Last month, Julie asked me for my log word problems (which include zombie attacks and other infectious issues). I usually use these after I have introduced logs, but ever-wise Julie used the problems to create the NEED for logs. Brilliant. Using Julie’s idea, I modified my original lesson similarly.
The students intuitively realized that they were using the inverse of an exponential function – several wrote the equation quickly (and then proceeded to use guess and check) or created a table. Those students who made tables saw that the answer was between 7 and 8 hours, but when I told them I needed a more specific answer, they, too, moved to guessing and checking. (This process led to a nice mini-lesson on the use of the TABLE and TBLSET functions on the graphing calculator.)
After reviewing the answer to the problem, I moved immediately to Kate Nowak’s Super Fun Log Puzzle. The kids worked through the ‘puzzle’ fairly quickly, and were able to articulate what they were doing. When I revealed that they could replace the word ‘power’ with ‘log’ – well, I love that moment – the moment that they realize that something they thought would be difficult wasn’t really that bad, or that their friends were wrong when they said logarithms were terrible and mysterious. [There is a moment in the lesson when I show them a log table from the back of a textbook – the kind I used back in the day. They are suitably horrified by life before epidemic calculator use. Another moment I love.] And the room was filled with math talk all day long – students helping students, students arguing with each other, students making sense of something new. A pretty easy day for me as well.
Today, I followed up with another of Kate’s activities (I owe you SO much, @k8nowak!) – discovering the properties of logs. The students began the work the moment they entered the room, and they were engaged in the process all period long. I steadfastly bounced all questions back to them, referring them to their tablemates as resources. I nipped calculator usage in the bud, except as a checking device.
And, as I tweeted earlier, all I did was walk around the room listening for 30 minutes, occasionally redirecting if I saw a table going seriously off track, or asking some pointed questions to spark some recognition. Towards the end of class, we recapped their discovery as a whole class, and the relationship of the ‘laws of logs’ to those governing the use of exponents were universally clear. And the concept of inverse functions was reinforced once more, when viewing the relationship between logarithms and exponential functions.
After the difficulties I’ve been having in my problem solving classes, hitting a home run three times in a row in the Algebra 2 classes felt as good as you might imagine it would.
At the end of the day, I found myself in a conversation with an officemate who teachers Honors Pre-Calculus; he is finding that his students are weak in the skills that they should have mastered in Algebra 2, surprisingly so for that level of class. These are the students who will be going into AP Calculus next year, one third of them into BC Calc in fact, and this teacher is trying to strike a balance between making sure that their requisite skills are strong enough to handle the difficult conceptual work to come. He cited logarithms as a particular area of mystery and antipathy for his classes. I shared these lessons with him (the lessons are quite unusual in my school in their level of discovery), and I could see that, like the zombies, the idea might be infectious.