1. Algebra 1 Predicting Patterns & Examining Experiments Unit 4: Twice the Change Section 3: Respect the General

2. What combinations of quarters and dimes add up to be $1.50? I have a bag of quarters and dimes and I need to pull out $1.50. What are all my options? I have a bag of quarters and dimes and I need to pull out $1.50. What are all my options?

3. What combinations of quarters and dimes add up to be $1.50? I have a bag of quarters and dimes and I need to pull out $1.50. What are all my options? I have a bag of quarters and dimes and I need to pull out $1.50. What are all my options? # quarters # dimes 015 210 45 60

4. What combinations of quarters and dimes add up to be $1.50? I have a bag of quarters and dimes and I need to pull out $1.50. What are all my options? Is this a linear relationship? I have a bag of quarters and dimes and I need to pull out $1.50. What are all my options? Is this a linear relationship? # quarters # dimes 015 210 45 60 xy

5. What combinations of quarters and dimes add up to be $1.50? I have a bag of quarters and dimes and I need to pull out $1.50. What are all my options? Is this a linear relationship? -YES What is the equation? I have a bag of quarters and dimes and I need to pull out $1.50. What are all my options? Is this a linear relationship? -YES What is the equation? # quarters # dimes 015 210 45 60 xy

6. What combinations of quarters and dimes add up to be $1.50? I have a bag of quarters and dimes and I need to pull out $1.50. What are all my options? Is this a linear relationship? -YES What is the equation? I have a bag of quarters and dimes and I need to pull out $1.50. What are all my options? Is this a linear relationship? -YES What is the equation? # quarters # dimes 015 210 45 60 xy y-intercept –5+2

7. What combinations of quarters and dimes add up to be $1.50? I have a bag of quarters and dimes and I need to pull out $1.50. What are all my options? Is this a linear relationship? -YES What is the equation? I have a bag of quarters and dimes and I need to pull out $1.50. What are all my options? Is this a linear relationship? -YES What is the equation? # quarters # dimes 015 210 45 60 xy y-intercept –5+2 y-int: 15 slope: –5/2

8. What combinations of quarters and dimes add up to be $1.50? I have a bag of quarters and dimes and I need to pull out $1.50. What are all my options? Is this a linear relationship? -YES What is the meaning of this equation? I have a bag of quarters and dimes and I need to pull out $1.50. What are all my options? Is this a linear relationship? -YES What is the meaning of this equation? # quarters # dimes 015 210 45 60 xy y-intercept –5+2 y-int: 15 slope: –5/2 why 15? why -5/2?

9. What combinations of quarters and dimes add up to be $1.50? I have a bag of quarters and dimes and I need to pull out $1.50. What are all my options? Is this a linear relationship? -YES What is the meaning of this equation? I have a bag of quarters and dimes and I need to pull out $1.50. What are all my options? Is this a linear relationship? -YES What is the meaning of this equation? # quarters # dimes 015 210 45 60 xy y-intercept –5+2 y-int: 15 slope: –5/2 why 15? why -5/2? y-int: 15 Start with 15 dimes = $1.50 slope: –5/2 Since 5 dimes = 2 quarters, If you drop 5 dimes, you can add 2 quarters y-int: 15 Start with 15 dimes = $1.50 slope: –5/2 Since 5 dimes = 2 quarters, If you drop 5 dimes, you can add 2 quarters

10. What combinations of quarters and dimes add up to be $1.50? # quarters # dimes 015 210 45 60 xy y-intercept –5+2 y-int: 15 slope: –5/2

11. What combinations of quarters and dimes add up to be $1.50? I have a bag of quarters and dimes and I need to pull out $1.50. What are all my options? Is this a linear relationship? -YES What is the equation? I have a bag of quarters and dimes and I need to pull out $1.50. What are all my options? Is this a linear relationship? -YES What is the equation? # quarters # dimes 015 210 45 60 xy Quarters are worth.25 and dimes are.10. If we need 1.50, then.25 times the number of quarters plus.10 times the number of dimes, should equal the 1.50 that we need. or Quarters are worth.25 and dimes are.10. If we need 1.50, then.25 times the number of quarters plus.10 times the number of dimes, should equal the 1.50 that we need. or

12. What combinations of quarters and dimes add up to be $1.50? I have a bag of quarters and dimes and I need to pull out $1.50. What are all my options? Is this a linear relationship? -YES What is the equation? I have a bag of quarters and dimes and I need to pull out $1.50. What are all my options? Is this a linear relationship? -YES What is the equation? # quarters # dimes 015 210 45 60 xy Quarters are worth.25 and dimes are.10. If we need 1.50, then.25 times the number of quarters plus.10 times the number of dimes, should equal the 1.50 that we need. or This is a linear relationship, but what’s the slope?.. what’s the y-intercept? Is this line the same thing as ? Quarters are worth.25 and dimes are.10. If we need 1.50, then.25 times the number of quarters plus.10 times the number of dimes, should equal the 1.50 that we need. or This is a linear relationship, but what’s the slope?.. what’s the y-intercept? Is this line the same thing as ?

13. What combinations of quarters and dimes add up to be $1.50? I have a bag of quarters and dimes and I need to pull out $1.50. What are all my options? Is this a linear relationship? -YES What is the equation? I have a bag of quarters and dimes and I need to pull out $1.50. What are all my options? Is this a linear relationship? -YES What is the equation? # quarters # dimes 015 210 45 60 xy Quarters are worth.25 and dimes are.10. If we need 1.50, then.25 times the number of quarters plus.10 times the number of dimes, should equal the 1.50 that we need. or Let’s look at the situation where we have 0 quarters: Quarters are worth.25 and dimes are.10. If we need 1.50, then.25 times the number of quarters plus.10 times the number of dimes, should equal the 1.50 that we need. or Let’s look at the situation where we have 0 quarters:

14. What combinations of quarters and dimes add up to be $1.50? I have a bag of quarters and dimes and I need to pull out $1.50. What are all my options? Is this a linear relationship? -YES What is the equation? I have a bag of quarters and dimes and I need to pull out $1.50. What are all my options? Is this a linear relationship? -YES What is the equation? # quarters # dimes 015 210 45 60 xy Quarters are worth.25 and dimes are.10. If we need 1.50, then.25 times the number of quarters plus.10 times the number of dimes, should equal the 1.50 that we need. or Let’s look at the situation where we have 0 quarters: Quarters are worth.25 and dimes are.10. If we need 1.50, then.25 times the number of quarters plus.10 times the number of dimes, should equal the 1.50 that we need. or Let’s look at the situation where we have 0 quarters: y-intercept (0,15)

15. What combinations of quarters and dimes add up to be $1.50? I have a bag of quarters and dimes and I need to pull out $1.50. What are all my options? Is this a linear relationship? -YES What is the equation? I have a bag of quarters and dimes and I need to pull out $1.50. What are all my options? Is this a linear relationship? -YES What is the equation? # quarters # dimes 015 210 45 60 xy Quarters are worth.25 and dimes are.10. If we need 1.50, then.25 times the number of quarters plus.10 times the number of dimes, should equal the 1.50 that we need. or Let’s look at the situation where we have 0 dimes: Quarters are worth.25 and dimes are.10. If we need 1.50, then.25 times the number of quarters plus.10 times the number of dimes, should equal the 1.50 that we need. or Let’s look at the situation where we have 0 dimes:

16. What combinations of quarters and dimes add up to be $1.50? I have a bag of quarters and dimes and I need to pull out $1.50. What are all my options? Is this a linear relationship? -YES What is the equation? I have a bag of quarters and dimes and I need to pull out $1.50. What are all my options? Is this a linear relationship? -YES What is the equation? Quarters are worth.25 and dimes are.10. If we need 1.50, then.25 times the number of quarters plus.10 times the number of dimes, should equal the 1.50 that we need. or Let’s look at the situation where we have 0 dimes: Quarters are worth.25 and dimes are.10. If we need 1.50, then.25 times the number of quarters plus.10 times the number of dimes, should equal the 1.50 that we need. or Let’s look at the situation where we have 0 dimes: y-intercept (0,15) # quarters # dimes 015 210 45 60 xy

17. Two forms of the same line The two lines below ARE the same, just in different forms. Slope-intercept form?

18. The two lines below ARE the same, just in different forms. Slope-intercept formGeneral form Two forms of the same line

19. The two lines below ARE the same, just in different forms. Slope-intercept form y = mx + b General form Ax + By = C Two forms of the same line

20. The two lines below ARE the same, just in different forms. Slope-intercept form y = mx + b m = slope b = y-intercept General form Ax + By = C x-int: x = C/A ( C/A, 0 ) y-int: y = C/B ( 0, B/A ) Two forms of the same line

21. The two lines below ARE the same, just in different forms. How do we change from one form to another? The two lines below ARE the same, just in different forms. How do we change from one form to another? Two forms of the same line

22. The two lines below ARE the same, just in different forms. How do we change from one form to another? The two lines below ARE the same, just in different forms. How do we change from one form to another? Two forms of the same line Solve for y.

23. The two lines below ARE the same, just in different forms. How do we change from one form to another? What happens to y in the equation? The two lines below ARE the same, just in different forms. How do we change from one form to another? What happens to y in the equation? Two forms of the same line Solve for y.

24. The two lines below ARE the same, just in different forms. How do we change from one form to another? What happens to y in the equation? 1. multiply by.10 2. add.25x The two lines below ARE the same, just in different forms. How do we change from one form to another? What happens to y in the equation? 1. multiply by.10 2. add.25x Two forms of the same line Solve for y. layerslayers

25. The two lines below ARE the same, just in different forms. How do we change from one form to another? What happens to y in the equation? 1. multiply by.10 2. add.25x The two lines below ARE the same, just in different forms. How do we change from one form to another? What happens to y in the equation? 1. multiply by.10 2. add.25x Two forms of the same line Solve for y.

26. The two lines below ARE the same, just in different forms. How do we change from one form to another? What happens to y in the equation? 1. multiply by.10 2. add.25x The two lines below ARE the same, just in different forms. How do we change from one form to another? What happens to y in the equation? 1. multiply by.10 2. add.25x Two forms of the same line Solve for y.

27. The two lines below ARE the same, just in different forms. How do we change from one form to another? What happens to y in the equation? 1. multiply by.10 2. add.25x The two lines below ARE the same, just in different forms. How do we change from one form to another? What happens to y in the equation? 1. multiply by.10 2. add.25x Two forms of the same line Solve for y.

28. The two lines below ARE the same, just in different forms. How do we change from one form to another? What happens to y in the equation? 1. multiply by.10 2. add.25x The two lines below ARE the same, just in different forms. How do we change from one form to another? What happens to y in the equation? 1. multiply by.10 2. add.25x Two forms of the same line same slope: - 5. same y-intercepts 2 = -2.5

29. What linear equation allows us to transform Fahrenheit temperature to Celsius? ºC to ºF: ºF to ºC:

30. What linear equation allows us to transform Fahrenheit temperature to Celsius? ºC to ºF: ºF to ºC: Solve for C What happens to C? 1) multiply by 9/5 2) add 32

31. What linear equation allows us to transform Fahrenheit temperature to Celsius? ºC to ºF: ºF to ºC: Solve for C What happens to C? 1) multiply by 9/5 2) add 32 What’s the opposite? 1) subtract 32 2) multiply by 5/9

32. What linear equation allows us to transform Fahrenheit temperature to Celsius? ºC to ºF: ºF to ºC: Solve for C What happens to C? 1) multiply by 9/5 2) add 32 What’s the opposite? 1) subtract 32 2) multiply by 5/9

33. Graph and Describe. What are the intercepts? x-int: y-int: What is the slope? slope: What is the slope-int form? What are the intercepts? x-int: y-int: What is the slope? slope: What is the slope-int form?

34. Graph and Describe. What are the intercepts? x-int: y-int: What is the slope? slope: What is the slope-int form? What are the intercepts? x-int: y-int: What is the slope? slope: What is the slope-int form?

35. Graph and Describe. What are the intercepts? x-int: (4,0) y-int: What is the slope? slope: What is the slope-int form? What are the intercepts? x-int: (4,0) y-int: What is the slope? slope: What is the slope-int form?

36. Graph and Describe. What are the intercepts? x-int: (4,0) y-int: (0,6) What is the slope? slope: What is the slope-int form? What are the intercepts? x-int: (4,0) y-int: (0,6) What is the slope? slope: What is the slope-int form? To find slope, use the two points above...

37. Graph and Describe. What are the intercepts? x-int: (4,0) y-int: (0,6) What is the slope? slope: What is the slope-int form? What are the intercepts? x-int: (4,0) y-int: (0,6) What is the slope? slope: What is the slope-int form? -6 +4

38. Graph and Describe. What are the intercepts? x-int: (4,0) y-int: (0,6) What is the slope? slope: What is the slope-int form? What are the intercepts? x-int: (4,0) y-int: (0,6) What is the slope? slope: What is the slope-int form?

39. Graph and Describe. What are the intercepts? x-int: (4,0) y-int: (0,6) What is the slope? slope: What is the slope-int form? What are the intercepts? x-int: (4,0) y-int: (0,6) What is the slope? slope: What is the slope-int form?

40. Graph and Describe. What are the intercepts? x-int: (4,0) y-int: (0,6) What is the slope? slope: What is the slope-int form? What are the intercepts? x-int: (4,0) y-int: (0,6) What is the slope? slope: What is the slope-int form?

41. Where are we? The equation below is graphed to its right. Label the axes correctly.

42. Where are we? The equation below is graphed to its right. Label the axes correctly. x-intercept y-intercept

43. Where are we? The equation below is graphed to its right. Label the axes correctly. x-intercept y-intercept 30 60

44. Where are we? The equation below is graphed to its right. Label the axes correctly. x-intercept y-intercept 30 60 20 10 10 20 30 40 50

45. Disclaimer All photos contained are used under creative commons rights. Day 181/365 - Big Change by kevharb http://www.flickr.com/photos/kevharb/3422792058/