Thinking with Technology
Module 1 - Targeting Thinking in the Classroom

Grade 3 Mathematics

The Case of Mrs. Chen's Grade 3 class

Mrs. Chen’s Grade 3 class are learning about multiplication. To encourage her learners to deepen their understanding of numbers, Mrs. Chen urged them to look for patterns in the chart and to make conjectures.

She asked them, “Do you see any patterns in the charts you are making? Can you make any guesses about any of those patterns?” The class worked in pairs to investigate the multiplication chart.

Jesse and Samantha noticed that they could compute 4 x 8 by doubling 2 x 8 and wondered if that worked for other problems. During whole class discussion, Mrs. Chen asked Jesse and Samantha to share what they had noticed with their classmates.

She asked several learners to restate the idea and then discussed this process with the class

Mrs. Chen: Do you think Jesse and Samantha’s way of multiplying by 4—by doubling and then doubling again—works with problems other than 4 x 8?”

Matt:     Because if you have 2 times 8 and 4 times 8, you’re doubling the answer. It works every time.

Lindsey: It has to be doubled because you’re doing the same thing over again. It’s like you did 2 times 8 is 16 and then you did 2 times 8 is 16 again, so it has to be 32.

Ben:     What you are doing is counting by 8s, so you’re counting ahead, you’re skipping some of the 8’s. You’re doing another two of them, so it’s like doubling them up.

Mrs. Chen:  What do others think about this idea? Does it always work?

Lindsey: I tried to see if it would work with triples, so I did 2 times 8 and 6 times 8, and it worked. You multiply it by 3 and the answer is tripled.

Mrs. Chen: I want you to use your multiplication charts to further explore this idea for tomorrow. I want all of you to see if you can find out if Jesse and Samantha’s method always works and explain why or why not.

The Case of Mr. Mollo's Grade 3 class

Mr. Mollo’s Grade 3 class have been studying multiplication. He gave his learners a partially constructed multiplication chart and asked them to complete it. The class worked individually to complete their multiplication charts.

He asked them, “Do you see any patterns in the chart you are making? Are there easy multiplication facts that can help you with the harder ones?”

Aisha said that she noticed, “The twos were easier than the fours.” Mr. Mollo began asking Aisha a series of questions:

Mr. Mollo: How can you use the twos to help you with the fours Aisha?

Aisha:      ummm, I don’t know.

Mr. Mollo: What is 2 times 8 Aisha?

Aisha:       16

Mr. Mollo: What is 4 times 8 Aisha?

Aisha:      ummm….(long pause)

Mr. Mollo: Can someone help Aisha with the answer to 4 times 8?  Trevor?

Trevor:      32

Mr. Mollo: Thank you Trevor. So, Aisha,  how much bigger is 4 than 2?

Aisha:       2 times bigger.

Mr. Mollo: Yes, or could we say that it is double? That 4 is double 2?

Aisha:       Uh, huh.

Mr. Mollo: What about 16 and 32 Aisha, how much bigger is 32 than 16?

Aisha:       (pause) ummm…double?

Mr. Mollo: Yes, very good. (turning to the class). Can you see this double rule and how it can help you with bigger multiplication problems? I want you to look at your multiplication charts and try out the double rule with other numbers.



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