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6th Grade

Dependent & Independent Variables

2015-11-25

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Table of Contents

Graphing Equations

Equations and Tables

Click on a topicto go to that section.

Dependent and Independent Variables

Translating to Equations

Glossary

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Translating to Equations

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An equation is a statement that shows that two

When writing equations, you want to pull the pieces of important information from the problem, and turn them into mathematical expressions.

mathematical expressions are equal.

These are examples of equations:

12 + a = 15 x / 5 = 10 y = 3x

Equation

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Mrs. Smith bought pencils for her class and spent $5.50 total. If each pencil costs $0.15, how many students are in the class? This tells us one pencil costs $0.15.This tells us the equation must equal $5.50.

Do we know how many students are in the class? No, so this will be our variable, so let s = students in the class.

Now, think about how you would calculate the total price of the pencils. You would multiply the cost of one pencil by the number of students in the class and get the total, so that is how you must set up your equation.

$0.15s = $5.50

Example

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Kristin is three years older than her sister, Sarah. If Kristin is 13 years old, how old is Sarah?

Try This:

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1 The Dolphins scored 9 more points than the Jets. The Jets scored 34 points. Which equation could be used to find the number of points p that the Dolphins scored during the game?

A p + 9 = 34B p + 34 = 9C p - 9 = 34D p - 34 = 9

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2 Frank earns $8.50 per hour for mowing his neighbors' yards. Last weekend Frank earned $51. Which equation can be used to determine the number of hours h Frank worked last weekend?A 8.5h = 51B 51h = 8.5C 8.5/h = 51D h/51 = 8.5

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3 Scott is working at a local restaurant. He earns $45 for each shift plus the tips he receives from his customers. Last night, he earned $73 for his shift. Which equation can be used to determine how much Scott earned in tips, t, last night?

A 45 + t = 73 B 73 + t = 45 C 73 - t = 45 D 45 - t = 73

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4 Gabby was trying to pay off her credit card bill. After her payment of $75, her credit card balance was $263. Which equation can be used to determine the initial balance, b, on Gabby's credit card?

A b - 263 = 75 B b - 75 = 263 C b + 75 = 263 D b + 263 = 75

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5 Melinda and her friends went to the theater and purchased 3 adult tickets and one large popcorn. One adult ticket costs $9, and they spent $33.75 at the theater. Which equation could be used to determine the cost c of the popcorn?

A 9c - 3 = 33.75B 33.75-9 = cC 3c + 9 = 33.75D 3(9) + c = 33.75

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Dependent and Independent Variables

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An independent variable is the variable that is subject to choice, or one that is not influenced by another variable.

Vocabulary

The value of a dependent variable relies on the values of the independent variable.

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Frank earns $8 per hour mowing his neighbors' lawns.

The amount of money he earns, m, depends on how many hours he works, h.

The more hours he works, the more money he earns.

Therefore the dependent variable is money, m,and the independent variable is hours, h.

The amount of hours he works does not rely on the money he earns.

Example

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Independent Dependent

how far you drive how much gas you use

weight of a sick child dosage of medicine given

your test score how you study for the test

Try to guess the missing variable.

With your group, try to think of at least three examples of independent and dependent variables.

Independent Dependent

Try This:

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6 The number of tickets I can buy depends on how much money I have.

True

False

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7 So which value is the independent variable?

A amount of moneyB number of tickets

The number of tickets I can buy depends on how much money I have.

Click for Question

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8 It costs $4.25 to rent a movie. The amount of money I spend depends on how many movies I rent. So the dependent variable is the number of movies I rent.

True

False

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9 The older I get, the taller I am. My height is the...

A Independent Variable

B Dependent Variable

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10 The more people I have at my party, the more brownies I need to bake. The number of people at my party is the...

A Independent Variable

B Dependent Variable

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Equations and Tables

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The relationship between dependent and independent variables can be

represented with a table.

Independent Dependent

Input Output

The independent variable is always in the left column, and the dependent variable is

always in the right column.

The relationship between independent & dependent variables and input & output works like a machine.

Tables

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The value of the output relies on Input

Output

Rule

The rule is the relationship between the input and the output.

It says what happens to the input inside the machine.

The value of the output always depends on the value of the input.

1. The value of the input

2. The rule

Rule Machine

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Let's Practice figuring out the rule.

Step 1. Assign a value to the input.

Step 2. Hit Enter to see the output.

The input and output

values will show

on this table.

Step 3. Once you have enough input/output values to figure out the rule, select + or * and the addend or factor.

Step 4. Check Your Rule

Click here for online practice.

Practice

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n 2n

20

40

100 200

80

40click

click

click

The value of n is the input.Given the value for n, find the output using the given rule.

Input Output

Practice

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x + 15

$53

$70

$115

The manager of the department store raised the price $15 on each video game. Can you find an expression (rule) that will satisfy the total

cost of the video game if given the original price?

$100

$38

x

Price after mark up

$55

Original price

click

click

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Practice

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g - 2

Kindergarten

8th grade

4th grade

A parent wants to figure out the differences in grade level of her two sons. The younger son is two years behind the older one in terms of grade level. Write an expression (rule) containing a variable which satisfies the difference in grade level of the two boys.

older son's grade level

younger son's grade level

6

10

2

g

click

click

click

click

Practice

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Tables can be used to represent equations.

The table below represents the equation y = x + 3.

x y

3 6

4 7

6 9

The output (y) depends on the input (x).

So x is the independent variable, on the left,and y is the dependent variable, on the right.

TablesM

ath

Prac

tice

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51

39

21tn

10

28

40

click

click

click

This table represents the equationt = n + 11

Find the values for t, given the values for n.

Practice

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120

60

20

yx 80

120

180

click

click

click

This table represents the equationy = x - 60

Find the values for y, given the values for n.

Practice

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x y20

40

100 200

80

40

click

click

click

This table represents the equationy = 2x

Find the values for y, given the values for x.

Practice

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Equations and tables can also be used to represent real-life information mathematically.

Natalie is going ice skating. The skating rink charges $6.25 per hour of skating.

We will let h represent the number of hours of skating and c represent the total cost.

Equation: c = 6.25 h

hours (h) cost (c)

1 $6.25

2 $12.50

3 $18.75

Real Life Example

Teac

her N

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&

Mat

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2j = m

48

28

24

Mary's age is twice the age of Jack.

Jack's Age

12

14

24

j

Mary's Age

Can you think of an algebraic equation which determines Mary's age (m), given Jack's age (j)?

click

click

click

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Real Life Example

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11 Henry downloads songs into iTunes. The amount of time it takes him to download a song depends on the song's file size.Which is the independent variable?

A Download time

B File size

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12 If it takes 50 seconds to download one megabyte, which equation represents this scenario ? Use the variable t for download time and s for file size.

A s = 50t

B t = 50s

C 50 - s = t

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13 Which table represents the equation t = 50s ?

A B

C D

s t200 4100 250 1

s t4 2005 2506 300

s t200 150100 5050 1

s t1 512 523 53

Ans

wer

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14 Find the missing value in this table.

It takes Jonathan 6 minutes to run a mile. Let t represent the number of minutes and d represent the number of miles.

(t) (d)

6 1

18 ?

Ans

wer

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15 Use the equation y = 5x to complete the table.

A

B

C

D

x y

20 ?

? 150

50 ?

?= 4 ?= 30 ?= 250

?= 4 ?= 30 ?= 10

?= 100 ?= 30 ?= 250

?= 100 ?= 3 ?= 10

Ans

wer

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Graphing Equations

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You have learned that equations and tables are two ways to represent real-life scenarios.

Equations and tables can also be graphed to represent a real-life scenario.

Example:

A cafeteria has an automatic waffle-making machine. The table shows the relationship between the time in hours (x) and the number of waffles the machine can make (y).

(x) (y)

1 50

2 1003 1504 2005 250

Tables to Graphs

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(x) (y)

1 502 1003 1504 2005 250

The equation for this scenario is y = 50x

Each hour, 50 waffles are made.

The value of y depends on the value of x.

y is dependent x is independent

Tables to Equation

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This scenario can be represented with a graph.When graphing:The independent variable is always across the x axis.The dependent variable is always up the y axis.

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y axis

(dependent)

x axis (independent)

Variables on Graphs

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0 1 2 3 4 5 6 7 8

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50

Once you have represented the equation in a function table, you can utilize the independent and dependent variable values as coordinates.

Plot the coordinates from the table below to graph the scenario.

Time (x) Waffles Made (y)

Coordinate (x,y)

1 50 (1, 50)

2 100 (2, 100)

3 150 (3, 150)

4 200 (4, 200)

5 250 (5, 250)

Function Table

Mat

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actic

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A bookstore is running a special and is charging $5 for any childrens' book.

1. Write an equation to represent the scenario. 2. Complete the table to represent the scenario. 3. Graph the function.

0 1 2 3 4 5 6 7 8

40

35

30

25

20

15

10

5

Number of Books

(x)

Total Cost (y)

Coordinate (x,y)

1 5 (1, 5)

2 10 (2, 10)

3 15 (3, 15)

4 20 (4, 20)

5 25 (5, 25)

Tables to Graphs

Ans

wer

&

Mat

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16 A plane descends at a rate of 50 feet per minute. Which graph represents this scenario?

A B

C

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17 Which scenario does the graph represent?A Mia earns $12 per hour.

B The river rose at a steady rate of 15 feet per hour.

C The stock value decreased by $.50 per minute

D The plane traveled 300 miles per hour.

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18 Which scenario does the graph represent?A Tia starts out with $30. Every hour Tia earns an additional $20.B Tia starts out with $50. Every minute she spends $5.C Tia runs a mile every 20 minutes.

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19 A dogwood tree grew at the rate of 4 feet per year. Which table represents the relationship between the height of the tree (h) and the number of years (t)?

A B

C D

t h

1 4

2 5

3 6

t ht ht h

1 4

3 12

5 20

t h

5 1

10 2

15 3

t h

4 1

8 2

12 3

Ans

wer

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20 The table and graph represent which equation?

A y = 50x

B 50y = x

C y = x -50

D y = x + 50

minutes (x) 1 3 4 7

# of words typed (y)

50 150 200 350

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21 Which equation represents the relationship between x and y shown in the graph?

A y = 3x

B y = x - 3

C y = 1/3x

D y = x + 3

From PARCC PBA sample test calculator #5

The graph shows the number of teaspoons of water, y, that have dripped from a leaky faucet at the end of x minutes.

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22 Based on the relationship in the graph, how many teaspoons of water will have dripped from the faucet at the end of 21 minutes?

From PARCC PBA sample test calculator #5

The graph shows the number of teaspoons of water, y, that have dripped from a leaky faucet at the end of x minutes.

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Glossary

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Back to

Instruction

Coordinates

A pair of values that show an exact position on a coordinate plane.

(x,y)(0,0)

x

y

(2,3)

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Back to

Instruction

Dependent Variable

The variable whose value varies based on the value of another variable.

The dependent variable is always along the y axis.y = x2

The value of y is determined by the value of x.

(x) (y)1 12 43 9

The dependent variable is always the output (y).

Function Tables

Equations Graphing

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87654321

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Back to

Instruction

Equation

Two expressions that are equivalent to each other. Equivalence is shown with an equal sign.

4x=8equivalent expressions

4 =x3

equivalent expressions

no equivalence

x3

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Back to

Instruction

ExpressionNumbers, symbols and operations

grouped together that show the value of something.

2 x 3 = 6 Expressions DO NOT have

equal signs.

3 2 + 12An

expression is one side of an equation.

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Back to

Instruction

Function Table

Organizes the special relationship between x and y values. Each of its input values gives

back exactly one output value.

The output (y) depends

on the input (x).

(x) (y)1 12 43 9

x is the independent variable, on the left,

and y is the dependent variable,

on the right.

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Back to

Instruction

The variable that holds its value and is not influenced by another variable.

Independent Variable

The independent variable is always across the x axis.y = x2

x holds its value, regardless of

what y is.

(x) (y)1 12 43 9

The independent variable is always the input (x).

Function TablesEquations Graphing

87654321

0 1 2 3 4 5 6 7 8

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Back to

Instruction

Input

The input is the independent variable in a function (y).

The output (y) depends on the

input (x).

(x) (y)1 12 43 9

y = x2

The value of y is determined by the value of x.

Input

Input

Rule

y = x2

3

Output

9

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Back to

Instruction

Output

The output is the dependent variable in a function (y).

The output (y) depends on the

input (x).

(x) (y)1 12 43 9

y = x2

The value of y is determined by the value of x.

Output

Input

Output

Ruley = x2

3

9

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Back to

Instruction

RuleDefines the relationship between the input and the output, using an algebraic equation. It says

what happens to the input in a function.Input3

Output

9

Rule:y = x2

(x) (y)1 32 63 9

Rule: y = 3x

(x) (y)1 62 103 14

Rule: y = 4x+2

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Back to

Instruction

VariableA letter or symbol that

represents a changeable or unknown value.

4x + 2variable

x = ?2x = 6x x

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Standards for Mathematical Practice

MP1: Making sense of problems & persevere in solving them.MP2: Reason abstractly & 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 & make use of structure.MP8: Look for & express regularity in repeated reasoning.

Additional questions are included on the slides using the "Math Practice" Pull-tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used.

If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull-tab.

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