1 Math CAMPPP 2011 Math at the Beach!!! Grade 5-8 Sandra Fraser Erik Teather

Proportional Reasoning Breakout 1 2

What time is it? The numbers were cut off of this clock. On your own, determine what time the clock is showing. After a few minutes, share your thinking with your table. 33

Discussion What mathematical concepts came out of that clock problem? Was there proportional reasoning involved? 4

The Palm Trees 5 This year, they were measured again. Now they are: Large Tree: 12 m tall Small Tree: 9 m tall The two palm trees were measured last year. Large Tree: 8 m tall Small Tree: 5 m tall Which palm tree grew more?

The Families 6 Which family has more girls? AB

The Life Preservers 7 A B Which group has more duck life preservers?

Proportional Reasoning There are 2 different ways to interpret these problems: –Additive Thinking –Multiplicative Thinking Discuss the differences between them and whether there’s an appropriate time to use each one. 8

Reflection How did your thinking change during these problems? Which questions were asked which helped promote your thinking? 9

Responding to Students in the Moment How do we respond to students in class? What questions do we ask to promote student thinking? Are our questions funneling (to arrive at the teacher’s thinking) or focusing (to focus students on their own thinking)? 10

Questioning Look through the DI Question Cards (or Mathematical Processes on Computers) Have you used any of these questions before? Which ones do you like? 11

The Marker Problem You have more than 20 markers. When you give them out in groups of 3, there is exactly one marker leftover. How many markers could there be? 12

Video – Marker Problem Group 6 Video –How did the group interact? –Were they thinking additively or multiplicatively? –What did they know? –Did the manipulatives help their thinking? –What question would you like to ask them? 13

Video – Marker Problem Group 4 Video –How did the group interact? –Were they thinking additively or multiplicatively? –What did they know? –Did the manipulatives help their thinking? –What question would you like to ask them? 14

Responding to Students We will now look at a few student samples. How do you think they solved the problem? –Additive Thinking –Multiplicative Thinking Decide what question you would want to ask this student if you were there while they were working on the problem. 15

Responding to Students 16 What question would you want to ask this student? 23, 26, 29, 32, 35, …

Responding to Students 17 What question would you want to ask this student? 19, 22, 25, 28, 31, …

Responding to Students 18 There are 22 markers. I put them into 7 groups of 3 and there is one more left over. What question would you want to ask this student?

Responding to Students 19 What question would you want to ask this student?

Responding to Students 20 What question would you want to ask this student? 3 x 3 = = = 22

Responding to Students 21 What question would you want to ask this student? 3 x 7 = = = 25

What did we learn? …about the mathematics? …about teaching mathematics? 22

Learning Goals (Did we meet them?) To explore the differences between multiplicative and additive thinking (within proportional reasoning) To reflect on the ways in which we respond to students in the moment. 23

Activity Make up your own proportional reasoning task. Create 2 solutions: –Additive thinking –Multiplicative thinking 24