Polygonal Numbers

I just finished up arithmetic series, and I wanted to push my Advanced Precalculus students to think hard. I usually provide them with enough scaffolding that I know they will be able to get from Point A to Point B. But today I decided I wanted to “be less helpful.”

I told the groups to mix themselves up — for a change of pace. And then I handed this out.

I told each group that they had one opportunity to call me over so I could give them A Big Hint. Then I let them go, giving them giant whiteboards to work on if they wanted.

In my two classes, I had: one group solve the problem explicitly (they had a formula that worked) and one group come up with a recursive solution that really impressed me! The other eight groups were at varying stages of understanding. Most all the groups were gung ho about working, and most all the groups started discovering all these patterns.

Only one of the ten groups asked for The Big Hint, which means my kids have perseverance! I did give varying degrees of mini-hints to kids as I saw them progress, to nudge them this way or that way.

At the end of the second class, a few kids said how they are now braindead, because they did so much thinking. They were exhausted. As a teacher, I call that winning!

I’m still at a bit of a loss as to what I am going to do tomorrow. Since kids hadn’t really finished, I thought I would have them work a bit more. I’m not good at debriefing. Also, many kids have different observations, and I don’t have time to really dwell on this. This wasn’t supposed to even take the whole day!

My plan is to give all my groups collectively A Big Hint, give each group 12 minutes to figure out how to find the nth pentagonal number, and then after those 12 minutes are up, I will give them this:

We will go over the triangular and square numbers together as a class. And then WHAM, I leave them with the pentagonal and p-gon figures to figure out on their own.

Wish me luck that it goes well tomorrow. I can see it crashing and burning, or being a good wrapup.



  1. This was audacious and I think your “teach less, learn more” approach went awesomely. But (in my humble opinion) I don’t think you need to drop the “big hint” so soon! If I had worked on this for an hour today and you gave me the second handout tomorrow I would be somewhere between mad at you for telling me, and mad at myself for not figuring it out before you told me. (Note: I make no assertions that many, or any, of your students are like me.) But, I think it would give me license to tidily put this problem away and never think about it again, instead of stewing on it for a while and possibly learning some other things. Polya is supposed to have said “A problem isn’t a problem if it can be solved in 24 hours”. I presume students in advanced precalc have had some prior success in math. Push them a bit more. I would have appreciated it in high school (or at least current-me would have appreciated past-me-in-HS having a few more cases of this). (And when you do let the other shoe drop, I wonder how it would go to let students do the dropping — and of course the critiquing and pressing for justification back, as seems to be the norm in your classes. Perhaps this is what you were describing.) My two cents…

    1. This gave me A LOT of food for thought. Especially since I have to decide what to do in class today.

      But the truth is, I need to close this out and move on. I know enough about many of my kids that they would just put this away and never think about it if I did say “okay, have fun thinking some more about it.” And unless I gave them a whole other (or more!) class days to work on it, almost 95% of them would just let it sit and never think about it again. Partly because I know at the end of the last class they were hitting a small note of frustration that probably wasn’t productive frustration, and partly because they have a ton of other school work for other classes, so “optional math” isn’t really an option.

      If only I *could* spend days on this. But I’m already behind schedule with my curriculum. This was probably one day I couldn’t afford to do, but I know it’s also one day I couldn’t afford not to do.

      Le sigh. I’m torn.

  2. Hi Sam- I know you are teaching Geometry this year, so I thought I’d mention that I give these same problems to my Honors Geometry class. But instead of approaching it from a numerical/sequences (algebra) perspective, I encourage them to use the patterns they see in the picture (a more geometric approach) to find a formula. The only hint I give them for pentagonal numbers, after they’ve suffered and struggled good and plenty, is “diagonals”. They look at me like I’m crazy, but always eventually figure it out. They are often amazed that approaching the problem geometrically (as opposed to solely looking for patterns in the numbers and trying to force them into some sort of algebra equation) not only makes it easier to understand and more accessible, but makes the formula actually make sense! Good stuff. I’m not entirely sure I’m making sense here, if you want me to explain in more detail, just let me know. Thanks for always sharing all your amazingness with the world. It is appreciated!

    1. WHOA! I have to think about this. I love this idea of a geometric approach! I am so flabbergasted that diagonals could be used for this. So curious. When I have a moment, I will become one of your students and try to figure this out. Thank you!

  3. Your method views each pentagonal number as the sum of an arithmetic sequence, but you could also view each pentagonal number as a term in a quadratic sequence since you have a pattern of constant second differences: 1 + 5 + 12 + 22 + … has a pattern where 5-1 = 4, 12-5 = 7, 22-12 = 10, and 4, 7, 10, … form an arithmetic sequence. So (1,1), (2,5), (3,12) and (4,22) are points that satisfy a quadratic equation, which of course turns out to be the same equation you get using the sum of an arithmetic sequence. I usually approach this type of sequence by looking for an explicit pattern that involves n^2. However, in this case, I am running into a brick wall (just tired or the pattern is not obvious) and so I am glad to know of another way to approach this problem. (I just did the problem using the sum approach you showed and it definitely works better for this problem. Why did I never think of that?)

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