**The Task**

This is another task from MathWorks 10.

**What Dave Major And I Did**

I don’t have any huge beef with this task. I like that students get to pick their own route. Those kind of self-determined moments are tough to come by in math class. Here, the buoys are pre-determined but students get to make their own path around them. So we get the motivation that comes with self-determination but feedback isn’t the chore it would be if students got to choose the placement of the buoys also.

**Establish a need for the bearing format**. We’re going to take a cue from the research of Harel, et al. Rather than just introducing the bearing format as the next new thing we’re doing in math class, we’ll put students in a position to see why it’s necessary.

**Offer an incentive for more practice.** We’re going to make it really easy and enticing for students to try different routes, learning more about degree measure and bearings with each new route they try.

**Raise the ceiling on the task.** Rather than moving along to another context and another question, let’s stay right here in this one and do more.

Show this image.

Ask students to write down some instructions that tell the boat’s blind skipper how to navigate around the buoys and return to its original position. Don’t let this go on all that long. Whenever we’d like students to learn new vocabulary or notation, it’s useful for them to experience what it’s like to communicate *without* that vocabulary and notation, if only briefly.

Write the notation “50 miles at 60Â° South of East” on the board and ask them what they think it means. After some brief theorizing, send them to this website where they can test out their theories.

Then they can create a series of bearings that carry them around the buoys.

We’ve timed the boat’s path. But this isn’t the kind of timing you find on timed multiplication worksheets that freaks kids out for no discernible benefit. The timer here gives students feedback on their routes. The feedback is also easy to remediate and change. Feel free to try again and do better than your previous time. Or, if you’re feeling competitive, perhaps you want to try for the best time in class. (Or the *worst* time. That isn’t simple.)

Move on to the next page where we give you a series of bearings and ask you where the boat will come to rest. I find it tough to get inside 10 miles worth of error here.

If we wanted to draw this out even further, we might have:

- featured multiple courses.
- let students create their own courses and challenge their classmates.

**What You Did**

- FrÃ©dÃ©ric Ouellet animated the boat in Geogebra. As with the work of a lot of expert Geogebraists, it seems as though the interesting mathematics is in
*making*the animations or the sliders and has been done by*the teacher*, not the student. - L Hodge offers another Geogebra applet, one that puts more of the math onto the student.
- Lindsay also asks her students to describe the path of the boat without yet knowing the vocabulary.

**2013 Aug 21**. It strikes me that some useful questions for provoking an understanding of degree measure would include:

- What do you think “-20Â° North of East” means? Is there another way to write it?
- What do you think “120Â° South of West” means? Is there another way to write it?

## 12 Comments

## Kyle Pearce

August 19, 2013 - 9:59 am -Awesome work, Dan & Dave!

Really like the boatrace testing website where students can really put their new terminology into practice, rather than let it fade away faster than a Justin Bieber song.

This is my first opportunity I’ve had to check out “Makeover Mondays.” I’ve gotta bust my butt and put something together to add to the community!

Keep up the great work, guys!

## Colin

August 19, 2013 - 2:28 pm -If you want to step back, then ask why buoys exist. Show a topographical map and 1) ask how you could indicate the safest route (buoys), and 2) ask where should they be (lat, long). Then you can ask the bearings a ship would use to navigate this safe channel from buoy to buoy if they couldn’t see the next one (e.g., fog). How long would it take to navigate the channel?

Add in wind and you more than double the complexity of the problem.

How does the, now, ubiquitous GPS affect this problem?

## Max Ray (@maxmathforum)

August 20, 2013 - 9:59 am -Okay folks, how did you get your boat race times down to under 3 minutes? I’m accepting hints, and even answers, at this point. Thanks!

## l hodge

August 20, 2013 - 1:12 pm -@Max, entering a big angle will use up a huge amount of time. Not a fan of allowing these super low times for this particular simulator â€“ very misleading if I am thinking about it correctly.

I really like how the angle is expanded out as the next bearing is set (maybe slow it down a bit). This activity is a nice way to introduce and get a little practice with bearing notation. Nice clean look as well.

Coming to an understanding that â€œ20 degrees South of Eastâ€ means to going mostly to the right and a little bit down is vocabulary or language learning, not math. The proportional reasoning with the 10 mile map key and the angle estimation introduces some relatively routine math (low floor). I would love to see a twist that leads to a mathematical puzzle or decision or strategizing – raise the ceiling as you say.

It is great that students get to pick their own routes. Why not make the optimal route less obvious in a straightforward way? Maybe the speed differs in different parts of the ocean due to currents/winds, so the shortest path is not always the quickest path. Give them an opportunity to do some reasonable analysis that will improve a route â€“ doesnâ€™t even matter if they find the optimal one.

I am completely sympathetic to the programming difficulties that may be involved in bringing in additional features. It is still a nice tool for working with the language of bearing notation and eyeballing distances and angles.

## William Carey

August 20, 2013 - 1:28 pm -> Coming to an understanding that â€œ20 degrees South of Eastâ€ means to going mostly to the right and a little bit down is vocabulary or language learning, not math.

Is there a difference between vocabulary and language learning and math? If so, what is it?

## l hodge

August 21, 2013 - 4:53 pm -@William,

Many students wouldnâ€™t be able say much if you asked them what they know about about canis lupus familaris. That doesnâ€™t mean they donâ€™t know anything about dogs. Learning words to describe dogs is not directly related to learning about dogs. Learning the meaning of notation or language that can be used in math is not always directly related to learning the underlying math. Too often what passes for math learning is actually language learning.

Suppose a student is able to follow these instructions: put a dot that is 2.5 units to the right and 4 units above the first dot. But cannot follow these: plot a point at (2.5, 4). This student needs to learn some language, not math, in order to get the second task done.

## William Carey

August 22, 2013 - 6:40 am -@Dan – any reason you and Dave removed the grid lines from the original problem? As a student, I might clamor for them by the time I’m trying to plot complex courses or figure out where the boat ends up.

@Ian – interesting. I’d agree that the student who can follow one of those sets of directions but not the other needs some linguistic work. What would the converse look like? Are there situations where a student grasps the language and vocabulary but not the underlying math? What would that look like for the same example?

## l hodge

August 22, 2013 - 1:19 pm -@William, The converse is quite common. An example might be a student that can graph y = 4x + 8, but not 2y = 4x + 8. They may see y = 4x + 8 as language meaning plot the y-intercept then go up 4 and over 1. They are not thinking of solutions to the equation as points.

Many short cuts are basically ways to think in terms of language instead of math.

## William Carey

August 22, 2013 - 1:28 pm -> An example might be a student that can graph y = 4x + 8, but not 2y = 4x + 8. They may see y = 4x + 8 as language meaning plot the y-intercept then go up 4 and over 1.

Is that also an example of a language problem? They have an incorrect understanding of what the notation for the equation *means*? (Genuinely curious about this; I often wonder whether math education couldn’t effectively pillage instructional techniques from the study of languages–inflected languages in particular.)

## Jake

August 28, 2013 - 6:38 am -This is a great tool for working with bearings and direction! Is there any way you could rewrite the page so that the directions are given in the other format, N20W (meaning 20 degrees west of north). That’s how I teach it since that’s how my book teaches it.

Thanks,

Jake

## Phil @liketeaching

August 29, 2013 - 9:21 am -I’ve been on holiday so I missed this MakeoverMonday, but weirdly enough, I’ve already had a go at doing it. I too went for a computer app, which can be found here: http://globalracing.overthefence.heliohost.org/

(Instructions on using the multiplayer aspect are on my blog here: http://liketeaching.blogspot.co.uk/2013/07/lesson-sketch-online-bearings-game-gets.html)

I really like how simple and uncluttered your version is, and I REALLY like the opening ‘blind sailor’ that builds a need for bearings.

From mine I prefer that you can watch you and your classmates racing against each other (increasing motivation to make a quicker time).

I also tried to get in ‘reverse’ bearings questions to raise the ceiling of the app, though I’m not entirely how successfully it worked.

I’m guessing bearings are written differently in the US to the UK (ours are a three digit integer angle measured clockwise from north).