Eureka Math Homework Helper 2015–2016 Grade 7 Module 1

Eureka MathTM Homework Helper 2015?2016

Grade 7 Module 1 Lessons 1?22

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A Story of Ratios 7?1 2015-16

G7-M1-Lesson 1: An Experience in Relationships as Measuring Rate

Rate and Unit Rates

Find each rate and unit rate.

1. $8.96 for 8 pounds of grapefruit

.

=

.

Rate: . dollars per pound

Unit Rate: .

I determine the cost of one pound of grapefruit in order to find the rate. To do this, I divide the cost by the number of pounds.

2. 300 miles in 4 hours

=

Rate: miles per hour

Unit Rate:

The label explains the numerical value of the rate.

Ratios and Rates

3. Dan bought 8 shirts and 3 pants. Devonte bought 12 shirts and 5 pants. For each person, write a ratio to represent the number of shirts to the number of pants they bought. Are the ratios equivalent? Explain.

The ratio of the number of shirts Dan bought to the number of pants he bought is : .

The ratio of the number of shirts Devonte bought to the number of pants he bought is : .

The order of the ratios is important. In this case, it is stated that the ratio is shirts to pants, which means the first number in the ratio represents shirts and the second number represents pants.

The

ratios

are

not

equivalent

because

Dan's

unit

rate

is

or

,

and

Devonte's

unit

rate

is

or

.

I know these are not equivalent ratios because they do not have the same unit rate.

Lesson 1:

An Experience in Relationships as Measuring Rate

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A Story of Ratios 7?1 2015-16

4. Veronica got hired by two different families to babysit over the summer. The Johnson family said they would pay her $180 for every 20 hours she worked. The Lopez family said they would pay Veronica $165 for every 15 hours she worked. If Veronica spends the same amount of time babysitting each family, which family would pay her more money? How do you know?

Calculating the unit rate helps compare different rates and ratios.

Veronica will earn $ per hour when she babysits for the Johnson family and will earn $ per hour when she babysits for the Lopez family. Therefore, she will earn more money from the Lopez family if she spends the same amount of time babysitting for each family.

Lesson 1:

An Experience in Relationships as Measuring Rate

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A Story of Ratios 7?1 2015-16

G7-M1-Lesson 2: Proportional Relationships

Proportional Quantities 1. A vegetable omelet requires a ratio of eggs to chopped vegetables of 2 to 7.

This means that I use 2 eggs and 7 chopped vegetables to make an omelet.

a. Complete the table to show different amounts that are proportional.

Number of Eggs

Number of Vegetables

b. Why are these quantities proportional?

Answers may vary, but I need to create ratios that are equivalent to the ratio 2: 7.

The number of eggs is proportional to the number of chopped vegetables since there exists a

constant number, , that when multiplied by any given number of eggs always produces the

corresponding amount of chopped vegetables.

2. The gas tank in Enrique's car has 15 gallons of gas. Enrique was able to determine that he can travel 35 miles and only use 2 gallons of gas. At this constant rate, he predicts that he can drive 240 more miles before he runs out of gas. Is he correct? Explain.

Once I calculate the unit rate, I use this to determine how many miles Enrique can travel with the gas remaining in his tank by multiplying both values by 15.

Gallons of Gas Used

Miles Traveled

.

.

Enrique can travel . more miles because has he can only travel . miles with gallons of gas, but he has already traveled miles. . - = . . Therefore, Enrique's prediction is not correct because he will run out of gas before traveling more miles.

Lesson 2:

Proportional Relationships

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G7-M1-Lesson 3: Identifying Proportional and Non-Proportional Relationships in Tables

Recognizing Proportional Relationships in Tables In each table, determine if is proportional to . Explain why or why not.

To determine if is proportional to , I determine if the unit rates, or value of each ratio, are equivalent.

1.

3

4

6

= = = =

8

5

10

No, is not proportional to because the values of all the ratios

6

11

: are not equivalent. There is not a constant where every

measure of multiplied by the constant gives the corresponding

measure in .

2. 6 9 12 15

2

= =

=

=

3

4

Yes, is proportional to because the values of the ratios :

5

are equivalent. Each measure of multiplied by this constant of

gives the corresponding measure in .

If

I

multiply

each

-value

by

1

,

the

outcome

3

will be the corresponding -value.

Lesson 3:

Identifying Proportional and Non-Proportional Relationships in Tables

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3. Ms. Lynch is planning a field trip for her class. She knows that the field trip will cost $12 per person.

a. Create a table showing the relationships between the number of people going on the field trip and the total cost of the trip.

Number of People

Total Cost ($)

I choose any value for the number of people, and then multiply this value by 12 to determine the total cost.

b. Explain why the cost of the field trip is proportional to the number of people attending the field trip.

The total cost is proportional to the number of people who attend the field trip because a constant value of exists where each measure of the number of people multiplied by this constant gives the corresponding measure of the total cost.

I know the relationship is proportional, so I can use the constant of 12 to determine the total cost of the field trip if 23 people attend.

c. If 23 people attend the field trip, how much will the field trip cost?

() =

If people attend the field trip, then the total cost of the trip is $.

Lesson 3:

Identifying Proportional and Non-Proportional Relationships in Tables

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A Story of Ratios 7?1 2015-16

G7-M1-Lesson 4: Identifying Proportional and Non-Proportional

Relationships in Tables

Recognizing Proportional Relationships 1. For his birthday, Julian received 15 toy cars. He plans to start collecting more cars and is going to buy 3

more every month. a. Complete the table below to show the number of toy cars Julian has after each month.

Julian has 15 toy cars when he decided to start collecting more. Therefore, at month 0 he already has 15 toy cars.

Time (in months)

Number of Cars

Julian has 18 toy cars after one month because he had 15 cars and then bought 3 more during the first month.

b. Is the number of toy cars Julian has proportional to the number of months? Explain your reasoning. The number of toy cars Julian has is not proportional to the number of months because the ratios are not equivalent. : is not equivalent to : .

If an additional explanation is needed, please refer to Lesson 3.

Lesson 4:

Identifying Proportional and Non-Proportional Relationships in Tables

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2. Hazel and Marcus are both training for a race. The tables below show the distances each person ran over the past few days.

Hazel:

Marcus:

Days

2

Miles

6

5

9

15

27

Days

3

Miles

6

6

8

11

20

a. Which of the tables, if any, represent a proportional relationship?

Hazel:

Marcus:

=

=

=

=

=

The number of miles Hazel ran is proportional to the number of days because the constant of is multiplied by each measure of days to get the corresponding measure of miles. There is not a constant value for Marcus's table, so this table does not show a proportional relationship.

=

These ratios do not have the same value, so the number of miles is not proportional to the number of days.

b. Did Hazel and Marcus both run a constant number of miles each day? Explain.

Hazel ran the same number of miles, , each day, but Marcus did not run a constant number of miles each day because the relationship between the number of miles he ran and the number of days is not proportional.

Lesson 4:

Identifying Proportional and Non-Proportional Relationships in Tables

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