1. Investigating Slope as Rates Guided Practice Finding Rate of Change (Slope) Guided Practice Speedy Rates

2. Finding Slope / Finding Rate of Change (Guided Practice) Up until now, we have been working with rate. Rate assumes you have linearity and a starting point at (0,0).

3. Finding Slope / Finding Rate of Change Since we will encounter other types of graphs, we will need to develop skills to find rate of change and find slope. We will also need to check that our rate of change is constant and that our graphs are linear. We’ll start by verifying our thinking about rate with a zero starting point.

4. Rachel gave a birthday party for her niece. She chose to prepare fruit punch for her niece and her guests. She thought 8 cups of fruit punch were good for 6 kids or 12 cups for 9 kids. How many cups of punch will each kid drink based on Rachel’s assumption? depends on Amount of punch the number of kids at the party. Dependent QuantityIndependent Quantity

5. Remember When…

6. Rate DependentIndependent depends on Which must happen 1 st ? Amount of punch The number of kids at the party.

7. Rachel gave a birthday party for her niece. She chose to prepare fruit punch for her niece and her guests. She thought 8 cups of fruit punch were good for 6 kids or 12 cups for 9 kids. How many cups of punch will each kid drink based on Rachel’s assumption? 8 8 cups 6 6 kids 12 9 12 cups 9 kids Eight cups of fruit drink will be divided among 6 kids at a child’s birthday party. == 4 3 cups kids ÷2 ÷3

8. Rate of Change Rate of Change = 3 cups kids 4 The difference from 8 to 12 is 4 The difference from 6 to 9 is 3

9. This lesson’s focus... between two points using the 3 Column Process Chart (3CPC). …find the rate of change

10. 6 8 12 9 6 9 4 4 3 2 1 12 cups for 9 kids 8 321 3 From 8 to 12 is a change of positive 4 in Y. Change in Y or “delta Y” or ∆Y ∆Y = 4 From 6 to 9 is a change of positive 3 in X. Change in X or “delta X” or ∆X ∆X = 3∆X ∆Y = 4 3 4 3 ( ) 9 6 xx y Rate of Change Using the 3 Column-Process-Chart = Since the # of cups depends on the # 8 cups for 66 8 cups 3 ( ) 4 3 4 3 # #

11. X Y 6 3 ∆X∆X = 8 4 ∆Y∆Y = 12 4 3 ∆X∆X ∆Y∆Y = Using the Graph y x 66 8 8 9 9 9 12 (6, 8 ) (9, 12 ) 4 3

12. ∆X∆X ∆Y∆Y ( ) XY ∆X ∆Y ∆X ∆Y

13. 222666 333999 x ΔxΔx Δ yΔ y 00 y ` (2, 6) (3, 9) Δx =1 3 Δ y = 3 = ΔxΔx Δ yΔ y = 33 1 ΔxΔx Δ yΔ y 3 2 3 ( ) 3 3 x = yx Rate of Change Or Slope Equation or a Rule 3 ( ) 0 3 Column-Process-Chart 3 1

14. 48 510 x 00 y (0, 0) (5, 10) (4, 8) 8 1 ` ` ` 10 22 4 5 1 Graph and 3 Column-Process-Chart

15. 48 510 00 (0, 0) (5, 10) (4, 8) 8 1 ` 10 2 2 4 5 1 2 1 ΔxΔx Δ yΔ y ΔxΔx Δ yΔ y ` ` = = =22 2( )2 2 2 xyx 4 5 4 0 ΔxΔx Δ yΔ y =2 Rate of Change Equation or a Rule x y Graph and 3 Column-Process-Chart

16. X Y ∆X∆X ∆Y∆Y 00 XY 5 1 10 2 15 3 1 5 = ∆Y∆Y == 1 5 ∆X∆X 1 5 1 5 1 5 ( 0 ) ( 5 ) ( 10 ) ( 15 ) 1 5 ( X ) = 1 5 ∆Y∆Y = =∆X∆X 1 5

17. X Y ∆X∆X ∆Y∆Y 00 XY 5 1 10 2 15 3 2 10 = ∆Y∆Y = ∆X∆X = 2 ÷ 2 ÷ 2 = 1 5 1 5 ( 0 ) 1 5 ( 5 ) 1 5 ( 10 ) 1 5 ( 15 ) 1 5 ( X ) = 2∆Y∆Y = 10∆X∆X = ∆X∆X ∆Y∆Y = 1 5

18. X Y 00 XY 5 1 2 15 3 2 10 ∆Y∆Y = ∆X∆X = 1 5 1 5 1 5 1 5 ( 0 ) ( 5 ) ( 10 ) ( 15 ) 1 5 ( X ) = 2∆Y∆Y = 10∆X∆X = ∆X∆X ∆Y∆Y = 2 ÷ 2 ÷ 2 = 1 5 ∆X∆X ∆Y∆Y = 1 5

19. Independent x ProcessDependent y 35 610 Let’s review. Suppose you had to find the equation of the line that contained the points (3, 5) and (6, 10). (3)(3) (x) Equation: y = 5353 x Now, suppose you had to find the equation of the line that contained any two points (x 1, y 1 ) and (x 2, y 2 ). 35 Independent x ProcessDependent y x1x1 y1y1 x2x2 y2y2 ∆x ∆y (x) (x 1 ) (x 2 ) Rate of change is commonly referred to as slope, m. (6)(6) 5353 Rate of change = ∆y = 10 – 5 = ∆x 6 – 3 5353 5353 5353 Equation: y = m (x)(x) Rate of change = ∆y = y 2 – y 1 = m ∆x x 2 – x 1 m m m As noted on your formula chart, the slope formula is m = y 2 – y 1. x 2 – x 1