the root of the matter, part one

This is a story in 2 parts.

Part one:

I always try be creative in my approach to teaching different mathematical topics, and often the nature of the material informs the decisions I make about the way I should organize the lesson. (Is this a lesson that suits itself to more direct instruction followed by guided practice? Should I start with an investigation and let students explore the big ideas of the topic organically? Do I start with an application problem to motivate the new content? etc…) Nevertheless, it’s hard to shake myself out of certain routines I have when it comes to planning lessons. When I’m not feeling particularly creative or brave, I find myself often relying too heavily on the direct instruction -> guided practice model, even when it might not be the best fit for a topic.

So the brutal month of February sapped all of my creative juices (really, February is the worst), softball season started this week (I am the assistant coach for the JV team), and I needed to figure out how I wanted to teach solving radical equations. Normally in this situation (tired, let’s just get something ready for class) I’d whip up a lesson in Word – some text/definitions/guided notes, example problems that we’d do together, then problems my students could practice on their own. I would transfer that to a flipchart to project on the ActivBoard for class, and there you go. Boring.

I don’t know whether to blame exhaustion or brilliance for what I decided to do instead. (A little of both?) Instead of creating a me-centered lesson, I did what felt to me like very little prep-work. I pulled a worksheet from the textbook’s supplemental resources that had medium/hard radical equations problems. The worksheet was nicely chunked into 4 sections. I decided to split the class into 3 groups, each group tackling 1 section (6 problems.)

Group 1: equations with variable expressions under square roots

Group 2: equations with variable expressions under nth roots

Group 3: equations with variable expressions raised to fractional powers

(I saved the 4th group of problems for us to talk about together: radical equations where the variable shows up in more than one place – under 2 different radicals, and the “square twice” kind of equations.)

I told my students that they were responsible, in their groups, for working through and figuring out their 6 problems. I brought up a variety of Algebra 2 level textbooks for them to use if they’d like, and directed them to use their calculators and/or computers to check their answers. I told them that after everyone in the group agreed to the answers for their problems, they needed to pick 1 or 2 problems that they thought best exemplified the type of problem they solved. Each group would have a chance to come to the front of the room to teach their problems to the class. I told them I expected them to try to teach the way I do: they should prompt their classmates to take notes, generalize the steps they used to solve their equations, write and explain the work step by step, and take questions from their classmates. I also warned them that I would be acting like a student while they were teaching – taking notes and asking questions. (When I told them this, one of the students who is notorious for asking “why?” at least 5 times every class period said: “Oh, so you’re going to be impersonating me?”)

So here’s what actually happened, after I explained the plan (and took just a moment to get over my fear that this was going to be a total disaster.)

The groups launched right in. They all talked to each other. They (to my surprise) didn’t even bother with the textbooks. In the process of solving some of the problems, they debated things like the need for a +/- symbol in some problems, the simplest way to represent their answer when it was irrational, whether they should bring imaginary numbers into things when they ended up with a negative number under an even root, remembering to check for extraneous solutions. I walked around the room and eavesdropped on their conversations, but I think I only made one or two mathematical comments to groups the entire time they were working. Mostly, my role in this process was to spend a bit of time with each group coaching them on what they should do when they were being the “teachers” and the “students.” I emphasized that they should treat their classmates with the same respect they give me – raise their hands, ask questions, be attentive. And when at the board, to have no fear! “I make mistakes at the board all the time, and you guys bail me out. Your classmates will do the same for you.”

I cannot even tell you how impressed I was in my students’ ability to lead the class through their examples. They spoke with mathematical clarity, and when they didn’t – their classmates asked the “why” questions before I even had a chance to. Each group’s work built upon the last, and at the board the “teachers” referred back to things that previous groups had already explained.

One of the best moments was in my Blue period class, when the third group (with the most challenging problems) presented this example:

I was planning to ask the question (is there an easier way to do the work in the middle) but one of my students beat me to it. Raised his hand, and asked: “Is there a shortcut here? Would it be possible for us to raise both sides of the equation to the reciprocal power and do both steps at once?” I could have hugged him.

We wrapped up class with me taking the reins once more, doing a recap of the techniques we had seen, and walking students through 2 examples from the 4th group of problems. Miraculously, the lesson wrapped up right when the bell rang. Students finished the problems they hadn’t done in their groups or on the board for homework.

Here are some reasons I think this lesson worked as well as it did:

My students already possessed a great deal of the knowledge needed to solve radical equations.

• They’ve solved equations of various types all year – linear, absolute value, quadratic, higher order polynomial (limited to ones that are factorable, or ones that only have the x^n and constant terms), and rational equations.
• They understand that the main goal of solving an equation is to isolate the variable, and that doing that often involves performing inverse operations on both sides of the equation to “peel away” the operations on x.
• They already understood how to represent radicals using rational exponents (and vice versa.)
• They had already encountered extraneous solutions before (when solving rational equations.)

Additionally, my students are GREAT at talking to each other, and have learned to trust their mathematical intuition. I’ve placed a high value on the ability to talk about math. There is a classroom climate of respect, and I think I’ve done a good job of creating a space where it is okay to make mistakes, and to ask questions!

The big takeaway for me is that I need to get out of my students’ way more often. This was one of the standout days of an already pretty amazing year.

Part two to come…