1. Direct and Inverse Variation/Proportion

2. Which is it?? Table, Equation, or Graph Linear Nonlinear
Not an inverse variation… just nonlinear
Direct Variation
Not a direct variation… just linear
Inverse Variation
11/14/2018

3. Q3 Grade 7 Content Overview
Vocabulary
Variation and proportion
Constant of proportionality
Direct Proportion
Indirect Proportion
Equation
Ratio
Atlanta Public Schools
Q3 Grade 7 Content Overview

4. What is Direct Variation/Proportion?
Direct Proportion Direct Variation Directly Proportional
A relationship between two variables (usually x and y) where one variable changes and the other variable changes in proportion to the first.
Atlanta Public Schools
Q3 Grade 7 Content Overview

5. Direct Variation/Proportion continued
As the x values increase at a constant rate, the y values will also increase at a constant rate.
As the x values decrease at a constant rate, the y values will decrease at a constant rate.
X and y will have a proportional relationship.
Atlanta Public Schools
Q3 Grade 7 Content Overview

6. Constant of Proportionality
When y is directly proportional to x (or is a direct variation), there will be a CONSTANT OF PROPORTIONALITY.
In equations, this constant is represented by the letter k.
y = kx
The constant of proportionality is found by dividing the y values by the x values.
y/x = k
The constant of proportionality, or k, is the exact same thing as the slope (rise over run) that we previously learned about.
Atlanta Public Schools
Q3 Grade 7 Content Overview

7. Explain: Direct Proportion
Consider the following direct proportion situation:
It takes 2 hours for a company to manufacture each ladder that it makes
Examine the table below:
Ladders(x) Hours(y) 2 4 6 8 10
©Microsoft Word clipart
Now look at the graph of the data
Atlanta Public Schools
Q3 Grade 7 Content Overview

8. Q3 Grade 7 Content Overview
Graph Of Ladders Made against Time Taken
Hours 2 4 6 8 10 12
Ladders
Atlanta Public Schools
Q3 Grade 7 Content Overview

9. Q3 Grade 7 Content OverviewL T
Key Facts about Direct Proportion
All direct proportion graphs are straight line graphs passing through the origin
Look at the table of values again:
Ladders(x) Hours(y) 2 4 6 8 10
If you divide the number of hours by the given number of ladders, what do you find each time?
The answer is 2 every time. As x increases, y increases at a constant rate of 2x. This is a linear equation in the form y = kx where 2 is the constant of proportionality (k). Note also that the ratio of Ladders to Hours remains constant at x:2x.
Atlanta Public Schools
Q3 Grade 7 Content Overview

10. Inverse Variation/Proportion
A relationship between two variables in which the product is a constant. When one variable increases the other decreases in proportion so that the product is unchanged.
As x increases, y decreases.
As x decreases, y increases.
xy = k (Remember, k is the constant of proportionality)
The graph of an inverse variation is not a straight line, but it is curved. The graph will never cross the x-axis.
Atlanta Public Schools
Q3 Grade 7 Content Overview

11. Explain: Indirect Proportion
A farmer has enough cattle feed to feed 64 cows for 2 days.
Examine the table below. As the number of cows increases the number of days of feed decreases.
Cows (x)
Days (y) 8 16 32 64
We are going to draw a graph of the table.
The scale allows each axis to go at least up to 65.
We estimate the position of the points (2,64) (4,32) etc. as accurately as possible.
Atlanta Public Schools
Q3 Grade 7 Content Overview

12. Q3 Grade 7 Content Overview10 20 30 40 50 60 70
Graph of Number of Cows against Days of Feed
(2,64)
Days
(4,32)
(8,16)
(16,8)
(32,4)
(64,2) 10 20 30 40 50 60 70
Cows
Atlanta Public Schools
Q3 Grade 7 Content Overview

13. Q3 Grade 7 Content Overview
The graph is a typical inverse or indirect proportion graph
Cows
Days
It shows us that as the number of cows increases, the number of days left of feed decreases. The reverse is also true. If we decrease the number of cows, we will increase the number of days of feed.
Cows (x)
Days (y) 8 16 32 64
This is a non-linear equation in the form y = k
where k is still the constant of proportionality. However, y varies inversely to x. In this situation, what is k? x
Atlanta Public Schools
Q3 Grade 7 Content Overview