1. Proportional Relationships
I can compare two proportional relationships represented in different ways.

2. What is a Proportional Relationship?
Proportional Relationships
What is a Proportional Relationship?
When two quantities always have the same size in relation to each other. In other words they have a constant rate of change.
Proportional relationships can be represented in different ways such as an equation, word problem, between two points, table, or graph.

3. What is a Proportional Relationship?
Proportional Relationships
What is a Proportional Relationship?
There are two types of proportional relationships:
Direct: as one object gets larger, the other object gets larger as well.
Inverse: as one item gets larger, the other gets smaller.

4. Direct Proportional Relationships - Equations
Equations of proportional relationships are written in the form y=mx, where m is the slope but can also be called unit rate, rate of change, or constant of proportionality.
Example: y=3x
3 would be the slope or rate of change which means each time the relationship increases by 3

5. Direct Proportional Relationships - Equations
You Try. In each of the equations below identify what the relationship is changing by each time.
y=5x
y=12x
y=8x
y=25x
y=3x

6. Direct Proportional Relationships - Equations
Which of these has the greatest rate of change? Which of these has the lowest rate of change?
y=5x
y=12x
y=8x
y=25x
y=3x

7. Direct Proportional Relationships - Word Problems
When we read a word problem that contains a proportional relationship we should be looking for a constant rate of change.
Example: Mary’s mom gives her $5 each week for allowance.
5 would be the constant rate of change. In this scenario it means each week Mary’s money would increase by 5 dollars so long as she did not spend it.

8. Direct Proportional Relationships - Word Problems
You Try. In each of the word problems below identify what the relationship is changing by each time.
Jack’s plant grows 5 inches every day.
Each ticket to see the movie is $8.00.
The recipe calls for 3/4 cup of milk for each batch of cookies.
Jessy walks 1 block every 10 minutes.

9. Direct Proportional Relationships - Word Problems
Which of these has the greatest unit rate? Which of these has the lowest unit rate?
Jack’s plant grows 5 inches every day.
Each ticket to see the movie is $8.50.
The recipe calls for 3/4 cup of milk for each batch of cookies.
Jessy walks 1 block every 10 minutes.

10. Direct Proportional Relationships - Between Points
When we are given two points we can use the slope formula y2-y1 to calculate the proportional relationship.
Example: What is the slope between (1,4) and (2,8)?
label the points and plug them in
(1,4) (2,8) = 4 = 4
x2-x1 x1 y1 x2 y2
2-1 1

11. Direct Proportional Relationships - Between Points
You Try. Find the slope for each pair of points listed below.
(6,8)(5,7)
(1,4)(3,7)
(10,5)(12,9)
(4,12)(8,15)

12. Direct Proportional Relationships - Between Points
Which of these has the greatest slope? Which of these has the lowest slope?
(6,8)(5,7) = 1
(1,4)(3,7) = 1.5
(10,5)(12,9) = 2
(4,12)(8,15) = 0.75

13. Direct Proportional Relationships - Tables
When we are given a table we can pick two points from the table and use the slope formula y2-y1 to calculate the proportional relationship.
Example: (1,2.50)(2,5.00)
= 2.50 = 2.50
x2-x1 x1 y1 x2 y2
Number of People
Cost 1
2.50 2
5.00
2-1 1

14. Direct Proportional Relationships - Tables
When we are given a table we can also use the unit rate which is the y-value when the x-value is 1.
Example:
The unit rate is 2.50
Number of People
Cost 1
2.50 2
5.00

15. Direct Proportional Relationships - Tables
You try. Find the rate of change for each of the tables below.

16. Direct Proportional Relationships - Tables
Which of these has the greatest unit rate? Which of these has the lowest unit rate? 12
0.40
4.24

17. Direct Proportional Relationships - Graphs
When we are given a graph we can pick two points from the graph and use the slope formula y2-y1 to calculate the proportional relationship.
Example: (1,20) (2,40)
40-20 = 20 = 20
x2-x1 x1 y1 x2 y2
2-1 1

18. Direct Proportional Relationships - Graphs
When we are given a graph we can also use rise over run which is how far we go up and how far we go over to get to our next point.
Example:
= 20
up 20
over 1
over 1
up 20

19. Direct Proportional Relationships - Graph
You try. Find the rate of change for each of the graphs below.

20. Direct Proportional Relationships - Graph
Which of these has the greatest rate of change? Which of these has the lowest rate of change?
0.75 80
0.6

21. Comparing Direct Proportional Relationships
To compare two proportional relationships identify the constant rate of change for each and then compare them.
Example:
a) Which has the greater rate of change?
y=2x y=5x
b) Which has the lowest constant of proportionality?
Dave bakes 3 pies each hour.
Marcy grills 10 hamburgers each hour.

22. Comparing Direct Proportional Relationships
Example Continued…
c) Which has the smaller slope?
(1,4)(2,5) (2,2)(4,5)
d) Which has the larger unit rate?
apples
cost 1
0.50 2
1.00 3
1.50
oranges
cost 1
0.75 2
1.50 3
2.25

23. Comparing Direct Proportional Relationships
Example Continued…
c) Which has lower rate of change?

24. Comparing Direct Proportional Relationships
Let’s mix it up!!!
a) Which has the greater rate of change?
y=5x

25. Comparing Direct Proportional Relationships
Let’s mix it up!!!
b) Which has the smaller unit rate?
apples
cost 1
0.50 2
1.00 3
1.50

26. Comparing Direct Proportional Relationships
Let’s mix it up!!!
c) Which has the greatest constant of proportionality?
oranges
cost 1
0.75 2
1.50 3
2.25
y=2x

27. Creating Direct Proportional Relationships
Now that you know how to identify proportional relationships we are going to learn how to create them!
Example: Start with a word problem: Jack’s plant grows 5 inches every day.
Write an equation to represent the proportional relationship: y=5x where y=height of the plant, x=number of days, and 5=constant rate of change

28. Creating Direct Proportional Relationships
Example continued
Create a table to represent the proportional relationship by substituting in values for x like 0,1,2,3 then filling in the table. Don’t forget your titles.
y=5(0)=0 y=5(1)=5
y=5(2)=10 y=5(3)=15
days
height 1 5 2 10 3 15

29. Creating Direct Proportional Relationships
Example continued
Create a graph to represent the proportional relationship by placing each of the coordinate points on a graph and connecting the points with a line. Don’t forget your titles.
height
days