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MathLinks: Ratios and Proportional Relationships (Student Packet) 7-CORE2 – SP

7-CORE2.1 Proportional Reasoning • Use sense-making strategies to solve proportional reasoning

problems. • Solve problems that involve whole numbers and decimals. • Solve rate problems.

1

7-CORE2.2 Best Buy Problems • Use different methods to determine the best buy, including

tables and graphs. • Write equations that represent relationships between the

quantity of an item purchased and the cost of the purchase. • Determine if quantities are in a direct proportional relationship.

8

7-CORE2.3 Complex Fractions • Identify rates as complex fractions. • Convert complex fraction rates to unit rates.

14

7-CORE2.4 Vocabulary, Skill Builders, and Review 20

7-CORE2 STUDENT PACKET

GRADE 7: MATHEMATICS COMMON CORE SUPPLEMENT RATIOS AND PROPORTIONAL RELATIONSHIPS

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Ratios and Proportional Relationships

MathLinks: Ratios and Proportional Relationships (Student Packet) 7-CORE2 – SP0

WORD BANK

Word or Phrase Definition or Explanation Example or Picture

complex fraction

cross-multiplication property

direct proportion

ratio

unit price

unit rate

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Ratios and Proportional Relationships 2.1 Proportional Reasoning


PROPORTIONAL REASONING

Ready (Summary) We will use sense-making strategies to solve proportional reasoning problems.

Set (Goals)

• Use sense-making strategies to solve proportional reasoning problems.

• Solve problems that involve whole numbers and decimals.

• Solve rate problems.

Go (Warmup)

Solve each pencil problem in two ways. Show your work.

1. If 1 pencil costs $0.17, then what is the cost of 4 pencils?

Method 1: Method 2: Answer: 4 pencils cost $ ________

2. If 5 pencils cost $1.15, then what is the cost of 1 pencil?

Method 1: Method 2: Answer: 1 pencil costs $ _________

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PENCIL PROBLEMS Solve each pencil problem in two ways. Show your work.

1. If 3 pencils cost $0.57, then what is the cost of 6 pencils?

Method 1: Method 2: Answer:



3. How much will 4 pencils cost if 6 pencils cost $1.68?


4. How much will 2 pencils cost if 7 pencils cost $1.82?


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PAPER PROBLEMS Solve each paper problem in two ways. Show your work.

1. How much will 50 copies cost if it costs $4.50 to make 100 copies?


2. If the cost of making 20 copies is $1.30, how much will 1 copy cost?


3. What will it cost to make 1,000 copies if the cost of 1 copy is $0.041?


4. If the cost of 100 copies is $3.10, how many copies can be made for $155?


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PETER AND POLLY PAINTING PROBLEMS 1. How many feet are in 1 yard?______ How many square feet are in 1 square yard? ______ Solve each painting problem in two ways. Show your work such as rates, proportions or pictures.

2. If Peter takes 20 minutes to paint a fence that is 3 feet tall and 10 feet long, what is Peter’s rate in square feet per minute (ft2/min)?

Method 1: Method 2:

Answer:

3. If it takes Polly 20 minutes to paint a fence that is 3 feet tall and 10 feet long, how long will it take her to paint a fence that is 6 feet tall and 20 feet long, assuming she continues at the same rate? Method 1: Method 2:

Answer:

4. If Peter paints 1 square yard in 2 minutes, how long will it take him to paint a fence that is 3 feet tall and 30 feet long?


5. If Polly paints 1 square foot in 1 minute 30 seconds, how many minutes will it take her (not counting breaks because she gets tired, and assuming she continues to paint at the same rate) to paint a fence that is 1 yard by 10 yards?

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WRITING PROPORTIONS IN DIFFERENT WAYS

1. The fractions (or ratios) 34

and 68

are equivalent.

a. How can you show that the proportion 3 6=4 8

is true by using the multiplication

property of 1?

b. Madeleine started with the proportion above and wrote a new equation by taking the reciprocal of both sides. Write Madeleine’s equation and explain why it is true. We will call this the fraction inverse property.

2. If 3 =

4 4x , then x =______. In a proportion, when the denominators are the same, the

_______________ must be the same. We will call this the numerator equality property.

3. The symmetric property of equality states that if a = b then b = a. Use this property to

rewrite 3 = 4 8

x with the variable on the left.

4. Match each statement in the solution of the proportion 3 4 =

5 x with an appropriate reason.

Statements Reasons

i 3 4 = 5 x

a fraction inverse property

ii 5 = 3 4

x b multiplication of fractions

iii 5 4 3 = 3 4 4 3

x• • c numerator equality property

iv 20 3 = 12 12

x d original equation

v 20 = 3x e multiplication property of 1

vi 3 = 20x f symmetric property of equality

5. How can you get from statement (i) to statement (vi) in one step?

We will call this the cross multiplication property.

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THE CROSS MULTIPLICATION PROPERTY

The cross multiplication property states that if = , then = ( 0, 0).a cb d

ad bc b d≠ ≠

Use the cross multiplication property to solve for x: (Remember that x does not need to be a whole number.)

1. 23 21= x

2 • 21 = 3 • x = 3x = x

2. 59 7=x

x • 7 = ______

3.

2 513

=x

4. 3 47=

x

5. 1036 40

=x 6. 2.55 12

= x

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SOLVING RATE PROBLEMS

Write a rate for each problem. Then, show work such as equivalent fractions, proportions, and pictures to solve each problem.

Problem Guide Rate Work


cost

# of pencils

$0.45

5 pencils 4 pencils = x

2. You drive at a rate of 65 miles per hour for 4 hours. How far will you go?

3. If you drive 25 miles at a rate of 50 miles per hour. How long will this take?

4. If 1 square yard can be painted in 30 seconds, how many minutes would it take to paint 42 square yards?

5. How many pencils can be purchased with $0.51 if the cost of 8 pencils is $1.36?

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Ratios and Proportional Relationships 2.2 Best Buy Problems


BEST BUY PROBLEMS

Ready (Summary) We will use tables, graphs, and symbolic representations to determine whether a relationship is proportional.

Set (Goals)

• Use different methods to determine the best buy, including tables and graphs.

• Write equations that represent relationships between the quantity of an item purchased and the cost of the purchase.

• Determine if quantities are in a direct proportional relationship.

Go (Warmup)

Suppose you are shopping for your favorite protein drink. You find there are two different size drinks available. The 12-ounce drink costs $4.80 and the 21-ounce drink costs $6.30.

1. Consider the ratio of dollars spent to ounces of drinks purchased with the Regular Size and

the Super Size. Are these equivalent ratios?

2. Which drink is the better buy? Explain.

$4.80 for 12 ounces

21 ounces for $6.30

Regular Size Super Size

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BAGELS

Two quantities that vary are directly proportional if one is a constant multiple of the other. One method of testing for a direct proportion is to write and compare rates. If the rates of one of the quantities to the other are all equal, then the quantities are directly proportional.

2. Write ratios for each shop using the data table above.

SH

ME

AR

‘N T

HIN

GS

cost in dollars # of bagels

34

68

12

Simplify 34

Unit rate (price per bagel)

HO

LE-Y

B

RE

AD

cost in dollars # of bagels

45

10

Simplify 45

Unit rate (price per bagel)

3. Which shop has better buy? ______________________ Circle entries in your tables to justify your answer and explain using words.

1. Complete the tables. Assume each shop will sell you any number of bagels at the rate shown above.

SHMEAR ‘N THINGS

HOLE-YBREAD

# of bagels (x) cost in dollars (y) # of bagels (x) cost in dollars (y)

4 5

8 10

12 15

16 20

20 25

SHMEAR ‘N THINGS 4 bagels for $3.00

HOLE-Y BREAD 5 bagels for $4.00

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TESTING FOR PROPORTIONAL RELATIONSHIPS Each table below shows a relationship between quantities. Write, simplify, and compare the rates. Then state whether the pairs in each table represent a proportional relationship. 1. The number of meals Jill served to the homeless and their cost.

Monday Tuesday Wednesday Thursday Friday # of meals 45 60 20 100 55 cost $135 $180 $60 $300 $165

2. The number of bags of feathers Jamie used to make pillows.

# of bags 9 24 3 18 4.5 # of pillows 6 16 2 12 3

3. The number of tables Joyce rented for a party and their cost.

# of tables 1 2 3 4 5

cost $20 $25 $30 $35 $40 4. The size of each room in Stanley’s house and the number of plants in each room.

bedroom living room den kitchen patio square feet 100 200 150 125 250 # of plants 4 8 6 5 10

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TORTILLAS

2. Find the unit price at each tortilla store. 3. Write equations to relate the number of

tortillas to the cost.

FLAT ‘N ROUND y = ______________ WRAP IT UP y = ______________

1. Complete the tables. Assume each shop will sell any number of tortillas at the rate shown.

FLAT ‘N ROUND

WRAP IT UP

# of tortillas

(x)

cost (y)

# of tortillas

(x)

cost (y)

0 0

3 4

6 8

4. Label and scale the grid. Graph the data using two different colors.

5. Identify the y-coordinate when x = 1 FLAT ‘N ROUND (1, ____) WRAP IT UP (1, ____) 6. Explain the meaning of these coordinate

pairs in this context.

7. How are these coordinates related to the linear function in the form y = mx?

FLAT ‘N ROUND 3 tortillas sell for $0.60

WRAP IT UP 4 tortillas sell for $0.40

These equations are linear functions in the form y = mx. This is called a direct proportion equation because y is directly proportional to (is a constant multiple of) x. SAMPLE

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PITA BREAD

1. Complete the tables and graphs. The graph for EAT-A PITA is provided. A partial table for PAPA’s PITA is provided. Use tables and graphs to complete the pricing information above. Assume each shop will sell any number of pitas at the rate shown.

2. Find the unit prices at both pita stores. 3. Write equations to relate the number of

pitas to the cost. PAPA’S PITA y = ____________ EAT-A PITA y = ____________ 4. Verify that the relationship between the

number of pitas and the price paid at EAT-A PITA are a direct proportion by creating ratios and checking for equivalent fractions.

PAPA’S PITA

EAT-A PITA

# of pitas (x)

cost (y)

# of pitas (x)

cost (y)

2 $1.00 0

5

10

5. Which graph illustrates a slower rise in

price? How do you know? 6. Notice that both graphs are straight lines

that go through the origin. What does this mean in the context of the problem?

PAPA’S PITA 6 pitas for $____

EAT-A PITA 5 pitas for $____

The graph of a proportional relationship will result in a straight line through the origin.

10

●

●

●

EAT-A PITA $5

$0

1 5

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PIZZA

1. Complete the tables and graphs. The graph for PIZZA PLACE is provided. Use tables and graphs to complete the pricing information above.

2. Using data from the table, explain/show

which relationship is proportional. 3. Complete the graphs. Do both graphs

represent proportional relationships? Why or why not?

DOOR-TO-DOOR

PIZZA PLACE

# of

pizzas (x)

cost (y)

# of pizzas

(x)

cost (y)

1 13 0

2 21 1

3 29 2

4 3

5 4

PIZZA PLACE

4. Using the data from the tables, determine

which is the better buy. Explain your reasoning.

5. Identify the coordinates of the point where

the two graphs intersect. 6. What does this point mean within the

context of the problem?

MOON’S 5 croissants for $3

More than 5, $1 each

PIZZA PLACE

$_____per pizza

$45

$0

1 5

DOOR-TO-DOOR PIZZA

$_____ per pizza

$5 for delivery

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Ratios and Proportional Relationships 2.3 Complex Fractions


COMPLEX FRACTIONS

Ready (Summary) We will learn how to compute the unit rate for various measurements. We will learn strategies for working with complex fractions.

Set (Goals)

• Identify rates as complex fractions. • Convert complex fraction rates to unit

rates.

Go (Warmup)

Solve each problem and show your work.

1. How many hours are in 360 minutes? 2. How many inches are in 2 feet?

3. •3 24 5

. ÷2 13 6

4

5. Wendy walks at a rate of 4 miles per hour. If Wendy walks for two hours, how far will she walk?

6. Cynthia travels 6 miles at a rate of 2 miles per hour. How long will it take her to arrive at her destination?

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STRATEGIES FOR SIMPLIFYING COMPLEX FRACTIONS

A complex fraction is a fraction whose numerator or denominator is a fraction. Here are two mathematical strategies for simplifying complex fractions. Fill in the missing parts of the fractions.

Strategy 1: Write the complex fraction as a division problem.

= = =

÷ •1

123 24

(step 1) (step 2) (step 3)

Strategy 2: Multiply by a form of the “big one” to create a denominator equal to one.

= =

••••

• =

1423

4 33 4

1234

(step 1) (step 2) (step 3) (step 4)

For strategy 1: 1. Draw a circle around the step where the complex fraction is written as a division problem.

2. What mathematical principle is applied in step 2? For strategy 2: 3. Draw a circle around the step where a form of “big one” is used to create a denominator

equal to one. Why do you think the numbers that form the “big one” were chosen?

4. What mathematical principle is applied in step 1?

5. What happens to the denominator of the complex fraction from step 2 to step 3? Connecting the strategies: 6. In what step is it clear that strategy 1 and strategy 2 will give the same result?

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SIMPLIFYING COMPLEX FRACTIONS Simplify each fraction by rewriting it as a division problem (strategy 1).

. 3515

1

. 2356

2

Simplify each fraction by using a form of “the big one” to create a denominator equal to one (strategy 2).

. 1458

3

. 5936

4

5. Which strategy do you prefer? Why? 6. Blakely said, “I think I know a shortcut for simplifying complex fractions,” and she drew this

picture. Is her work correct? Explain what she did.

= =1

4 223 6 34

Simplify each complex fraction. Use a strategy of your choice.

. 53110

7

. 25215

8

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WATER SPORTS

Michael likes to surf. It takes him 10 minutes to paddle from the beach 18

of a mile out into the

ocean to catch a wave. He rides the wave for one minute back to the beach. 1. Write ratios that express his rate as he paddles out into the ocean.

miles miles miles

1 hour minutes 1 hour6

→ = =

2. Write ratios that express his rate as he rides the wave back to the beach.

miles miles miles

minutes hour 1 hour

→ = =

Jody likes to water ski. Jody skis for 1

2 of a mile at a speed of 20 miles per hour.

3. What is her rate of speed? ________

4. How many hours does it take Jody to travel 1

2 mile at this rate? Use the formula:

Distance = (Rate)(Time)

D = (R)(T)

5. How many minutes does it take Jody to travel 12

mile at this rate?

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MORE WATER SPORTS

Many sports involve traveling from start to finish. Find the missing information.

Remember: Distance = (Rate)(Time)

Sport Rate miles

per hour (mph)

Time Distance Show work here

Minutes Hours 1. canoeing 4 mph

12

____miles ( ) ⎛ ⎞⎜ ⎟⎝ ⎠

•

• 12

=

= 4

= ______ miles

D R T

D

D

2.

swimming

1 mph 14

mile

3. rowing 90 min 7.5 miles

4.

jet skiing 50 mph 20 miles

5.

skiing*

131 48 miles

*Longest ski record set in 1974.

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QUIRKY WORLD RECORDS

Solve each problem and show your work.

1. The mile record for one person carrying another on their shoulders was set at a rate of four miles per hour. If the race was one mile long, how many hours did it take them to complete the race?

2. While piggyback racing, a man completed a 1 1

2 mile race in 1

3of an hour. What was

his rate in miles per hour?

3. The record for running a race while balancing a baseball bat on one finger was set at a rate of 10 miles per hour. If the race was one mile long, how many minutes did it take the runner to complete the race?

4. The fastest sack race was set at a rate of 3 miles per hour. If the sack racer finished the race in 20 minutes, what was the distance of the race?

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Ratios and Proportional Relationships 2.4 Vocabulary, Skill Builders, and Review


VOCABULARY, SKILL BUILDERS, AND REVIEW

FOCUS ON VOCABULARY Match the words to the clues.

Words Clues 1. _______ proportional relationship 2. _______ direct proportion 3. _______ proportion 4. _______ rate 5. _______ ratio 6. _______ unit price 7. _______ unit rate 8. _______complex fraction

9. _______ cross-multiplication property

a. A ________ is a comparison of two numbers by division.

b. A price for one unit of measure.

c. A rate for one unit of measure

d. A _______ is a ratio in which the numbers have units attached to them.

e. A _________ is a statement that two ratios

are equal.

f. The _____________________ states that if = , then = ( 0, 0).a cb d

ad bc b d≠ ≠

g. A fraction that has one or more fractions in

the numerator and/or denominator. h. A linear functions in the form y = mx is

also called a ____________________. i. A ___________________ can be shown

by two equivalent ratios and will result in the graph of a straight line through the origin.

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SKILL BUILDER 1

Solve each proportion using any method.

1.

86 12

= n

2.

10 5 50

=n

3. 10 6 4

=n

4. 15 3 6

= n

5. 30 12 5

= n 6. 5 8 12

= n

Solve for n. The first step is given.

7. 76 9=n

• •

3 7 26 3 9 2

=n

8. 55 6=n

• •

6 5 55 6 6 5

=n

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SKILL BUILDER 2 Write a rate and write a proportion for each problem. Then, use equivalent fractions or the cross multiplication property to solve each proportion.

Problem Guide Rate Proportion

1. How much will 27 copies cost if it costs $4.50 to make 150 copies?

2. If the cost of making 25 copies is $1.30, how much will 110 copy cost?

3. What will it cost to make 1,580 copies if the cost of 50 copies is $2.05?

4. If it takes 2 12

hours to

paint 50 square feet, how long will it take to paint 110 square feet?

5. How many square feet of lawn can be mowed in 12 minutes if 215 square feet take 22 minutes?

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SKILL BUILDER 3

In the following charts, simplify each rate if possible. Then determine which charts represent examples of proportional relationships. Explain answers. 1. Mason likes to hike. This chart shows her distances and the number of hours hiked.

miles 3 6 8 10 12 hours 1 2 4 5 6

2. Ramsay bought an annual pass to the zoo. This chart shows his trips and the cost.

# of trips 0 1 2 3 4 cost $35 $35 $35 $35 $35

3. Graham ran a race. Here are his times as the race progressed

Distance (miles) 0.10 0.5 2 3 4

times 1 min. 5 min. 13

hr. 12

hr. 40 min.

. 4. Camille went to an amusement park. This chart shows the cost for tickets.

# of tickets 1 2 3 4 5 cost $5 $6 $7 $8 $9

Solve the rate problems using any method.

5. If 12 pencils cost $1.68, how much money will you need to purchase 9 pencils?

6. You ride your bicycle at 10 miles per hour (mi/hr). How many hours will it take you to ride 6 miles?

7. Lawrence finished painting 8 square yards in 28 minutes. At that rate, how long would it take him to paint 1 square yard?

8. How long would it take Simone to completely color a 792 square inch poster board if she colors a smaller 88 square inch board in 5 minutes?

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SKILL BUILDER 4

1. Fill in the table. Assume each shop will

sell any number of pairs of socks at the rate shown. Which is the better buy? Use entries in the tables to explain your reasoning.

SOCKS ‘R WE HOSIERY HUT

# of pairs (x)

Cost (y)

# of pairs (x)

cost (y)

2. Label and scale the grid. Graph the data using two different colors. Explain which graph illustrates a slower rise in price.

3. Find the unit price for pairs of socks at both shops. Use these numbers to explain which has the better buy.

4. Does there appear to be a proportional relationship between the number of pairs of socks and the cost of the socks at either shop? Use data from the tables and/or reference the graphs.

SOCKS ‘R WE 4 pairs of socks for $6.00

HOSIERY HUT 6 pairs of socks for $7.80

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SKILL BUILDER 5

Below are relationships between two quantities that vary. In each pair of relationships, circle the one that is proportional. Explain your reasoning.

1.

x y x y 0 2 0 0 1 3 1 2 2 4 2 4 3 5 3 6 4 6 4 8

2.

3. y = 5x y = 5x + 3

4.

x y x y

2 3 5 15 6 11 3 9 8 15 7 21 -1 -3 1 3 3 5 -2 -6

Circle the better deal. Show work.

5.

8 oz can, $ 0.80

6 pack of 8 oz cans, $ 4.80

6.

12 oz can,

$ 1.05

6 pack of 12 oz cans, $ 5.50

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SKILL BUILDER 6 Use any method to solve for the missing number in each proportion.

1. 312 4

= n

2. 4

5 35= n 3. 16 12

1 =

n

4. 169 22.5

= n

5. 1216 2

5 =

n 6.

317 4124

= n

7. Four fruit smoothies use 2 12

cups of bananas. How many cups of bananas are needed

for 6 fruit smoothies?

8. A serving of popcorn is 34

cup. How many servings of popcorn can be made from 5 cups

of popcorn?

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SKILL BUILDER 7 Solve each problem by using a ratio to express each rate.

1. Cary is towed up a ski hill. The hill is 15

of

a mile long and takes 6 minutes to get to the top. What is Cary’s rate as she travels up the hill in miles per hour?

2. Cary skis down the 15

mile hill in 10

minutes. What is her rate as she travels down the hill in miles per hour?

3. Aaron likes to snow ski on the faster runs. He skis down a hill 1

2 mile in 1

6 hours.

What is his rate in miles per hour?

4. Jonathan is a ski racer. He slaloms down an 880 ft. course in 8 seconds. What is his rate in miles per hour?

5. Greg runs a marathon at a rate of 6 miles per hour. If a marathon is 26.2 miles, will he finish in 4 1

2hours or less? Explain.

6. You water ski for 4 minutes and traveled 2 miles. What is your rate of speed in miles per hour?

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TEST PREPARATION

Show your work on a separate sheet of paper and choose the best answer.

1. If 3 pencils cost $0.78, then how much will 7 pencils cost? A. $0.26 B. $0.82 C. $1.26 D. $1.82

2. Office Plus and The Office Supply Store both sell notebooks. Office Plus sells 8 notebooks for $7.12. The Office Supply Store sells 5 notebooks for $5.25. Which store offers the better buy?

Office Plus # of

notebooks (x) 8 16 24 32 40

cost (y)

The Office Supply Store # of

notebooks (x) 10 20 30 40 50

cost (y)

A. Office Plus B. Office Supply C. The prices are the same D. Can’t tell from information given

3. How many miles per hour is equivalent to 9 miles per 90 minutes?.

A. 110

mph B. 6 mph C. 10 mph D. 13 12

mph

4. For the proportion 5 = 4 12

n, solve for n.

A. n = 141 B. n = 3 C. n = 13 D. n = 15

5. Which graph best represents a direct proportion? A.

B. C. D.

6. If 1 35

yards of fabric weighs 110

pounds, how many yards would weigh a 12

pound?

A. 8 yards of fabric B. 10 35

yards of fabric

C. 4 yards of fabric D. 16 yards of fabric

y

x x

y

x

y

x

y

x

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KNOWLEDGE CHECK

Show your work on a separate sheet of paper and write your answers on this page. 2.1 Proportional Reasoning Solve the pencil problem in two ways. Show your work. 1. If it costs $3.24 to buy 9 pencils, then how many pencils can be purchased for $6.48?

2. Find the missing number (n) in each proportion.

a. 325 5

= n b. 30 310

= n

c. 20 8

4 =

n 2.2 Best Buy Problems T-Shirt Mania and Shirts R’ Us sell souvenir t-shirts. T-Shirt Mania charges $18 for three t-shirts and Shirts R’ Us charges $25 for four t-shirts. 3. Find the unit rates for t-shirts at both stores. Use the numbers to explain which store has

the better buy. 4. Complete the two tables below and graph the result. Use different colors for each store.

Does the relationship between t-shirt cost to number of t-shirts represent direct proportions for either shirt shop? Why or why not?

2.3 Complex Fractions Ivan drinks 2 1

2 protein drinks for every 1 3

4 hours he spends at the gym.

5. At this rate, how many protein drinks will Ivan drink if he spends 4 hours at the gym? 6. If Ivan drank 20 protein drinks, how many hours did he spend at the gym?

T-Shirt Mania Shirts R’ Us # of t-shirts

(x) 3 6 9 12 # of t-shirts (x) 4 8 12 16

cost (y) cost (y)

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HOME-SCHOOL CONNECTION Here are some questions to review topics from these lessons with your young mathematician. 1. If 4 pencils cost $0.92, how much will 9 pencils cost? 2. Cookies n’ Things charges $3.20 for 8 cookies.

a. Complete the following table. Then label the axes and graph the ordered pairs.

# of cookies (x) 8 16 24 32 40

cost (y)

b. Does the table demonstrate a direct proportion? Why or why not? c. Does the graph demonstrate a proportional relationship? Why or why not?

3. Simplify 1356

Parent or Guardian Signature ________________________________

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Ratios and Proportional Relationships


COMMON CORE STATE STANDARDS – MATHEMATICS

STANDARDS FOR MATHEMATICAL CONTENT 7.RP.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas, and other

quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour.

7.RP.2a Recognize and represent proportional relationships between quantities. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.

7.RP.2b Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

7.RP.2c Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.

7.RP.2d Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.

7.NS.3 Solve real-world and mathematical problems involving the four operations with rational numbers.

7.EE.3 Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.

STANDARDS FOR MATHEMATICAL PRACTICE

MP1 Make sense of problems and persevere in solving them.

MP2 Reason abstractly and quantitatively.

MP3 Construct viable arguments and critique the reasoning of others.

MP4 Model with mathematics.

MP5 Use appropriate tools strategically.

MP6 Attend to precision.

MP7 Look for and make use of structure.

MP8 Look for and express regularity in repeated reasoning. © 2013 Center for Mathematics and Teaching, Inc. SAMPLE

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