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. I don’t quite understand the problem. Here is my attempt at a paraphrase. Is this right?

    The “true bearing” is the angle turning clockwise from north.

    Draw a straight line from the ship to the closest buoy. The bearing of that line is a true bearing.

    Draw a line from that buoy to the one to the far right. The bearing of that line is the second true bearing.

    Draw a line from that buoy to the ship. The bearing of that line is the third true bearing.

    The green triangle has zero effect on the problem.

  2. The problem reminded me of something called “vector racing” — can’t remember where I read about it. Have the student provide the bearing and speed in order to move the ship. Make it tougher by limiting how often or by how much the speed or direction can be changed. Here is a rough start:


    Don’t like the original question. What was the book?

    Am I supposed to trace this including the grid lines? Then draw three line segments. Tedious. Doubly tedious because not providing the segments makes for a lengthier and more convoluted passage to read.

    Nit-picky, but it says the “vertical lines point north”. The lines don’t have arrows on them. They are not “pointing” anywhere. And there is a compass on the page and a green arrow that indicate north. Also, what does it mean to go “just past” a buoy? Why not just say “go to the buoy”.

  3. @l hodge: The “just past” a buoy is part of what threw me for a loop when I read it.

    Thinking about this further, I don’t think the confusion with the text (it took me about five read-throughs to get it) is really worth it; I’d just put the arrows in showing the route. Not just the ELLs but all of my landlocked students would have trouble with this one.

    I’m guessing Dave Major has a cool interactive thing that makes all of this moot, anyway.

  4. This problem seems like a pseudocontext aimed at getting students to repeatedly compute bearings from a points on a grid to another point on that grid. There’s absolutely no modeling or abstraction here besides trying to unpack the incoherent directions. If the goal is drill, it’d be a lot more honest to give the students a bunch of numbered points on a grid and just tell them to compute a bunch of bearings.

    Right now, what the problem teaches students is that computing a bearing is difficult if the two points involved are disastrously underspecified. This, while true, is perhaps not so important as to teach it to young people. There’s time enough for that sort of disappointment when they’re a bit older.

    An interesting modeling path would be starting the kids out with a protractor and no coordinate grid. Straight up measuring and writing down angles. The next step up the ladder is probably the imposition of some sort of coordinate system to aid measuring. The next step up might be creating tables for and defining the inverse trigonometric functions (arctangents for everyone!) in terms of the results of the first couple of rungs. The top rung is probably the creation of a program that encapsulates the abstraction of computing a bearing from one point to another.

  5. @William, I don’t understand how this is pseudocontext. Computing bearings is something ship captains actually do. How are you defining the term?

  6. Why is the boat traveling around these red bouys? Why does it need to end up by the black marker? It seems rather arbitrary and contrived. Why should I (as the skipper) calculate bearings in this situation? Why can’t I just aim my ship at the first bouy, go around it, aim for the second, go around it, then head to the last spot? There’s no scale on this map, so the bouys might all be within sight.

  7. William Carey

    August 16, 2013 - 7:16 pm -

    > Computing bearings is something ship captains actually *do*.

    In principle that’s true. According to the problem, I’m a ship captain. Why am *I* computing bearings? Context that aims the student’s perplexity at the *context* instead of the *content* is a type of insidious pseudocontext. Here are the questions that popped into my head while I read that question:

    – If I want to end up at the black boat, why don’t I just sail directly at the black boat?
    – Why do I have to go clockwise?
    – What does “just past” mean?
    – Why is there a green triangle in the upper left of the map? Does it mean anything?
    – Where do I measure the position of my boat from? The front? The back? The middle? That *totally* matters for the answer.
    – Is recording the bearings different from writing them down? What do I record them in? My log book? My phone’s voice recorder?

    The last one might be directed at the content (though it’s really just about clarifying an ambiguous diagram). The rest are perplexity about the *context*.

    I think there’s something like the uncanny valley for context in math problems. Good context embeds the mathematics in a narrative that fosters perplexity. That’s what the three act videos do so well. Total absence of context can work too. Are there as many fractions as whole numbers is a *really* perplexing question without any context at all.

    My experience is that uncanny-valley-context is asking to be taken to task by my students. If the narrative is incoherent, they’ll figure out how and hammer away at that.

    I’d say that “just enough context to make the context itself the focus of the perplexity” is a third type of pseudo context in addition to the two you point out here:


    Does that make sense?

    (This is a salvageable pseudocontext. Actual ship captains do *measure* bearings and draw them on their charts, and we’ve made computer programs that compute them. There is interesting stuff going on there, but this problem doesn’t naturally lead to it.)

  8. I don’t have as much problem with the question, poorly worded as it may be, as I do with the technological limitations of the textbook which the question appears in. Placing a piece of paper over the text book will not under normal classroom lighting make a traceable image and thus you will have students holding a textbook up to the light with one hand and trying to trace the boats with the other which is silly. This is a question which would make a lot more sense if the students were able to write on it directly.

  9. My only idea for context help is to plan a boat race. Students could either be the race event planners–trying to pick a long/short, easy/hard, sharp-/gentle-turn course–or the races–trying to plan the most direct route around the race course.

    Putting a finish line somewhere else could allow some students to be the white boat while other are the black boat, allowing for different answers, at least for the first leg of the ‘race’.

  10. I am considering using something like this as an introduction to Vectors. Give the the students the image and explain a course that the boat needs to take (in words like the original problem) and then ask them how they could describe their exact course to someone else to follow. I would leave off the grid lines. My hope is they would then start measuring angles and a discussion could follow about convention of direction and polar coordinates vs. navigator method, etc.

  11. What about a cruise, …isn’t that exactly what they do, sail an circle? Identify points of interest and create the path to visit those points and have to port in (the red dots would be the “port” (I think I am a little off in terminology but I have never taken a cruise).

  12. This would be a cool competition between students if they could then actually race their boats: program the vectors into a game that moves the boat as directed by the student at a set speed with a timer. The fastest student boat wins.
    Any thoughts on what program to use? scratch? geogebra?