Seeing structure in expressions. What does that standard mean? I can think of a bunch of pretty cool examples, but it’s pretty hard to nail down the more general skill of seeing structure in expressions and using it to reason mathematically. In particular, this standard is a tough one:
Pretty intimidating. There’s definitely a lot more to this than converting annual interest rates to monthly interest rates, but I thought I’d take a stab at that as an attempt at getting students to see exponential structure. This standard mostly intimidates me because I want my students to see authentic opportunities to see structure, not to have it forced on them. While my kids are pretty good with compound interest and exponential growth at this point, I was pretty skeptical they could figure out how to rewrite annual interest rates as monthly interest rates, but decided to go for it, and see what I could do to emphasize seeing structure in expressions.
I got some pretty interesting answers. Almost all kids got #1 based on what we’ve already done with compound interest. On #2, a bunch of kids were frozen, and a bunch more said 2%, as I figured they would. But I got other answers — some kids found that Jahleil’s account would have $14,658 after 5 years, and then divided that by 12.
So kids shared their ideas, and I grabbed a few papers to project and discuss. The point of this discussion was pretty simple — it might seem intuitive that, if the annual interest rate is 24%, the monthly interest rate should be 2%. But for some reason, Jahleil ends up with $14,658 using the annual interest rate, and $16,405 with the monthly interest rate. Doesn’t make sense. I showed them this graph to reinforce the point.
At this point, I had plenty of kids with me, but I could tell that a few were a little confused — some still didn’t see the connection between 1.24 and 1.02 as 24% and 2%, and others had trouble making the connection between the interest rates and the money in the accounts. I’m sure there were more misconceptions I didn’t catch. And here we were done with the discovery.
Discovery is great. But only if the kids can really, genuinely discover something. We shifted into some direct instruction — I showed them how to transform the function as written in the standard, talking about how we want the exponent to be 12x because every increase in x should increase the number of months by 12. To make sure the functions are equivalent, we need the reciprocal of 12, 1/12, on the inside. This was a tough concept, and while I asked some questions and solicited questions from them, I’m sure some kids weren’t looking for the structure here. We also got a bit caught up figuring out exactly how to calculate fractional exponents accurately on graphing calculators.
Anyway, we had hit the key skill-based points I wanted to hit. We did a bit of practice as a class, and some practice on their own, rewriting functions, converting annual interest rates to monthly ones, and pushing their knowledge converting 10-year rates to annual rates, and talking about exactly what the domain of this function means, and when to substitute the number of years vs the number of months.
At this point, we were getting a little lost in the sauce. My questions were focused on pushing their understanding of the structure, but there were lots of roadblocks. Trouble with graphing calculators, mistakes converting percents, trouble seeing 1.08 as an increase of 8%, and trouble seeing 1.00643 as 0.643% (the equivalent monthly rate). Algebra is hard, and kids were getting caught up in the calculation, instead of thinking about the structure. So it was time to shift gears.
I gave kids this graph:
I gave them a few minutes to answer some questions — starting simple, with how much money is in the account to start, then the difference between the functions and which account they would rather have. Kids wrote first, then discussed with a partner before we shared with the whole class. I was really impressed with the quality of their analysis — obviously, you want the monthly function, because it almost always has more money in it than the annual function. I think kids really appreciated taking a break from the algebra to step back and look at a visual, and they asked some good questions about what that floor thing means in the equation.
This was a great end to class. Kids left on a positive note — and I told them they were awesome for sticking with a tough lesson and asking great questions. I’m not sure if I did any magic of my students’ ability to see structure in expressions, but they had a bunch of opportunities to stretch their thinking. Most importantly, this class was less focused on what kids could do, what kids could produce, than focused on what they could understand. It’s a tough shift for kids — they’re always looking for the steps and the quick fix to solve a problem with as little thinking as possible. And while some kids were still looking for that today — “I just do the 1/12th power?” — there were serious opportunities for them to pause and reason about the math they were seeing.
I want to find some more chances to teach lessons like this that pause and look deeply at some functions and the structures that relate them. What am I missing that can make this lesson better? Where are more opportunities to see structure in expressions?