Exploring Rates Using a Three-Column Process Chart (Module 6) Warm Up Spring Extensions

1. The Apollo Curriculum Project Exploring Rates Using a Three-Column Process Chart (Module 6) Warm Up Spring Extensions Guided Practice Finding Rate of Change (Slope) Guided Practice Speedy Rates

2. Spring Extensions (Warm Up) KEY Students in a science class collected the data below during a spring stretching experiment. Write an equation to model the data. Use the equation to predict what stretch distance would be produced from 95 pounds of force. y = .2x or y = (1/5)x 9

3. 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).

4. 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.

5. Rachel provided 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 are good for 6 kids or 12 cups for 9 kids. How many cups of punch will each kid drink based on Rachel’s assumption? the number of kids at the party. depends on Amount of punch Dependent Quantity Independent Quantity

6. Remember When…

7. Rate Dependent Independent Amount of punch The number of kids at the party. depends on Which must happen 1st?

8. 12 cups 8 cups 9 kids 6 kids Rachel provided 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 are good for 6 kids or 12 cups for 9 kids. How many cups of punch will each kid drink based on Rachel’s assumption? Eight cups of fruit drink will be divided among 6 kids at a child’s birthday party. Rate 8 cups 12 ÷3 ÷2 4 = = 3 kids 9 ÷3 6 ÷2

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

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

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

12. = = ∆Y ∆X (9 , 12 ) 6 6 8 8 (6 , 8 ) 12 12 9 9 ∆Y = ∆X Using the Graph 3 3 9 Y x y 4 4 4 6 8 12 Slope Rate of Change X 3

13. ∆Y ∆Y ∆Y ∆X ∆X ∆X 3 Column-Process-Chart Slope ( ) X Y Graph ∆X ∆Y Rate of Change

14. 3 3 3 3 ( ) ( ) ( ) ( ) Δy = 3 Δx = 1 Δy Δx 3 Column-Process-Chart x y Equation or a Rule ( ) x x y = 0 0 0 2 2 2 2 6 6 6 3 1 1 3 3 3 3 9 9 9 (3, 9) 3 (2, 6) ` Δy Δy = = 3 3 Δx Δx Rate of Change Or Slope

15. Graph and 3 Column-Process-Chart y 0 0 4 4 8 8 2 2 1 1 5 5 10 10 ` (5, 10) ` (4, 8) (0, 0) x `

16. Δy Δx Δy Δx 2 2 2 2 ( ) ( ) ( ) ( ) Graph and 3 Column-Process-Chart Equation or a Rule x x y = y 0 0 0 4 4 4 4 8 8 1 2 1 5 5 5 10 10 ` (5, 10) 2 ` (4, 8) Δy 2 = = 2 2 Δx 1 = 2 Rate of Change (0, 0) x `

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

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

19. 1 5 ÷ 2 1 1 1 1 1 1 ÷ 5 5 5 5 5 5 2 ( X ) = X Y Y 0 ( 0 ) 0 1 5 ( 5 ) 2 = ∆Y ∆X = 10 ∆Y 10 ( 10 ) 2 = ∆X 15 3 ( 15 ) ∆X = 10 ∆Y 2 = = ∆X 10 ∆Y = 2 X

20. Speedy Rates (Guided Practice) Now it is time for you to develop accuracy and speed when finding rates using the 3-column process charts (3CPC). Examine the tables that follow. Determine the rates and create an equation that models the data in the tables. Plot and label the points. Graph the line. Finally, verify your equations using the graphing calculator.

21. Problem #1 1(x) ∆y1 ∆x 1 1 = = 1(0) 1 1 1 1 1 1 1 1 1 1 1(1) 1(2) ∆x = 1 ∆y = 1 1(3) 1(4) 1(5) y = x Equation _____________

22. ½(x) = = ∆y31 ∆x 6 2 ½(0) ∆x = 3 6 6 6 3 3 3 ½(6) ∆y = 6 ½(12) ½(18) y = ½x Problem #2 Equation _____________

23. ¼(x) = = ¼(0) ∆y31 ∆x 12 4 12 12 12 3 3 3 ∆x = 3 ¼(12) ∆y = 12 ¼(24) ¼(36) y = ¼x Problem #3 Equation _____________

24. ∆x = 3 4(x) = = ∆y12 ∆x 3 4 4(0) ∆y = 12 3 3 12 12 4(3) 4(6) y = 4x Problem #4 Equation _____________

25. 6(x) = = 6(6) ∆x = 3 3 3 3 18 18 18 ∆y = 18 6(9) 6(12) ∆y18 ∆x 3 6 y = 6x Problem #5 Equation _____________

26. –3(x) –3 = = –3(6) ∆y–9 ∆x 3 –9 –9 3 3 –3(9) –3(12) ∆x = 3 ∆y = 18 y = –3x Problem #6 Equation ____________

27. (x) = = (–18) ∆y93 ∆x 6 2 9 9 6 6 (–12) ∆x = 3 ∆y = 9 (–6) y = x 3 2 Problem #7 Equation ____________

28. –5 ∆y–30 ∆x 6 –5(x) ∆y = 9 –5(–9) = = –30 –30 –30 6 6 6 –5(–3) ∆x = 3 –5(3) –5(9) y = –5x Problem #8 Equation ____________

29. 3 ∆x ∆y 5 As noted on your formula chart, the slope formula is m = y2 – y1 . x2 – x1 Equation: y = 5 3 x Equation: y = m (x) The Slope Formula Now, suppose you had to find the equation of the line that contained any two points (x1, y1) and (x2, y2). Let’s review. Suppose you had to find the equation of the line that contained the points (3, 5) and (6, 10). 5 3 m (x) (x) 5 3 (3) m (x1) 5 3 (6) m (x2) Rate of change = ∆y = y2 – y1 = m ∆x x2 – x1 Rate of change = ∆y = 10 – 5= ∆x 6 – 3 5 3 Rate of change is commonly referred to as slope, m.

30. The Apollo Curriculum Project The End