Co-constructing the relationship between multiplication and division

At the beginning of each math lesson, we have an activity called “10 minute math.” It is designed to get students thinking mathematically in a fun way, and it is usually designed as some sort of game or puzzle.
Last Thursday, the arrangement to the right was shown on the board. The students were to make a division statement which matched the numbers shown. Everyone who had an answer was asked to write it on the board for comparison.

Many people came up to the board, and many wrote the statements 84÷ 12=7 or 84÷7=12.
We all agreed that this was accurate. Emma, however, wrote that 84÷14=6. Several people told her that she had made an error, that she should have used 12. Emma insisted that she was correct. So we factored 84. We discovered that it has factors of 7, 2, and 6, which would make both statements correct. I wrote this on the board:

84______

7×2 6 x 2 7 x 6

After a moment to think, everyone agreed that this was true. Then Evan said- wait, you can show how any two numbers go together this way, not just those that are factors of each other. So, you can show a relationship between 26 and 27 by writing 26/27, or 27/26, depending on which one you want to show. 26 is 26/27 of 27 and 27 is 27/26 of 26. This was intriguing, and we moved on to the main body of the lesson. I was interested to see where this would go, so I decided that we would have a very similar problem the next day. So this relationship is being seen by way of division.

We followed the same procedure the next day. Everyone was invited to come up and show a division statement for the arrangement of 72 dots shown. Many people showed 72= 8 x9, or 72= 9×8, which were accurate. Kate’s list, however is to the left:

She used the division sign to show 72 as a combination of factors. (She also started with a different combination.) Everyone agreed with her right away this time. This time, Kate is using division, and multiplication, as I did, but she is moving toward representing division as multiplication. Everyone is becoming more comfortable with decomposing numbers and rearranging the parts. This is a really critical mathematical skill.

On Tuesday of this week, the challenge for the day was to come up with efficient division strategies, based on different starting problems. Kate was able to take the thinking we had been doing further. She decided to take the dividend (first number), and factor it down to the prime factors. She removed the factors which would produce the divisor (second number). The remaining factors, multiplied together, produce the quotient (answer). The class discovered that this always works IF the dividend is a multiple of the divisor. Kate is trying to see if she can come up with a method which works regardless of the two numbers. She is also representing division completely as multiplication without using the “multiplying up” method. She has progressed toward a more sophisticated way of seeing multiplication as division.

We are trying to help her think about it. This problem is moving the group as a whole toward a more complete understanding of the relationship between multiplication and division. This is an intriguing problem, which I am not even sure of the answer to yet. We will all have to do some testing and research. But that is what real math is- it is a language that you use to discover new things and explain them so that other people can understand them too. So we are on the way.

The post Co-constructing the relationship between multiplication and division appeared first on Sabot at Stony Point.

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