sixths

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e. Eighths

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f. Twelfths

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Problem 2 Who’s Correct? 1. Carmen designed a flag and shaded half of it. Did she correctly label her flag?


2. Katy wondered if she could make a flag using fifths. Katy had an idea and drew these

lines on her flag and said her flag was divided into fifths. Is she correct?


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3. What do you think? Can you make fifths in the rectangular flag shown? If so, draw the

flag. Explain your reasoning.

4. Represent each fraction on the 10 3 10 grid by shading.

a. 1 __ 2

b. 1 __ 4

c. 3 __ 4

d. 1

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Problem 3 Think Backward

1. Use these flag pieces to answer each question.

A

B

C

D

a. If A 5 1, then B 5 , C 5 , D 5 .

b. If B 5 1, then A 5 , C 5 , D 5 .

c. If C 5 1, then A 5 , B 5 , D 5 .

d. If D 5 1, then A 5 , B 5 , C 5 .

As you answer each question, think

of the given shape as "one whole." Are the other shapes smaller or larger

than the whole?

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Problem 4 Fractions of a Set

1. The seventh grade class is collecting books for the local children’s hospital library.

They have collected:

● 16 books about animals

● 8 books about sports

● 20 books about different cultures

● 10 biographies

● 6 mysteries

● 12 books about ancient civilizations

Represent each book type in the collection of books as a fraction.

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Problem 5 Fundraising Goals

The eighth-grade class at Carnegie Middle School decided to sell pom-poms at all the

sporting events. The money from the sales will be donated to the local children’s hospital.

Each homeroom set a different goal, displayed at the top of each thermometer. The total

money raised so far by each homeroom, including today’s donations, is shaded.

GOAL: $180 $120 $100 $80 $120

HR804 HR805 HR806 HR807 HR808

Yolanda is in charge of announcing the progress of the pom-pom fundraiser during the

morning announcements. She announces that Homeroom 805 (HR 805) has raised the

most money so far.

1. Do you think Yolanda is correct? Explain why or why not. If you think Yolanda is

incorrect, determine which homeroom has raised the most money.

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2. Complete the table shown using the information from the fundraising thermometers.

Homeroom

FractionalPartoftheGoal

CompletedasofToday

MoneyRaised(indollars)

HR 804

HR 805

HR 806

HR 807

HR 808

a. Is it better to announce the fractional part of money raised or the actual amount of

money raised? Why?

b. If HR 804 raises 11 ___ 12

of their goal, how much money would they have raised?

c. If HR 807 raises 3 __ 5 of their goal, how much money would they have raised?

d. If HR 805 raises 7 ___ 12

of their goal, how much money would they have raised?

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3. Let’s consider the same thermometers, but this time all the homerooms have the

same goal. How much money has each homeroom raised?

a. Homeroom 804

b. Homeroom 805

c. Homeroom 806

d. Homeroom 807

e. Homeroom 808

Be prepared to share your solutions

and methods.

Don't forget you just calculated the fractional part of

each thermometer's shaded region in the

previous table.

$300 $300 $300 $300 $300

HR804 HR805 HR806 HR807 HR808

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3.2 Fractional Representations • 113

Pattern blocks can be used to show fractions. Pattern blocks are a relatively

new mathematical model that was invented in the 1960s. You will use pattern

blocks in this lesson. What shapes are the pattern blocks? Have you ever used

pattern blocks before?


Create different fractional representations using pattern blocks.

Write fractional statements for different representations given the whole.

Determine fractional representations given the whole.

Determine fractional representations given parts of the whole.

YouMeanThreeCanBeOne?Fractional Representations

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Problem 1 Hexagonal Fractions

1. Complete the table shown. Use your yellow hexagon to represent the whole, or 1.

Yellow

ShapeNameofShape

FractionalPartofWhole

Number of Fractional Parts to

Make a Whole

red

blue

green

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2. Create different representations for the yellow hexagon.

Follow the example shown.

● Start with the yellow hexagon.

● Cover the yellow hexagon with other pattern blocks.

● Record your designs.

● Write a fraction sentence to describe your design.

● Repeat the process to create as many representations as possible.

You can place different pattern blocks on top of the yellow

hexagon to create another representation for the whole. For

example, you can create the design shown.

red

green

green green

As you saw from the table you completed, a red trapezoid

covers 1 __ 2

of the hexagon, and a green triangle covers 1 __ 6

of

the trapezoid. So, in this example, the red trapezoid

covers 1 __ 2

of the hexagon, and 3 triangles cover 1 __ 2

of the

hexagon. The fraction sentence for this representation would

be 1 5 1 __ 2 1 1 __

6 1 1 __

6 1 1 __

6 .

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3. How did you know you had determined all the combinations?

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Problem 2 You Mean Three Can Be One?

Recall the title of this lesson: “You Mean Three Can Be One?” You will now determine the

parts of a whole when the whole is more than one hexagon.

1. The three hexagons shown represent the whole, or 1. Determine what fractional part

each pattern block shape represents. Explain your reasoning.

a.

Yellow 1 hexagon

Keep in mind, these 3 hexagons

represent the whole. Are the new figures given smaller or larger than

the whole?

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b.

red 1 trapezoid

c.

blue 1 rhombus

d.

green 1 triangle

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2. Complete each statement.

a.

If red 5 1, then Yellow 5 .

green 5 .

blue 5 .

b.

If blue

5 1, then Yellow 5 .

green

5 .

red

5 .

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c.

If

green


blue

5 .

red 5 .

d.

green

Yellow If 5 1, then Yellow 5 .

green

5 .

red

5 .

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e.

If

green

green


green5 .

red

5 .

f.

red

blue

If 5 1, then Yellow 5 .

green

5 .

red

5 .

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3. Build and sketch each representation for the description given.

a. A triangle that is two-thirds red, one-ninth green, and two-ninths blue.

b. A parallelogram that is three-fourths blue and

one-fourth green.

c. A trapezoid that is two-thirds blue and one-third green.

Now, think in reverse. What will the whole look for each description?

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d. A parallelogram that is one-half red and one-half blue.

e. A triangle that is one-third green and two-thirds red.

f. Create a puzzle for your partner to solve using three pattern block types. Make

sure it is possible to create.

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4. Determine each fractional representation by drawing a model.

a.

If is 1, what is 3 __ 4

?

b.


?

c. If is 1, what is 3 __ 4

?

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d.

If is 2 __ 3

, what is 1?

e.

If is 1 1 __ 2 , what is 1?

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f.

If is 1

1

__

2

, what is 1?

g. If is 1 1 __ 2

, what is 1?

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h.


?

Talk the Talk

1. Describe a method for determining the value of a fractional part of any set.

Be prepared to share your solutions and methods.

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3.3 Dividing a Whole into Fractional Parts • 129

Key Terms unit fraction

equivalent fractions


Create equal parts of a whole.

Determine if fractions are equal.

Graph fractions on a number line.

Is there more news in a newspaper or are there more advertisements? In

addition to supplying news stories, newspapers routinely sell advertisement space

on each page. With your partner, take a section from a newspaper and measure

the size of each article in the section’s first four pages. Then, measure the size of

each advertisement in the first four pages. What do you notice?

RocketStripsDividing a Whole into Fractional Parts

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Problem 1 Newspaper Column Preparation

You signed up to participate in the school newspaper club. During the first meeting, faculty

advisors Ms. Foster and Ms. Shu showed everyone copies of last year’s publication of the

Rocket. The teachers have already planned out the sections for this year’s Rocket.

Matthew volunteered to create the “Random Acts of Kindness” section. The section

will appear along the right side of the paper’s back page. The newspaper is printed on

8 1 __ 2 -inch by 11-inch paper.

Matthew plans to put a box in each homeroom and ask students to nominate classmates

for the monthly recognition of random kindness acts. Students must tell what nice act their

nominee performed on a nomination slip. In preparation for completing his section, help

Matthew plan the layout of the column; do not worry about the top or bottom margin of

the page.

1. To begin, cut eight strips of paper the length

of a newspaper page. Remember, the

Rocket is printed on 8 1 __ 2 -inch by 11-inch

paper. Each strip of paper should be 1 inch

wide. The strip represents one whole column.

Do not fold the first strip, and label it as 1 whole.

2. Take one of your paper strips and fold it carefully in half to divide the

strip into two equal parts like the one shown. Label each folded part

of the paper strip with the appropriate fraction, and draw a line to

mark your fold. The strip shown will represent a column that

recognizes two students.

12

12

The strips are provided for

you at the end of this lesson.

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3. Take another paper strip and fold it carefully in half two times. Unfold and draw lines to mark

your folds. Then, label each folded part of the paper strip with the appropriate fraction.

How many students can be recognized in this column?

4. Take another paper strip and fold it in half three times. Be very careful to fold

accurately. Unfold and draw lines to mark your folds. Then, label each folded part of

the paper strip with the appropriate fraction.


5. Take another paper strip and fold it very carefully in half, four times. Unfold and draw

lines to mark your folds. Then, label each folded part of the paper strip with the

appropriate fraction.


6. Take another paper strip and fold it carefully into three equal sections. Unfold and

draw lines to mark your folds. Then, label each folded part of the paper strip with the

appropriate fraction.


7. Take the next paper strip and fold it into thirds, and then fold the strip in half. Unfold

and draw lines to mark your folds. Then, label each folded part of the paper strip with

the appropriate fraction.


8. Finally, take your last paper strip and fold it into thirds. Then, fold in half, and then fold

in half once more. Unfold and draw lines to mark your folds. Then, label each folded

part of the paper strip with the appropriate fraction.


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Arrange your strips in a column so that all of the left edges are lined up and the strips are

ordered from the strip with the smallest parts to the strip with the largest parts.

9. As the number of students who can be recognized in the column increases, describe

what happens to the space for each student.

A unitfraction is a fraction that has a numerator of 1 and a denominator that is

a positive integer.

10. List the unit fractions for each strip you

created in ascending order.

11. Explain how understanding the size of a

unit fraction helps you determine the size of

the whole.

To list a set in ascending order

means to list the set from least to greatest.

To list a set in descending order means to list the set from

greatest to least.

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If you folded the paper strips carefully, you will notice that some of the folds line up with

each other. Fractions that represent the same part-to-whole relationship are equivalent

fractions.

12. Show that 1 __ 2

is equivalent to 6 ___

12 . Draw on the paper strips to represent halves and

twelfths. Then, shade the strips to represent 1 __ 2 and 6 ___

12 .

13. Make a collection of equivalent fractions using your fraction strips. Then, complete

the graphic organizer by writing all the equivalent fractions for each.

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2__3

1

1__2

3__4

EquivalentFractions

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14. What do you notice in the collection of equivalent fractions? Give an example to

justify your answer.

Talk the Talk

1. What do you notice about the numerator and denominator of the equivalent

fractions?

2. What do you need to do to both the numerator and the denominator of a fraction in

order to write another equivalent fraction?


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3.5 Equivalent Fractions • 153

Key Terms Multiplicative Identity Property

simplest form


Determine equal portions of a whole.

Determine equivalent fractions.

Calculate equivalent fractions using

a form of 1.

Simplify fractions.

Determine equivalent fractions

in context.

Order fractions.

What’sMyCut?Equivalent Fractions

Today, you can take your coins to the grocery store to have them counted by a

machine. You can exchange your coins for a cash voucher, gift cards, or you can

even donate your coins to charity.

Careful, though. Unless you’re donating your money, these machines usually take

a “cut” of about 10 cents for every one of your dollars that it counts.

One of the largest transactions recorded for one of these machines was in San

Dimas, California. A customer turned in over $8000 worth of coins! What was the

machine’s “cut”?

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Problem 1 Pizza Cuts

1. Coach Finley buys pizzas to share with everyone who participated in the volleyball

intramural program. She buys 21 pizzas to share among the 28 students who are

sitting at tables of four.

a. How many pizzas should each table receive if she wants each table to receive the

same amount of pizza? Explain your reasoning.

b. Coach Finley suggests dividing each pizza into fourths. Draw a diagram that

represents the problem. Then, explain her reasoning.

c. What fractional part of a pizza will each student receive?

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d. Mario suggests dividing the pizzas in half. Will Mario’s method work? Why or why

not? If Mario’s suggestion is not possible, explain what he can do to make his

suggestion work. Draw a diagram that supports your explanation.

e. Sydney tries to divide his group’s pizzas into thirds. Will Sydney’s method work?

Why or why not? If Sydney’s method is not possible, explain what he can do to

make his method work. Draw a diagram that supports your explanation.

f. Natalie tries to divide her group’s pizzas into eighths. Will Natalie’s method work?

Why or why not? If it is not possible, explain what she can do to make her method

work. Draw a diagram that supports your explanation.

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g. Name the student(s) whose method did not require your suggestion. Then, name

the fractional part of a pizza each student at the table will receive.

h. Juanita claims that dividing the pizzas into any number of equal-sized pieces will

work to divide the pizzas equally among the 4 students in her group. Is she correct?

Explain your reasoning and offer a solution if Juanita’s method does not work.

2. Casey and Jamal are talking about pizza parties they had in each of their classes.

Casey said, “In my class, four people shared three pizzas.” Jamal said, “In my class,

three people shared two pizzas.”

a. Which student’s classmates each got more pizza? Use a drawing or diagram to

explain your reasoning. Assume the pizzas are the same size.

All this talk of pizza

is making me hungry!

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Problem 2 Equivalent Fractions

Throughout this chapter, you have encountered equivalent fractions. Recall that equivalent

fractions are fractions that represent the same part-to-whole relationship.

When determining equivalent fractions, you must multiply the numerator and the

denominator of a fraction by the same

number. This process is the same as

multiplying the given fraction by a

fraction with the same numerator and

denominator, such as 3 __ 3

.

Recall that any fraction whose

numerator and denominator are the

same number is equivalent to 1.

Multiplying any number by 1 does not

change that number.

1. Complete each number sentence with the correct fraction to make it true. Explain

your reasoning. Use your fraction strips if you need help determining the missing

fraction to make equivalent fractions.

a. 5 __ 6

3 ( )

_____ ( )

5 15 ___ 18

b. 3 __ 4

3 ( )

_____ ( )

5 15 ___ 20

c. 5 __ 8 3

( ) _____

( ) 5 15 ______

( ) d. 9 ___

27 3

( ) _____

( ) 5

( ) _____

27

To change a fraction to an equivalent

fraction with a larger numerator

and denominator, you multiply

the fraction by a form of 1.

For example,

5 __ 8

3 3 __ 3 5 15 ___

24 .

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2. You just multiplied each fraction by the same number. What number was it?

The MultiplicativeIdentityProperty states: a 3 1 5 a, where a is a nonzero number.

3. Complete each equation to make the fractions equivalent. Explain your reasoning.

Use your fraction strips if you need help determining equivalent fractions.

a. 1 __ 4

5 ( )

_____ 16

b. 2 __ 3

5 ( )

_____ 6

c. 7 ___ 16

5 ( )

______ 32

d. ( )

_____ 12

5 3 __ 4

e. 1 ___ 18

5 1 ______ ( )

4. Write the first 10 equivalent fractions of each using what you know about equivalent

fractions. The first example is done for you.

a. 1 __ 4

, 2 __ 8

, 3 ___ 12

, 4 ___ 16

, 5 ___ 20

, 6 ___ 24

, 7 ___ 28

, 8 ___ 32

, 9 ___ 36

, 10 ___ 40

b. 3 __ 5

,

c. 2 __ 3 ,

d. 5 __ 8 ,

e. How did you determine the order to list your equivalent fractions?

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Whenever you determine an equivalent fraction whose numerator and denominator are

smaller than the original fraction’s numerator and denominator, the new fraction is simpler.

When a fraction cannot be simplified further, the fraction is in simplest form, or completely

simplified. Simplestform is a way of writing a fraction so that the numerator and

denominator have no common factors other than 1.

You can say a fraction is in a simpler form in several ways:

“ 1 __ 2 is simpler than 4 __

8 .”

“ 1 __ 2 is a simplified form of 4 __

8 .”

“ 4 __ 8 in simplest form is 1 __

2 .”

Sometimes it is read:

“ 1 __ 2 is the simplified form of 4 __

8 .”

“ 1 __ 2 is in lowest, or simplest, terms.”

5. Determine if each fraction is simplified completely. If the fraction is not simplified

completely, write the fraction in simplest form.

a. 9 ___ 24

5 9 4 3 _______ 24 4 3

5 3 __ 8

b. 6 ___ 12

5 6 4 3 _______ 12 4 3

5 2 __ 4

c. 18 ___ 27

5 18 4 3 _______ 27 4 3

5 6 __ 9

To change a fraction to an equivalent fraction with a smaller numerator and

denominator, you must divide the numerator and denominator by a form of 1.

For example, 15 ___ 24

4 3 __ 3

5 5 __ 8 .

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6. Circle the fractions that are simplified completely. How do you know?

9 ___13

10 ___ 32

9 ___ 12

12 ____ 144

7 ___ 12

33 ___ 55

Recall that the factors of a number are those numbers that divide into the number with

no remainder.

7. How can you use factors of a number to simplify fractions?

8. Write each fraction in simplest form.

a. 6 __ 9

b. 9 ___ 12

c. 8 ___ 24

d. 12 ___ 15

e. 18 ___ 36

f. 14 ___ 42

g. Explain how you simplified each fraction.

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9. Ms. Glick asked her students to simplify 24 ___ 36

. Jose, Sara, and Clifton’s methods are

shown. Analyze each method and solution to determine if it is correct. If one of the

methods is incorrect, what would you tell the student to do to correct his or

her method?

Jose:

Isimplified24___36byfirstdividingboththenumeratoranddenominatorby2.Ithen

continueddividingby2untilthefractioncouldnotbeevenlydividedby2__2.

24___36=

24:_2______36:_2=

12___

18=

12:_2______18:_2=

6__9

Sara:

IdividedboththenumeratoranddenominatorbytheGCFof12and

gotaquotientof2__3.

24__36=24:_12______36:_12=

2__3

Clifton:

Iwrotetheprimefactorizationofthenumeratoranddenominator,andthenIdividedoutthecommonprimefactors.

24___36=2x2x2x3_________2x2x3x3=2__3

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10. Simplify each fraction using prime factorization.

a. 15 ___ 20

b. 8 ___ 24

c. 24 ___ 28

d. 20 ___ 24

e. 24 ___ 30

f. 8 ___ 15

11. Simplify each fraction using the GCF. State the GCF used.

a. 24 ___ 28

b. 45 ___ 56

c. 33 ___ 77

d. 16 ___ 32

e. 63 ___ 72

f. 72 ___ 99

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Problem 3 Vegetable Lasagna Recipe

Shauna has invited three of her friends to come to her house to watch the big game. She

decides to make a pan of her famous vegetable lasagna for the four of them to enjoy. The

ingredients Shauna uses in her lasagna are shown.

1. The only measuring cups that Shauna owns are a 1 __ 8

-cup measuring cup and a 1-cup

measuring cup. How can Shauna use the 1 __ 8 -cup measuring cup for:

● the chopped green pepper?

● the chopped onion?

● the parmesan cheese?

VegetableLasagna

16 ounces lasagna noodles 52 ounces pasta sauce

1 cup fresh mushrooms 1 1 __ 3

teaspoon dried basil

3 __ 4

cup chopped green pepper 15 ounces ricotta cheese

1 __ 4

cup chopped onion 4 cups shredded mozzarella cheese

3 cloves of garlic 2 eggs

2 tablespoons vegetable oil 1 __ 2

cup parmesan cheese

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2. Shauna’s friend Gustavo brought two cheese quesadillas to share among the four

friends. Each quesadilla is cut into 8 pieces.

Shauna

If we divide the quesadillas evenly,

each of us will receive 4 __ 16 of the

total amount.

GustavoWe will each have 2 __

8 of the

quesadillas if we split them up

evenly.

Explain why both Shauna and Gustavo are correct.

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Problem 4 Frac-O

Frac-O is played with two players. The object of the game is to be the first player to

arrange five fraction cards on the game board in ascending order. To begin, cut apart the

16 Frac-O fraction cards.

Directions:

1. Shuffle the fraction cards. Deal one card face down on each of the five spaces on

each player’s game board.

2. Put the remaining cards face down in a pile. Turn the top card over and place it in a

discard pile.

3. Each player turns over the five cards on his or her game board. You may NOT change

the order of the cards at any point during the game.

4. Players take turns as follows:

a. The first player takes either the top card from the face down pile or the top card

from the discard pile.

b. The player decides whether to keep the card or put it face up in the discard pile.

c. If the player keeps the card, he or she must replace one of the five cards on the

game board with the card drawn. The replaced card now goes face up on the

discard pile.

5. If all the facedown cards are used, then shuffle the discard pile and continue.

6. The winner is the first person to have all five cards in ascending order.

Good Luck!


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3 __ 5

1 ___ 12

3 __ 4

1 __ 5

1 __ 3

11 ___ 12

4 __ 7

5 __

6

3 __ 8

1 __ 4

7 __ 9

2 __ 3

4 __ 9

3 __ 7

4 __ 5

5 __ 8

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Fra

c-O

Gam

e B

oar

d

Clo

sest

to 0

Clo

sest

to 1

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