# Town Squares problem

A friend of mine, a father, recently posted this item on his Facebook feed. It’s from Pearson, and he was struggling figuring it out. I also had to read it several times to figure it out.

This is a large part of why people hate story problems. The table and the premise make it sound as if this will be some sort of meaningful, real world problem: Mathville is situated between a bunch of towns, the table has a bunch of distances. Any vaguely critical reader will be primed to do something with relative distances.

Any vaguely critical reader will be wrong. What’s actually wanted: Draw lines in each square so that the two letters in each subsection of the square will have the same sum. The miles and the back story are completely irrelevant. This is the problem, stripped of all the confusing mess:

Given A = 359, B = 430, C = 110, D = 119, E = 249, F = 229, G = 158, H = 407, I = 178. Pair each set of letters so the pairs have the same sum: (1) A, B, E, I; (2) A, C, G, H; (3) A, D, E, F; (4) A, B, F, G.

The defense offered by Pearson would likely be two-fold.

First, this is intended for students developing algebraic thinking. One purpose of the exercise is to get students used to the idea of letters standing for things. However, as Jo Boaler of Stanford has argued, this is misleading. A in this problem doesn’t stand for Allentown, it stands for the distance between Mathville and Allentown. This is a crucial distinction. What if we then wanted to explore the distance between Bensonville and Allentown? Would that be 430? 359? Something else?

Second, we’re trying to get students to develop critical thinking skills by showing how to model real life problems in mathematics. Except… this doesn’t. The mileages don’t add anything at all to the actual problem, they just provide a distraction for any adults trying to figure it out. My revised problem is actually more difficult because it requires students to try more pairs of numbers (or to develop strategies, such as “try the largest and the smallest”). A student who completely guesses on the Pearson version has a 50% chance of getting the first three right and, assuming “divide” implies using a straight line, a 100% chance of getting the last one right.

There is a way to improve the problem: Joe’s Trucking is creating routes for its two drivers. A route consists of starting in Mathville, going to another city, and returning to Mathville. Each day, a driver has to complete two routes. The dispatcher wants to make sure that the drivers get the same number of miles. For each set of four cities listed, pair up the cities so the day routes are equal. Students could be provided with a hub-and-spoke diagram with nine spokes. There. Visual. Critical reasoning. Real-world modeling. The “real world” is still a world that only exists in math books, but at least it’s closer to reality than what Pearson has on display here.