1. Plenary 4 Summer Institute Thunder Bay

2. 2 Consider this relation…

3. Contrast Open: Describe a relation sort of like this one. 3 Not Open: What type of relation is this?

4. The underlying idea To which big idea do you think this question might relate? 4

5. Contrast Open: A graph passes through the points (2,4) and (3,8). Describe the relation the graph represents. 5 Not open: A line passes through the points (2,4) and (3,8). Write the equation of the line.

6. 6 Contrast

7. What made it open? What made the first question open? 7

8. Contrast Open: Write an equation and solve it. 8 Less Open: Solve. 3x-2 = 8. Describe your strategy.

9. Using open questions for assessment for learning An open question, as a minds-on activity, provides valuable information about how to proceed with your lesson. 9

10. Might tell you that there are giant missing prerequisites that will get in the way of new learning. 10 OR assessment for learning

11. Might tell you that your students already know what you were planning to teach. 11 OR assessment for learning

12. Comparing parallel tasks and open questions 12 Look at this pair of parallel tasks. Choose one and try it.

13. 13 Jennifer drew a graph of y = 2x – 8. She said there was no more work to do to draw a graph of y =. What might she mean? Jennifer drew a graph of y = 2x 2 – 8. She said there was no more work to do to draw a graph of y =. What might she mean?

16. 16 What would the original graph look like? How do you know? What would the second graph look like? How do you know? Common questions

17. 17 What do you think Jennifer meant when she said there was no more work to do? Do you think she was right? Common questions

18. 18 What big idea was implicit in both of the parallel tasks? What do you think?

19. 19 With a partner, reframe the two parallel tasks into a single open question. Once you do, decide if you think the parallel tasks or the open question might be better to use and why. What do you think?

20. 20 Open questions can work in all three parts of a three-part lesson. Where to use open questions

21. 21 For example, a minds-on open question might be: Create two linear growing patterns that you think are really similar. Minds-on

22. 22 You could ask: What makes them similar? How are their pattern rules similar? Open Minds-on

23. 23 How might such an open question be effective in diagnosing student differences? Do you see it more as exposing or evoking thinking? Purpose Open Minds-on

24. 24 For a lesson on inequalities: Create a mathematical statement where any number greater than 10 is a possible solution. More examples – Open Minds-on

25. 25 For a lesson on quadratics: Which two graphs do you think are alike? Why? Y = 3x 2 -2y = -3x 2 -2 Y = 3x 2 +2y = 2x 2 + 3 More Open Minds-on

26. 26 How would you open up this question (or is it open now)? Fill in the missing values: Your turn – Open Minds-on 123678 8132328

27. 27 Action How might you use open questions for action?

28. 28 Choose two different values for the missing amounts. Make them different kinds of numbers: 3x – = 4 +  x Draw a diagram that would help someone understand how to solve your equation. Open Action

29. 29 How could this activity help the struggler? How could it help the strong student? What math is learned? Open Action

30. 30 What strategy do you think I used to make this question open? Is it a strategy that could be used in other situations? Open Action

31. 31 A linear growing pattern has 50 as the 25 th term. Create a bunch of possible patterns. How does the starting value for your pattern relate to the rate at which it grows? More examples – Open Action

32. 32 Two lines intersect at (1,3). One is much steeper than the other. What could the pair of lines be? Give several possibilities. More Open Action

33. 33 Compare the roots of these three equations. What do you notice? Why does it happen? 4x 2 – 17x + 4 = 0 6x 2 – 37x + 6 = 0 8x 2 – 65x + 8 = 0 More Open Action Add another question that works the same way.

34. 34 Make up other sets of quadratic equations that have something in common? More Open Action

35. 35 A trig function goes through the point (π,7). What could it be? List a bunch of possibilities. More Open Action

36. 36 What combination of $2 coins and $5 bills have a total value of $100? Your turn – Open Action

37. 37 Open questions are also very appropriate for consolidation. Consolidation questions

38. 38 Imagine that students have learned how to solve linear equations (e.g., 4x – 2 = 9). Open Consolidation

39. 39 Without solving this equation, how do you know that the solution has to be positive? -4x + 7 = -5x + 30 Open Consolidation

40. 40 How are these equations alike and different? 3x – 2 = 6 + 9x 3x – 2 = 6 + 9x – 6x -8 Open Consolidation

41. 41 students have learned the quadratic formula for solving quadratic equations and many other traits of quadratics. Imagine…

42. 42 Besides the solutions to ax 2 +bx + c = 0, what else does the quadratic formula tell you? Open Consolidation

43. 43 How is solving a quadratic equation like solving a linear equation? How is it different? Open Consolidation

44. 44 One parabola is very narrow and one parabola is very wide. Is there any way you be sure about how their equations differ? Or…

45. 45 For a lesson on trigonometric functions: You must graph a trig function. In what situation might you want a y-axis going from -100 to 100? Open Consolidation

46. 46 Create an open consolidation question that you might use having taught students a lesson about how to solve equations of the form ax +b = cx + d (e.g., 2x – 4 = 3x +1) Your turn – Open Consolidation

47. 47 Often you can start from an existing lesson and open up parts of it. Starting from a source

48. 48 One strategy is to start with an answer and create a question. For example, a growing pattern has 20 as the 9 th term. What could the pattern be? Fail-safe strategies

49. 49 Another is to ask for similarities and differences. For example, how is factoring x 2 + 5x + 6 like and different from factoring 3x 2 - 2x – 8. Fail-safe strategies

50. 50 Another is to let the student choose values. For example, ask students to choose values for and  and graph x +  y = 8. Fail-safe strategies

51. 51 From the other examples we have seen, you can see that these are not the only three strategies but they are helpful. Other strategies

52. 52 Now you try. With a partner, choose a lesson. Your own lesson

53. 53 Open up: - the minds-on, - the action, - the consolidate. Your own lesson

54. 54 Interact with others at your tables. Select one minds-on, one action, and one consolidate question. Post them in the right spot. Share time

55. 55 Think about why you might use an open question in each part of the lesson. Would your reasons be the same? Discuss this in your group. Consolidate

56. 56 Assessment for learning is your focus. An open question should be accessible to all students. Open Minds-on

57. 57 This is really an unscaffolded investigation, whether large or small. Students who need scaffolding receive it, but not everyone needs it. Open Action

58. 58 I had a lesson goal tied to a big idea. Did I achieve my lesson goal? By opening it up, I continue to allow for exposure of student thinking. Open Consolidate