The Tree Problem

1. The Tree Problem 1st measure Now B A B 6 m 5 m A 3 m 9 m Which tree grew more? 1

2. Your Task with the Tree Problem Think – Pair - Share • Individually, determine which Tree grew more • You have 1 min • In pairs, labelled Partner A and Partner B: • Partner A convinces B that tree A grew more • Partner B convinces A that tree B grew more • You have 1 min each

3. What reasoning did A people use? What reasoning did B people use? 9 m B B A 5 m A 6 m 3 m 3

4. Debriefing the Tree Problem • This is an example of a Proportional Reasoning question • The different partners were arguing two different perspectives: • Additive or Absolute growth • Multiplicative or Relative growth

5. 5-3 9-6 A student who is functioning at the additive stage might respond that Tree B grew more as it grew 3 m and Tree A grew 2 m. A student who can function at both additive and multiplicative stages might respond that Tree A grew more since 2/3 > 3/6 or 5/3 > 9/6 or 6:3 > 9:5 9 m B B A 5 m A 6 m 3 m 5

6. Consolidating Additive vs Multiplicative The use of these approaches is developmental Additive thinking develops first Multiplicative thinking needs to be refined for proportional reasoning to develop

7. Things to Notice The Tree Problem was selected as a problem to illustrate Additive vs Multiplicative thinking The Tree Problem is an Open Question. Multiple solution paths are possible Rich discourse to promote reasoning and sense making

8. Tree Problem Questions What do you notice about the numbers chosen for the question? If you hear an approach that is additive what might you say?

9. Proportional Reasoning involves the deliberate use of multiplicative relationships to compare quantities and to predict the value of one quantity based on the values of another. Proportional Reasoning – What is it?

10. The essence of Proportional Reasoning is the consideration of number in relative terms, rather than absolute terms. Proportional Reasoning – What is it?