Figurate Numbers, and the Unsummables
I decided to have my middle school group follow up our look at Gauss’ method for summing arithmetic series with a look at the triangle numbers and other figurate numbers. I started with the material from the bottom of this Number Patterns Activity, but added a section on the square numbers in between the triangle numbers and the pentagonal numbers. I then asked them to look at the patterns formed by the triangle, square and pentagonal numbers, and predict what the pattern would look like for the hexagonal numbers. I had them drawing charts for each set of numbers showing the differences between subsequent numbers, and the differences between those differences, like this one for the square numbers:
I had introduced this at the end of class on Wednesday and assigned it for homework. The kids had good success with the triangle and square numbers, however most had trouble drawing the next pentagonal number. Only one recognized the pattern and went with that, correctly computing the next several pentagonal numbers without trying to draw them. This one of the younger kids in the group, and a girl who lacks confidence in her mathematical ability, so it was great to be able to have her explain what she did to the rest of the class. No one got the hexagonal numbers on their own; but when we did it together in class, they all knew what to do.
I wanted to have a chance to look over everyone’s homework before we discussed it, so after I collected the homework, I introduced Dave Marain’s “unsummables” investigation. Dave has recently posted an extension to this investigation, which reminded me of how perfectly it would fit in with the work I’ve been doing with this group. (Be sure to read the comments, which contain some more great explorations of these ideas!)
I gave them about 25 minutes to work on making their charts of ways to express different numbers from 3 to 35 as the sum of two or more consecutive positive numbers (and start on their conjectures and observations, if they finished that). I had them work alone, though I may let them pair up to work on observations during our next meeting. They were quite productive, and I think most of them either finished the charts, or got close, and seemed to have the “right” numbers left blank, from brief observations. (I did not allow calculators for this.) I didn’t collect their work, but left it with them to work on when they have odd free moments between now and when we meet again in 2 weeks. So, I’ll follow up more after that. But I wanted Dave to know someone had tried it on some middle schoolers. 🙂
I’ll probably split the group next time, and have the younger end of the group go through a Handshake Problems exercise that the others did last year, and handhold the other kids through some part of Dave’s extension activity.