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|
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 , the “y-coordinate” column is , and the “ratio” column is , 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 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.