Lately, I have been obsessed with children’s literature across K-5. My most recent obsession is the book One Hundred Hungry Ants. I did this in Kindergarten and this in 4th grade and today I invaded a 3rd grade classroom with it!
I followed the same pattern I usually do, I read the story aloud and did a notice/wonder. These are all of the things they noticed:
The last one led perfectly into asking about the ways the ants rearranged themselves. I wrote the combinations they recalled from the book and asked them to chat with a neighbor about patterns they see.
The discussion started with the 50+50=100, 25+25=25 and 10+10=20. Another student said they had the same things but it sounded different because she saw 50 was half of 100. They moved away from that and went to divisibility by the numbers that did not show up like 3,6,7,8, and 9 and pointed out that all of the second factors were multiples of 5. At this point they were focusing primarily on the second factor until someone pointed out the increasing and decreasing pattern happening. Then we got into the doubling and halving, quadrupling and dividing by 4 and multiplying and dividing by 10 of the factors.
I asked them if that would work with any number I gave them. They were quiet so I threw a number out there for them to think about, 24. They had to move into another activity so I left them with that thought. Before I left, however, one student said yes for 24 because 2×12, 4×6,8×3. Another student said it could be sixteen 1 1/2s and then thirty-two 3/4s! Wow!
Tomorrow they are going to investigate this further to see if they can come up with a conjecture about this work! So excited!
This makes me think of a recent numberphile video on “anti-primes”/highly composite numbers. These are numbers that have more factors than any smaller natural number. Two examples: 1 and 2 are highly composite because their number of factors (1 and 2 respectively) are bigger than the smaller numbers (empty set and 1, respectively).
So, 100, with 9 factors works nicely for this story, but is it highly composite? In other words, is it the smallest number that the author could have chosen for this number of arrays?
The numberphile video is here.