Taking Students’ Money for Fun(ction) and Profit

Inspired by this post, an in the vein of painless ways to improve instruction, I thought I’d share how I introduce functions. I throw up a slide of some function, usually represented by one of those circley-arrowy thingies, and tell students we’re going to play a game. I pick a domain value, and they have to tell me which range value I’m thinking of. If they guess right, I’ll give them \$10, but if they guess wrong, they pay me \$10. No money ever changes hands, of course, but it raises the stakes nicely. One brave student volunteers to play, and in short order, takes all my money. “Just a practice round! Just showing you the rules! Doesn’t count!” Next slide goes up, and just like that, they are hooked.

“Who wants to play next?”

Next slide is a relation with one domain going to twelve different ranges and I try my best to wheedle some student into playing, but they know something’s up. I go through each domain value, asking “Do you know what range I’m thinking of?”  I usually get someone with a relation that has a bunch of one-to-one values and a single domain that goes to two range pairs. The student often justifies playing by saying the odds are in their favor, only to have me break character and pick that same value time and time again — “oooh, this time I was thinking of the 5, not the 2! You almost got me! Let’s keep playing!” After taking all that student’s money (again: not really), everyone is more wary. Each new slide prompts a burst of hurried strategizing, followed by a decision to play with me or stay away. The first slide that has multiple domains going to a single range value always elicits some good discussion, as students initially suspect a trap, but then figure out that they will rob me blind.
After they are okay on maps, we move on to ordered pairs, tables, points on a coordinate grid, continuous graphs (where brief mention is made of the Vertical Line Test), and eventually equations. Students eventually realize that in theory, they should be able to clean house on ANY equation.

Theory, meet Practice

I usually save the equation part for a few weeks later, because once students realize that the game is no fun with equations, we start playing backwards. Now I give them range and they have to determine which domain I used, and all of a sudden, we’re talking about which functions are invertible and which are not. We can push it further and put up equations that have values which “break” the game like radical or rational functions, and now we’re talking about restricted domains.

The advantage to this approach is that it lets students engage functions using their already well-honed intuitions about fair vs. unfair games. Since adopting this approach, students have a much better (if at times more casual) language for what makes a function a function (not that “oh, there’s a repeated x-value, so it’s not a function” BS), and are much more able to explain why a single “problematic” value is all it takes to lose function-hood, and why it’s totally okay to have multiple domains going to the same range value.

And it sticks. First year I did the lesson like this, I had a student mis-identify a function literally MONTHS later, and another kid chimed in with: “Nooooooo, he’s gonna take your money!”

Teaching Absolute Value

Sometimes making teaching work is all about quality materials, innovative lessons, and all that jazz. Other times, it’s the small stuff that counts. In Algebra 2, I like to dive right into linear equations for the first few weeks, then skip back and start talking absolute value. This year, I did the same stuff I usually do, but with one tiny change in language, and the kids went from:

to:

Here it is, and I realize probably everybody already does this, but I didn’t, and it is SO MUCH better.

I do everything on a number line and make students interpret the meaning of this as “the distance from x to -5 is equal to 8.” We solve some problems just like that, talking through them on a number line, but when I go to show the algebra “trick”, I teach them to drop the absolute value and set it equal to 8, or drop the absolute value and set it equal to … (here it comes)…. (seriously this is awesome)…. the OPPOSITE of 8.

I know, I know, everybody does this already. Shut up, I’m excited.

So when the right-hand side has two terms, like  |x – 2| = 2x – 12, the very first example of that type we look at, everyone already knows that either x – 2 = 2x + 12 or x – 2 = -2x + 12 and when I called on a student to explain why, she was like: “Duuuhhh, it’s the OPPOSITE.” Didn’t have to mention distributive property or anything and 100% of the class was ahead of me.

Then when we get to inequalities, it’s all very natural that something like |x – 2| > 7 will split into one branch of x – 2 > 7, and the other branch should be x – 2 < -7 because duuuhh… it’s the OPPOSITE!

And now we’re on to absolute value in two variables, and after some good old table of values graphing to convince them that they really didn’t want to have to do that all the time, I went back and talked about splitting up the absolute value equations into 2 linear equations, and wow… did not feel like I was earning my paycheck.

“So when we look at the second branch of y = |x + 4| – 2, how do you think we should write that? Should it be y = -x + 4 – 2 or y = -x – 4 – 2 or y = -x – 4 + 2?”
Usually questions like this spark an intense debate, or at least a lot of students with that scared “I ain’t picking anything” look.
But this year, all I got was eyerolls as 25 kids all picked the middle option. WHY?
“Because x +4 is in the absolute value, so you just take the opposite.”
“And why not take the opposite of -2?”
“Well, it’s not in the absolute value, so it doesn’t represent a distance where the answer could be in either direction. You’re just subtracting 2, like it says.”

So, 3 days of absolute value and we hit equations, inequalities, graphing two variables with tables and linear equation, including writing out interpretations referring back to the number line, no solution & all real number solution situations, and everyone’s on board. So if anyone out there is still teaching absolute value without using THE OPPOSITE, come give it a try. The most remarkable thing about this profession is how those tiny, tiny changes can have such an impact on students.

Socratic Questioning

To introduce points, lines, planes and just the general notion of undefined terms in Geometry, I like to start with a game of Socratic questioning. Usually students give the background on Socrates, and we talk a little about his life and how he operated.

So annoying, an entire city voted to make him drink poison

Then I divide the class into pairs, one as “the explainer” and one as “the questioner” and write a question on the board such as: “What is a rectangle?”

Here are the rules:
1) the explainer must make a good-faith attempt to explain the concept at hand

2) the questioner must ask a question designed to expose a flaw in the definition or to frustrate the explainer

3) Neither player can give up. No matter how good the explanation, the questioner must find something to question, and no matter how annoying the question, the explainer must try to answer it.

After a few minutes, I have the class vote on which role they thought did a better job, then we share some of the more frustrating questions that were asked. This is usually awesome, as the questions will range from the completely off-the-wall to the remarkably-apposite-to-the-lesson. “What is space?” “What is a point?” “What is a thing?” “What is a shape?”

We usually play four rounds, switching roles each time. The first two rounds I use actual defined objects (rectangle, circle, square, etc.), while the last two rounds I go with “What is a line?” and “What is a point?” The proportion of the class who thinks the explainer “won” goes sharply down in rounds 3 and 4.

I use this to talk about the four basic types of mathematical statement: Undefined Term, Definition, Axiom/Postulate, and Theorem, and show how definitions are based on undefined terms.

I played a game with a student where I asked: “What is a point?” and he said, “Don’t make me answer that, give me something else.” I let him pick what he wanted to explain, and showed how in a few questions, I could get him right back to “What is a point?”

Try this out with your classes — loads of fun, starts the year off well, and really gets students thinking about definitions in a more precise way.