## Our Library’s Summer Math Contest

Every summer, our local library runs a contest called The Great Big Brain Game. Young patrons who solve all of the weekly puzzles receive a prize. The second puzzle for Summer 2017 looked like a typical math competition problem:

Last weekend, the weather was perfect, so you decided to go to Cherry Hill Park. When you got there, you saw that half of Falls Church was at the park, too! In addition to all the people on the playground, there were a total of 13 kids riding bicycles and tricycles. If the total number of wheels was 30, how many tricycles were there?

A tricycle has 3 wheels. (Duh.)

1. I dislike using “you” in math problems. I believe it’s a turn-off to students who can’t see themselves in the situation described. There are enough reasons that kids don’t like math. Why give them another reason to shut down by telling them that they went somewhere they didn’t want to go or that they did something they didn’t want to do?
2. Word problems are not real-world just because they use a local context, and this one is no exception. This problem attempts to show an application for a system of linear equations, but true real-world problems don’t have all the information neatly packaged like this.
3. Wouldn’t the person posing this problem already have access to the information they seek? That is, if she counted the number of kids riding bikes and the total number of wheels, couldn’t she have just counted the number of bicycles and tricycles instead? It has always struck me as strange when the (implied) narrator of a math problem wants you to figure out something they already know.

All that said, this was meant to be a fun puzzle for a summer contest, and I don’t mean to scold the library. I don’t know that I’d use this puzzle in a classroom — at least, not presented exactly like this — but I love that kids in my town have an opportunity to do some math in June, July, and August.

Now, I’ll offer some comments on the solution. In particular, the solution provided by the library was different than the method used by one of my sons. Here’s what the library did:

Imagine that all 13 kids were on bicycles with 2 wheels. That would be a total of 26 wheels. But since 30 wheels are needed, there are 4 extra wheels. If you add each of those extra wheels to a bicycle, that’ll create 4 tricycles, leaving 9 bicycles. So, there must have been 4 tricycles at Cherry Hill Park.

And here’s what my son did:

If you can’t see what he wrote, he created a system of two equations and then solved it:

2a + 3b = 30
a + b = 13

a + 2b = 17
13 – b + 2b = 17
b = 4

2a + 12 = 30
2a = 18
a = 9

That’s all well and good. In fact, it’s perfect if you want to assess my son’s ability to translate a problem and solve a system of equations. But I have to admit, I was a little disappointed. What bums me out is that he went straight to a symbolic algorithm instead of considering alternatives.

I think I know the reason for this. This past year, my son was in a pull-out math program, in which he studied math with someone other than his regular classroom teacher. In this special class, the teacher focused on preparing him to take Algebra II in sixth grade when he enters middle school. Consequently, students in the pull-out class spent the past year learning basic algebra. My fear is that they focused almost exclusively on symbolic manipulation and, as my former boss liked to say, “Algebra teachers are too symbol-minded.”

A key trait of effective problem solvers is flexibility. That type of flexibility comes from solving many problems and filling your toolbox with a variety of strategies. My worry — and this isn’t just a concern for my son, but for every math student in the country — is that students learn algorithms at the expense of more useful problem-solving heuristics. What happens when my son is presented with a problem that can’t be translated into a system of linear equations? Will he know what to do when he doesn’t know what to do?

The previous pull-out teacher said that when she presented my sons with problems that they didn’t know how to solve, their eyes would light up. They liked the challenge of doing something they hadn’t done before. I’m hopeful that this enthusiasm isn’t lost as they proceed to higher levels of mathematics.

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• 1. Alden Bradford  |  July 14, 2017 at 8:52 am

I’ll mention a similar problem, which I found in “Propositions for Sharpening Youths”, a math puzzle book written in the year 782:

“A certain buyer said: ‘I want to buy 100 pigs with 100 denarii in such a way that a mature boar is bought for 10 denarii; a sow for five denarii; and two small female pigs for one denarius.’ Let him say, he who knows, How many boars, sows, and small female pigs should there be so that there are neither too many nor too few of either [pigs or denarii]?”

This is in the same vein as the bicycle/tricycle problem, but I found the library’s method more useful on this one than setting up a system of equations.

• 2. venneblock  |  July 14, 2017 at 12:40 pm

Thanks for sharing, Alden! I’ve seen variations of this problem, but not the “original” involving boars, sows, pigs, and denarii. The difference between this problem and the library’s problem is that this one results in a system of two equations with three unknowns:

b + s + p = 100
10b + 5s + 0.5p = 100

And it’s “impossible” to solve that directly using just symbolic manipulation. But as you’ve figured out, you can find an answer with some guess-and-check, or by spending some time with Diophantus.

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