# A Double Progression: Exponential Functions and Logarithms

I’m teaching Algebra II, and just completed a unit on exponential functions and logarithms. Students should have seen exponential functions before, but many have forgotten the big ideas. In Algebra I they likely only wrote, interpreted, and evaluated exponential functions. In Algebra II, we’ll be solving a wider variety of problems with exponential functions, and one major new idea is using logarithms to solve for a variable in the exponent.

I taught the unit with a double progression, with two ideas working in parallel. The first several days were focused on writing and interpreting exponential functions. We spent some time working on interest and other applications with money, as well as a variety of other contexts requiring students to construct exponential functions and evaluate them at different values.

At the same time, we were doing short activities introducing logarithms either at the start or end of class. I started with this puzzle from Kate Nowak

We then formalized that knowledge, spent some time playing log war to practice and get some muscle memory behind logs, and sprinkled log problems generously through homework. The bulk of class time was spent on exponential functions, but this parallel progression both helped to mix things up and proved to be enough to communicate the big idea of what a logarithm actually means — log base 2 of 16 means 2 to what power is equal to 16. We didn’t get into more advanced log rules at this point, because those rules are rarely necessary to solve problems with exponential functions and I saw as an unnecessary distraction from my key mathematical goals.

Then we moved into problems like this one from Illustrative Mathematics that require logarithms to solve.

We had done some guess and check earlier in the unit, but with a solid base to work from, it seemed like a natural step to connect between the two strands of this progression and make our calculations more precise and efficient using logarithms.

I really liked this approach. It helped that I teach 90 minute blocks, so it’s manageable to have multiple things going on in a simple class compared with shorter daily blocks. I also think it made the math we were learning feel more purposeful, as well as having relevant ideas fresh in students’ minds when they became necessary. I’m curious now — for what other topics would a double progression like this be effective?

## One thought on “A Double Progression: Exponential Functions and Logarithms”

1. Amy Zimmer

Right Triangle Trig and Inverse Trig Functions at the same time count? I don’t understand how to “untwine” them and write curriculum to view the two forces separately. This was a Geometry class, I think the next time I teach Trig/PreCalc, I will “double” progress the unit circle AND graphing Trig Functions. Thanks for sharing!