[Makeover] Shipping Routes

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.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. More here.


  1. Don’t really like the original question. The makeover only slightly better. Sorry, the question ‘when return at same time’ doesn’t do it for me, dare I say it even has s.t. pseudocontextual (which, as you pointed out earlier, doesn’t mean the context can’t be imaginary, after all ‘realistic’ doesn’t come from ‘realistic’ but ‘to realize’). Compared to some great 3act movie the animation here, in my opinion, is less convincing, as well as the simulator.

    Wrt the topic. How about connecting the topic to LCM and gcd? Or jumping kids on a trampoline :-)

  2. Does it matter which shore they go to? What about the other one?

    Similar patterns:
    — Turn-signals of cars waiting for the stop light
    — “Beats” when turning musical instruments (and waves in general)
    — Pendula (this is a current favorite: http://www.youtube.com/watch?v=yVkdfJ9PkRQ)

    Maybe take several different examples and find a general way of determining the “match point” instead of a single instance.

  3. Same idea, but with a car crash simulator:


    I like your question that asks students if they can pick times so the boats never return at the same time. I think you can push a lot on having the students find time intervals that produce a requested result (asking the original question backwards basically):

    1) Can you set the time intervals so that the boats arrive simultaneously every 42 seconds? Can you do it with the time interval for one boat less than 10 and the other between 10 & 20? If so, can you find a second way to make it happen? A third? Is there a largest time interval you can use for a boat and still make this happen? Is there a smallest? Etc. etc.

    2) Is it possible to set the time intervals so that every 3rd boat on the left arrives simultaneously and every 5th boat on the right arrives simultaneously (yes).

    The boat and the car crash simulator clarify the task and provide a means for experimenting. But, they really do not provide intuition for the answer or the role of the least common multiple. Anyone got an idea on how to do a better job of that?

    @wwndtd, that pendulum video is terrific.

  4. (Elementary school perspective) Love the video simulation and the additional depth to the problem provided by your question: “Could you create two boat times so that the boats would never return to shore at the same time? Prove it.” I’d also provide “first departure”/”last arrival” times so the kids wouldn’t try to take the “never” into infinity.

  5. While the constraint built into the simulator that the times must be multiples of .1 is practical, it is misleading as far as the extension: “Could you create two boat times so that the boats would never return to shore at the same time? Prove it.”

    Take for example, a boat that takes pi minutes and a boat that takes 1 minute. The pi minute boat will never return at an integer interval.

    I realize this is getting beyond the Middle School expectations surrounding irrationals, but perhaps worth noting.

  6. Cynthia Nicolson

    August 13, 2013 - 9:45 am -

    As an island dweller who rides ferries all the time, I was intrigued with this makeover challenge and the ideas for helping students visualize this situation. A few of the earlier comments touched on the important idea that ferries need to spend time loading and unloading each time they berth, i.e. no instantaneous turnarounds. Previous discussions on this blog have helped me see the need to be clear about the varying levels of abstraction at which a problematic real world situation matches or does not match introduced examples and representations – and the dangers of jumping up the “ladder of abstraction” too quickly. To prompt a discussion and make these ideas more explicit in this case, I’m wondering if it would be useful to give a quick pendulum demo and then ask something along these lines – Is a swinging pendulum a good analogy for the ferry situation? Why or why not? Some students would probably focus on similarities and differences in the concrete physical situations while others would (we hope!) recognize and share their thoughts about the common characteristic of periodicity that makes the math applicable here.