11. Appendix: visualizing algebra - sample lesson plans

Teaching the “visualizing algebra” material can clearly be sliced up in many different ways. I have taught it several times, and I always followed a similar logistical format: class length of 1 hour and 15 minutes, and new material once/week, with catch-up sessions twice/week.

But the order in which I teach the various pieces, and the choice of which pieces to teach, depends largely on the makeup of the group of students I work with.

Similarly, the amount of extra background material and the choice of “visions into advanced material” anecdotes depend on the group of students I have.

In this appendix I will start collecting some of the specific lesson plan sequences I have chosen.

11.1. Fall 2024

We had a total of 10 lessons, from 2024-10-07 to 2024-12-16.

Opening lesson 1:

  • Introduce myself, discuss how researchers see mathematics

  • Discussion of logistics and what I expect of students

  • Section “The why of messy exponents”

  • Section “An example to whet your appetite”

Lesson 2:

  • Review from OpenSTAX book: “A nostalgic romp through coordinates and plotting” - do the exercises suggested in the section.

  • Review from OpenStax book: “What are functions?” - do the exercises suggested in the section.

  • Section “Special powers of binomials”

  • If we have time, start with sympy and introduce expand((a + b) ^ 7) and so forth.

Lesson 3:

  • Start with sympy, using the expressions in Section 4

Lesson 4:

Return to the “review of prerequisites” with the following sections:

  • Reviewing fractions - what, seriously??

  • Getting comfortable with visualizing functions

  • Quadratic equations

  • Some heavy emphasis on how functions are constraints

Remember to really lay it thick on how each equation reduces the dimensionality of the space.

Lesson 5:

We are now well beyond the review section, and we can spend a brief amount of time on:

  • Visualizing functions, using Section 5 – the purpose here is to get used to 2D and 3D plots in Desmos or whatever other online graphing calculator we use.

Then we spend most of our time on:

Lesson 6:

We continue with The Pantheon of Functions, moving from polynomials to rational functions - using Section 6

Lesson 9:

Fitting functions through points - Section 10

First show how fitting works, with constant reminders of how we impose that a function with free parameters must pass through a collection of points, and how this gives us a system of equations that we can solve.

After working through this chapter we move to the book on programming mini courses and look at how fitting works for a collection of data points, and I discuss how overfitting can go wrong.

Lesson 10:

We discuss, with a lot of examples, how dimensional reduction works - Section 5