1. Plenary 3

2. So how do we decide what to teach? You might be inspired by an approach you see potential in. Here’s an example very popular in some places right now.

3. A Video Problem We’ll do a “Dan Meyer” type problem. Students watch a video and raise mathematical questions that arise from the video. They would use the video to solve the problems they propose.

4. I’d like you to… Think of a proportional reasoning problem evoked by this video. Then think about what math ideas could come out of the investigation of that problem and why the math would be worth investigating. http://www.youtube.com/watch?v=H9Vpw mtnOZc

5. Now let’s think …. about what happens if you start with curriculum instead.

6. Where are you now? You read an expectation. You have to decide what to focus on in a lesson. Choose one of the expectations on the handout. Talk to a colleague about how you currently decide how to focus your lesson and how you figure out what closing question you’d want to use to see if students had learned the main idea.

7. Grade 2 expectation: Estimate, count, and represent (using the ¢ symbol) the value of a collection of coins with a maximum value of one dollar. What matters?

8. Numbers are compared in many ways. Related big idea

9. Lesson goal Students will be able to determine how to represent a cent amount using only one type of coin

10. You use 8 nickels to show a certain amount of money. Will it take more or fewer dimes? Why? How many dimes would it take? How do you know? Consolidating question

11. What do you think? Talk to a partner. If students were co-constructing criteria for a good answer, what do you think they would focus on? What would a great answer look like?

12. There is a relationship to either skip counting by 5s or the pattern in counting by 5s. There is a recognition that there would be fewer dimes since they are worth more. There is an understanding of a 2 for 1 trade even without calculating the 40. What would a great answer look like to you?

13. an answer that recognizes that odd numbers are not possible (and why) is good, but the why is important an answer that takes a known multiple of 4 and keeps adding 4 to it for other answers shows good insight What would a great answer look like to you?

14. More open? A more open question might have been: Choose two different amounts that you could show with either only nickels or only dimes. How did the number of nickels compare to the number of times? Was it the same both times? Why?

15. -The focus is on the quality of the mathematical thinking and not the details of how many responses, organization of material, how closely directions are followed, etc. Assessing responses

16. Grade 4 expectation: Demonstrate an understanding of simple multiplicative relationships involving unit rates, through investigation using concrete materials What matters?

17. Classifying numbers provides information about characteristics of the numbers in the relationship. Related big idea

18. Lesson goal Students will recognize what costs are possible or impossible based on a particular unit rate.

19. Jane had to buy lots of packages of hamburger buns for a school picnic. Tell several prices over $100 she could have spent and several she could not have spent and how you know. Consolidating question $4

20. What do you think? If students were co-constructing criteria for a good answer, what do you think they would focus on? What would a great answer look like?

21. There should be some values in each category, but the focus should be on the how they know. an insight focusing on why, for example, two consecutive numbers are both not possible or numbers not four apart are not both possible etc. are particularly encouraging What would a great answer look like to you?

22. an answer that recognizes that odd numbers are not possible (and why) is good, but the why is important an answer that takes a known multiple of 4 and keeps adding 4 to it for other answers shows good insight What would a great answer look like?

23. At this point… Talk to a partner about what makes you feel good, but what makes you feel nervous, about this approach of filtering expectations through big ideas and starting with consolidating questions. Prepare two points to make or questions to ask. You will get to bring these up a bit later.

24. What matters? Grade 6 expectation: Solve problems that arise from real- life situations and that relate to the magnitude of whole numbers up to 1 000 000

25. Big Idea There are many equivalent representations for a numerical relationship. Each one may be useful in a different circumstance.

26. Lesson goal Students will recognize that changing the form of a relationship using proportional thinking can be useful to solving a problem.

27. Consolidating question A cat’s heart might beat 500 times in 4 minutes. You want to know how long it would take for the heart to beat 1 000 000 times. What might be a better, but equivalent, way to describe the rate 500 times in 4 minutes to start your thinking and why? What other equivalent rates might you use to solve the problem?

28. What do you think? If students were co-constructing criteria for a good answer, what do you think they would focus on? What would a great answer look like?

29. The student needs to speak to why the new description is equivalent to the old one using concepts and not just rules. The student needs to explain why the new form is useful. What would a great answer look like?

30. The student recognizes that converting minutes to days involves an equivalence, too, e.g. 60 minutes:1 hour = 600 minutes: 10 hours or 200 minutes: about 3 1/3 hours or 800 minutes at about 13 hours What would a great answer look like?

31. What matters? Grade 9 expectation: Solve problems involving ratios, rates, and directly proportional relationships in various contexts using a variety of methods

32. Big Idea There are many equivalent representations for a numerical relationship. Each may be more useful in a particular situation.

33. Lesson goal Students will recognize that determining correct distances on a scale diagram or real distances from a scale diagram involves creating equivalents to the scale ratio or rate

34. Consolidating question Tara is looking at a scale diagram of her house. She noticed that the entrance to the kitchen and the front door that are really 6 m apart are 12 cm apart on the map. She said that’s all you need to know to figure out how far apart any two locations on the diagram really are. Do you agree or disagree? Explain why.

35. What would a great answer look like? What do you think? If students were co-constructing criteria for a good answer, what do you think they would focus on?

36. What would a great answer look like? The student would need to recognize that no matter what the distance was on the map, there would be an equivalent rate (6m /12 cm) or ratio (600:12) and would explain how to get it.

37. What would a great answer look like? They would also realize that if it is a scale diagram, you can’t have a different ratio in one part of the diagram as another and explain why.

38. At this point… Remember the points you brought up or questions you had earlier. Discuss them one more time to see if any of your ideas/concerns have changed. Now is the time to share a number of these ideas/concerns.

39. So which path would you take? On the BLM you will receive there are choices of lesson goals and possible consolidating questions. With a partner, choose one of the questions thinking about the following criteria:

40. So which path would you take? - Is it focused on important math? - Will it tell me if the student learned what I was hoping for? - What would I hope to see to tell me this?