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


  1. SanchezTemple

    August 8, 2013 - 12:59 pm -

    I believe this is a similar problem with the “Taco Cart”. I would try to incorporate a purpose for wanting to know why they will be returning at the same time. Perhaps a refueling schedule works best when both Ferry’s are back at the same time. In fact that could be the task. They must create a schedule to highlight when the will return at the same time.

  2. Two things jump to mind when students have devices:

    1. Use the Ruler tool in Google Maps or Google Earth to calculate actual shipping distances between actual ports. Ports can be assigned to students (all same or all different), or students can be allowed to choose.

    2. Research different shipping boats (big/small, fast/slow) to see at what rates they would cover different distances. This allows a chance to attend to units and to make sure that km/hr or mi/hr match with measuring distances in km or in mi. It also allows for conversion practice if speeds are researched in knots.

  3. One struggle I have making over these tasks is maintaining the content of the original. The original task requires factoring to figure out when the two boats will meet at the same time. You’re basically finding the least common multiple of the two shipping times.

    So adding in shipping research and Google Earth and units may even be more fun than any makeover I could come up, but you aren’t making over the task. You’re making a different task. Which, again, might be awesome but it isn’t the assignment.

  4. William Carey

    August 8, 2013 - 4:33 pm -

    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.

  5. Ah, gotcha; thanks for setting me on the slightly narrower path…

    Maybe the red and blue ships could be two members of the fleet of Deadliest Catch. One is going to get several boatloads of fuel (to share), and the other is going to get several boatloads of food (also to share). In a remix of SanchezTemple’s insightful idea, at what time can they meet to exchange supplies?

    As a modification/extension, maybe each ship has different sized cargo. For example, fuel is of arbitrary size 40, and food is of arbitrary size 25. What size should the red and blue ships be so that they can best hold supplies received in exchange from the other ship?

    Just some ideas. Thanks for the feedback, everyone!

  6. The real problem here is the manpower issue of loading two ferries at the same time. That should be the focus of the problem.

    Providing a portion of a daily schedule, students could pick out the key information–when both ferries are in the dock together, what the interval is between departure times.

    I think the entry point here is fairly low–to raise the ceiling:

    One ferry can return to Hyannis, I mean Ferryport, in 19 minutes. The other can return in 24 minutes. Each ferry takes four minutes to unload and load. Write a schedule that doesn’t require both ferries to be loading and unloading at the same time, while running 29 ferries to place one and 24 ferries to place two over twelve hours. Is it possible? Is it possible while keeping the intervals between ferries constant?

  7. I agree with Dylan. Let’s not tell the students that the ferries will return at the same time. Ask if it will ever happen.

  8. What if we simply put the times for the trips for each ferry on the map (ie. Red 25 min. and Blue 30 min.) and state that after they leave the port at the beginning of the day only one ferry can be at the port at a time. Then ask, “Is this reasonable?”

    This creates a discussion and brainstorming for reasons why the two ferries cannot be at the port at the same time. It also asks students to look at the “reasonableness” of a situation without giving them a specific time (ie. the original 300 minutes).

    As an extension, we could ask whether or not the ferries being at the port at the same time occurs only once in a day or multiple times.

  9. Ferryport has a terminal that can only service one ferry at a time, but there are two ferries that need to operate out of it. Cathy, who is in charge of scheduling, thinks that she has come up with a solution: “one ferry could return every 25 minutes and one ferry could be on a 30 minute return schedule”.
    Denise agrees, but Kelly, who is second in charge doesn’t believe it will work.
    Can you help them solve the dilemma ?

  10. I would specify the starting time for the ferry service, say 6AM. So “starting at 6am, two ferry boats leave…..”.

    Question: What time does the last ferry leave from Ferryport that would get you to Lobster Island in time for a 12:15 appointment?

    I see some good scheduling problem solving issues here. Converting to hours from minutes and vice-versa.

  11. Regarding making a new assignment instead of doing a makeover, Dan, I think Luke, Kate, and I were sort of doing that with the last one. The original task was just to see that periodic functions exist. Extending that to bring triangle trig into it is a big extension, but it is how I plan to use it.

  12. Follow up question: when should a passenger leave island A to go to island B throught ferry port and have the shortest wait time?

  13. My big question on this one is, “Why don’t they label the islands as Martha’s Vineyard and Nantucket (and ‘Ferryport’ as Hyannis), which they clearly are? Are those names trademarked somehow?”

  14. I teach upper elementary, so my modification is simpler than most of yours. Instead of using “300 minutes”, I’d set a start time, such as 8:00am, and then use “5 hours”, and ask students if the ferryboats would be in port at the same time and ask for the answer as clock time(s), rather than elapsed time(s).

  15. @courtney How does problem get designed so that it begs that question from the students? I’m not sure yet, I need more time to think it through.

  16. I find myself wondering what the underlying principle of engagement or learning is with a lot of different suggestions here. For instance, Terry Jacks seems to have taken the original task and added some kind of argument to it. Is the underlying theory that students will be more engaged, or that they’ll do more productive work, if they perceive themselves to be settling a debate? Same question to everybody else.

  17. Elaborating on William’s idea:

    Create a video with a split screen. The image on the left side appears every n seconds. The image on the right appears every m seconds. The images should be chosen so that a natural question is: when will the two images appear at the same time, creating a ‘complete’ picture or message.

    Alternatively, you could have a split screen with two Dan’s doing something periodically (maybe trying to touch the ‘other’ Dan). The question is, when will they touch?

  18. @Burton 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.

  19. How about saying that both captains take off as soon as they arrive to work at 6am and generally like to take their lunch hour together. Will that be possible if one of them gets stuck in traffic on the way to work?

  20. I am drawn to the same task as Andy Brown, but making it the main question: how long does it take to get from island A to island B?

    They have to make assumptions about transit time vs how long is spent at the docks, but roughly 1/2 the round trip time seems reasonable. The students can come to the idea of whether or not they have to wait at Ferryport and how much time that adds.

  21. jimPa21:

    @Burton 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.

    Right there with you. I gave that a shot awhile ago. I don’t like how it turned out, I think, because the answer comes along too quickly. I like that the boats are on 25 and 30 minute intervals, which pushes the answer out to 150 minutes.

  22. George Bigham

    August 9, 2013 - 9:31 am -

    Depending on where students are from, ferries might be very familiar or totally foreign. Since the core of the problem is about common multiples I might change the objects to something students are more familiar with, such as local subway trains or delivery trucks from a local business or anything familiar to the specific group of students.

    Two BART trains leave 16th street station at 6:00 am, one going to Richmond and returning every 25 min, one going to Concord and returning every 30 min. If they both return at the same time they will crash. Will they crash? If so at what time?

    I guess settling a debate or averting disasters might provide more motivation and engagement, but that’s just my hunch. If the math is settling a debate that students actually have with each other or others then I’m certain it will increases engagement.

  23. @George, I like the collision suggestion.

    Perhaps an intersection. Every 30 seconds a car passes through going N. Every 25 seconds a car/dog/ball crosses going E. Pretty easy to animate on geogebra.

  24. 2 kids swinging in swings with different periods. If that’s not possible, motivate with swings and switch to pendula. Ask the same question: when will they be at the start at the same time?

    Then show first 4 min if this vi hart video


    And get out 2 tuning forks. Tell them the frequency of each fork and ask if they will sound good or bad together. Repeat for more pairs of forks

  25. George beat me to the idea of modeling this situation with public transportation.

    I live on the north side of Chicago near the Howard Street station on the CTA, which serves as a hub for different lines connecting nearby Evanston with downtown Chicago. The Purple Line runs north into Evanston, the Red Line runs south into downtown.

    Also, there’s a plaza just outside the station with a Dunkin’ Donuts.

    So, suppose that northbound Purple Line trains depart the station every 12 minutes, and southbound Red Line trains depart the station every 15 minutes. You see a Purple Line train leaving the station just as you arrive. You MUST catch the next Red Line train in order to get to work on time. You want a coffee from Dunkin’ Donuts, and it will take 5 minutes to get a coffee and get onto the station platform.

    Do you have time to get a coffee?

    (The time at which trains start leaving at these intervals is purposefully omitted. Also, I just kinda picked the numbers out of my head because I’m in a bit of a hurry at the moment.)

  26. I think there is an assumption that students will do more thoughtful work if there is a debate to be settled. In my experience, these types of questions can lead students to consider all sides, more carefully explain or justify their positions, and have more natural conversations about math with peers. I’m not sure if they are always more engaging but the promise of a good debate moderated by an enthusiastic and respectful teacher can draw many students in.

  27. The first sentence in the original task creates some confusion. Other than that it is an ok question. The context is not super compelling, but helps to clarify the mathematical task.

    Show a few passes of this car crash simulation, but stop before there is a crash:


    Time intervals of 8 & 11 make for some close calls. Hopefully some disagreement about if or when there will be a crash. Repeat with time intervals that have a common denominator (like 10 & 25). Hopefully some think the first crash will be at 250 (10 x 25).

    You could ask the same question again. But you could also ask things like:

    1) Choose some time intervals (less than 20 sec) that create frequent crashes. Choose some time intervals (less than 20 sec) that create few crashes.

    2) Set the time intervals so that the third crash happens after 24 seconds.

    3) Can you set the time intervals so that the time between crashes isn’t constant? (no you can’t)

    4) Set the time intervals so that every 3rd car from the right crashes and every 5th car from the top crashes.

    5) Can you set the time intervals so that every 4th car from the right crashes and every 8th car from the top crashes (no you can’t).

  28. I’ve just watched your talk on TED. I am really impressed by it! Thanks for sharing such great approach!