expressions and equations 6.ee - math...

Copyright © 2013 Hilda Ratliff

Expressions and Equations 6.EE.9


Teacher Notes

Common Core State Standard

Expressions and Equations 6.EE

Represent and analyze quantitative relationships between dependent and independent variables. 9. Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

1st Activity – Notes and Examples

A variable could be a shape such as a or a , but by 6th grade it will

be a letter.

A quantity is a number of things – 25 days, 20 miles, 3 inches, 6 books, etc.

As the teacher defines variable, independent variable and dependent

variable, students will fill in blanks with the correct word.

Guided practice may be used on #5 and #6. Questions 7 – 10 may be used

as independent practice and checking for understanding. By the time

students work problems 9 and 10 they should remember to define variables

before writing an equation.

Throughout all activities question students about identifying which variable is

the independent variable and which variable is the dependent variable.

2nd Activity – Tables, Graphs and Equations

In this activity students will be given a table of values representing a

situation and they will be required to graph the data and write an equation.

Students will be given a graph of data and must determine an equation and

a table of values using the data in the graph.


Students will be given a word problem and must determine all three ways to

represent the situation – equation, table and graph.

Teach the students when the situation requires a scatter plot (graph points

and do not draw a line through them). This is when fractional parts of a

quantity make no sense. If the independent variable is number of

hamburgers to be purchased, fractional parts make no sense. You cannot

buy half a hamburger.

If the independent variable is number of hours worked, it is possible to work

part of an hour, so a line may be drawn through the points to represent that.

When students make a graph on their own they must label the axes with the

quantities they represent. The dependent variable customarily is on the

vertical axis and the independent variable is on the horizontal axis.

Students should learn to use a scale to match the data. Make sure students

have equal spaces representing the same value.


3rd Activity – Group Activity

Each group will receive a card with instructions. The group will complete one

of the following tasks.

Given an equation, construct a table and a graph representing the

situation.

Given a table of values, construct a graph and write an equation

representing the same situation.

Given a graph, construct a table and write an equation representing

the same situation.

Materials:

6 cards provided with this resource

Grid Chart Paper and Meter Stick OR Graph Paper and Straight Edge

Markers

The groups may make their graphs and tables on grid chart paper to be

displayed on the wall.

Extension Activity – Check for Understanding!

A single page extension activity is included. The students are presented with

a word problem and must represent the problem in three different ways.


Writing Equations Name _____________________

1. A _________________________ is a symbol (usually a letter) that

represents a number.

2. In the expression 2a – 5, ____________ is the variable.

3. In the equation 15 = 3d , ____________ is the variable.

4. We describe how two quantities are related using __________________.

You will identify two kinds of variables: independent and dependent.

The _________________________ variable depends on the

_________________________________ variable.

Example: The total cost, t, of hamburgers which cost $1.95 each

depends on the number of hamburgers, h to be bought.

Independent variable: ___________

Dependent variable: ___________

5. Coach Tally will buy basketball shoes for each person on the team. Each

pair of shoes will cost $98.00.

Write an equation showing the relationship between the number of

players and the total cost to be spent on the team.

Define variables. _______ = number of players

_______ = total cost

Identify the independent variable and the dependent variable.

Independent Variable _________ Dependent Variable _________

Write an equation for this situation.

Find the total cost of buying shoes for 15 basketball players.


6. Mario is selling magazines for a school fundraiser. The school will receive

$2.10 for each magazine he sells.

The variable quantities are number of magazines sold and total

money for the school.

Assign a variable for each quantity.

________ = number of magazines ________ = total money

______________________ depends on _______________________,

so ________________________ is the dependent variable.

Write an equation showing the relationship between these two

variables.

The school made a profit of $105.00 on the magazines that Mario sold.

Use the equation to determine the number of magazines Mario sold.

7. Heather plans to read three books each month.

Write an equation relating the total number of books Heather reads

and the number of months.

Define variables. (Tell what each variable represents.)

Independent Variable ___________ Dependent Variable __________

Write the equation.

Use the equation to find the number of books Heather will read in one

year.


8. Sam will mow lawns for people in his neighborhood.

He will charge $12.00 for each lawn.

Write an equation to show the relationship between the number of lawns he

mows and the total money he will earn.

Define variables.

Independent Variable ___________ Dependent Variable __________

Write the equation.

How many lawns must Sam mow to have enough money to buy a

video game that costs $64.00?

9. John gets an allowance of $8.00 per week. He plans to save half his

allowance each week.

Write an equation to show the relationship between the number of weeks he

saves and the total amount he will save.

(Don’t forget to define the variables.)

10. Olivia’s mom will bake four cookies for each guest that is invited to

Olivia’s birthday party.

Write an equation relating the total number of cookies her mom will bake to

the number of guests invited.


Writing Equations Name __ANSWER KEY__

1. A __variable________ is a symbol (usually a letter) that

represents a number.

2. In the expression 2a – 5, ____a________ is the variable.

3. In the equation 15 = 3d , ___d_________ is the variable.

4. We describe how two quantities are related using _variables___.

You will identify two kinds of variables: independent and dependent.

The ___dependent_______ variable depends on the

______independent______________ variable.

Example: The total cost, t, of hamburgers which cost $1.95 each

depends on the number of hamburgers, h, to be bought.

Independent variable: _h or number of hamburgers__

Dependent variable: __t or total cost_________

5. Coach Tally will buy basketball shoes for each person on the team. Each

pair of shoes will cost $98.00.

Write an equation showing the relationship between the number of

players and the total cost to be spent on the team.

Define variables. __n_(or any letter)____ = number of players

_c__(or any letter)____ = total cost

Identify the independent variable and the dependent variable.

Ind. Variable n or no. of players Dep. Variable c or total cost

Write an equation for this situation. c = 98n

Find the total cost of buying shoes for 15 basketball players.

c = 98n c = 98(15) c = 1470 Total cost = $1470


6. Mario is selling magazines for a school fundraiser. The school will receive

$2.10 for each magazine he sells.

The variable quantities are number of magazines sold and total

money for the school.

Assign a variable for each quantity.

__n______ = number of magazines ___t_____ = total money

Total money depends on number of magazines sold,

so total money is the dependent variable.

Write an equation showing the relationship between these two

variables. t = 2.10n

The school made a profit of $105.00 on the magazines that Mario sold.

Use the equation to determine the number of magazines Mario sold.

t = 2.10n

105 = 2.10n Divide both sides of equation by 2.1.

50 = n Mario sold 50 magazines.

7. Heather plans to read three books each month.

Write an equation relating the total number of books Heather reads

and the number of months.

Define variables. (Tell what each variable represents.)

b = number of books m = number of months

Independent Variable __m_____ Dependent Variable ___b_______

The number of books read depends on the number of months she reads

books.

Write the equation. b = 3m

Use the equation to find the number of books Heather will read in one

year. 1 year = 12 months b = 3m b = 3(12) b = 36 books


8. Sam will mow lawns for people in his neighborhood.

He will charge $12.00 for each lawn.

Write an equation to show the relationship between the number of lawns he

mows and the total money he will earn.

Define variables. n = number of lawns m = total money earned

Independent Variable ___n____ Dependent Variable _m_______

Write the equation. m = 12n

How many lawns must Sam mow to have enough money to buy a

video game that costs $64.00?

m = 12n

64 = 12n He cannot mow 5 3

1lawns.

64 = 12n

12 12 He must mow 6 lawns to have enough money.

5 3

1= n

9. John gets an allowance of $8.00 per week. He plans to save half his

allowance each week.

Write an equation to show the relationship between the number of weeks he

saves and the total amount he will save.

(Don’t forget to define the variables.) w = no. of weeks t = total amount

t = 4w

10. Olivia’s mom will bake four cookies for each guest that is invited to

Olivia’s birthday party.

Write an equation relating the total number of cookies her mom will bake to

the number of guests invited.

g = number of guests c = total no. of cookies c = 4g


Tables, Graphs and Equations Name_______________________

1. The Hamburger Hut charges $2.50 for each hamburger. Mrs. Shannon’s

6th grade class will eat lunch at the Hamburger Hut next week.

Fill in the table to show the total cost for the number of hamburgers in the

table.

Number of

Hamburgers h

Total Cost

c

1 $2.50

2 $5.00

3

4

5 $12.50

6

20

n

Write an equation showing the relationship between the total cost and the

number of hamburgers.

Equation:

2

4

6

8

10

12

14

16

6

Total Cost

0 2 4 6 8 10 0

No. of Hamburgers

On the grid, graph the first six

entries in the table above.

The independent variable is graphed

on the horizontal axis.

The dependent variable is graphed

on the vertical axis.


2. The graph shows the distance Marla walked in a certain amount of time.

Make a table of values for the data in the graph.

Time

t

Distance

d

0 1 2 3 4 5

200

400

600

800

1000

Distance (feet)

Time (minutes)

Write an equation describing the

relationship of time and distance.

What distance could Marla walk in 12

minutes?

What distance could she walk in 4 ½

minutes?


3. Callie charges $5.00 per hour to babysit.

Fill in the table to show the relationship of hours worked and total earnings.

Hours

h

Earnings

e

1

2

3

4

5

Graph the data in the table on

the grid.

Write an equation relating

hours worked to total earnings.

How much money will Callie

earn if she babysits 3 ½ hours?

How many hours must Callie

work to buy a computer which

costs $487.00?

0 1 2 3 4 5

5

10

15

20

25

Hours

Earnings


Tables, Graphs and Equations Name__ANSWER KEY__________

1. The Hamburger Hut charges $2.50 for each hamburger. Mrs. Shannon’s

6th grade class will eat lunch at the Hamburger Hut next week.

Fill in the table to show the total cost for the number of hamburgers in the

table.

Number of

Hamburgers h

Total Cost

c

1 $2.50

2 $5.00

3 $7.50

4 $10.00

5 $12.50

6 $15.00

20 $50.00

n $2.50n

Write an equation showing the relationship between the total cost and the

number of hamburgers.

Equation:

c = 2.50n

2

4

6

8

10

12

14

16

6

Total Cost

0 2 4 6 8 10 0

No. of Hamburgers

On the grid, graph the first six

entries in the table above.

The independent variable is graphed

on the horizontal axis.

The dependent variable is graphed

on the vertical axis.


2. The graph shows the distance Marla walked in a certain amount of time.

Make a table of values for the data in the graph.

Time

t

Distance

d

0 0

1 200

2 400

3 600

4 800

5 1000

0 1 2 3 4 5

200

400

600

800

1000

Distance (feet)

Time (minutes)

Write an equation describing the

relationship of time and distance.

d = 200t

What distance could Marla walk in 12

minutes? d = 200(12)

d = 2400 feet

What distance could she walk in 4 ½

minutes? d = 200 (4 ½)

d = 900 feet


3. Callie charges $5.00 per hour to babysit.

Fill in the table to show the relationship of hours worked and total earnings.

Hours

h

Earnings

e

1 $5.00

2 $10.00

3 $15.00

4 $20.00

5 $25.00

Graph the data in the table on

the grid.

Write an equation relating

hours worked to total earnings.

e = 5h

How much money will Callie

earn if she babysits 3 ½ hours?

e = 5(3 ½)

e = $17.50

How many hours must Callie

work to buy a computer which

costs $487.00?

e = 5h

487 = 5h

487 = 5h

0 1 2 3 4 5

5

10

15

20

25

Hours

Earnings

5 5 97.4 = h

She must work

97.4 hours to earn

$487.00.


Card A

Mr. Jones drives on the interstate at a constant rate of 70 miles per hour.

The distance he travels for a given time may be represented by the equation

d = 70 t .

d = distance (miles) t = time (hours)

Make a table with at least 6 entries to represent this situation.

Graph the points from your table on a grid. Use a reasonable scale and

label the horizontal and vertical axes with the words “time” and “distance.”

Card B

Jolie wants to download different ringtones in her cell phone. The table

below shows the total cost of downloading a particular number of ringtones.

Represent this data as a graph. Show the scale and label the horizontal

and the vertical axes with “Number of Ringtones” and “Total Cost”.

Write an equation to represent the data using “r” and “c”.

Number of

Ringtones r

Total Cost

c

1 $1.50

2 $3.00

3 $4.50

4 $6.00


Card C

The graph below shows the number of pounds of meat needed to feed a

given number of lions in one day.

Card D

Cathy will decorate cupcakes for a party. She will place 6 jelly beans on the

top of each cupcake.

The number of jelly beans she needs may be represented by the equation

y = 6x .

x = number of cupcakes y = total number of jelly beans

Make a table with at least 6 entries to represent this situation.

Graph the points from your table on a grid. Use a reasonable scale and

label the horizontal and vertical axes with the words “cupcakes” and “total

number of jelly beans.”

1 2 3 4 5 Lions

30

60

90

120

150

Pounds

Make a table to represent the data.

Write an equation to represent this

situation.


Card E

The Corner Bookstore sells used books and makes a profit of $2.00 per

book. The table below shows the profit on a given number of books.

Number

of Books

b

Total

Profit t

1 $2.00

2 $4.00

3 $6.00

4 $8.00

Represent this data as a graph. Show the scale and label the horizontal

and the vertical axes with “Number of Books” and “Total Profit”.

Write an equation to represent the data using “b” and “t”.

Card F

The graph below shows the total cost of renting a car for a given number

of days.

(1, 42)

(2, 84)

(3, 126)

(4, 168)

1 2 3 4 5 Days

40

80

120

160

200

Cost

Make a table to represent the

data.

Write an equation to represent

this situation.


Group Activity Answer Key

Card A

Card B c = 1.5r

Total Cost

0 1 2 3 4 5 6

Ringtones

Time (hrs)

Distance (miles)

1 70

2 140

3 210

4 280

5 350

6 420

0 1 2 3 4 5 6

Time (hours)

100

200

300

400

Distance (miles)

1

2

3

4

5

6


Card C p = pounds of meat n = number of lions

p = 30 n

Card D

Jelly Beans

0 1 2 3 4 5 6

Cupcakes

No.

of lions

Pounds

of meat

1 30

2 60

3 90

4 120

No. of

cupcakes

No. of jelly

beans

1 6

2 12

3 18

4 24

5 30

6 36

5

10

15

20

25

35

40

30


Card E

b = number of books t = total profit

t = 2b

Total Profit

0 1 2 3 4 5

Number of books

Card F c = 42d

Days d

Cost c

1 42

2 84

3 126

4 168

2

4

6

8


Check for Understanding! Name ________________________

Mike is a brick mason. He averages laying 175 bricks each hour.

Fill in the table below to show how many total bricks Mike can lay for a given

number of hours.

Number

of Hours h

Total

Bricks b

1

2

3

4

5

Define variables. h = ____________________ b = _________________

Independent Variable ___________ Dependent Variable ___________

Write an equation representing this situation.

Graph the points from the table on the grid. Since Mike may work a

fractional part of an hour, draw a line

through the points.

Label the axes with names of the

quantities.

Mark the scale with equal values for

each space.

Use a scale appropriate for the

numbers in the table.


Check for Understanding! Name __ANSWER KEY_____

Mike is a brick mason. He averages laying 175 bricks each hour.

Fill in the table below to show how many total bricks Mike can lay for a given

number of hours.

Number

of Hours h

Total

Bricks b

1 175

2 350

3 525

4 700

5 875

Define variables. h = _number of hours_ b = _total bricks

Independent Variable __h_______ Dependent Variable ___b_______

The number of bricks he lays depends on the number of hours he works.

Write an equation representing this situation.

Graph the points from the table on the grid. Since Mike may work a

fractional part of an hour, draw a line through the points.

Graph is a sample answer.

Label the axes with names of the

quantities.

Mark the scale with equal values for

each space.

Use a scale appropriate for the

numbers in the table.

b = 175h

(1, 175)

(2, 350)

(3, 525)

(4, 700)

(5, 875)

1 2 3 4 5 Hours

200

400

600

800

1000 Bricks


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