1. Write The amount of money in my pocket is directly related to how many hours I work. The number of cheese sandwiches I can afford is inversely related to how much a cheese sandwich costs. –What do I mean by this? What’s the difference? –What do the general forms of direct and inverse variation equations look like?

2. Direct and Inverse Variation

3. Direct Variation A simple linear relationship between two variables where the two increase (or decrease) together. –Constant rate of increase. –When one variable equals zero, the other one does as well.

4. Direct Variation # of bricks vs. height of stack # of hours vs. amount of paycheck # of minutes vs. distance from starting line # of years old vs. # of days old # of candy bars you buy vs. total cost

5. Direct Variations y=kx 1. A constant rate of change. 2. When x is zero, y is also zero. –That is, there is no starting condition. –This is the same as saying that the ratio of y/x is constant. Contrary to popular belief, k can be negative. That is, it’s not true that as x increases, y increases.

6. Direct Variation Another way to look at things… k is called the constant of variation. –if y/x is not constant, it’s not a direct variation.

7. Direct Variation Skills Is it a direct variation? xy 00 26 412 618 826 1030

8. Direct Variation Skills Is it a direct variation? xy -315 5 00 2-10 4-20 6-30

9. Direct Variation Skills What’s the constant of variation? (12, 4)

10. Direct Variation Practice Find the mising value in the DV (4, 32) (x, 56) What is the constant of variation? (4, 22) Is this a direct variation? If not, change the table so it is, and write an equation. xy -6 212 316 530 1166

11. Inverse Variation Mr. Bowen is having a stupid pizza party for jerks. Suppose there are 16 pieces in a pizza. How many pieces will each student get if…. –…only 1 student shows up? –…2 students show up? –…4 students show up? –…10 students show up? –…1,000,000 students show up?

12. Inverse Variation As x gets farther from zero, y gets closer to zero. There are two ways to talk about these: In a direct variation, the ratio of x and y is constant. In an indirect variation, the product of x and y is constant.

13. For Example Suppose the number of absences you have in a semester relates to your grade in the following way: 1 absence100 % 8 absences12.5 % We test whether or not this is a indirect variation by asking whether xy is constant. If so, this is the constant of variation.

14. Inverse Variation So a direct variation is sort of an upside down direct variation: –There are two quantities in an indirect variation that react to one another in a very predictable manner. –There is a constant of variation that describes this relationship. –The difference is simply that in an indirect variation, an increase in one quantity causes a decrease in the other.

15. Inverse Variation Skills Identifying IV’s, and writing equations xy 218 49 123

16. Inverse Variation Skills Identifying IV’s, and writing equations Cost # of Items 1100 1010 205

17. Inverse Variation Skills Finding the missing value for an IV. (5, 8) (80, x)

18. 1 minute practice Ms. Roberts teaches yoga in the mornings, and has some mats. Ms. Roberts teaches yoga in the mornings, and has some mats. –If 9 students were to show up, they’d each be able to use two mats. –If 36 students showed up, they’d each only get half a mat.

19. 1 minute practice We won the jackpot! Wait – how many people were in our lottery pool? – If 2 of us share the jackpot, we each get $250. – If 40 of us share the jackpot, we each get $12.50.

20. Inverse Variation Problems They tend to be about sharing something. This is a good way to recognize them in word problems. –Generally, there’s a limited amount of something – pizza, cake, time, money, whatever – to be split among people.

21. Practice Identify which of these are inverse variations and which are direct, and write the equation that describes the relationship. xy 418 627 1045 xy 310 56 215 xy -48 2 7-14 xy 221 410.5 67

22. Practice Find the missing values in these indirect variations. (4, 4)(20, ½ ) (x, 8) (2, x) (x, 8) (2, x)

23. Summary: Direct Variation Another way to look at things… k is called the constant of variation. –if y/x is not constant, it’s not a direct variation.

24. Summary As x gets farther from zero, y gets closer to zero (and vice versa). There are two ways to talk about these: In an inverse variation, the product of x and y is constant.