Translating to Equations and Tables Equations Graphing Equations - - PDF document

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6th Grade

Dependent & Independent Variables

2015-11-25 www.njctl.org

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Graphing Equations Equations and Tables Click on a topic to 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 Slide 6 / 65

• 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

• f 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?

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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 = 34 B p + 34 = 9 C p - 9 = 34 D 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 = 51 B 51h = 8.5 C 8.5/h = 51 D 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 = 33.75 B 33.75-9 = c C 3c + 9 = 33.75 D 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 Slide 16 / 65

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:

Math Practice

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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 money B number of tickets The number of tickets I can buy depends

• n 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 Slide 24 / 65

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

• n

this table. Step 3. Once you have enough input/output values to figure

• ut the rule, select + or * and the addend or factor.

Step 4. Check Your Rule

Click here for online practice.

Practice Slide 26 / 65

n

2n 20 40

100

200 80 40

click click

click

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

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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 click click

Practice Slide 28 / 65

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.

• lder son's

grade level younger son's grade level

6 10 2

g

click click click click

Practice Slide 29 / 65

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.

Tables

Math Practice

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51 39 21

t

n

10 28 40

click click click

This table represents the equation t = n + 11 Find the values for t, given the values for n.

Practice

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120 60 20

y x

80 120 180

click click click

This table represents the equation y = x - 60 Find the values for y, given the values for n.

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x

y

20 40 100

200 80 40

click

click click

This table represents the equation y = 2x Find the values for y, given the values for x.

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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

• f 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

Teacher Notes & Math Practice

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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 click

Real Life Example Slide 36 / 65

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 t 200 4 100 2 50 1 s t 4 200 5 250 6 300 s t 200 150 100 50 50 1 s t 1 51 2 52 3 53 Answer

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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 ?

Answer

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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 Answer

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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 2 100 3 150 4 200 5 250

Tables to Graphs

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(x) (y) 1 50 2 100 3 150 4 200 5 250 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 Slide 44 / 65

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 Slide 45 / 65

0 1 2 3 4 5 6 7 8 400 350 300 250 200 150 100 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) Coordinat e (x,y) 1 50 (1, 50) 2 100 (2, 100) 3 150 (3, 150) 4 200 (4, 200) 5 250 (5, 250)

Function Table

Math Practice

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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.

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Number

• f Books

(x) Total Cost (y) Coordinat e (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

Answer & Math Practice

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

0 1 2 3 4 5 6 7 8 400 350 300 250 200 150 100 50 0 1 2 3 4 5 6 7 8 400 350 300 250 200 150 100 50 0 1 2 3 4 5 6 7 8 400 350 300 250 200 150 100 50

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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.

0 1 2 3 4 5 6 7 8 80 70 60 50 40 30 20 10

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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 h t h t h 1 4 3 12 5 20 t h 5 1 10 2 15 3 t h 4 1 8 2 12 3 Answer

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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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Answer

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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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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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Dependent Variable

The variable whose value varies based

• n 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 1 2 4

3 9 The dependent variable is always the

• utput (y).

Function Tables

Equations Graphing

0 1 2 3 4 5 6 7 8 8 7 6 5 4 3 2 1

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Equation

Two expressions that are equivalent to each

• ther. Equivalence is shown with an equal sign.

4x=8

equivalent expressions

4 =x3

equivalent expressions

no equivalence

x3

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Expression

Numbers, symbols and operations grouped together that show the value of something.

2 x 3 = 6

Expressions DO NOT have equal signs.

3 2 + 1

2

An expression is

• ne side of

an equation.

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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

• n the

input (x).

(x) (y)

1 1 2 4 3 9

x is the independent variable, on the left, and y is the dependent variable,

• n the right.

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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 1 2 4

3 9 The independent variable is always the input (x). Function Tables

Equations Graphing

8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8

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Input

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

The

• utput (y)

depends

• n the

input (x).

(x) (y) 1 1 2 4

3 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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Output

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

The

• utput (y)

depends

• n the

input (x).

(x) (y) 1 1 2 4

3 9

y = x2

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

Input

Output

Rule

y = x2

3 9

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Rule

Defines the relationship between the input and the output, using an algebraic equation. It says what happens to the input in a function.

Input

3

Output

9

Rule:

y = x2

(x) (y) 1 3 2 6

3 9

Rule:

y = 3x

(x) (y) 1 6 2 10

3 14

Rule:

y = 4x+2

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Variable

A letter or symbol that represents a changeable or unknown value.

4x + 2

variable

x = ?

2x = 6

x 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

• thers.

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.