Penny Problem

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I always struggle with teaching divisibility (including factors, multiples, primes and composites). I think the concept is really important in grade 6, and foundational to mathematics in the higher grades, so I introduce it early and refer to it often. But no matter how often I circle back around to divisibility, many students seem to have a hard time understanding what it means on an intuitive level.

This year, I’m trying to break down the big idea into smaller pieces and making each lesson as tangible as possible. To start, I held up two pennies, one in each hand, and asked how many different groupings were possible. The class quickly determined that there were only two possibilities: 1 group of 2 or 2 groups of 1. Then, I gave each group 3 pennies and a large sheet of paper. They had to write or draw the possible groupings for 3 pennies and then 4 and then 5 and so on…

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As I circulated, distributing pennies to each group as they finished one number and were ready to go on to the next, I was able to check that they were on-track. I was also able to see the different notation that students used and was interested to see which students used numerical notation and which students used diagrams. I was also surprised by how engaged the students were and how long they persisted with the task. In the end, I had to stop them in order to leave enough time for a concluding discussion.

To conclude, I asked students what this had to do with divisibility and how they could tell which numbers were more divisible than others. When they had made that connection, they quickly made the connection to prime and composite numbers. Finally, we talked about factors. By the end of the lesson, students understood what I needed them to know about divisibility, primes, composites and factors and they had a concrete example to refer back to. Time will tell whether this activity has fostered the enduring understanding that I was hoping for, but so far, I’m pleased with the results!

 

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