In Multivariable Calculus today, I had my first long block. I wanted students to understand how in the course we’re going from 2D to higher dimensions, but often times to *get* what we’re doing, you just have to make an analogy.
So I devised an activity that has students go from Pascal’s Triangle to Pascal’s Tetrahedron.
Unfortunately I was running out of time. I started by having students remember what Pascal’s Triangle was all about — and interesting properties in it.
First good thing? The kids I taught in Advanced Precalculus two years ago remembered a bunch of stuff.
Second good thing? One kid who I didn’t teach saw the triangular numbers in the triangle: 1, 3, 6, 10, 15, …
And then I said “okay, let’s look at the “next” diagonal” and had kids conjecture what it meant… 1, 4, 10, 20, 35, … and asked them what that might represent.
They bandied about different ideas. And then someone said maybe it’s a higher dimension. I had blocks, and had them play around to figure it out. They saw 1, 4, 10, 20, 35, … were the tetrahedral numbers. It was awesome. They built each layer of the tetrahedron to see this in action.
And then I said, we understand a few things. Let’s use what we see — 2D triangles, 3D triangles — to understand more. So we looked at a different diagonal. We agreed it was the 1D triangular numbers.
What?! But we defined a 1D triangle, and viola, it made sense. (First we had to understand 1D.)
And of course we talked about 0D triangles too…
So when at the end of class I asked kids what the following diagonal represented, their minds were blown:
Yup. 4D triangular numbers.
So we didn’t get to building Pascal’s Pyramid, and seeing how many of the properties that existed in Pascal’s Triangle have analogies when building Pascal’s Pyramid… But we still got to the driving point. We can use the idea of analogy to understand things in different dimensions. Different path, same result. Awesomesauce.
I also asked kids to fill out an exit ticket at the end of class about three takeaways. This is what I got, which made me happy: