At a recent professional development session on multiplication and division, my colleague and I asked participants to examine some student solutions to the problem: 1564 ÷ 36. Before looking at the student work below, think about how you would solve the problem.
The following solution from a fifth grade student seemed to either amaze participants or they had a hard time making sense of it. I’ve been thinking a lot about their reactions. Take a moment to look at this work. What do you notice? Is there anything that surprises you or that you have questions about?
The more I look at this piece of work, the more I see. Nora’s recording of her solution suggests that first she divided 720 by 36 and got 20. She then doubled 720 and got 1440, so the answer to 1440 ÷ 36 is double 20, or 40. Next she subtracted 1440 from 1564 and found out that she had 124 left to divide. She multiplied 36 x 3, which got her as close as possible to 124, and figured out that 124 ÷ 36 was 3 R 16. Combining her answers to 1440 ÷ 36 and 124 ÷ 36 she came up with the answer 43 R 16.
I still have questions about how Nora came to her solution—how did she decide to start with 720 ÷ 36? Did she choose this problem, because of its relationship to 1440 ÷ 36 or did she start with 720 ÷36 and think about how to build up to 1,564 from there? Why did she choose to multiply 36 x 3? Did she know this would get her close to 124? Was it a product she already knew?
Whatever the answer to these questions, I understand why participants were amazed by her solution—there is a lot of knowledge of number, number relationships, and the operation of division evident in Nora’s work. I think Nora knew that:
- if she doubled the 720 in 720 ÷ 36, the quotient would be doubled
- when she found the solution to 1440 ÷ 36 she then had to figure out how much of the 1564 she had left to divide (124) and divide that by 36
- her answer was the combined quotients of 1440 ÷ 36 and 124 ÷ 36
- while she could break up the 1564, she had to keep the 36 whole
- she could use things she already knew (e.g. 720÷ 36 = 20), to figure out things she did not
I continue to wonder about what aspect of Nora’s solution confused or challenged some participants. I wonder whether it was the fact that she used division notation to show her solution that made it difficult for participants to make sense of how she solved the problem. Would it have seemed less confusing if she had written: 36 x 20 = 720 and 36 x 40 = 1440 to show her first steps? Below is another piece of student work that participants looked at, along with Nora’s work. I see a lot of similarities between Nora’s work and Sophia’s work and yet participants did not seem perplexed by Sophia’s work.
In any case, puzzling over participants’ reactions to this piece of student work has made me think more about the operation of division and how students and adults understand it. Looking more carefully at the work highlighted all that is involved in solving division problems with understanding. It is pretty amazing what digging into one solution can reveal about a student’s understanding and about mathematics.