Mod 1 MMA Study Guide Acc 19_20.pdf

Grade 7 Module 1 –Ratio and Proportional Relationships Page 1 NAME:____________________ BLOCK: _______ DATE:____________

CONSTANT OF PROPORTIONALITY

Example: Courtney earns $17 for babysitting for 2 hours, and $51 for babysitting for 6

hours. Show that the relationship between the number of hours Courtney babysits and

the money she earns is a proportional relationship.

Using a Table:

Constant of

proportionality

(unit rate): k =

𝑥𝑥

Write an

equation:

1.) Determine the independent variable

Which makes more sense: Cost per 1 hour or Hours per $1

“Dependent (y)” per “Independent (x)”

2.)Find the constant of proportionality (constant unit rate),

k

k =

𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑

,

𝑦𝑦

𝑖𝑖𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑

,

𝑥𝑥

𝑐𝑐𝑐𝑐𝑐𝑐𝑑𝑑 ℎ𝑐𝑐𝑜𝑜𝑜𝑜 =

17 2 = 8.5

3.)Define the variables:

Let x represent the # hours babysitting, H (independent) Let y represent the cost, C (dependent)

4.)Apply the formula:

y = kx,

Hours (x) 2 6

Cost (y) $17 $51

Think of “per” as a fraction bar

17 2 = 8.5

51 6 = 8.5

Make a graph:

Q: What does (0,0) mean? A: She earns no money ($0) if she babysits for 0 hours. Q: Where is the unit rate located on the graph? A: Where x = 1, so (1, 8.5)

Graphs all have:  Labeled axes  Equal intervals

Proportional means =  Straight line

 Goes through (0,0)  Has constant unit

rate

0 8.5 17 25.5 34 42.5 51 59.5

0 1 2 3 4 5 6 7

Co

st

, C

($)

dependent

var

iabl

e

Hours, H

independent variable

Babysitting Earnings

Unit rate (1, 8.5)

Name: _________________________________ Period: _________ Date Due: ____________

Proportional & non- Proportional relationships

Worksheet 2-2

“To be or not to be proportional”

Dylan makes $336 for 32 hours of work, and Angela makes $420 for 42 hours of work. 1] How much do Dylan and Angela each make per hour?

2] Is Dylan’s wage for 25 hours proportional to Amber’s wage for 42 hours? Why or why not?

To determine proportionality between two ratios or rates,

Conclusion: __________________________________________________

__________________________________________________.

Find the ratio of y to x for Table 1 and Table 2, simplify the fraction to simplest form, and answer the questions that follow.

Table 1: Table 2:

3] Which table shows a proportional relationship?

4] What makes it a proportional relationship?

Conclusion: To determine proportionality from a table,

____________________________________________________.

NUMBER

OF HOURS

TOTAL

COST ($) RATIO:

y x

1 $75

2 $120

3 $165

4 $210

5 $255

NUMBER

OF HOURS

TOTAL

COST ($) RATIO:

y x

1 $45

2 $90

3 $135

4 $180

Below are the graphs for the tables in the previous section. Use the graphs to determine

proportionality.

Table 1: Table 2:

5] Which graph shows a proportional relationship?

6] What makes it a proportional relationship?

To determine proportionality from a graph,

Conclusion: ________________________________________________________

_______________________________________________________

_______________________________________________________.

Determine which of the following tables represent proportional relationships.

1) 8) 9) 10)

Number of Hours

To ta l C os t ($ )

0 2 4 6 8 10 12 14 300 270 240 210 180 150 120 90 60 30

Number of Hours

To ta l C os t ($ )

0 2 4 6 8 10 12 14 300 270 240 210 180 150 120 90 60 30

x y 1 2 3 4 5

x y 0 0 2 4 4 8

x y 0 1 2 0 3 4

x y 1 1.5 3 4.5

5 7.5 7 10.5

3 6 9 12 15 16 8 20 10 5 3

4 7

Determine which of the following graphs represent proportional relationships. Circle the appropriate response.

11.

12.

13.

14.

15.

16.

17. Is the following relationship proportional? Explain.

Number of Movie Tickets (x)

Total Cost of

Tickets (y)

_x

y

1 6

2 12

3 18

4 24

18. How is a proportional relationship different from a non-proportional relationship?

Proportional non-proportional Proportional non-proportional Proportional non-proportional

Proportional non-proportional Proportional non-proportional Proportional non-proportional

• •

1-8 88 75 63 50 38 25 13 0

Ex) _{Phone Sold (x) 9 4 6 5 3}

Money Earned (y) 369 164 246 205 y/x = 41 , 3 x 41 = 123

1) _{Pieces of Chicken (x) 5 9 4 10 8} Price in dollars (y) 5 9 4 10 y/x =

2) _{Enemies Destroyed (x) 9 5 6 4 7} Points Earned (y) 297 165 198 132

y/x =

3) _{Time in minute (x) 2 6 8 10 9}

Distance traveled in meters (y) 34 102 136 170 y/x =

4) _{Tickets Sold (x) 8 3 6 2 10}

Money Earned (y) 112 42 84 28 y/x =

5) _{Votes for Bianca (x) 9 10 4 5 3} Votes for Luke (y) 198 220 88 110 y/x =

6) _{Glasses of Lemonade (x) 4 10 9 3 6} Lemons Used (y) 12 30 27 9

y/x =

7) _{Chocolate Bars (x) 7 4 5 3 10} Calories (y) 1,869 1,068 1,335 801

y/x =

8) _{Boxes of Candy (x) 8 3 2 6 10} Pieces of Candy (y) 120 45 30 90

y/x =

Find the constant of proportionality for each table.Then use it to find the missing quantity.

Identifying Constant of Proportionality (Tables)

Math

Name:

A n s w e r s

Ex.

y =

123

1.

y = 1x

2.

y =

33x

y =

17x

y =

14x

y =

22x

y =

3x

y =

267x

y =

15x

1-8 88 75 63 50 38 25 13 0 Ex) _{Concrete Blocks (x) 3 8 10 6 7}

weight in kilograms (y) 30 80 100 60 70 Every concrete block weighs 10 kilograms.

1) _{Cans of Paint (x) 5 10 6 9 2}

Bird Houses Painted (y) 15 30 18 27 6 For every can of paint you could paint 3 bird houses.

2) _{Votes for Faye (x) 9 7 6 8 3}

Votes for Victor (y) 342 266 228 304 114 For Every vote for Faye there were 38 votes for Victor.

3) _{Chocolate Bars (x) 6 4 10 3 8} Calories (y) 1,212 808 2,020 606 1,616

Every chocolate bar has 202 calories.

4) _{Pieces of Chicken (x) 7 8 6 10 2} Price in dollars (y) 14 16 12 20 4

For each piece of chicken it costs 2 dollars.

5) _{Boxes of Candy (x) 2 5 9 7 10} Pieces of Candy (y) 32 80 144 112 160

For every box of candy you get 16 pieces.

6) _{Lawns Mowed (x) 7 6 10 3 4}

Dollars Earned (y) 301 258 430 129 172 For every lawn mowed 43 dollars were earned.

7) _{Time in minute (x) 9 2 7 3 10}

Distance traveled in meters (y) 117 26 91 39 130 Every minute 13 meters are travelled.

8) _{Pounds of Beef Jerky (x) 7 8 5 6 10} Price in dollars (y) 84 96 60 72 120

For every pound of beef jerky it cost 12 dollars.

Ex.

y = 10x

1.

y = 3x

2.

y = 38x

3.

y = 202x

4.

y = 2x

5.

y = 16x

6.

y = 43x

7.

y = 13x

8.

y = 12x

Determine the constant of proportionality for each table. Express your answer as y = kx

Identifying Constant of Proportionality (Tables)

Math

Name:

A n s w e r s

1-8 88 75 63 50 38 25 13 0 1) 0 9 18 27 36 45

0 3 6 9 12 15

2) 0 30 60 90 120

0 6 12 18 24

3) 0 25 50 75 100 125

0 5 10 15 20 25

4) 0 6 12 18 24 30

0 1 2 3 4 5

5) 0 8 16 24 32 40

0 2 4 6 8 10

6) 0 21 42 63 84 105

0 7 14 21 28 35

7) 0 90 180 270 360

0 10 20 30 40

8) 0 16 32 48 64

0 8 16 24 32

1.

y = 3x

2.

y = 5x

3.

y = 5x

4.

y = 6x

5.

y = 4x

6.

y = 3x

7.

y = 9x

8.

y = 2x

Identify the constant of proportionality. Write your answer as y = kx

Identifying Constant of Proportionality (Graphs)

Math

Name:

A n s w e r s

1-6 83 67 50 33 17 0 1)

Pieces of Chicken

Price

A

(1 , 1.25)

Every piece of chicken costs

$1.25.

2)

Sodas Drank

Calories Consumed

A

(1 , 100)

For every soda drank 100 calories

are consumed.

3)

Minutes Pages Printed A(1 , 100)

Every minute 100 pages are

printed.

4)

Hours

Distance (miles)

A

(1 , 60)

Every hour 60 miles are travelled.

5)

Pounds of Meat Price A_{(1 , 3.79)}

Every pound of meat costs $3.79.

6)

Glasses of Lemonade

Lemons Used

A (1 , 8)

Every glass of lemonade requires

8 lemons.

Determine what the value of A means in each problem.

Explaining X and Y with Proportionality

Math

Name:

1-10 92 83 75 67 58 50 42 33 25 17 11-12 8 0

1) _{X Y}

4 5 0.80

3 6 0.50

2 7 0.29

1 8 0.13

2) _{X Y}

6 2 -3.00

7 1 -7.00

8 0 0.00

9 1 9.00

3) _{X Y}

70 10 7.00

56 8 7.00

14 2 7.00

7 1 7.00

4) _{X Y}

6 3 2.00

7 4 1.75

8 5 1.60

9 6 1.50

5) _{X Y}

1 10 0.10

2 20 0.10

7 70 0.10

10 100 0.10

6) _{X Y}

2 2 1.00

4 4 1.00

8 8 1.00

10 10 1.00

7) _{X Y}

20 32 -0.63

15 24 -0.63

10 16 -0.63

5 -8 -0.63

8) _{X Y}

70 10 -7.00

63 9 -7.00

35 5 -7.00

21 3 -7.00

9) _{X Y}

2 7 0.29

6 21 0.29

18 63 0.29

20 70 0.29

10) _{X Y}

-12 -32 0.38

-9 -24 0.38

-6 -16 0.38

-3 -8 0.38

11) _{X Y}

9 3 3.00

36 6 6.00

64 8 8.00

81 9 9.00

12) _{X Y}

2 4 0.50

3 6 0.50

4 12 0.33

7 21 0.33

1.

no

no

yes

no

yes

yes

yes

yes

yes

yes

no

no

Determine if the values in the table are proportional (yes) or not (no). If yes, state the rule.

Determining Proportionality with Tables

Math

Name:

A n s w e r s

Grade 7 Module 1 –Ratio and Proportional Relationships Page 2

1) The graph to the right shows the relationship of the amount of time (in seconds) to the distance (in feet) run by a jaguar.

a. What does the point (5, 290) represent in the

context of the situation?

b. What does the point (3, 174) represent in the context of the situation?

c. Is the distance run by the jaguar proportional to the time? Explain why or why not.

d. Write an equation to represent the distance run by the jaguar. Explain or model your reasoning.

Grade 7 Module 1 –Ratio and Proportional Relationships Page 3

2) Coach George went to a store to buy mouth guards. The following packages of mouth guards are available at this store.

• 10 mouth guards for $14.50 • 15 mouth guards for $22.50

Coach George needs to buy 60 mouth guards. How much money will he save by purchasing 60 mouth guards in packages with the lowest unit price compared to the highest unit price?

3) Underarmour t-shirts sell for $22 each.

a. What point(s) must be on the graph for the quantities to be proportional to each other?

b. What does the ordered pair (5, 110) represent in the context of this problem?

Name ________________________________________ Block___________ Date:____________

Quiz Review

1. Find the unit rate, _𝑥𝑥𝑦𝑦, the constant of proportionality, and write an equation for the table. Use your equation to find the cost for 3 pounds?

Unit rate:________ Equation:________________ Constant of Proportionality:________ Cost for 4 pounds:___________

2. Find the unit rate, _𝑥𝑥𝑦𝑦, the constant of proportionality and write an equation for the table. Use your equation to find the pay for 6 hours?

Unit rate:________ Equation:________________ Constant of Proportionality:________ Pay for 6 hours:___________

3. Find the unit rate, _𝑥𝑥𝑦𝑦, the constant of proportionality and write an equation for the table. Use your equation to find the pay for 6 hours?

Unit rate:________ Equation:________________ Constant of Proportionality:________ Cost for 9 notebooks:___________

pounds 2 3 5 6

cost ($) 6 9 15 18

hours 1.5 2 3.5 5

pay($) 10.50 14 24.50 35

notebooks 2 4 5 7

4. Find the unit rate, _𝑥𝑥𝑦𝑦, the constant of proportionality and write an equation for the table. Use your equation to find the cost for 3 pounds?

Unit rate:________ Equation:________________ Constant of Proportionality:________ Cost for 3 pounds:___________

5. For each graph write the unit rate as _𝑥𝑥𝑦𝑦, then determine the ordered pair that represents the unit rate. Plot and label this point on the graph.

pounds 7.5 10 17.5 20

Name __________________________________________ Block _____ Date _____

Illustrative Mathematics 7.RP.2 Tasks

1) BeatStreet, TunesTown, and MusicMind are music companies. BeatStreet offers to buy 1.5 million shares of TunesTown for $561 million. At the same time, MusicMind offers to buy 1.5 million shares of

TunesTown at $373 per share. Who would get the better deal, BeatStreet or MusicMind? What is the total price difference?

Name __________________________________________ Block _____ Date _____

2. The students in Ms. Baca’s art class were mixing yellow and blue paint. She told them that two mixtures will be the same shade of green if the blue and yellow paint are in the same ratio.

The table below shows the different mixtures of paint that the students made.

A B C D E

Yellow 1 part 2 parts 3 parts 4 parts 6 parts

Blue 2 part 3 parts 6 parts 6 parts 9 parts

a. How many different shades of paint did the students make?

b. Some of the shades of paint were bluer than others. Which mixture(s) were the bluest? Show work or explain how you know.

c. Carefully plot a point for each mixture on a coordinate plane like the one that is shown in the figure.

d. Draw a line connecting each point to (0,0). What do the mixtures that are the same shade of green have in common?

e. Write an equation that relates y, the number of parts of yellow paint, and b, the number of parts of blue paint for the most common shades of paint the students made.

Name __________________________________________ Block _____ Date _____

3. Coffee costs $18.96 for 3 pounds. What is the cost for one pound of coffee? At this store, the price for a pound of coffee is the same no matter how many pounds

you buy. Draw a complete graph of the relationship between the number of pounds of coffee and the total cost. Where can you see the cost per pound of coffee in the graph? What is it?

Name __________________________________________ Block _____ Date _____ 4. Nia and Trey both had a sore throat so their mom told them to gargle with warm salt water.

 Nia mixed 1 teaspoon salt with 3 cups water.

 Trey mixed 1/2 teaspoon salt with 1 1/2 cups of water.

Nia tasted Trey’s salt water. She said, “I added more salt so I expected that mine would be more salty, but

they taste the same.”

a. Explain why the salt water mixtures taste the same.

b. Write an equations that relates s, the number of teaspoons of salt, with w, the number of cups of water, for both of these mixtures?

Name __________________________________________ Block _____ Date _____ 5. Carli’s class built some solar-powered

robots. They raced the robots in the parking lot of the school. The graphs below are all line segments that show the distance d, in meters, that each of three robots traveled

after t seconds.

a. Each graph has a point labeled. What does the point tell you about how far that robot has traveled?

b. Carli said that the ratio between the number of seconds each robot travels and the number of meters it has traveled is constant. Is she correct? Explain.

Name __________________________________________ Block _____ Date _____

6. Julianna participated in a walk-a-thon to raise money for cancer research. She recorded the total distance she walked at several different points in time, but a few of the entries got smudged and can no longer be read. The times and distances that can still be read are listed in the table below.

a. Assume Julianna walked at a constant speed. Complete the table and plot Julianna’s progress in the coordinate plane.

Time in hrs Miles walked

1

2 6.4

8 5

b. What was Julianna’s walking rate in miles per hour? How long did it take Julianna to walk one mile? Where do you see this information on the graph?

c. Write an equation for the distance d, in miles, that Julianna walked in n hours.

d. Next year Julianna is planning to walk for seven

Name __________________________________________ Block _____ Date _____

7. In January, Georgia signed up for a membership at Anytime Fitness. The plan she chose cost $95 in start-up fees and then $20 per month starting in February. Edwin also signed up at Anytime Fitness in January. His plan cost $35 per month starting in February, and his start-up fees were waived.

a. Create tables for both Georgia and Edwin that compare the number of months since January to the total cost of their gym memberships. Continue this table for one year.

b. Plot the points from the two tables in part (a) on a coordinate plane. c. Decide if either or both gym memberships

are described by a proportional relationship, and write an equation representing any such

relationship. Explain how parts (a) and (b) could be used to support your answer.