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Chapter 8Algebra: Ratios and Functions

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Chapter 8Algebra: Ratios and Functions

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Lesson 8-1 Ratios and Rates

Lesson 8-2 Problem-Solving Strategy: Look for a Pattern

Lesson 8-3 Ratio Tables

Lesson 8-4 Equivalent Ratios

Lesson 8-5 Problem-Solving Investigation: Choose the Best Strategy

Lesson 8-6 Algebra: Ratios and Equations

Lesson 8-7 Algebra: Sequences and Expressions

Lesson 8-8 Algebra: Equations and Graphs

88Algebra: Ratios and Functions



Five-Minute Check (over Chapter 7)

Main Idea and Vocabulary

California Standards

Example 1

Example 2

Example 3

8-18-1 Ratios and Rates

Ratios and Tangrams



• I will express ratios and rates in fraction form.

• ratio

• rate

• unit rate



Preparation for Standard 6NS1.2 Interpret and

use ratios in different contexts (e.g., batting

averages, miles per hour) to show the relative

sizes of two quantities, using appropriate

notations .


Write the ratio in simplest form that compares the number of scooters to the number of unicycles.


=410

25

unicyclesscooters



Answer: The ratio of unicycles to scooters is ,

2 to 5, or 2:5. This means that for every

2 unicycles there are 5 scooters.

25



Write the ratio in simplest form that compares the number of singers in a duet to the number in an octet.

A. 14

D. 24

C. 18

B. 28



Several students were asked to name their favorite kind of book. Write the ratio that compares the number of people who chose sports books to the total number of responses.

7 students preferred sports out of a total of 7 + 9 + 4 + 5 or 25 responses.



725

sports responsestotal responses

Answer: The ratio in simplest form of the number of

students who chose sports to the total number

of responses is , 7 to 25, or 7:25. So,

seven out of every 25 students preferred

sports.

725



Several students were asked to name their favorite kind of movie. Choose the ratio that compares the number of people who chose thriller movies to the total number of responses in simplest form.

A. 12:18

B. 2:3

C. 2:5

D. 12:30


Find the cost per ounce of a 16-ounce jar of salsa that costs $2.88.

Answer: So, the salsa costs $0.18 per ounce.


$2.8816 ounce

$0.181 ounce

=



A 4 pound package of ground beef costs $3.56. What is the cost per pound?

A. $0.99

B. $0.88

C. $0.98

D. $0.89



Five-Minute Check (over Lesson 8-1)

Main Idea


Example 1: Problem-Solving Strategy

8-28-2 Problem-Solving Strategy: Look for a Pattern



• I will solve problems by looking for a pattern.



Standard 5MR1.1 Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, sequencing and prioritizing information, and observing patterns.

Standard 5NS2.1 Add, subtract, multiply, and divide with decimals; … and verify the reasonableness of results.


Emelia is waiting for her friend Casey to arrive. It is 1:15 P.M. now, and Casey said that he would be on the first bus to arrive after 6:00 P.M. Emelia knows that buses arrive every 30 minutes, starting at 1:45 P.M. How much longer will it be before Casey arrives?



Understand

What facts do you know?

• It is now 1:15 P.M.

• The first bus arrives at 1:45 P.M.

• Casey will be on the first bus after 6 P.M.

What do you need to find?

• How much longer will it be before Casey arrives?



Plan

Start with the time the first bus arrives and look for a pattern.



Solve

Answer: So, the first bus to arrive after 6:00 P.M. is the 6:15 P.M. bus. Since it is now 1:15 P.M., Casey will not arrive for another 5 hours.



Check


Look back at the problem. Continue adding 30 minutes to the previous arrival time until you reach 6:15 P.M.

Then add up the 30-minute periods.






Example 1

Example 2

Example 3

Example 4

8-38-3 Ratio Tables


8-38-3 Ratio Tables

• I will use ratio tables to represent and solve problems involving equivalent ratios.

• ratio table

• equivalent ratio

• scaling


8-38-3 Ratio Tables

Standard 5MR2.3 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning.

Preparation for Standard 5AF1.5 Solve problems involving linear functions with integer values; write the equation; and graph the resulting ordered pairs of integers on a grid.


A recipe calls for 5 cups of water for each cup of pinto beans. Use the ratio table to find how many cups of water should be used for 4 cups of pinto beans.

8-38-3 Ratio Tables


One Way: Find a pattern and extend it.

For 4 cups of beans, you would need a total of 5 + 5 + 5 + 5 or 20 cups of water.

8-38-3 Ratio Tables

2 3

10 15 20


Another Way: Multiply each quantity by the same number.

Answer: So, for 4 cups of pinto beans, you will need 20 cups of water.

8-38-3 Ratio Tables

20


8-38-3 Ratio Tables

The recipe for rice calls for 3 cups of water for each cup of rice. How many cups of water should be used for 6 cups of rice?

A. 18 cups

B. 9 cups

C. 12 cups

D. 16 cups


There are over 50,000 species of spiders. Use the ratio table below to find how many legs a spider has.

8-38-3 Ratio Tables

Answer: So, a spider has 8 legs.

2

168


8-38-3 Ratio Tables

A marathon runner can run 24 miles in 3 hours. How many miles can he run in 1 hour?

A. 16 miles

B. 8 miles

C. 12 miles

D. 4 miles


Coco used 12 yards of fabric to make 9 blouses. Use the ratio table to find the number of blouses she could make with 24 yards of fabric.

Answer: So, with 24 yards of fabric, Coco could make 18 blouses.

8-38-3 Ratio Tables

18

18

13.5


8-38-3 Ratio Tables

Mrs. Stine can grade 48 papers in 96 minutes. How many can she grade in 24 minutes?

A. 6

B. 12

C. 24

D. 96


It takes a worker 70 minutes to pack 120 cartons of books. The worker has 14 minutes of work left. Use a ratio table to find how many cartons of books the worker can pack in 14 minutes.

Answer: So, a worker can pack 24 cartons in 14 minutes.

8-38-3 Ratio Tables

24


8-38-3 Ratio Tables

It takes Sarah 60 minutes to walk 4 miles. How far will she have walked after 30 minutes?

A. 1 mile

B. 2 miles

C. 3 miles

D. 4 miles




Main Idea


Example 1

Example 2

Example 3

Example 4

Example 5

8-48-4 Equivalent Ratios



• I will determine if two quantities are equivalent.



Preparation for Standard 5AF1.5 Solve problems involving linear functions with integer values; write the equation; and graph the resulting ordered pairs of integers on a grid.


Determine if the pair of rates is equivalent. Explain your reasoning.


42 people on 7 teams; 64 people on 8 teams

Answer: These rates are not equivalent since they are not the same.

42 people7 teams

= 6 people per team

64 people8 teams

= 8 people per team



Determine if the pair of rates are equivalent.2 chapters in one day; 18 chapters in 9 days

A. Yes, both are 2:1.

B. Yes, both are 18:9.

C. No.

D. not enough information to solve



Determine if the pair of rates is equivalent. Explain your reasoning.

20 rolls for $5; 48 rolls for $12

Answer: These are equivalent because the rates are the same.

20 rolls$5

= $4 per roll

48 rolls$12

= $4 per roll



Determine if the pair of rates is equivalent. $12 for 3 hours; $15 for 5 hours

A. Yes, both are $4 an hour.

B. Yes, both are $5 an hour.

C. No, they are not the same.

D. not enough information


One day Jafar sold 21 pizzas in 3 hours. The next day he sold 35 pizzas in 5 hours. Are these selling rates equivalent? Explain your reasoning.


Write each rate as a fraction. Then find its unit rate.


Answer: Since the rates have the same unit rate, they are equivalent. So, Jafar’s selling rates are equivalent.


21 pizzas3 hours

=7 pizzas1 hour

35 pizzas5 hours

=7 pizzas1 hour



Paella sold 27 magazine subscriptions in 3 hours. The next day she sold 32 magazine subscriptions in 4 hours. What are the selling rates for each day? Are they equivalent?

A. 9, 9; yes

B. 9, 8; no

C. 8, 8; yes

D. 8, 9; no


Determine if the pair of ratios is equivalent. Explain your reasoning.

Answer: Since the fractions are not equivalent the ratios are not equivalent.


5 laps swam in 8 minutes; 11 laps swam in 16 minutes

Write each ratio as a fraction.

The numerator and denominator do not multiply by the same number. So, they are not equivalent.

5 laps8 minutes

11 laps16 minutes

=?


B. No.


Determine if the pair of ratios is equivalent.

15 pages read in 30 minutes; 22 pages read in 40 minutes

D. not enough information

A. Yes, they are both . 12

C. Yes, they are both . 23


Determine if the pair of ratios is equivalent. Explain your reasoning.


8 corrals with 56 horses; 4 corrals with 28 horses

Write each ratio as a fraction.

8 corrals56 horses

1 corral7 horses

=?


Answer: Since the fractions are equivalent, the rates are equivalent.


4 corrals28 horses

1 corral7 horses

=?



Determine if the pair of ratios is equivalent.

7 barnyards with 49 cows; 9 barnyards with 63 cows

A. Yes, both are 1 barnyard per 7 cows.

B. Yes, both are 7 barnyards per 1 cow.

C. Yes, both are 1 barnyard per 9 cows.

D. No.




Main Idea


Example 1: Problem-Solving Investigation

8-58-5 Problem-Solving Investigation: Choose the Best Strategy



• I will choose the best strategy to solve a problem.



Standard 5MR2.3 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning.

Standard 5SDAP1.1 Know the concepts of mean, median, and mode; compute and compare simple examples to show that they may differ.


JUWAN: I took my dog to the veterinarian’s office. While waiting, I noticed that there were more dogs than cats in the waiting room. The vet said that for about every 5 dogs he sees, he sees 2 cats.

YOUR MISSION: Find about how many dogs the vet will see if 21 total pets come into the office.



Understand

What facts do you know?

• You know that the ratio of dogs to cats is about 5:2.

What do you need to find?

• You need to find about how many dogs the vet will see.



Plan

Use counters to act out how many dogs the vet will see.



Solve

Use red counters to represent the dogs and yellow counters to represent the cats. Since the ratio of dogs to cats is 5:2, place 5 red counters and 2 yellow counters in a group. Make groups of 7 counters until you have 21 counters total.



Solve

After three groups there are 21 counters, so you can stop making groups. Find the number of red counters to find how many dogs the vet will see. 5 + 5 + 5 = 15.


Answer: So, if the vet sees 21 pets, about 15 of them will be dogs.


Check


Find the ratio of red counters to yellow counters. If the ratio is equivalent to the original ratio, 5:2, then the answer is correct.




Main Idea


Example 1

Example 2

Example 3

8-68-6 Algebra: Ratios and Equations

Example 4

Example 5

Example 6



• I will solve equations using equivalent fractions.



Standard 5AF1.1 Use information taken from a graph or equation to answer questions about a problem situation.

Standard 5AF1.2 Use a letter to represent an unknown number; write and evaluate simple algebraic expressions in one variable by substitution.



Since 4 × 7 = 28, multiply the numerator and denominator by 7.

Solve = .45

28x

45

2835

= 5 × 7 = 35

45

28x

=

Answer: So, x = 35.



A. x = 36

B. x = 32

C. x = 40

D. x = 2

Solve = .58

20x



Answer: So, b = 4.


THINK What number multiplied by 4 equals 16? The answer is 4.

b5

1620

=

45

1620

=

Solve = .b5

1620



A. h = 12

B. h = 5

C. h = 6

D. h = 1

Solve = . h6

636



Solve = .1938

n22

Since = , then n = 11, which would make the

equation true because is the same as .

1938

12

12

1122

Answer: So, n = 11.



Solve = .1845

m5

A. m = 2

B. m = 9

C. m = 3

D. m = 7



Solve = .2860

7t


THINK What number multiplied by 4 equals 60? The answer is 15.

7t

2860

=

715

2860

=

Answer: So, t = 15.



Solve = .6480

4y

A. y = 3

B. y = 6

C. y = 5

D. y = 4



Out of the 40 students in a gym class, 12 say soccer is their favorite sport. Based on this result, predict how many of the 4,200 students in the community would rate soccer as their favorite sport.

Write and solve an equation. Let s represent the number of students who can be expected to prefer soccer.



Class School

12 s40 4200

prefer soccer

total students

prefer soccer

total students=

The denominators 40 and 4,200 are not easily related by multiplication, so simplify the ratio 12 out of 40. Then solve using equivalent fractions.



1240

=s

4200=

310

Since 10 × 420 = 4,200, multiply the numerator and denominator by 420.

Answer: So, about 1,260 out of 4,200 students in the school can be expected to prefer soccer.



A. 264 girls

B. 300 girls

C. 260 girls

D. 284 girls

Out of the 30 kids in Mrs. Ankrum’s class, 12 are girls. Based on this result, predict how may of the 660 students in the school are girls.



Cedric earned $184 for 8 hours of work. At this rate, how much will he earn for 15 hours of work?

Step 1 Set up the equation. Let a represent the amount of money to be earned.

184 dollars

8 hours=

a dollars

15 hours



Step 2 Find the unit rate.

184 dollars

8 hours=

23

1=

$345

15 hours

Answer: So, Cedric will earn $345 for working for 15 hours.



A. $174

B. $870

C. $850

D. $445

Julio earned $145 for mowing 5 lawns. At this rate, how much will he earn for 30 lawns?






Example 1

Example 2

Example 3

Example 4

8-78-7 Algebra: Sequences and Expressions



• I will extend and describe arithmetic sequences using algebraic expressions.

• sequence

• term

• arithmetic sequence



Standard 5AF1.2 Use a letter to represent an unknown number; write and evaluate simple algebraic expressions in one variable by substitution.

Standard 5AF1.5 Solve problems involving linear functions with integer values; write the equation; and graph the resulting ordered pairs of integers on a grid.



Use the words and symbols to describe the value of each term as a function of its position. Then find the value of the tenth term in the sequence.

Notice that the value of each term is 7 times its position number. So the value of the term in position n is 7n.



Now find the value of the tenth term.

7n = 7 × 10

= 70

Replace n with 10.

Multiply.

Answer: So, the value of the tenth term in the sequence is 70.



Use the words and symbols to describe the value of each term as a function of its position. Then find the value of the fifth term in the sequence.

A. 7 times its position number; 7n, 35

B. 6 times its position number; 6 + n, 11

C. 6 times its position number; 6n, 30

D. 3 times its position number; 3n, 15




Notice that the value of each term is 2 more than its position number, so the rule is n + 2.



Now find the value of the tenth term.

n + 2 = 10 + 2

= 12

Replace n with 10.

Add.

Answer: So, the value of the tenth term in the sequence is 12.




A. 2 less than its position number; 2 – n, 8

B. 2 less than its position number; n – 2, 8

C. 3 less than its position number; 3 – n, 7

D. 3 less than its position number; n – 3, 7


MEASUREMENT There are 60 seconds in 1 minute. It takes Panya 9 minutes to walk to school. Make a table, and then write an algebraic expression relating the number of seconds to the number of minutes. Find how many seconds it takes Panya to walk to school.



Notice that the number of seconds is 60 times the number of minutes.



Now find the ninth term.


60n = 60 × 9

= 540

Replace n with 9.

Multiply.

Answer: So, it will take Panya 540 seconds to walk to school.



There are 60 minutes in an hour. It takes Mr. Daugherty 5 hours each week to grade all of his fifth graders’ papers. Choose an expression and correct answer that represents the amount of minutes Mr. Daugherty spends grading papers each week.

A. 60h; 300 C. 60 – h; 55

D. 60 = h; 5B. ; 1260h


The table to the right shows the number of plants in a garden, based on the number of rows. Write an expression to find the number of plants in n rows.




Answer: So, 3n + 1 gives the number of flowers in n rows.

The number of plants increases by 3, so the rule contains 3n. If the rule were simply 3n, then the valuefor 1 row would be 3. Notice that adding 1 to the number of rows multiplied by 3 gives the number of plants.



Choose the expression to find the number of cars in each row.

A. n + 7

B. n × 8

C. 3n × 8

D. 6n + 2




Main Idea


Example 1

Example 2

Example 3

8-88-8 Algebra: Equations and Graphs

Example 4

Example 5

Example 6



• I will write an equation to describe a linear situation.



Standard 5AF1.1 Use the information taken from a graph or an equation to answer questions about a problem situation.

Standard 5AF1.5 Solve problems involving linear functions with integer values; write the equation; and graph the resulting ordered pairs of integers on a grid.


Write an equation to represent the function displayed in the table.


Each output y is equal to 5 times the input x.



Answer: So, the equation that represents the function is y = 5x.



Choose the equation that represents the function displayed in the table.

A. x = 6y

B. y = 3y + 3

C. y = 6x

D. y = x + 6


Javier sells handmade notebooks. He charges $25 for each book. Make a table to show the relationship between the number of b books sold and the total amount Javier earns t.

The total earned (output) is equal to $25 times the number of books made (input).




Jean sells dream catchers. She charges $15 for each one. Which table correctly shows the relationship between the number of b dream catchers and the total amount Jean earned?

A.




B.




C.




D.


A.




Write an equation to find the total amount earned t for selling b books.


Study the table.


The total earned equals $25 times the number of books Javier sells.


Answer: So, the equation is t = 25b.



Choose the equation to find the total amount t Jean earned.

A. 15t = b C. t + 15 = b

D. 15 × 5 = bB. = b15t


How much will Javier earn if he sells 7 books using the equation t = 25b?


t = 25b Write the equation.

t = 25(7) Replace b with 7.

t = 175 Simplify.

Answer: So, Javier will earn $175.



How much will Jean earn if she sells 9 dream catchers? Use the equation t = 15b.

A. $140

B. $135

C. $160

D. $155


The table below shows the amount that a kennel charges for grooming a dog. Write a sentence and an equation to describe the data. Then find the total cost of grooming 11 dogs, 12 dogs, and 13 dogs.



The cost of getting a dog groomed is $12 for each dog. The total cost t is $12 times the number of dogs d. Therefore, t = 12d.




The table below shows the amount that a Girl Scout troop charges for a box of cookies. Choose the correct equation to describe the data. Then find the total cost for 12, 13, and 14 boxes of cookies.



A. 3t = c; $30, $40, $50

B. 3c = t; $30, $40, $50

C. c + 3 = t; $36, $39, $42

D. 3c = t; $36, $39, $42


Graph the results from Example 5 on a coordinate plane.



Step 1 Make a coordinate place with the d values along the x-axis and the t values along the y-axis.


Step 2 Using the (d, t) values from Example 5, plot the coordinate plane.



Use the information in the table to make a graph on a separate sheet of paper. Choose the best description of the line that is formed.

A. curved, descending line

B. steep, straight, upward line

C. straight, horizontal line

D. “u” shaped line




Five-Minute Checks

Ratios and Tangrams


Lesson 8-1 (over Chapter 7)

Lesson 8-2 (over Lesson 8-1)








A. 8

B. 7

C. 12

D. 9

(over Chapter 7)

Solve 8p = 56.


A. 10

B. 3

C. 12

D. 15

(over Chapter 7)

Solve 30 = 3f.


A. 5

B. 4

C. 3

D. 6

(over Chapter 7)

Solve –15v = –45.


(over Chapter 7)

A. –7

B. 10

C. 7

D. –10

Solve 7y = –70.


C. 23

(over Lesson 8-1)

Write the ratio as a fraction in simplest form.

16 apples out of 24 pieces of fruit

A. 46

B. 28

D. 86


C. 15

(over Lesson 8-1)

Write the ratio as a fraction in simplest form.

18 dogs out of 90 pets

A. 45

B. 810

D. 910


A. $0.50/1 folder

(over Lesson 8-1)

Write the following as a unit rate.

$5 for 10 folders

B. $0.10/1 folder

C. $5.00/1 folder

D. $2.00/1 folder


D. 15 chairs/1 row

(over Lesson 8-1)

A. 12 chairs/1 row

Write the following as a unit rate.

48 chairs for 3 rows

B. 8 chairs/1 row

C. 16 chairs/1 row


B. multiply by 3; 1,458; 4,374; 13,122

(over Lesson 8-2)

C. multiply by 3; 972, 1,844; 3,688

Solve. Luis saw the numbers below in a science report. Describe the pattern. Then find the next 3 numbers in the pattern.

A. multiply by 3; 1,548; 4,747; 13,122

6, 18, 54, 162, 486, __, __, __


A. $26

B. $32

C. $20

D. $24

(over Lesson 8-3)

Use the ratio table to solve the problem. A dozen roses sell for $18. How much will 16 roses cost?


A. $30

B. $36

C. $12

D. $18

(over Lesson 8-3)

Use the ratio table to solve the problem. How much will 24 roses cost?


A. No

B. Yes

(over Lesson 8-4)

Determine if each pair of ratios or rates are equivalent.

$18 in 3 days; $42 in 6 days


A. No

B. Yes

(over Lesson 8-4)


50 desks in 2 rooms; 75 desks in 3 rooms


A. No

B. Yes

(over Lesson 8-4)


8 fruit drinks for $20; 9 fruit drinks for $24


(over Lesson 8-4)

A. No

B. Yes


12 dogs out of 18 pets; 10 dogs out of 15 pets


A. 35 rings

B. 19 rings

C. 34 rings

D. 21 rings

(over Lesson 8-5)

Solve. Tell what strategy you used. Maria is building chains. She uses 1 ring on the first chain, 6 rings on the second, 11 rings on the third, and 16 rings on the fourth. If she continues the pattern, how many rings will be on the next chain?


A. 15

B. 25

C. 100

D. 30

(over Lesson 8-6)

Solve. = w40

58


A. 18

B. 20

C. 15

D. 25

(over Lesson 8-6)

Solve. = 611

n33


A. 18

B. 22

C. 60

D. 15

(over Lesson 8-6)

Solve. = 300a

503


A. 90

B. 88

C. 15

D. 45

(over Lesson 8-6)

Solve. = 1511

x66


A. 5n + 5; 45

B. 15n + 5; 60

C. 15n + 10; 80

(over Lesson 8-7)

Use words and symbols to describe the value of each term as a function of its position. Then find the value of the ninth term in the sequence.

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