## Tuesday, September 28, 2010

### Equations Three Ways

I've never really been satisfied with how I teach students to solve equations.  No matter what, it ends up being one big algorithm and kids have no idea why one side of the equation is equal to the other.  Here's what I'm doing to try to fix that.

Modeling

Strength:  It's not math.  It's a puzzle.
Weakness:  Dealing with negatives is a real pain in the butt.

Guess and Check

I actually really like this method.  Guess and check is probably my most under-used problem solving strategy, but using it to solve equations has been really helpful.  I've noticed a greater understanding of rate(s) of change, using information from wrong answers to help find right ones and checking answers--something most kids don't want to do--is embedded in the process.

We've gotten to the point where we can nail the answer on the third guess by using the information gained in the first two--even for equations with non-integer solutions.

Strength: Students understand that simplified expressions on each side of the equal sign end up looking the same every time (ax + b = cx + d).  Rate of change is very useful.  Being wrong helps you to become right.  Did I mention they are checking their answers?
Weakness:  Leave 'em in the comments.

From Construction to Deconstruction

We spend a lot of time teaching kids how to break things down whether it's reducing fractions, simplifying radicals or solving equations but we rarely (read: I rarely) have taught them how to construct things that may eventually need to be deconstructed.

Constructing a more complicated equation from a simple equation has helped my students understand that, no kidding, the two expressions on opposite sides of the equal sign are equivalent.   I've used the just-unwrap-the-present illustration many times, but we really need to teach the students to wrap one up first.  Having them list their steps for construction makes the process for actually solving the equation seem much more natural.  When I say, "just use the inverse order of operations" --or whatever completely abstract thing I've been known to throw out there in order to make myself feel better when they keep screwing it up--it makes no sense to them.  This helps.

Strength: Kids get a grasp of which operation to tackle first while solving for x﻿.

Weakness: Very complicated equations with variables on both sides don't seem so natural when you begin with x = 2.

I've heard rumors that there are some teachers who actually teach solving equations by graphing.  Never seen it in the wild, though.

Andrew said...

I have had some success with "construction to deconstruction" by asking students to make up problems to try to stump their colleagues. This prompt motivated at least some of them to try to cook up very complicated things.

CalcDave said...

For modeling, the negatives could be balloons?

David Cox said...

Nice call, Andrew.

Dave,
Yeah, we could use negatives but it gets a bit confusing when you need to "take away negatives" from one side when there aren't any on the other. You'd need to add "zeroes" in the form of pairs of positives and negatives. That's not quite as intuitive as I'd like.

Unknown said...

At the very beginning, I teach solving by graphing. They make a table for y = 4x + 5 (or whatever) and graph it. Then we look at the graph and I ask "What is y when x is 2?" questions for a while, then switch to "What is x when y is 1?"
I motivate other methods of solving by "there's got to be a better way to get the answer to that question..."

I like doing it this way because it really stresses variables as representing all different values, rather than having students think that x always represents a single number, which you just have to find.

Anonymous said...

The way we teach our kids with calculator accomodations is to put one side of the equal sign in Y1 and the other side in Y2, graph the lines, and find the intersection. The biggest weakness (besides not learning how to manipulate equations) is that they have to learn how to manipulate the graphing window.

Example: Y1=x+7
Y2=9

grace said...

While I really like guess-and-check as an introductory strategy, it makes me wonder how we can help students codify the intuition that they develop in using it; even if students are able to get the right answer fairly efficiently once they start to develop a sense of how it works, I'd want them to be able to explain why their guesses are good guesses rather than just saying "it felt right."

I'm not clear from your post, but it feels like this set of activities, in the order in which you describe them, is actually a strong sequence of scaffolding as students progress from easier equations to more challenging ones and can revert back to other methods and graphing to check their work (and to reinforce that there's more than one way to solve a problem).

David Cox said...

Alison
We played around with graphing the expressions set equal to y. We plugged them into GeoGebra and just looked around for things they thought were interesting. They were like, "hey, the x value for the point of intersection is our answer and the y value is the number we get when we know our guess was right."

Grace
I couldn't agree more. Sorry I wasn't more clear in the post.

The modeling comes first as a way of just playing with the idea of equivalent expressions. They solve without really "solving."

The guess and check gets them to simplify expressions without worrying about solving and has checking embedded. (I have always felt like it was a fight to get kids to check their solutions)

As a class, we kind of decided that 0 was a good first guess because it's easy to use (also sets the table for finding y intercepts later) and 1 is a pretty good second guess because it gives us the rate of change between the answers. Once they have locked in on rate of change, they only need a third guess.

Normally, when I teach guess and check, I encourage guessing too high and then too low (or vice versa), but the type of information we gain from two guesses one unit apart is too important to pass up.

Once we have those two dialed in, we can go to construction and kids can get all kinds of crazy (As Andrew suggested) with their equations and guarantee that the solution is an integer. It also helps formalize the "steps to solving an equation."

I didn't quite plan it out as well as it seems, but after some reflection, I can't wait to get to equations with my 7th graders. In fact, we have already started playing with modeling.