On Thursday, in Algebra 2, I aimed for understanding… and came up short. It was one of those lessons I went in excited to teach, because I knew I could get my kids to “get” it. And then, I didn’t. It wasn’t a bad lesson, and I don’t think my kids are the worse for it. But it just wasn’t that killer lesson I had hoped for.
I had 50 minutes to teach absolute value inequalities. I definitely had to get through “less than” absolute value inequalities (like , but I wanted to at least introduce “greater than” absolute value inequalities (like ).
1. Start off lesson with a fundamental question raised by an image (found at 360).
2. Generate a basic understanding of what a “less than” absolute value inequality is, using a simple question to illustrate why there are many solutions. We would then fill in a number line from our solutions and then talk specifically about why our answer is not [-12,2], but actually (-13,3) (for the question below).
3. Point out the basic geometric interpretation, and tie that back to the initial question we wanted to answer.
4. Show how we can use this geometric interpretation, and a compound inequality, to help us come up with a method to solve these sorts of inequalities. (Work backwards.)
5. Formalize our method of solution.
6. Practice, practice, practice.
I only was able to cover “less than” absolute value inequalities, and I doubt that my students have a good understanding of why our solution method works. I do think my students will be able to follow the procedure though.
Where exactly did I fail? I failed in part 3 and part 4 of my plan. For part 3, I should have found a better way to explain the geometric meaning. My students didn’t “get” it totally. You can see part 4 of my plan executed here on my SMARTBoard slide:
As you can see, I tried to start from the compound inequalities and work our way to the absolute value inequality. At the end, there is this “ta da” moment which was actually more like “ta WHA?” They didn’t get what I was trying to show them.
And I don’t blame them.
A huge part of me doesn’t want to teach something without proving — or at least deriving by example — why something works. I feel like a fraud, like I’m teaching ’em magic instead of math, when I teach a method of solution first and then show where it comes from. But in this case, it would have gone over so much better if I had shown the method of solution, and then after practicing it a few times, took a moment to look at our work backwards to see why it worked. Talking about what each step means — algebraically and geometrically — backwards might have clarified things a bit.
I could have also designed the lesson in a totally different way. I could have worked off of our understanding of absolute value equations (e.g. the equation ). Then we could have had a great discussion on how to find solutions to , focusing on why we use open dots at the solutions to the equality, and why we shade inbetween those dots. Now that I’m thinking about it, maybe I should remember to try this out next year.
If you want to see my entire SMARTBoard for the lesson, look below the jump.