The Tree Problem 1st measure Now Which tree grew more? B A 9 m 6 m 5 m Think about this problem for a moment. After 30 s to 1 min, ask if there are questions about the problem. Clarify that Tree A on the left needs to be compared to Tree A on the right and the same with Tree B. Some might begin to compare Tree A on the left to Tree B on the left … Be sure all participants are solving the same problem. When people ask questions, it is ok to say to the collective, “How do WE want to answer that?” The actual problem with instructions and protocols is on the next slide. 1 1

Your Task with the Tree Problem 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 Please solve this problem individually. (1 min) Then we will engage with an elbow partner. Have then Partner B explains how B grew more and have Partner A explain how A grew more. 2 min = 1 min for B and 1 min for A Think – Pair - Share 2

What reasoning did A people use? What reasoning did B people use? Elicit response from the participants… You might want to type straight into the PPT as people give responses. Students in primary and grades 4 and 5 would likely respond additively and they have shown their mathematical thinking. By grade 6 students, students should be starting to think multiplicatively. Teachers will have to listen for thinking expressed by students in the junior grades and prompt them to think multiplicatively. Below are some examples of reasoning you might hear. The next few slides summarize these ideas. One answer is that the taller tree (B) grew more because it grew 3 m and 3 m > 2 m – the growth of the smaller tree (A). So, if we consider only absolute growth, i.e. actual growth unrelated to anything else, B grew more. Another perspective, relates the growth of each tree to their original heights: Tree A grew 2 m or 2/3 of its original height. Tree B grew 3 m or 3/6 of its original height. Which tree grew more? 2/3 > 3/6 so tree A grew more from a relative perspective. Tree A grew from 3 m to 5 m and Tree B grew from 6 m to 9 m so relatively 5/3 represents a comparison of (the height of Tree A after growing) to (its original height). And 9/6 represents( the height of Tree B after growth) to (its original height). Now 5/3 = 1 2/3 = 1 4/6 and 9/6 = 1 3/6. Since 1 4/6 > 1 3/6 then Tree A grew more. Also 3/6 = 1/2 and 5/9 > 1/2 then in the second situation, Tree A’s height is more than 1/2 of Tree B’s height, then Tree A gained more on Tree B and thus can be considered to have grown more. 6:3 > 9:5 B starts out double A’s height and then it is less than double A’s height so A grew more 9 m B B A 5 m A 6 m 3 m 3 3

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 You might point out that there is a Glossary in the coming package and these words are defined there. During proportional reasoning lessons, it will be very just in time to focus on reasoning and proving as a math process. Having students doing the talking is essential. They need time to reason through these questions themselves. 4

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 5-3 9-6 Again, some answers you might hear… One answer is that the taller tree (B) grew more because it grew 3 m and 3 m > 2 m – the growth of the smaller tree (A). So, if we consider only absolute growth, i.e. actual growth unrelated to anything else, B grew more. Another perspective, relates the growth of each tree to their original heights: Tree A grew 2 m or 2/3 of its original height. Tree B grew 3 m or 3/6 of its original height. Which tree grew more? 2/3 > 3/6 so tree A grew more from a relative perspective. Tree A grew from 3 m to 5 m and Tree B grew from 6 m to 9 m so relatively 5/3 represents a comparison of (the height of Tree A after growing) to (its original height). And 9/6 represents( the height of Tree B after growth) to (its original height). Now 5/3 = 1 2/3 = 1 4/6 and 9/6 = 1 3/6. Since 1 4/6 > 1 3/6 then Tree A grew more. Also 3/6 = 1/2 and 5/9 > 1/2 then in the second situation, Tree A’s height is more than 1/2 of Tree B’s height, then Tree A gained more on Tree B and thus can be considered to have grown more. 6:3 > 9:5 B starts out double A’s height and then it is less than double A’s height so A grew more 9 m B B A 5 m A 6 m 3 m 5 5

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

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 Consolidating the Open Question Usage It is considered an Open Question because it is open-routed. There are many solution paths converging on the same correct answer. Some Open Questions are truly open-ended will have students arrive at different solutions. Then there is an opportunity to talk about the different points of view and again clarify thinking. The discourse that occurs during discussion of various solutions, helps students clarify their own reasoning and make sense of the mathematics. The purpose of all of these questions, is to get the students talking. If you talk you learn, if they talk they learn. Throughout the presentations, we will be outlining Instructional Strategies (referred to as I.S. on the agenda) that we ask presenters to make explicit. Please use a strip of chart paper to make labels that says “Think – Pair – Share ” and one for “Building an Argument” and post it on your Instructional Strategies wall. 7

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? Participants turn and talk Presenters ask for comments 1 at a time Some comments you might hear… The numbers were small – the difference in growth was 2 m and 3 m but in comparison to the original heights the 2 turned out to be more “powerful” than the 3. There were at least 3 or 4 different approaches – according to those you discussed. Only 3 are printed on slide 6. You would confirm that the amount stated is a correct measure of the difference in the heights of the trees between first measure and second. You might ask, “Would your answer be different if you looked at it comparatively, like the height of Tree A after the first measure to the height of Tree A after the “Now” measure? Think about it.” And go on to gather other responses. If you have a strong Math-talk community in your classroom and the students are used to listening to many points of view and adapting their thinking accordingly, you could just pass the question on to the collective. 8

Proportional Reasoning – What is it? 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. Notes: Proportional reasoning is more about the use of number sense than formal, procedural solving of proportions. Students use proportional reasoning in early math learning, for example, when they think of 8 as two fours or four twos rather than thinking of it as one more than seven. They use proportional reasoning later in learning when they think of how a speed of 50 km/h is the same as a speed of 25 km/30 min. Students continue to use use proportional reasoning when they think about slopes of lines and rate of change. 9

Proportional Reasoning – What is it? The essence of Proportional Reasoning is the consideration of number in relative terms, rather than absolute terms. Notes: Students are using proportional reasoning when they decide that a group of 3 children growing to 9 children is a more significant change than a group of 100 children growing to 150, since the number tripled in the first case; but only grew by 50%, not even doubling, in the second case. 10