Solving for x, a Common Sense Approach

1. Solving for x, a Common Sense Approach

2. Purpose • Solving for an unknown variable is a common theme in Algebra Curriculums.

3. Purpose • Solving for an unknown variable is a common theme in Algebra Curriculums. • We can build a method for solving many equations based on a common sense approach.

4. Purpose • Solving for an unknown variable is a common theme in Algebra Curriculums. • We can build a method for solving many equations based on a common sense approach. • The way we will tackle this common problem is related to the way the content is taught in traditional mathematics courses.

5. Purpose • Solving for an unknown variable is a common theme in Algebra Curriculums. • We can build a method for solving many equations based on a common sense approach. • The way we will tackle this common problem is related to the way the content is taught in traditional mathematics courses. • The approach we will take is different in the sense that we will not seek to make a new algorithm for every solving situation.

6. What you will need to begin. • A knowledge of how to navigate through a PowerPoint presentation.

7. What you will need to begin. • A knowledge of how to navigate through a PowerPoint presentation. • A piece of paper to take notes on and complete the exercises contained in this presentation.

8. What you will need to begin. • A knowledge of how to navigate through a PowerPoint presentation. • A piece of paper to take notes on and complete the exercises contained in this presentation. • A four function calculator (add, subtract, multiply and divide)

9. Lets get started! • We are going to think about the process you would typically go through to get to school.

10. Getting to School • On a piece of paper write down a list of steps that you go through to get to school. • Keep your list between four and eight items in length. • Make sure it is in the order that you went through.

11. Getting to School Here is an example of a list. • Woke up. • Showered. • Got dressed. • Ate breakfast. • Drove to school. • Walked into the school.

12. Getting to School • Now that you have you list, consider what it would look like if you watched your morning in reverse (“un”going to school). • Write down a list what it would look like to go backwards.

13. Going to School Woke up and got out of bed. Showered Got dressed. Ate breakfast. Drove to school. Walked into the school. “Un”going to School Walked out of school Drove in reverse from school Regurgitated breakfast (nasty) Undressed Unshowered. (also nasty) Got back in bed and went to sleep. “Un”going to School

14. “Un”going to School • Aside from some of the disgusting ideas in your “un”going to school list there are some important things to notice about it.

15. “Un”going to School • Aside from some of the disgusting ideas in your “un”going to school list there are some important things to notice about it. • Every item should be the opposite action from the original list.

16. “Un”going to School • Aside from some of the disgusting ideas in your “un”going to school list there are some important things to notice about it. • Every item should be the opposite action from the original list. • The order it is written in should start at the end of the original list and go backwards to the start of it.

17. So what’s the point??? • If we have a process that contains a series of steps that are in order, we can think about how we would work backwards from the end undoing each item as we go. • If we do this we arrive at the beginning, where we were to start with.

18. No really, what’s the point??? • Let’s look at some “math” • Consider the linear function y=5x+3. • Let y=-3, so we have -3=5x+3. What x-value (input) would make this function give a y-value of -3?

19. y=5x+3 • How would we go about finding y if we knew what the x was? In other words what are the calculation steps that take x and give us a y? • We take x and multiply it by 5. Then add 3 to this answer and we would get y.

20. With y=-3 we have -3=5x+3 • We take x and multiply it by 5. Then add 3 to this answer and we would get y. • But we know y and need to figure out x. This is like being at school and needing to figure out how we got there. • We need to “un”go to school.

21. Forward Start with x Multiply by 5 Add 3 We get y “un”Forward Start with y Subtract 3 Divide by 5 We would be back to x. With y=-3 we have -3=5x+3

22. With y=-3 we have -3=5x+3 • Forward • Start with x • Multiply by 5 • Add 3 • We get y • “un”Forward • Start with y • Subtract 3 • Divide by 5 • We would be back to x. • We know y=-3 and we need to know x. • Which list will do this for us?

23. “un”Forward Start with y Subtract 3 Divide by 5 We would be back to x. Start with -3 -3-3=-6 -6/5=-1.2 We would have x so, x=-1.2 With y=-3 we have -3=5x+3

24. With y=-3 we have -3=5x+3 and we just found that x=-1.2 • Lets check our result. We think that when x=-1.2 we will get a y value of -3 when we plug it in. • Does 5(-1.2)+3=-3? • Well 5*-1.2=-6 and -6+3 is -3 so yes it does.

25. Try this example. • What x would make -9x+8=53 ? • Write down the problem before you go on. • We know the answer and need to know where we started. • If we knew x how would we find y? • Click on the list that describes this.

26. No, that’s not the one. • Take x • Divide by -9 • Subtract 8 • This gets us y • Remember at this point we only want to write a list that takes us from x to y. We are not trying to undo anything. • Did you follow order of operations correctly (PEMDAS)? Click here to try again

27. No, that’s not the one. • Take x • Add 8 • Multiply by -9 • This gets us y • Remember at this point we only want to write a list that takes us from x to y. • Did you follow order of operations correctly (PEMDAS)? Click here to try again

28. No, that’s not the one. • Take x • Subtract 8 • Divide by -9 • This gets us y • Remember at this point we only want to write a list that takes us from x to y. These operations don’t do that. • Did you follow order of operations correctly (PEMDAS)? Click here to try again

29. You have identified the steps that take us from x to y for the equation y=-9x+8 Take x Multiply by -9 Add 8 This gets us y Great job!

30. Let’s keep going! You have identified the steps that takes us from x to y for the equation y=-9x+8 Take x Multiply by -9 Add 8 This gets us y What is the process that will take us from y to x? Write down the steps before going on. y x

31. Which process did you write down? y Divide by -9 Subtract 8 x y Multiply by 9 Add -8 x y Subtract 8 Divide by -9 x y Add -8 Multiply by 9 x

32. You got it! You have identified the steps that takes us from x to y for the equation y=-9x+8 Take x Multiply by -9 Add 8 This gets us y So if we knew a y-value we can figure out what x-made it using this list y Subtract 8 Divide by -9 x

33. Oops, that’s not it. • Remember that the order we go in matters. • We need to undo the operations. • Negative numbers stay negative and positives stay positive. • y • Divide by -9 • Subtract 8 • x Click here to try again

34. Oops, that’s not it. • Remember that the order we go in matters. • We need to undo the operations. • Negative numbers stay negative and positives stay positive. • y • Multiply by 9 • Add -8 • x Click here to try again

35. Oops, that’s not it. • Remember that the order we go in matters. • We need to undo the operations. • Negative numbers stay negative and positives stay positive. • y • Add -8 • Multiply by 9 • x Click here to try again

36. We know the y-value (y=53) and need to know x and we have a list that does just that thing! y Subtract 8 Divide by -9 x y is 53 Subtract 8, 53-8=45 Divide by -9, 45/-9=-5 x is -5 -9x+8=53

37. Let’s Check • Does -9x+8=53 if x=-5? • Input -5 for x and check to make sure it gives an output of 53. • -9(-5)+8=? • 45+8=? • 53

38. Lets look at an example that really shows why this is such a powerful way to “solve for x” Write down the equation and the steps to go from x to y. This list will be longer since there are five operations. Take x To get y. One More!

39. The list you made should look like this. • Take x • Multiply by 5 • Subtract 2 • Divide by -27 • Multiply by 3 • Add 46 • To get y

40. The list you made should look like this. Take x Multiply by 5 Subtract 2 Divide by -27 Multiply by 3 Add 46 To get y But we know y and we need x. We need the other list again. Take y To get x

41. The list you made should look like this. Take x Multiply by 5 Subtract 2 Divide by -27 Multiply by 3 Add 46 To get y But we know y and we need x. We need the other list again. Take y Subtract 46 To get x

42. The list you made should look like this. Take x Multiply by 5 Subtract 2 Divide by -27 Multiply by 3 Add 46 To get y But we know y and we need x. We need the other list again. Take y Subtract 46 Divide by 3 To get x

43. The list you made should look like this. Take x Multiply by 5 Subtract 2 Divide by -27 Multiply by 3 Add 46 To get y But we know y and we need x. We need the other list again. Take y Subtract 46 Divide by 3 Multiply by -27 To get x

44. The list you made should look like this. Take x Multiply by 5 Subtract 2 Divide by -27 Multiply by 3 Add 46 To get y But we know y and we need x. We need the other list again. Take y Subtract 46 Divide by 3 Multiply by -27 Add 2 To get x

45. The list you made should look like this. Take x Multiply by 5 Subtract 2 Divide by -27 Multiply by 3 Add 46 To get y But we know y and we need x. We need the other list again. Take y Subtract 46 Divide by 3 Multiply by -27 Add 2 Divide by 5 To get x

46. This list takes us from y to x. Take y Subtract 46 Divide by 3 Multiply by -27 Add 2 Divide by 5 To get x Lets follow the steps on our y value, 39. 39 39-46=-7 -7/3=-2.3333333 -2.33333333*27=-63 -63+2=-61 -61/5=-12.2 So x=-12.2

47. We found that x=-12.2, Lets check to make sure it works. Plug it in for x. • If you follow order of operations you will find that it does work.