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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
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 2321
16
3 ∗ 7 = = 8 ∗ 2 2 3 6918
18
9 ∗ 2 = = 3 ∗ 6 2 8 10420
32
= 8 ∗ 4 10 ∗ 2 = 1 4 20520
20
= 4 ∗ 5 20 ∗ 1= 4 8 10 1680
64
8 ∗ 10 = =16 ∗ 4 6 9 12872
72
12 ∗ 6 = = 9 ∗ 8 5 8 10 1280
60
= 12 ∗ 5 8 ∗ 10 = 5 7 1521105
105
21 ∗ 5 = = 7 ∗ 15 3 5 101545
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
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_ cc
=
9
3. _m 20 = _2 8m
=
5
4. _2 10 = 5_zz
=
25
5. _15 9 = _12kk
=
20
6. 10_12 = d_9d
=
7.5
7. _2 9 = _54 tt
=
12
8. _4 10 = 26_zz
=
65
9. _3 4 = _r 28r
=
21
10. 16_ p = 128_ 40p
=
5
11. _51 102 = _6 hh
=
12
12. _8 j = _72 192j
=
3
4 6 p 15 15 ∗ 4 = = 6 ∗p 278_323_EMCS_S_G6_U08_576442.indd 288 2/21/11 4:41 PMMath 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
Solving Proportions with Cross Products
continuedLESSON
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 AMMath Journal 2,
p. 289
Student Page
Date Time LESSON8
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
continuedLESSON
8
3
x
euros
∗
$25
=
20 euros
∗
$80
x
euros
∗
$25
=
$1,600
∗
euros
x
euros
=
_
$1,600 $25∗ eurosx
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 ∗flightss
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]
PROBLEM
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SOLVING
704-709_EMCS_T_TLG2_G6_U08_L03_576922.indd 706
3/10/11 3:42 PM
Date Time
Solving Proportions with Cross Products
continuedLESSON
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
_
3d
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
_
848
108.5
1
_
14m
=
12
∗
5
1
_
14m
=
60
m
=
_
60 1 _ 14m
=
48
m
minutes
=1
_
14miles
5 miles
12 minutes
12
p
=
186
∗
7
12
p
=
1,302
p
=
_
1,30212p
=
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 AMMath 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 _ 22
3 1
_
448
96
4
_
1 2 278_323_EMCS_S_G6_U08_576442.indd 287 2/26/11 1:15 PMMath 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.
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
poundsb. 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 150c. If he continues to eat at this rate, how many pounds of food will the whale eat in 5 days?
250
pounds249A-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 _ 2Sample 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.
Teaching Master
LESSON
8
3
Name Date Time
Ingredients for Peanut Butter Fudge
1. The list at the right shows the ingredients usedto 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 _ 2Peanut 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
_
41
_
1 21
_
8160
160
246-284_EMCS_B_G6_MM_U08_576981.indd 250 2/28/11 1:02 PMMath Masters,
p. 250
Name Date Time
Double Number Lines
continued LESSON8
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 4a. How many dogs can he wash with one bottle of shampoo?
6
dogsb. How many bottles of shampoo should he order if he expects to wash 30 dogs this week?
5
bottles3. 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 $5a. 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 0a. On average, how long does it take Katie to ride one mile?
5
minutesb. At that rate, how long will it take her to ride 11 miles to get from work to her sister’s house?
55
minutes249A-249B_EMCS_B_MM_G6_U08_576981.indd 249B 4/1/11 11:45 AM
