Intrigued by Matt Parker’s tweet yesterday, I decided to have a go at it.
Arrange the numbers 1-17 so that adjacent numbers sum to a perfect square.
It’s the kind of problem that makes students who struggle at math, or hate puzzles, shutter. I decided to take this problem on, in my lowest level remedial math class. This class has about 20 kids, 9th-12th grade, all who have failed a previous math class. These kids range in ability as there is a large variety of classes failed.
I started off by having them number little pieces of paper with the numbers 1-17. I think this helped in setting up the task in a low pressure way. Numbers 1-17, how hard could it be?
Then we talked about what numbers were square numbers. Those who thought they could begin were off.
Others who still couldn’t wrap their head around how the numbers could be arranged, together, came up with some examples of pairs of square numbers. Then, all students were able to make triples with adjacent square sums. They built on those smaller sets and began to come up with strings of 4 or 5 or 6 numbers. One student noticed that the highest square number that could be made from the numbers 1-17 was 25, so 16 and 17 needed to be on the ends of the arrangement since only one other number each would sum to 25 from the list.
Toward the middle of the task, I saw students getting frustrated that they only had a few left and they couldn’t seem to place them. We talked about how 1 could pair with multiple numbers on the list. That discovery seemed to re-energize them to rearranging more numbers and persevere in solving the problem.
The students had differentiated themselves at this point and some were working alone and some together. What I found very interesting is that when one of them solved it, they weren’t immediately drawn to that person to show them the answer. They wanted to figure it out on their own. They weren’t rushed either as students finished. Sometimes when students finish quickly, others become frustrated and just want the answer.
The students also developed some interesting strategies, like grouping pairs that totaled 16 and 25. By the end of the 30 minutes, every single student had arrived at the correct solution. I’m not sure if it was the physical manipulative or the puzzle-like feel of the task, but I was so proud of this group of kids. These are students who have already failed at math and have convinced themselves that they are inherently bad at it. Today they proved that not only is the latter completely false, but also that success is in math is achievable with perseverance and resilience.