Once in a while I have to hold back my chuckles when I’m tutoring.
Here’s a classic situation: a student is looking at part of a problem that has the expression: x times x.
He says he thinks the answer is (in order):
Then “2x squared”
… and on and on.
I call this the “shotgun” approach to algebra. If you don’t know the answer, fire off a whole slew of possible answers, and one of them is bound to hit the bullseye.
I’ve finally developed a successful way to help students who slip into “shotgun mode.” I require them to consider how real numbers behave in similar expressions.
So with the student who says that x times x = 2x, I’ll initiate a conversation like this:
Josh: Interesting! Let’s check that out with real numbers.
Student: Like how?
J: Well, let’s replace the x’s with actual numbers. Why don’t you pick a whole number between 2 and 9.
S: Is 6 ok?
J: Sure. So you thought that possibly x times x = 2x. When we replace both of those x’s with 6, that what would we get?
S: So it would say: 6 x 6 = 2(6)
J: So is that true?
S: I don’t know.
J: Well, what is 2(6)?
S: Is it 12?
J: Yes, because that’s just multiplication between the numbers.
S: So it says 6 x 6 = 12.
J: Is that true?
S: Um … no.
J: So what does that tell you about the idea that x times x = 2x?
S: That it’s not true?
J: Exactly. But let’s stick with the numbers for a while. What is 6 x 6, anyhow?
J: And do you see any way to make 36 using the number 6, other than just 6 x 6?
S: Um, not really.
J: Well, isn’t 6 x 6 the same thing as 6 squared?
S: Yes …
J: So can you say that 6 x 6 = 6 squared?
S: I guess. No, it is. That’s true.
J: Why don’t you try that and see if it’s true with another number, like 4. Substitute 4 for each 6 in
6 x 6 = 6 squared, and let’s see if that’s true, too.
Student writes: 4 x 4 = 4 squared.
S: That’s true too. 4 squared is just 16.
J: Right. So do you think this pattern would be true for any number?
J: Want to try one more number?
S: Can I use 10?
S: Let’s see. That would be 10 x 10 = 10 squared … That’s true. Yes, it has to be true. Squaring is the same as multiplying the number by itself.
J: Exactly! So if this is true for any number, then you can go the other way. You can replace the number with the letter x, to show it’s true for all numbers. What would you get if you take 6 x 6 = 6 squared, and you replace the 6 with x?
S: You’d get … x times x = x squared.
J: Right. Does that help you with the original problem we were working on?
S: I forgot. What was it?
J: We came to a spot where you had to simplify x times x.
S: Oh yeah. So I can just put x squared?
J: Yes, since you just figured out that this is true for all numbers.
S: Oh, ok. I get it.
The point is that students get so used to apparently weird things happening in algebra, they lose their intuitive connection to the rules of algebra. One example is the idea that x to the zero power equals 1. This makes no sense to at least 90% of the students I work with — at first. And that kind of “not making sense” leads students to more or less give up on the idea that algebra has to make sense. Students start to believe that in algebra, virtually anything can happen. Hence, the “shotgun approach.”
But relating the rules of algebra back to real numbers … helping students see that real numbers DO behave in accordance with algebra’s rules … helps students re-connect to their intuitive sense that things should and do make sense. I find that re-establishing this kind of belief helps students “stay in the algebraic saddle” when topics get difficult. With this faith that things do ultimately make sense, students have faith that if they keep trying to understand algebra’s rules, they will be able to understand them. And that trust gives them the perseverance they need to stick with the course.
I suggest that you try this approach — substituting real numbers for algebraic variables — the next time you see students being altogether mystified by an algebraic rule.