Kepler’s Laws

In my multivariable calculus class, we spent last Friday reading the textbook as a group, trying to understand the section on Kepler’s Laws. We got done showing that if there is a sun-like object and another object with a particular initial position and velocity, it will either fall into the sun, be an circle, be an ellipse, be a parabola, or be a hyperbola.

Today we were going to move onto using this result to derive the three Keplerian laws of planetary motion.

But then I decided to scrap that. Because even though we read the book and followed the text, line by line and equation by equation, we lost sense of what we were doing. We lost sense of the conceptual underpinnings for each equation. We didn’t know what motivated the book to make the moves it made. It’s largely the book’s fault, which is really unclear — if you’re a high school student and not used to having your book say “we leave this as an exercise to the reader.” (Seriously, it did that.)

One of the things you’ve heard me say is that I want to foster the skill of students learning to communicate math well.

So, I decided to scrap the plan of moving forward, and we’re devoting two or three days to


We started out the class outlining a basic structure to it (Part I: What we want to show; Part II: Initial Conditions; Part III: Gravitational Pull; etc.). Then the four students started talking about what they wanted to say. (One agreed to draw the diagrams we’re going to include in our text.) I just sat up front, and when they decided, I typed it up in my LaTeX editor — projected so that students could tell me to fix or reorder something. Sometimes I prompted them (“you told me to write \vec{v} but you never told the reader what that is” or “does it matter if the initial velocity weren’t orthogonal to the position vector?”). And it took us 50 minutes to get about a third of the way done.

But you know what? It is working. They’re talking, they’re thinking, they’re arguing with each other, they’re asking questions. And they’re learning to work through things, and explain them to someone else.

I was so pleased. Hopefully the next few days go as well.



  1. I can’t even tell you how great it is. It’s so relaxed and students are really engaging with the ideas and the text and presentation.

    It helps that the students were the ones that wanted to do the Kepler section (I gave them the option to skip it), so they had this inherent desire to do it.

    But still. I can’t believe how well this is working. The only downside is that so far we’re only about halfway done, and we’ve taken 2 days to do this so far. I was wondering if it was worth 4 days writing this, on top of the day we spent going through the book.

    However, after watching them work — after saying “we should submit this for publication somewhere” and “we should send this to [our teacher last year, who retired],” I know it is worth it.

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