I teach a class called Math Analysis/Trigonometry, which is close enough to Precalc that students who get A’s in my class are allowed to skip to calc. I’m spending the first several class periods covering a combination of Carol Dweck mindset stuff and the idea of distributing. Distributing? In Math Analysis/Trigonometry? Yes. I actually think many students, even post-Algebra 2, don’t understand it that well, and I think it’s what leads them to make some of their most common mistakes.
Later in this post, I’ll describe how I’m trying to talk about distribution in a way that’s conceptual and allows for Accountable Talk–I certainly don’t want to lead off the year with a bunch of drill-and-kill. And to my surprise, distribution seems to be fertile ground for conjecture and discussion. But first, here are the top 2 mistakes. Recognize them?
- In (x + 2)/x , you can cancel out the x’s.
- (x + 5)2 is just x2 + 25
With mistake #1, students just don’t see that it’s distribution. If it said x(x+2), they wouldn’t give the answer of x2 + 2, but they don’t see that distribution happens with division, too. Somewhere in their brains, they know it, but that knowledge must be only loosely connected to everything else.
With mistake #2, students mistakenly think that you can “distribute” an exponent across addition.
There are other big mistakes relating to distribution, too. Have you ever seen someone try to distribute the 2 in 2(3 * x)? Usually, they won’t do it for 2(3x), but if I put a multiplication sign and some big spaces inside the parentheses, I usually get more than a handful who distribute. This whole topic is closely connected to the ability to see whether an expression has one term or many, but the number-of-terms idea doesn’t unify it completely, because you can “distribute” the square in (3x)2, even though 3x is a single term. So let me share what I’ve done. Curious how you would have done it–surely other teachers must have to do battle with the same misconceptions. And you may object to the use of the term “distributing” when talking about an exponent. I used to discourage students from talking about it like that, but I now think allowing it as long as they have a deep understanding of distribution is better.
Accountable Talk Discussion Plan
Discussion was grounded in this student handout. The question of why the distributive property works led to some crickets chirping in class, so I had to take a student’s example from #1 — the example was 3(x + 2) — and ask WHY does it equal 3x + 6. The first 3 students to respond gave variations on the theme that you have to multiply the 3 by both terms inside. Right, but WHY is that rule correct? I think I ended up priming the pump by writing “3x = x + x + x” and asking how you could write a similar statement for 3(x + 2). A student explained that (x + 2) + (x + 2) + (x + 2) = 3x + 6. Still, lots of students were either tuning out (“Isn’t it obvious that it’s 3x + 6? I mean, you just distribute.” Grrr) or not getting it. After 3 more students paraphrased and small groups checked in with each of their members, everyone seemed to understand the point we were making.
So then, the set-up for the big discussion: #3 on the handout. To clarify the question, I said, “There are other situations where you can distribute. For example, what could you put inside the parentheses in the expression ( )2 so that you could distribute the square. And 90-100% of the class wrote something like (x + y)2. Success! This was exactly the misconception I wanted to address, and instead of just presenting it from out of nowhere, we’ve created a natural context in which to debate it. I grabbed a bunch of student responses and put them under the document camera. One student actually wrote out his work: (x + 5)2 = x2 + 52 = x2 + 25, I showed that response last, and a handful of students recognized that it was incorrect, but the bell was about to ring.
Bell rings, end of class, next class begins. We picked apart what was wrong with (x + 5)2 = x2 + 25, and I asked them to try #3 again. What can you put in the parentheses so you can distribute the square in ( )2. Now a couple of students tried a monomial in the parentheses. We proved that it worked using associativity and commutativity, and I asked several students to summarize. So far, the class thinks the rule is that you can distribute an exponent across different things as long as there’s no addition or subtraction between those things. That’s not the rule I want them to eventually come up with, but it’s a good enough start to let us move on to #4.
Check out #4 on the handout (here it is again). I really hope we get some good mistakes there, and some good debate. The rule I want them to generalize is that you can distribute what’s outside the parentheses if the operation/function outside is the repeated version of the operation inside. Since raising to a power is repeated multiplication, an exponent distributes across multiplication but not addition. Since multiplication is repeated addition, it distributes across addition (or subtraction) but nothing else.
Actually, I’m not sure if this is technically correct. Distribution is a property of rings from abstract algebra, and there may be some exotic versions of rings where the “multiplication” operation doesn’t represent repeating the “addition” operation. I got some help on Twitter from Dave Radcliffe (@daveinstpaul), who was telling me about power sets forming a commutative ring. If anyone can help me understand whether I’m making a big mistake talking about distribution this way, I’d really appreciate it!
Khan Academy for Review Practice?
Meanwhile, my students have a bunch of factoring and quadratics exercises to do on Khan Academy. (Yes, on Khan Academy. These kids won’t be harmed by boring videos about factoring. They learned factoring years ago. They just need to gain fluency, and I need data on who needs remediation with what.)
UPDATE 9/11/13: When we did #4a today, students’ initial answers on whether to distribute in 2(3 * x) were evenly split. Half the students got 12x, and half the students got 6x, until we discussed it. Should get to the final summarization of the principle I’m looking for tomorrow. (We don’t spend all class on this stuff, so it’s getting spread out over more days).