As the summer winds down and #MakeoverMonday comes to an end, we’re going to crank up the difficulty around here. For the final three makeovers, I’ve commissioned work from some of the best people I know working in math, education, and technology: Dave Major, Evan Weinberg, and the team from Desmos.

The Task

This task is from McDougal Littell Middle School Math Course 1, 2005. [via Chris Robinson]

What Dave Major & I Did

TLDR: Here’s the 101questions page.

Lower the floor. The task currently jumps straight to the question of calculation. We should head in that direction but start with other interesting, easier questions also.

Enable pattern-matching. I could tell students what to look for here and how to approach the problem. I could show a few worked examples. For example, ones where:

• one boat’s time is a factor of the other. (eg. 2 seconds and 4 seconds.)
• the boats’ times are coprime. (eg. 3 seconds and 11 seconds.)
• the boats’ times have a common factor. (eg. 6 seconds and 10 seconds.)

Two problems there:

1. Some students will need more than just three examples to determine a pattern.
2. My selection of those particular examples — that is, my decomposition of the entire solution space into just three categories — did a lot of the intellectual heavy lifting for my students. They need to decide on those three categories and come up with a rule that takes them all into account.

That isn’t to say I’d just “let them figure it out.” If a student just tries the first example and says, “It’s easy. It’s always the longer of the two times.” I can then say, “Great. But try that on several more examples and make sure it works.” (It won’t.) Or I can suggest one of the other two categories. But I’d rather not offer those categories before the student has even considered why she might need them, or even the fact that there are different categories.

Raise the ceiling. Our textbooks need fewer tasks and they need deeper tasks. The second fix would enable the first. Rather than jumping to another arbitrary context for another arbitrary example of cofactors, let’s stay in this context and extend it, developing the concept more for students who are ready for it.

Prove math works. It’s one thing to solve the original task for 150 seconds and find that answer in the back of the book. It’s another thing to watch the answer play out in front of you.

I’d ask students to watch this video.

I’d ask them, first, if they thought the boats would ever return to shore at the same time. The task gives that answer away but, me, I’d rather get every possible conception on the table so long as it doesn’t cost me too much time. “If you think they’ll return at the same time,” I’d then ask them, “write down how long you think it’ll take to see that happen.”

Then I’d send them over to Dave Major’s Shipping Route Simulator™. “Make up some boat time examples for yourself. Watch what happens. Make a table. Tables are useful for organizing data like this.” (We’ve intentionally set up the simulator so the domain maxes out at 10 minutes.) Then I’d tell them to pick two boat times and try to figure out what the answer will be before they check it by running the simulator. Whenever they’re ready, I’d ask them to tell me how long it’ll take the original boats to return to shore together and how they know.

I’d ask students who finished quickly:

• Could you create two boat times so that the boats would never return to shore at the same time? Prove it. (Incidentally, this is one way I try to “be less helpful” — an expression that drives a certain set of math educators and mathematicians up the wall. Why give away the fact that the boats have to meet again? That’s an interesting question. Don’t be so helpful.)
• What if you had three boats? Four?
• What if the boats didn’t have whole number shipping times? What if one boat made its route in 2.5 minutes and another boat made it in 8 minutes?

Then I’d show the answer:

What You Did

In the preview post, most commenters seemed content to add elements to the word problem itself — adding a sentence about refueling schedules for motivation or turning the whole thing into a debate between two people about whether or not the boats will both return within the hour.

I’m sure that’ll have some effect on motivation and cognition but I’m not sure how large of an effect it’ll have or in which direction.

William Carey took a different approach:

I wonder whether a video of two bouncing balls or two oscillating springs or two swinging metronome hands would capture the idea of factoring to figure out when two cyclic phenomena will be in sync? That seems like it’s the perplexing bit of the problem.

Jim Pardun:

It reminds me of sitting in the left hand turn lane trying to figure out how often the turn signals will match up on the cars.

Not for nothing, I gave Jim’s example a shot some time ago and abandoned it. With most cars, the frequencies are so close that they converge again rather quickly. Part of the appeal with the ships is that it takes a really long time for them to converge.

If you’d like to see what goes into my rubbish bin, here’s what would have been “Turn Signals,” with two cars, three cars, and eleven cars.

2013 Aug 13. I don’t say this enough, but students should walk away from this lesson with a definition of “coprime” and “cofactor” written in their notes and, more ideally, stuck in their heads. Those definitions should come in the debrief of this conceptualizing activity, though, not in its introduction.