It’s a big day for my freshmen, as today is their second unit test – this one on sequences, series and the binomial theorem. For the last few days, we have been knee-deep in the world of arithmetic and geometric sequences, so much so that you’d think they are the only types of numerical patterns. But in today’s opener I want to expose them to something deeper, without freaking them out before the test –

My fear is that if I dive too deep into these problems now, I’ll worry students who are studying for the test. So instead I provide about 60 seconds to discuss what we see, and will let the problem marinate. If students finish the test early, they’ll have something to think about and tackle. Hopefully we will have a chance to come back to these problems next week, but sometimes I feel bad when I start a new problem and don’t find the time to re-visit it. For now, I think I’ll leave the problems on the board, and see if anyone volunteers information.

If you are playing along at home, here is information about the two problems above. The first is the harmonic series, which diverges (approaches infinity), while the second (the sum of the reciprocals of the triangular numbers) surprisingly approaches 2.

Upon reflection, there were some natural places to flip instruction in this chapter. In the video below, students took notes on the sum of an arithmetic series. As a number of students in my last period class leave for sports, this was an effective way to keep everyone on the same page. Enjoy!

I always liked the schoolboy Gauss method, write the series backwards, add corresponding terms – they are all the same as first+last and there are n of them, but we have twice the total.
Also, the formulation of the total as n x (first+last)/2 sheds a different light on the process, and the whole thing looks like the formula for the area of a trapezium.
Just a few thoughts !

Before kids had this video as their assignment, I actually told them the Gauss story, which led to the first+last method. One of the problems I find with teaching sequences/series is that it’s easy for students to become fixated on formulas…it’s nice to be able to circle back to the first+last method.

I always liked the schoolboy Gauss method, write the series backwards, add corresponding terms – they are all the same as first+last and there are n of them, but we have twice the total.

Also, the formulation of the total as n x (first+last)/2 sheds a different light on the process, and the whole thing looks like the formula for the area of a trapezium.

Just a few thoughts !

Before kids had this video as their assignment, I actually told them the Gauss story, which led to the first+last method. One of the problems I find with teaching sequences/series is that it’s easy for students to become fixated on formulas…it’s nice to be able to circle back to the first+last method.