CONSTANT OF PROPORTIONALITY A.K.A. VARIATIONA.K.A. SLOPE

WHAT IS IT? A constant ratio in any proportional relationshipReally just another name for unit rate!

Remember, to be constant means it never changes!

WITH RATIO TABLES

Miles 50 100 150 200

Gallons

2 4 6 8

502

=?1

Find the

unit

rate!÷2

÷2

25 miles per gallon

This means our constant of proportionality is 25, so if we divide the miles by gallons we should always get 25.Let’s check!

𝟏𝟎𝟎÷𝟒=𝟐𝟓𝟏𝟓𝟎÷𝟔=𝟐𝟓𝟐𝟎𝟎÷𝟖=𝟐𝟓

Find the constant of proportionality/variation/slope between the gallons and the miles.

TRY THIS ONE!

Number of Apples

9 27 36

Cost $3.00 $9.00 $12.00

Find the constant of proportionality/variation/slope between gallons and miles.

Find the

unit

rate!

93=?1

÷3

÷3

Constant of proportionality = 3

Let’s Check!

𝟐𝟕÷𝟗=𝟑

𝟑𝟔÷𝟏𝟐=𝟑

WITH GRAPHS

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

20406080

100120140160180200

Hours

Miles

(0,0)

(1,45)

(2,90)

(3,135)

(4,180)

Y 45 90 135 180

X 1 2 3 4

Find the constant of proportionality.

To find our constant of proportionality we have to

divide!45

1= 45 90

2= 45

135

3= 45 180

4= 45

So, our constant of proportionality is 45.

We could write this as:

y=45(x)

IMPORTANT!We will ALWAYS be able to write our constant of

proportionality as an equation that looks like

this: y=kx

In our last example we had:

y = 45x

And “k” will always be our constant of proportionality/variation/slope!

Unit RateConstant of

ProportionalityConstant of Variation

Slope

FIND THE CONSTANT OF PROPORTIONALITY AND WRITE IT AS AN EQUATION: Y=KX

U-Swirl Frozen Yogurt

Weight (oz)

Cost

9 $2.25

11 $2.75

13 $3.25

𝑦=0.25 𝑥Weight (oz)

Cost

FIND THE CONSTANT OF PROPORTIONALITY AND WRITE IT AS AN EQUATION: Y=KX

MineCraft

Minutes Blocks

5 80

12 192

35 560

𝑦=16 𝑥

MinutesBlocks

FIND THE CONSTANT OF PROPORTIONALITY AND WRITE IT AS AN EQUATION: Y=KX

Baking

Minutes Cookies

10 12

20 24

30 39

There is no constant of

proportionality because there isn’t

a constant rate!

FIND THE CONSTANT OF PROPORTIONALITY AND WRITE IT AS AN EQUATION: Y=KX

𝑦=3 𝑥

Weight (lb.)

Cost ($)

Cost

($

)

Weight (lb.)2 4 6 8 10 12

6

12

18

24

30

36

42

FIND THE CONSTANT OF PROPORTIONALITY AND WRITE IT AS AN EQUATION: Y=KX

𝑦=5 𝑥

GallonsCost ($)

Cost

($

)

Gallons of Gas2 4 6 8 10 12

6

12

18

24

30

36

42

FIND THE CONSTANT OF PROPORTIONALITY AND WRITE IT AS AN EQUATION: Y=KX

Cost

($

)

Gallons of Gas2 4 6 8 10 12

6

12

18

24

30

36

42

There is no constant of

proportionality because there isn’t

a constant rate!

PROPORTIONAL VS. NON-PROPORTIONAL

If two quantities are proportional, then they have a constant ratio. To have a constant ratio means two quantities have the same unit rate.

If the ratio is not constant, the two quantities are said to be non-proportional.So, the two quantities do not have the same unit rate.

PROPORTIONAL RELATIONSHIPS

Will always go through the origin on a graph. (0,0)

Graph will always be a straight line.

In order to tell if a graph is proportional the line must go through the origin. Tell if the following graphs represent a proportional relationships.

1 2 3 4 5

1

2

3

4

5

x

y

1 2 3 4 5

1

2

3

4

5

x

y

Proportional ? _________ Proportional ? _________

Why? Line goes through the origin

Why? Line does notgo through the origin

Yes No

Let’s ReviewGuided Practice

State in words the proportional relationship shown here.(There are many correct answers!)

x

y

Dis

tance

(ft

.)Time (min.)

2 feet per min

Let’s ReviewQuick Quiz

State in words the proportional relationship shown here.(There are many correct answers!)

Cost

($

)Weight (ounces)

You Try

5oz for $2

You try: The following chart shows how much money Alex earns for mowing lawns. Is the amount of money he earns proportional to the number of hours that he spends mowing?

Earnings ($)

Hours (h)

Unit Rate (

)

14 1

28 2

42 3

56 4

1

$14

2

$28

1

$14

3

$42

1

$14

4

$56

Since the simplified ratios were equal, this was a proportional relationship.

hr

$

1

$14

We typically put time (hours) on the x-axis, and the earnings ($) on the y-axis.

Set up the graph paper to fit the data in the chart.

x

y

Hours worked

Earn

ings

($

)

1

14

28

42

56

2 3 4 5

Hours (h)

Earnings ($)

Point (x, y)

1 14 (1, 14)

2 28 (2, 28)

3 42 (3, 42)

4 56 (4, 56)

Plot points (x, y) from the table.

Connect the points.

Describe the graph of this proportional relationship.

Ticket Express charges $7 per movie ticket plus a $3 processing fee per order. Is the cost of an order proportional to the number of tickets ordered? Explain .

Cost ($) 10 17 24 31

Tickets Ordered

1 2 3 4

1

$10

ticketsof no.

($)cost 1

$8.5

2

17

1

$8

3

$24

1

$7.75

4

$31

Since all of the simplified ratios are not equal, there is no constant ratio, so this is NOT a proportional relationship.

Tickets ordered will be on the x-axis, and the cost ($) will be on the y-axis.

x

y

Tickets ordered

Cost

($)

1

4

24

32

2 3 4

Tickets Earnings ($)

Point (x, y)

0 0 (0,0)

1 10 (1, 10)

2 17 (2, 17)

3 24 (3, 24)

4 31 (4, 31)

Plot points (x, y) from the table.

Connect the points.

Describe the graph of this nonproportional relationship.

8

12

16

20

28

It passes through the origin,but it is not a straight line.