Solving Proportions by Cross Multiplication Objective To introduce and use cross multiplication to solve proportions.

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Lesson 8

3

703

Advance Preparation

Teacher’s Reference Manual, 

Grades 4–6

pp. 64 – 68, 295 – 297

Key Concepts and Skills

• Apply multiplication and division facts

to find cross products. 

[Operations and Computation Goal 2]

• Multiply whole or decimal numbers. 

[Operations and Computation Goal 2]

• Use cross products to write open

number sentences. 

[Operations and Computation Goal 6]

• Describe rules for patterns and use them

to solve problems. 

[Patterns, Functions, and Algebra Goal 1]

• Use a method to solve equations. 

[Patterns, Functions, and Algebra Goal 2]

Key Activities

Students use the cross-products rule

to determine whether two fractions are

equivalent. They solve rate problems

by writing proportions and using cross

multiplication.

Ongoing Assessment:

Informing Instruction

See page 705.

Ongoing Assessment:

Recognizing Student Achievement

Use journal page 288. 

[Operations and Computation Goal 6]

Key Vocabulary

cross products

cross multiplication

Materials

Math Journal 2,

pp. 286, 288–289B

Study Link 8

2

calculator (optional)

Playing 

Fraction/Whole

Number Top-It

Student Reference Book,

pp. 319

and 320

per partnership: 4 each of number

cards 1–10 (from the Everything

Math Deck, if available), calculator

(optional)

Students practice calculating and

comparing products of fractions and

whole numbers.

Math Boxes 8

3

Math Journal 2,

p. 287

Students practice and maintain skills

through Math Box problems.

Study Link 8

3

Math Masters,

pp. 249 and 251

Students practice and maintain skills

through Study Link activities.

READINESS

Solving Equations (

ax

=

b

)

Students practice solving simple equations.

ENRICHMENT

Using Double Number Lines

Math Masters,

pp. 249A and 249B

Students use double number lines to solve

rate problems.

EXTRA PRACTICE

Calculating Ingredient Amounts

Math Masters,

p. 250

Students practice solving rate problems by

calculating ingredient amounts for a recipe.

ELL SUPPORT

Illustrating Terms

posterboard

markers

Students make posters illustrating how to

use cross products to solve open proportions.

Teaching the Lesson

Ongoing Learning & Practice

1

3

2

4

Differentiation Options

Solving Proportions

by Cross Multiplication

Objective

To introduce and use cross multiplication

to solve proportions.

t

Common

Core State

Standards

703_EMCS_T_TLG2_G6_U08_L03_576922.indd 703

2/22/11 10:43 AM

(2)

Date Time

Math Message

For Part a of each problem, write = or ≠ in the answer box. For Part b, calculate the cross products.

1. a. 3 _ 5

=

6 _ 10 2. a. 7 _ 8 _ 23 b. b. 3. a. 4. a. 6_9 _12 8 b. b. 5. a. 2_8 _10 4 6. a. 10_12 _58 b. b. 7. a. 1_4 _20 5 8. a. 5_7 _1521 b. b. 9. a. 10_16 _48 10. a. 3 _ 5 10 _ 15 b. b.

11. What pattern can you find in Parts a and b in the problems above?

If the fractions are equivalent, the cross products are equal.

3 5 106

30

30

10 ∗ 3 = = 5 ∗ 6 114 115 7 8 23

21

16

3 ∗ 7 = = 8 ∗ 2 2 3 69

18

18

9 ∗ 2 = = 3 ∗ 6 2 8 104

20

32

= 8 ∗ 4 10 ∗ 2 = 1 4 205

20

20

= 4 ∗ 5 20 ∗ 1= 4 8 10 16

80

64

8 ∗ 10 = =16 ∗ 4 6 9 128

72

72

12 ∗ 6 = = 9 ∗ 8 5 8 10 12

80

60

= 12 ∗ 5 8 ∗ 10 = 5 7 1521

105

105

21 ∗ 5 = = 7 ∗ 15 3 5 1015

45

50

= 5 ∗ 10 15 ∗ 3 =

Sample answer:

=

2 _ 3 6_9

=

=

=

Equivalent Fractions and Cross Products

LESSON

8

3

278_323_EMCS_S_G6_U08_576442.indd 286 2/26/11 1:15 PM

Math Journal 2,

p. 286

Student Page

Adjusting the Activity

704

Unit 8 Rates and Ratios

Getting Started

1

Teaching the Lesson

Math Message Follow-Up

WHOLE-CLASS

DISCUSSION

(Math Journal 2, p. 286)

Algebraic Thinking

Go over the answers to Problems 1–10. Review

the following:

Cross products

are found by multiplying the numerator of each

fraction by the denominator of the other fraction.

Cross multiplication

is the process of finding cross products.

To support English language learners, demonstrate how an

X

is used

to

cross

out a word or number. Relate this

X

to the terms

cross products

and

cross multiplication.

A U D I T O R Y

K I N E S T H E T I C

T A C T I L E

V I S U A L

Discuss Problem 11. While there are several possible patterns, one

stands out: If the fractions in Part a are equivalent, then the cross

products in Part b are equal. If the fractions in Part a are not

equivalent, then the cross products in Part b are not equal.

Point out that this pattern provides a way to test whether two

fractions are equivalent. Have students use this rule to test

several pairs of fractions for equivalence.

Suggestions:

3

_

4

?

_

12

9

Cross products: 3

12

=

36; 4

9

=

36. The cross

products are equal; therefore, the fractions are equivalent.

5

_

6

?

_

8

9

Cross products: 5

9

=

45; 6

8

=

48. The

cross products are not equal; therefore, the fractions are

not equivalent.

3

_

8

?

_

1

4

Not equivalent

16

_

20

?

_

12

15

equivalent

Pose additional problems as needed.

ELL

_

3

5

=

_

12

20

_

5

6

>

_

5

8

_

12

7

>

_

11

6

_

6

7

<

_

8

9

_

14

22

<

_

6

8

_

10

13

>

_

11

7

Math Message

Complete the problems on journal

page 286.

Study Link 8

2 Follow-Up

Briefly go over answers. Have students share

strategies for solving Problem 4.

Mental Math and Reflexes

Students compare fractions using

<

,

>

, or

=

.

Suggestions:

If time permits, have students share the strategies they used to compare

the fractions.

Mathematical Practices

SMP1, SMP2, SMP4,

SMP6, SMP7,

SMP8

Content Standards

6.RP.3,

6.RP.3b, 6.RP.3d,

6.EE.5, 6.EE.7

704-709_EMCS_T_TLG2_G6_U08_L03_576922.indd 704

3/20/12 10:52 AM

(3)

Date Time

114 115

Solving Proportions with Cross Products

LESSON

8

3

Use cross multiplication to solve these proportions.

Example: _ 4 6 = p _ 15 15 ∗ 4 = 6 ∗p 60 = 6p _ 60 6 =p 10 =p 1. _3 6 = _10 y

y

=

5

2. _7 21 = 3_ c

c

=

9

3. _m 20 = _2 8

m

=

5

4. _2 10 = 5_z

z

=

25

5. _15 9 = _12k

k

=

20

6. 10_12 = d_9

d

=

7.5

7. _2 9 = _54 t

t

=

12

8. _4 10 = 26_z

z

=

65

9. _3 4 = _r 28

r

=

21

10. 16_ p = 128_ 40

p

=

5

11. _51 102 = _6 h

h

=

12

12. _8 j = _72 192

j

=

3

4 6 p 15 15 ∗ 4 = = 6 ∗p 278_323_EMCS_S_G6_U08_576442.indd 288 2/21/11 4:41 PM

Math Journal 2,

p. 288

Student Page

Lesson 8

3

705

Using Cross Products

WHOLE-CLASS

ACTIVITY

to Solve Proportions

Algebraic Thinking

Write the following proportion on the board:

5

_

6

=

x

_

18

. Ask volunteers to explain how to solve the proportion

using cross products. Students may suggest using the Identity

Property of Multiplication, as was done in Lessons 8-1 and 8-2.

This is correct; however, remind students that they are supposed

to find a solution using cross products. If no one is able to do so,

demonstrate the following approach:

Step 1

Cross multiply. Note that the cross product of 6 and

x

is

written as 6

x,

or 6

x.

6 P x or x P 6

18 P 5

90

5

––

6

18

––

x

Step 2

Because we want the two fractions in the proportion to

be equivalent, we also want the two cross products to be

equal; that is, we want the product 6

x

to equal the

product 18

5.

Step 3

Solve the equation from Step 2.

18

5

=

6

x

90

=

6

x

_

90

6

=

x

15

=

x

Step 4

Write 15 in place of

x

in the proportion:

_

5

6

=

_

15

18

. Use

cross multiplication to check that the two fractions

are equivalent.

6

15

=

90; 18

5

=

90

Ongoing Assessment:

Informing Instruction

Watch for students who doubt the need to apply and practice the cross-products

method because they can solve many of the problems in this lesson more

quickly using other methods. Explain that the advantage of the cross-products

method is that it works for all proportions, not just those with convenient

numbers. To prove your point, pose a problem such as the following:

_

8.4

t

=

_

11.2

6.8

6.8

8.4

=

t

11.2

57.12

=

11.2

t

5.1

=

t

Adjusting the Activity

Have students use pencil and

paper or a calculator to calculate products

as needed.

A U D I T O R Y

K I N E S T H E T I C

T A C T I L E

V I S U A L

(4)

Solving Proportions with Cross Products

continued

LESSON

8

3

Date Time

114 115

For Problems 13–16, set up a proportion and solve it using cross multiplication. Show how the units cancel. Then write the answer.

Example: Jessie swam 6 lengths of the pool in 4 minutes. At this rate,

how many lengths will she swim in 10 minutes?

Proportion:

10 minutes ∗ 6 lengths = 4 minutes ∗n lengths 60 minutes ∗ lengths = 4 minutes ∗n lengths

=n lengths 15 lengths=n lengths Answer: Jessie will swim

15

lengths in 10 minutes.

13. Belle bought 8 yards of ribbon for $6. Solution: How many yards could she buy for $9?

Answer: Belle could buy yards of ribbon for $9.

$9

8 yards

=

$6

n

yards

$72

yards

=

$6

n

yards

$72 ∗ yards

_

$6

=

n

yards

12 yards

=

n

yards

=

8 yards

n

yards

$6

$9

12

6 lengths 4 minutes n lengths 10 minutes = Solution: 6_4 = _ 10 n 6 4 n 10 10 ∗ 6 = = 4 ∗n 60 minutes ∗ lengths 4 minutes 278_323_EMCS_S_G6_MJ2_U08_576442.indd 289 3/9/11 11:11 AM

Math Journal 2,

p. 289

Student Page

Date Time LESSON

8

3

14. Before going to France, Maurice Solution: exchanged $25 for 20 euros. At that

exchange rate, how many euros could he get for $80?

Answer: Maurice could get euros for $80.

15. One gloomy day, 4 inches of rain Solution: fell in 6 hours. At this rate, how

many inches of rain had fallen after 4 hours?

Answer: inches of rain had fallen in 4 hours.

16. Adelio’s apartment building has Solution: 9 flights of stairs. To climb to the

top floor, he must go up 144 steps. How many steps must he go up to climb 5 flights?

Answer: Adelio must climb steps.

= =

9 flights

5 flights

144 steps

s

steps

=

4 inches

p

inches

6 hours

4 hours

64

2.

_

6

80

$25

$80

x

euros

20 euros

Solving Proportions with Cross Products

continued

LESSON

8

3

x

euros

$25

=

20 euros

$80

x

euros

$25

=

$1,600

euros

x

euros

=

_

$1,600 $25∗ euros

x

euros

=

64 euros

4 hours

4 inches

=

6 hours

p

inches

16 hours

inches

=

6 hours

p

inches

16 hours

__

6 hours ∗inches

=

p

inches

2.

_

6 inches

=

p

inches

s

steps

9 flights

=

144 steps

5 flights

s

steps

9 flights

=

720 steps

flights

s

steps

=

720 steps

__

9 flights ∗flights

s

steps

=

80 steps

289A_289B_EMCS_S_G6_MJ2_U08_576442.indd 289A 3/9/11 11:13 AM

Math Journal 2,

p. 289A

Student Page

Adjusting the Activity

706

Unit 8 Rates and Ratios

Guide students in solving a few more proportions using Steps 1–4.

Example:

_

48

6

=

_

8

n

Cross multiply:

n

∗ 6 = 48 ∗ 8

Solve: 6

n

= 384

n

=

_

384

6

n

= 64

Replace the

variable:

_

48

6

=

8

_

64

Check: 64

∗ 6 = 48 ∗ 8

Suggestions:

6

_

9

= x

_

12

_

15

20

=

_

9

r

_

3

z

=

_

1

5

x

= 8

r

= 12

z

= 0.6

Use a quick common denominator (QCD) or multiplicative inverses to

explain why cross multiplication works:

3

_

8

=

_

12

32

Find the QCD: Multiply both sides by 8

32.

32 º 8 º 3

8

8 º 32 º 12

32

32 º 3

8 º 12

cross products

Multiplicative inverses: Rewrite the proportion as 3

_

1

8

= 12

_

32

1

and multiply

both sides by the multiplicative inverses of

_

1

8

and

_

32

1

.

(3

_

1

8

)

8

32

=

(12

_

32

1

)

8

32

3

32 = 12

8

cross products

A U D I T O R Y

K I N E S T H E T I C

T A C T I L E

V I S U A L

Solving Problems Using

PARTNER

ACTIVITY

Cross Multiplication

(

Math Journal 2,

pp. 288, 289, 289A, and 289B)

Algebraic Thinking

Assign journal page 288. When most students

have completed the problems, bring the class together and go over

the answers.

Ongoing Assessment:

Journal

Page 288

Problems 1– 6

Recognizing Student Achievement

Use journal page 288, Problems 1–6 to assess students’ ability to use cross

products to write an open number sentence. Students are making adequate

progress if they are able to write open number sentences for Problems 1–6.

Some students may be able to solve mentally for missing variables.

[Operations and Computation Goal 6]

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SOLVING

704-709_EMCS_T_TLG2_G6_U08_L03_576922.indd 706

3/10/11 3:42 PM

(5)

Date Time

Solving Proportions with Cross Products

continued

LESSON

8

3

Set up a proportion for each problem and solve it using cross multiplication.

17. Sarah uses 5 scoops of coffee beans to brew Solution: 8 cups of coffee. How many scoops of beans

does Sarah use per cup?

Answer: Sarah uses scoop(s) of beans per cup of coffee.

18. Jeremiah ran 1 1_

4 miles in 12 minutes. At this Solution:

pace, how long would it take him to run 5 miles?

Answer: It would take Jeremiah minutes to run 5 miles.

19. It took Zach 12 days to read a book that was Solution: 186 pages long. If he read the same amount

each day, how many pages did he read in one week?

Answer: Zach read pages in one week.

20. At sea level, sound travels 0.62 mile in 3 seconds. Solution: What is the speed of sound in miles per hour?

(Hint: First find the number of seconds in 1 hour.)

Answer: Sound travels at the rate of miles per hour.

3,600

0.62

=

3

d

2,232

=

3

d

2,232

_

3

=

3

_

3

d

744

=

d

=

0.62 mile

d

miles

3 sec

3,600 sec

744

1

5

=

8

s

5

=

8

s

5

_

8

=

s

=

5 scoops

s

scoops

8 cups

1 cup

5

_

8

48

108.5

1

_

14

m

=

12

5

1

_

14

m

=

60

m

=

_

60 1 _ 14

m

=

48

m

minutes

=

1

_

14

miles

5 miles

12 minutes

12

p

=

186

7

12

p

=

1,302

p

=

_

1,30212

p

=

108.5

p

pages

=

12 days

7 days

186 pages

114 115 289A_289B_EMCS_S_G6_MJ2_U08_576442.indd 289B 3/9/11 11:13 AM

Math Journal 2,

p. 289B

Student Page

Math Boxes

2. A boat traveled 128 kilometers in 4 hours.

Fill in the rate table.

At this rate, how far did the boat travel in 2 hours 15 minutes?

72 km

LESSON

8

3

Date Time

5. Insert parentheses to make each number sentence true. a. 0.01 ∗ 7 + 9 / 4 = 0.04 b. _ 4 5 ∗ 25 - 10 / 2 = 15 c. _64 / 5 + 3 ∗ 3 = 3 d. 5 ∗ 102+ 102 2 = 700 1. Which rate is equivalent to 70 km in

2 hr 30 min? Fill in the circle next to the best answer.

A. 35 km in 75 min

B.70,000 m in 230 min

C. 140 km in 4 hr 30 min

D. 1,400 m in 300 min

3. A bag contains 1 red counter, 2 blue counters, and 1 white counter. You pick 1 counter at random. Then you pick a second counter without replacing the first counter.

a. Draw a tree diagram to show all possible counter combinations.

b. What is the probability of picking 1 red counter and 1 white counter (in either order)?

4. Add or subtract. a.-303 + (-28) = b. = 245 - 518 c. =-73 + 89 d. 280 - (-31) = 110 111 156 109–111 95 96 247

( )

( )

( )

( ) ( )

R

W B1 B2

R

B1 B2

R

W B2

R

W B1

B1

W

B2

-

331

-

273

16

311

2

_

12

, or

1

_

6 distance (km) 24 72 144 hours 3 _ 4 1 1 _ 2

2

3 1

_

4

48

96

4

_

1 2 278_323_EMCS_S_G6_U08_576442.indd 287 2/26/11 1:15 PM

Math Journal 2,

p. 287

Student Page

Links to the Future

Lesson 8

3

707

Work through the example at the top of journal page 289 with the

class. Show students how the units in the problem function much

like numbers when they are included in the computation. Just as a

number divided by itself is equal to 1, a unit divided by itself is

also equal to 1. It is sometimes said that the units “cancel” and

they can simply be crossed out as shown below.

60 minutes

__

4 minutes

lengths

=

n

lengths

NOTE This is a simple example of a strategy called

dimensional analysis

. Students

will use dimensional analysis in future mathematics and science courses. It is not

necessary to introduce the term at this time.

Have students complete journal pages 289 and 289A, showing how

the units cancel.

After most students have finished these problems, ask them why

keeping track of the units is a useful strategy.

Sample answers: It

helps ensure that I have set up the proportion correctly. It helps

me see the correct unit to use in my answer.

Tell students that it

is not necessary for them to include units every time they solve a

proportion with cross multiplication, but it is a good strategy to

use to check their work or help them on more difficult problems.

Have students solve the problems on journal page 289B. It is not

necessary for them to include units in their work on these

problems, but they may do so if they wish.

Students will apply their knowledge of cross products in future algebra and

science courses. It is important that they be able to use cross products to write

open number sentences.

2

Ongoing Learning & Practice

Playing

Fraction/Whole Number

PARTNER

ACTIVITY

Top-It

(

Student Reference Book,

pp. 319 and 320)

Distribute four each of number cards 1–10 (from the Everything

Math Deck, if available) to each partnership.

Students use cards to form whole numbers and fractions.

They then find and compare the products.

Math Boxes 8

3

INDEPENDENT

ACTIVITY

(

Math Journal 2,

p. 287)

Mixed Practice

Math Boxes in this lesson are paired with

Math Boxes in Lesson 8-1. The skills in Problems 4 and 5

preview Unit 9 content.

(6)

Teaching Master

Name Date Time

LESSON

8

3

Double Number Lines

Howie is making tamales. He used 8 cups of filling to make 4 dozen tamales. How much filling does he need to make 10 dozen tamales?

The double number line below can be used to help solve this problem. Notice that the scale at the top of the number line is labeled in dozens of tamales. The scale at the bottom of the number line is labeled in cups of filling. Find the mark for 8 cups of filling. Notice how it lines up with 4 dozen tamales. This represents the information given in the problem. The per-unit rate is also shown on the number line: Howie uses 2 cups of filling per 1 dozen tamales, so the mark for 1 dozen tamales lines up with the mark for 2 cups of filling. This information was used to complete the double number line.

Dozens of tamales Cups of filling 0 0 1 2 2 4 3 6 4 8 5 10 6 12 7 14 8 16 9 18 10 20

The problem asks how much filling Howie needs to make 10 dozen tamales. Find the mark for 10 dozen tamales. Then find the number on the cups-of-filling scale that lines up with this mark. It’s 20, so Howie needs 20 cups of filling to make 10 dozen tamales.

Use double number lines to help you solve the problems.

1. A marine animal trainer noted that the aquarium’s newest beluga whale ate 150 pounds of food in 3 days. The whale was fed the same amount of food each day.

a. How many pounds of food does the whale eat per day?

50

pounds

b. Use your answer to Part a to fill in the blanks on the top scale of the double number line below.

350

250

200

100

50

Pounds of food Days 0 0 1 2 3 4 5 6 7 300 150

c. If he continues to eat at this rate, how many pounds of food will the whale eat in 5 days?

250

pounds

249A-249B_EMCS_B_MM_G6_U08_576981.indd 249A 3/9/11 12:10 PM

Math Masters,

p. 249A

py g g p STUDY LINK

8

3

Calculating Rates

Name Date Time

If necessary, draw a picture, find a per-unit rate, make a rate table, or use a proportion to help you solve these problems.

1. A can of worms for fishing costs $2.60. There are 20 worms in a can. a. What is the cost per worm?

b. At this rate, how much would 26 worms cost?

2. An 11-ounce bag of chips costs $1.99.

a. What is the cost per ounce, rounded to the nearest cent?

6. A 1-pound bag of candy containing 502 pieces costs 16.8 cents per ounce. What is the cost of 1 piece of candy? Circle the best answer. 1.86 cents 2.99 cents 0.33 cent cent

7. Mr. Rainier’s car uses about 1.6 fluid ounces of gas per minute when the engine is idling. One night, he parked his car but forgot to turn off the motor. He had just filled his tank. His tank holds 12 gallons.

About how many hours will it take before his car runs out of gas? Explain what you did to find the answer.

Sources: 2201 Fascinating Facts; Everything Has Its Price

b. What is the cost per pound, rounded to the nearest cent?

3. Just 1 gram of venom from a king cobra snake can kill 150 people. At this rate, about how many people would 1 kilogram kill?

4. A milking cow can produce nearly 6,000 quarts of milk each year. At this rate, about how many gallons of milk could a cow produce in 5 months?

5. A dog-walking service costs $2,520 for 6 months. What is the cost for 2 months? For 3 years?

111–116 Try This

150,000 people

$2.88

$0.18 per oz

$3.38

625 gallons

$840

$15,120

1 _ 2

Sample answer: 128 oz

=

1 gal; 12 gal

=

1,536 oz;

16 hours

$0.13 per worm

1,536 oz

_

1.6 oz per min

=

960 min;

960 min

_

60 min per hour

=

16 hours

246-284_EMCS_B_G6_MM_U08_576981.indd 249 2/28/11 1:02 PM

Math Masters,

p. 249

Study Link Master

708

Unit 8 Rates and Ratios

Study Link 8

3

INDEPENDENT

ACTIVITY

(

Math Masters,

pp. 249 and 251)

Home Connection

Students solve rate problems on

Math Masters,

page 249.

If you haven’t already done so, review the instructions for

Math

Masters,

page 251 with the class. Students may postpone

completing Parts B and C of the table until after they have

completed Lesson 8-4. If a grocery store posts a unit price, ask

students to check that the price is accurate.

3

Differentiation Options

READINESS

SMALL-GROUP

ACTIVITY

Solving Equations (

ax = b

)

5–15 Min

To provide experience solving equations of the form

ax

=

b,

have

students review and practice solving equations using the method

of their choice.

Suggestions:

6

g

=

54 g

=

9

9

m

=

15

12 m

=

20

10

y

=

35 y

=

3.5

35

k

=

70(125) k

=

250

180

=

15

t

t

=

12

90

=

12

j

j

=

7.5

5

x

=

80

10 x

=

160

3

f

=

0.62(300) f

=

62

29(3)

=

3

p

p

=

29

_

5

1

w

=

20

_

1

2

w

=

102.5

ENRICHMENT

PARTNER

ACTIVITY

Using Double Number Lines

15–30 Min

(

Math Masters,

pp. 249A and 249B)

Students explore an alternative way to solve rate problems by

using double number lines. Tell students that a

double number line

is a number line that has two scales: one above the line and one

below the line. Have students look at the double number line at the

top of

Math Masters,

page 249A. Point out the two scales: dozens of

tamales above, and cups of filling below.

Have students read the top of

Math Masters,

page 249A with a

partner. Ask them to locate the per-unit rate on the number line

and discuss how it can help to determine the scales on each side of

the number line. Then have partnerships solve the problems on

Math Masters,

pages 249A and 249B.

(7)

Teaching Master

LESSON

8

3

Name Date Time

Ingredients for Peanut Butter Fudge

1. The list at the right shows the ingredients used

to make peanut butter fudge but not how much of each ingredient is needed. Use the following clues to calculate the amount of each ingredient needed to make 1 pound of peanut butter fudge. Record each amount in the ingredient list.

Clues

Use 20 cups of sugar to make 10 pounds of fudge.

You need cups of milk to make 5 pounds of fudge.

You need 15 cups of peanut butter to make 48 pounds of fudge. (Hint: 1 cup = 16 tablespoons)

An 8-pound batch of fudge uses 1 cup of corn syrup.

Use 6 teaspoons of vanilla for each 4 pounds of fudge.

Use teaspoon of salt for each 4 pounds of fudge.

2. Suppose you wanted to make an 80-pound batch of fudge. Record how much of each ingredient you would need.

Use the following equivalencies and your ingredient lists to complete each problem.

3 teaspoons = 1 tablespoon

16 tablespoons = 1 cup

3. cups of peanut butter are needed for 80 pounds of fudge.

4.

10

cups of corn syrup are needed for 80 pounds of fudge.

5.

40

tablespoons of vanilla are needed for 80 pounds of fudge.

25

Ingredient List for 80 Pounds of Peanut Butter Fudge

cups of sugar tablespoons of corn syrup

cups of milk teaspoons of vanilla

tablespoons of peanut butter teaspoons of salt

60

400

120

10

3 _ 34 1 _ 2

Peanut Butter Fudge (makes 1 pound) cups of sugar cup of milk tablespoons of peanut butter tablespoons of corn syrup teaspoons of vanilla teaspoon of salt

2

5

2

3

_

4

1

_

1 2

1

_

8

160

160

246-284_EMCS_B_G6_MM_U08_576981.indd 250 2/28/11 1:02 PM

Math Masters,

p. 250

Name Date Time

Double Number Lines

continued LESSON

8

3

For Problems 2–4, fill in the blanks on the double number lines and use them to help you solve the problem.

2. Jamie is ordering supplies for his dog-washing business. Last week, he washed 24 dogs and used 4 bottles of shampoo. Jamie uses the same amount of shampoo for each dog he washes.

42

7

36

6

30

5

24

18

3

12

2

6

1

Dogs Bottles of shampoo 0 0 4

a. How many dogs can he wash with one bottle of shampoo?

6

dogs

b. How many bottles of shampoo should he order if he expects to wash 30 dogs this week?

5

bottles

3. A craft store has skeins of yarn on special. They are selling 2 skeins for $5.

7

6

5

4

3

2

1

$17.50

$12.50

$2.50

$7.50 $10

$15

Skeins Cost 0 $0 $5

a. What is the cost per skein of yarn?

$2.50

b. Holly needs 6 skeins of yarn to make an afghan. How much will the yarn cost?

$15

4. Katie rode her bicycle to work today. The 8-mile ride took her 40 minutes.

11

55

10

50

9

45

40

7

35

6

30

5

25

4

20

3

15

2

10

1

5

Miles Minutes 0 8 0

a. On average, how long does it take Katie to ride one mile?

5

minutes

b. At that rate, how long will it take her to ride 11 miles to get from work to her sister’s house?

55

minutes

249A-249B_EMCS_B_MM_G6_U08_576981.indd 249B 4/1/11 11:45 AM

Math Masters,

p. 249B

Teaching Master

Lesson 8

3

709

EXTRA PRACTICE

INDEPENDENT

ACTIVITY

Calculating Ingredient Amounts

15–30 Min

(

Math Masters,

p. 250)

Students practice solving rate problems by calculating

how much of each ingredient is needed to make 1 pound

and 80 pounds of peanut butter fudge.

ELL SUPPORT

SMALL-GROUP

ACTIVITY

Illustrating Terms

15–30 Min

To provide language support for solving proportions, have students

create a poster that features the steps for using cross products to

solve proportions. Their poster should include the terms

cross

products

and

cross multiplication.

Planning Ahead

Remind students to collect nutrition labels from containers of

food, such as cans of soup, cups of yogurt, and cereal boxes. They

will need to bring these labels to school for use in Lesson 8-5.

If you haven’t already done so, provide students with a copy of

Study Link 8-4 (

Math Masters,

page 251) and remind them to

collect data about the cost and weight of the listed items. They

may postpone the calculations of unit price until after they have

completed Lesson 8-4.

(8)

Copyright © Wright Group/McGraw-Hill

Name Date

Time

249A

LESSON

8

3

Double Number Lines

Howie is making tamales. He used 8 cups of filling to make 4 dozen tamales. How much

filling does he need to make 10 dozen tamales?

The

double number line

below can be used to help solve this problem. Notice that the scale

at the top of the number line is labeled in dozens of tamales. The scale at the bottom of the

number line is labeled in cups of filling. Find the mark for 8 cups of filling. Notice how it lines

up with 4 dozen tamales. This represents the information given in the problem.

The per-unit rate is also shown on the number line: Howie uses 2 cups of filling per 1 dozen

tamales, so the mark for 1 dozen tamales lines up with the mark for 2 cups of filling. This

information was used to complete the double number line.

Dozens of tamales

Cups of filling

0

0

1

2

2

4

3

6

4

8

5

10

6

12

7

14

8

16

9

18

10

20

The problem asks how much filling Howie needs to make 10 dozen tamales. Find the mark

for 10 dozen tamales. Then find the number on the cups-of-filling scale that lines up with this

mark. It’s 20, so Howie needs 20 cups of filling to make 10 dozen tamales.

Use double number lines to help you solve the problems.

1.

A marine animal trainer noted that the aquarium’s newest beluga whale ate

150 pounds of food in 3 days. The whale was fed the same amount of food

each day.

a.

How many pounds of food does the whale eat per day?

pounds

b.

Use your answer to Part a to fill in the blanks on the top scale of the

double number line below.

Pounds of food

Days

0

0

1

2

3

4

5

6

7

300

150

c.

If he continues to eat at this rate, how many pounds of food will the whale

eat in 5 days?

pounds

(9)

Copyright © Wright Group/McGraw-Hill

Name Date

Time

249B

Double Number Lines

continued

LESSON

8

3

For Problems 2–4, fill in the blanks on the double number lines and use them

to help you solve the problem.

2.

Jamie is ordering supplies for his dog-washing business. Last week,

he washed 24 dogs and used 4 bottles of shampoo. Jamie uses the

same amount of shampoo for each dog he washes.

Dogs

Bottles of shampoo

0

0

4

a.

How many dogs can he wash with one bottle of shampoo?

dogs

b.

How many bottles of shampoo should he order if he expects to wash

30 dogs this week?

bottles

3.

A craft store has skeins of yarn on special. They are selling 2 skeins for $5.

Skeins

Cost

0

$0

$5

a.

What is the cost per skein of yarn?

b.

Holly needs 6 skeins of yarn to make an afghan. How much will the

yarn cost?

4.

Katie rode her bicycle to work today. The 8-mile ride took her 40 minutes.

Miles

Minutes

0

8

0

a.

On average, how long does it take Katie to ride one mile?

minutes

b.

At that rate, how long will it take her to ride 11 miles to get from work

to her sister’s house?

minutes

Figure

Updating...

References