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**www.everydaymathonline.com**

**Lesson 8**

**3 **

_{703}

_{703}

**Advance Preparation**

**Teacher’s Reference Manual, **

**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 **

**Fraction/Whole**

**Number Top-It**

**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}

_{3}

*Math Journal 2, *

### p. 287

### Students practice and maintain skills

### through Math Box problems.

**Study Link 8**

_{3}

_{3}

*Math Masters, *

### pp. 249 and 251

### Students practice and maintain skills

### through Study Link activities.

**READINESS**

**Solving Equations (**

**ax**

**ax**

**=**

**b**

**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**

**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}

_{3}

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

**Math Journal 2, **

**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 = 6*p*
* * _{_ }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 = _12

*k*

*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, **

**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 }

_{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 **

**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}

_{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, **

**Math Journal 2,**

**p. 289**

**Student Page**

Date Time
**LESSON**

**8**

_{3}

_{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}

_{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, **

**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 }

_{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 }

_{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

Date Time

**Solving Proportions with Cross Products **

**continued****LESSON**

**8**

_{3}

_{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

### _

1_{4 }

*m*

### =

### 12

### ∗

### 5

### 1

### _

14*m *

### =

### 60

*m*

### =

### _

60 1 _ 14*m*

### =

### 48

*m*

### minutes

=### 1

_{ }

### _

1_{4 }

### miles

### 5 miles

### 12 minutes

### 12

*p*

### =

### 186

### ∗

### 7

### 12

*p *

### =

### 1,302

*p*

### =

### _

1,302_{12 }

*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, **

**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_{+}_{ 10}2_{∗}_{ 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, **

**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**

**Fraction/Whole Number**

**PARTNER **

**ACTIVITY**

**Top-It**

**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

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

pounds249A-249B_EMCS_B_MM_G6_U08_576981.indd 249A 3/9/11 12:10 PM

**Math Masters, **

**Math Masters,**

**p. 249A**

py
g
g
p
**STUDY LINK**

**8**

_{3}

_{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, **

**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**

**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}

_{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}

_{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, **

**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

minutes249A-249B_EMCS_B_MM_G6_U08_576981.indd 249B 4/1/11 11:45 AM

**Math Masters, **

**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.

### 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

### Copyright © Wright Group/McGraw-Hill

### Name Date

### Time

**249B**

**Double Number Lines **

**continued**

**continued**