Bringing the Problem to Physical Reality: Trigonometry

This post is part of a series of lessons about immersing students in an environment so that they can ask questions of their “surroundings” instead of their teacher.  See my other posts and an introduction.

The night before I wanted to teach trigonometry to my students I gave them the following diagram.

This diagram took up an entire side of a piece of paper.  On the other side of the sheet I gave a huge, 4-column, 37-row chart.  The first four rows are below.

Angle x-coordinate y-coordinate ratio of y/x
0 degrees
10 degrees
20 degrees

and so on, down to 360 degrees.

I included the instructions below.

“This is a diagram showing a circle with a radius of one.  There is a 40-degree angle drawn on top of it.  Please check now to see that the angle ends at (approximately) the point (0.76, 0.64).  We can approximate to the hundredths place from this diagram.

“On the opposite side of this sheet you will see a table with rows for every angle between 0 and 360 degrees, in increments of ten degrees.  In the row for 40 degrees, please enter 0.76 under the x-coordinate, because 0.76 is the x-coordinate of the end of the 40-degree angle on the diagram.  Please enter 0.64 under the y-coordinate, because 0.64 is the y-coordinate of the end of the 40-degree angle.  Finally, please enter 0.84 under “ratio of y/x,” because 0.64/0.76 is approximately 0.84.

“Your homework is to fill out this chart completely, for all 35 of the other angles listed.  The coordinates you get will vary as you choose other angles.  You will need a protractor to draw angles – please actually draw the angles you need to measure, and do not attempt to estimate angles without a protractor.  If you find a logical shortcut, you may use it.”

Of course, the “x-coordinate” column is what we call cos(\theta), the “y-coordinate” column is sin(\theta), and the “ratio” column is tan(\theta), but the students don’t need those names yet, so I didn’t introduce them.  I wanted to meet the students at their own fluency level, and they are perfectly comfortable with x- and y-coordinates, protractors, angles, and ratios (this is a precalculus class).

Why would I start with “there’s a function called sin(x) that gives you the y-coordinate of a point at the end of an angle inside a unit circle?”  We focused on the calculation of cosine and sine before we had the names, and calculated it many times.  In our discussion of our values, the students were getting frustrated with saying “the x-coordinate of the point at the end of the angle where it intersects the circle,” and so I innocently mentioned that this is what mathematicians call “cos(x).”  Pretty shorter, huh?  They dug it.  Some of said, “oh, this is how you calculate cosines?”

The work they did at home varied.  The clever idea here is that you can actually fill out the entire chart with only nine measurements by using various symmetries. I would say that most students made 18 measurements – the x- and y-coordinates in the first quadrants – and then filled out the rest of the chart.  Some students came in with fully half of the measurements made but their chart incomplete, giving variations of “I can’t be bothered to do this kind of grunt work” (and I don’t grade homework).  Some students made all 70 measurements.  All of my students made some attempt at the homework – I think that somehow it’s kind of a fun activity!

This simple circle, with its highly-structured instructions and almost no student initiative, has the benefits of an immersive physical environment.  The students measure values directly off of the circle, and when they see the symmetries involved they can check them for themselves.  During class, when we’re talking about various properties of cosine and sine, they can check values directly.  When I eventually ask about 45 degrees and 15 degrees, they are fluent in this calculation and can easily adjust what they are doing.  When I ask about circles with different radii, they have intuitive guesses (some right and some wrong: great!) about what will happen to the x-coordinates and y-coordinates.  They are so comfortable with the measurements at this point that they can see the logic in using 2\cdot cos(x) instead of making a new cosine function for a circle of radius two.

Learning happens when you’re comfortable enough with a situation that you can experiment.  You have to be able to change the initial conditions a little bit, see what happens, and use that to find a pattern, to form a generalization in your mind.  This circle gives students something they can experiment with in a way that triangles don’t.  While they could certainly construct right triangles, it would be too hard for them, and take too long – they would be able to quickly try different angles, or to see all of the angles at once like they can with the circle.  The symmetry is hidden with the triangles, but it’s glaring with the circle.  This circle, even though it’s just a circle on a piece of paper, is a whole environment which the students can explore.  And, importantly, they are already good at the skills they need to explore it.  They can find new information with old skills.  That’s what this series is about!

Update 2/1/2011: Changed evaluation of sin(40) from 0.72 to 0.64.  I’m that powerful.

18 thoughts on “Bringing the Problem to Physical Reality: Trigonometry”

  1. Intriguing! I would’ve written that off as feeling like busy work, but now I’m not so sure.

    Had your kids seen sin, cos, and tan earlier in a right-triangle context? Our curriculum here has the right-triangle stuff come up a few years before the unit circle and the full real-value function. When I hit this stuff I focused on how the unit-circle definition has the right-triangle in it but lets us find values beyond 0-90 degrees.

  2. “Learning happens when you’re comfortable enough with a situation that you can experiment.”

    I like this idea, Riley. I really do. It’s much better than the traditional approach to teaching the unit circle. Here’s some stuff….go memorize it. The assignment itself may be memorable enough for SOME students to remember the “why” behind this integral part of trig. With all of that said, my question for you is this….

    Why would/should a student care to do all of this angle measuring and coordinate writing? What’s the hook? Because Mr. Lark told me to do it?

    Disclaimer: I’m only asking these questions, because we’re in this blogging/commenting thing to push each other, not because I have the answers or could have come up with anything better on my own. 🙂

    1. I like the questions being raised about this assignment – where’s the line between “a lot of busy work” and “a lot of work that’s necessary for understanding?”

      Please critique this technique: in this case the motivation was “Riley is giving out a lot of work. Yes, it is actually required.” I dunno, I made it sound bad ass. I don’t normally give a lot of homework (I usually don’t give any homework, actually), and for this assignment I played up how much work it was going to be. I tried really hard to make it sound like a big project (in reality it can be completed in ten minutes). And… it worked! There was no hook. I didn’t have a driving question. I just asked for all of those measurements and implied in my diction that it would be an impressive feat to do all of them, and they ate it up. They all like me and want to please me, and they trust me not to waste their time. Everything I do is fun or interesting in the end. Sorry, I know that’s not useful – I guess my technique could only be described as “motivation by sufficient showmanship and social capital.” Don’t disregard the technique of using a grave, urgent voice when describing a sort of boring assignment!

      Now, a couple of the students did not finish all of the measurements, and they did not find the shortcuts that would enable them to skip all the hard work. But those students were my favorite case, I think: they gave up on the assignment at home (rightfully so, perhaps, I mean who wants to do 70 measurements for no reason), but then in class could see a significant shortcut that would make the work 8 times faster. The shortcuts seemed useful, even though they were shortcuts for a meaningless exercise.

      Do you see why in this case it didn’t really matter that there was no motivation? It was sort of OK if some kids didn’t do it. We did it in class the next day, and saving time became the motivation. Which is a good motivation, even if you’re saving time doing something you don’t care about. It’s just interesting! The fact that this was a real structure (the circle) reinforced this idea – they could do real measurements and screw around with it, collecting as much data as they wanted.

      The real value of this approach was in their understanding of sines and cosines, later, and the ease with which they could translate between circles and triangles. This activity helped make them fluent in these trig functions.

      In many ways this lesson was less of an environment than my other posts on the subject, but it has all of the important qualities (see intro page), so it still fits.

      Thanks for the critiques – keep them coming! I’d love to hear any ideas for making this compelling in more ways than just “Riley got serious and he has a lot of social weight around here so we should really do it.”

      1. I think one major difference between this and a busy work assignment, is that this type organizied mathematical data collection is something that mathematicians actually do. When trying to figure out a pattern, one often uses a brute force approach to collect some more infomation so it becomes easier to see the pattern.

  3. I like this a lot. It is so simple. I tried a similar idea with one advanced group of kids that I have, except instead they made geometric constructions of equilateral triangles of varying sizes and came up with the different ratios of side lengths and realized that there ratios seemed to be the same no matter how big or small their triangle was, as long as the angle was the same. It was a good idea, but took too much guidance to be really effective or to have them make any nice discoveries. Plus, the unit circle is more of a struggle for comprehension, so maybe I will try this next year.

    1. One year I gave this circle out without the grid lines, and had the kids construct triangles themselves to find the coordinates. My thinking was, “omg, this will make it soooo easy to translate between circles and triangles later.” There may be a way to highlight the triangles that you’re reading off of the Cartesian plane right in this assignment, but I wanted to make it as simple as possible. I’ll be interested to hear about what you do next year!

        1. You know, that could be a great idea. The directions could get much shorter, accuracy would skyrocket, etc.

          My biggest worry would be about losing the measuring. I like these “reality” problems because the students are given essentially infinite information and have to query the environment (this circle) to get the specific information they need.

          Also, I can’t defend this at all, because it makes no rational sense, but if the lines were already there and I just had the kids read the coordinates off of their ends it would seem more like busy work to me. Hilarious, since it’s LESS work, and the only work I’d lose is measuring 36 angles with a protractor…

          What do you think?

          1. I like them measuring the angles, because their understanding of what an angle even is, is dicey. If they were pre-printed, many of them would not even think to wonder, why is she saying this radius is 30 degrees, and that one is 100 degrees – they would just be reporting on coordinates to get it done. I just also know that protractor use tends to be error-prone. It’s not a bad thing to make an error and then reconcile it though.

  4. Great idea :-)) May want to check the sin of 40. It’s .64 and not .72. Interestingly that’s actually a neat idea to have students verify first something is true (when it’s not) and watch how quickly they’ll be explaining why it’s not true.

    1. Ah, thanks for catching that, Kris. I did not make that error on purpose, but genuine errors are great sources of learning for transparent teachers. If I were going to make the error on purpose I’d provide at least one fully-correct example first 😉

  5. I wonder if it would make sense to have some kids with a circle of radius 1, but give a couple kids a circle of radius 2 and radius 3.

    Then the class could have a discussion about how the y/x stays the same regardless of the size of the circle. You said you had that discussion via intuition, but it might be nice to have someone having done the grunt work to really show it. (Then the question can be WHY?!)

    1. Good idea. Then you’d also have data for the general soh cah toa formulas, instead of just hypotenuse = 1!

      All this time later, the only thing I don’t like about this lesson is how difficult it is to operate the ruler correctly and accurately. I do like kids to use their hands, but maybe a geogebra applet that pumps out numbers automatically and perfectly would be better…

  6. I wonder if more students would have done the work at home if you had given two sheets of paper – one with the circle, one with the table. With only one sheet of paper student would have been constantly flipping over the page to measure and record data – which might get tedious for 70 measurements.

    Thanks for the idea btw – will be using this year!

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