Rethinking Logarithms
I am teaching logarithms in Algebra II for the first time since in three years and I am excited to finally get to this important concept! Covid and then Covid learning gaps prevented me from teaching it or even getting to it, so like I mentioned in a previous post, I rearranged my curriculum to make sure students didn’t leave my class without learning about logarithms. I have found that by the end of this unit, many students still cannot properly explain WHAT a logarithm is. I want to fix that this year!
I recently discovered math teacher blogs, and I have been obsessively reading blog posts about different methods of introducing them and collecting resources to pull from. Meaning, I was up until past 11pm last night because I got so wrapped up in lesson planning (not sure if this happens to anyone else haha…). So with all of these new ideas about how to better introduce this topic, I’m finding myself rearranging and reordering my old curriculum, which seems stale now after 3 years of collecting dust in a binder.
Math teacher blogging seemed to hit a peak in 2015, especially during the Twitter Math Camp era and MTBoS tags that connected hundreds of math teachers. Some extremely helpful blog posts are actually from 2015, but the methods described are still amazing. The best part is that once you find a helpful blog post, you will most likely find a link to another great blog post that the author used for reference, and soon you’ll have 10 different math blog tabs open.
A few that have been especially helpful and inspiring to me have been:
Mimi’s blog post describing her “SECRET” method to teach logarithms, which Meg Craig linked in one of her posts
Sara VanDerWerf’s QUICK THOUGHTS on logarithms post
Sarah Carter’s LOG WAR blog post
My goal this month will be to continually go back to the basic definition and idea of what a logarithm actually is. Here are my two favorite descriptions:
MIMI’S DIALOGUE: “Each time they worked on a new type of problem and they needed help, they had to laboriously say out loud what the question is that log is asking. "What exponent is required to go from base of ___ to reach a value of ___?" and they then had to identify, based on the equation given, whether that question being posed had already been answered or not. Once they said all of this out loud, they were able to figure out on their own what x was fairly easily, without any help from me.”
MEG CRAIG’S DIALOGUE: ““Hey, what power of ___ gives you ____?”
First I will give them the following statements to look at to discover that logarithms are really just POWERS (exponents). I missed out on her blog back in the early 2000’s, but I read that Kate Nowak (a former math blogging legend) used to introduce logarithms like this and loved it. It’s worth a try!
PDF HERE for the above photo.
Next, I’ll have them play Logarithm War to practice mentally evaluating logarithms. I found a few different versions of this activity, but none that fit exactly with what I was looking for, so I made my OWN VERSION. I printed out 12 sets of this PDF onto colorful cardstock. My initial plan was to laminate all the papers and then cut them into individual cards… After printing out all 12 sets and fully appreciating just how much laminating and cutting this would be, I scaled back my plans and decided not to laminate. That left a LOT of paper cutting to do! I am so thankful that several of my seniors hanging out in my room at this time were more than happy to help cut out all of the sets into cards. This would have taken me hours to do on my own! I found three papercutters from our teacher lounges and they got this done in less than an hour.
To motivate the study of logarithms more, I’ll go through portions of THIS POWERPOINT of applications of logarithms, and I also want to discuss with them WHY logarithms are really helpful -> they make numbers much more manageable! If you have a set of numbers that vary greatly in size, such as the distances between the sun and different planets (in our solar system or others), instead of plotting those huge values (which would be indistinguishable on a single sheet of paper), we can plot their logarithm value. This EXCERPT from “Functions Modeling Change” was very helpful to me my first year teaching and trying to fully understand how logarithms are helpful.
For example, instead of plotting 81, 243, and 729, we could plot 3, 4, and 5 (their respective powers of 3). Obviously, these numbers aren’t that difficult to plot on a single number line, but other sets of numbers would be much more difficult to scale properly.
After that, my goal is to revisit inverse functions, specifically Exponential and Logarithmic functions, which will help us graph logarithms.
This will be following by solving logarithmic equations, which I’ll tackle in another blog post!