A few years ago I started introducing trigonometry using circles. It started in Algebra 2 and moved down into Geometry. By basing all trig off the unit circle, we make an easy transition from Geometry trig (finding missing sides and angles of triangles) to Algebra 2 trig (sine and cosine functions and transformations). After a couple of years, I'm never going back.
| We start by watching this video: | |
Q: If you were the person taking this video, what would the graph of your height look like? Sketch a graph.
I get all sorts of mountain looking graphs or semi-circles. It's what I want. We are now going to get some intellectual dissonance. After students committed to a graph, we break out the hula hoops. I borrow them for a couple of days from our PE teacher. As an aside, I borrow stuff from the PE teacher and the science teachers all the time. Believe it or not, they are excited that your doing fun stuff in math class. They want you to borrow their stuff. Not kidding.
I get all sorts of mountain looking graphs or semi-circles. It's what I want. We are now going to get some intellectual dissonance. After students committed to a graph, we break out the hula hoops. I borrow them for a couple of days from our PE teacher. As an aside, I borrow stuff from the PE teacher and the science teachers all the time. Believe it or not, they are excited that your doing fun stuff in math class. They want you to borrow their stuff. Not kidding.
We give a two-minute talk through regarding how we are going to collect data. Something like: Put some tape on the hula hoops and put a dot on the side of the hula hoop. This dot is you. Throughout the experiment, you will always measure the height of the dot. You will roll the hula hoop 10 cm and then measure the height of the dot. In cm. When you get to the end of your meter stick, keep your fingers on the bottom point and slide it back to the beginning of the meter stick. Do this for one complete revolution of the hula hoop/ferris wheel.
Most groups finish with the data gathering in 30 minutes or less.
As another aside, I have students start at the bottom, like they are loading onto a Ferris wheel. Technically, this will look like a cosine function with an a-value of -1. However, we're measuring height, which is a sine function. Whatever. In four years of doing this, I've never had it throw students off the scent of the math we're doing.
Most groups finish with the data gathering in 30 minutes or less.
As another aside, I have students start at the bottom, like they are loading onto a Ferris wheel. Technically, this will look like a cosine function with an a-value of -1. However, we're measuring height, which is a sine function. Whatever. In four years of doing this, I've never had it throw students off the scent of the math we're doing.
| Student Handout: 
| Basic Teacher stuff, including some links:
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Here's some of the resulting graphs. I take pictures of some of the student work and we have a conversation about it.
Q: What do you like or dislike about this graph?
Q: What do you like or dislike about this graph?
Students tend to not like the "lumpy" portions of the graph.
I put the following graph on the screen in response. Students like this one much better. As if almost on cue, some student always says we should connect the points. Then we get to have a good discussion about whether the graph should be continuous or discrete.
I put the following graph on the screen in response. Students like this one much better. As if almost on cue, some student always says we should connect the points. Then we get to have a good discussion about whether the graph should be continuous or discrete.
Students generally settle on the following as their "ideal" graph. We then talk about rate of change in table groups. How do you talk about rate of change with this graph? After flailing about wildly, we usually are able to give a decent explanation about whey when the point we are measuring is at the bottom or the top of the circle it doesn't have much vertical change in distance as opposed to when it is on the sides.
Day 1 usually ends here and students finish their 7 questions on the handout. This year, Day 1 was half eaten by questions over a previous assignment and we gave up half of Day 2 to finish gathering data, graphing, and having this discussion.
Day 2 begins with a discussion about the unit circle.
The following is the definition of "sine" from Wolphram Alpha. Notice the first part of the definition.
The following is the definition of "sine" from Wolphram Alpha. Notice the first part of the definition.
Forget the "common schoolbook definition". As a class, we define 'sine' as the vertical distance of a point on the circle to the x-axis and the 'cosine' as the horizontal distance of a point on the circle to the y-axis. It's *expletive* amazing! We talk about what this distance looks like when we move the point around the circle. Here's some of the web-based visualizations I use:
Sine and Cosine as vertical and horizontal distances in the unit circle
http://www.analyzemath.com/unitcircle/unit_circle_applet.html
https://www.mathsisfun.com/algebra/trig-interactive-unit-circle.html
https://www.desmos.com/calculator/v7x6br3w6a
A visualization of Sine and Cosine in the Unit Circle:
(If you want to moving one, click on the link below)
http://i.stack.imgur.com/p8O4P.gif
Sine and Cosine as vertical and horizontal distances in the unit circle
http://www.analyzemath.com/unitcircle/unit_circle_applet.html
https://www.mathsisfun.com/algebra/trig-interactive-unit-circle.html
https://www.desmos.com/calculator/v7x6br3w6a
A visualization of Sine and Cosine in the Unit Circle:
(If you want to moving one, click on the link below)
http://i.stack.imgur.com/p8O4P.gif
This is THE move that sets up everything! It sets up talking about sine and cosine as functions. It sets up a point on the unit circle being (cos, sin). It sets up the Law of Cosines not being some weird abnormality that's never really understood by students. It sets up that what we commonly refer to as trig ratios are lengths of lines in the unit circle. It also sets up any triangle with a hypotenuse other than 1 being a dilation of a triangle on the unit circle, with a scale factor of the length of the hypotenuse. Of course we end up deriving the shortcut methods for finding sine, cosine and tangent without having to go back to the unit circle every time. However, the fact that these are real lengths and not just abstractions goes a long way with my concrete learners. Sine is a vertical distance. Cosine is a horizontal distance. Tangent is the distance from the tangent to the point on the circle to the x-axis. Which is why tangent of 90 degrees and 270 degrees is undefined.
This also helps discuss Cotangent, Secant, and Cosecant when they come up later in the year. They are all actual lengths in the unit circle!
This also helps discuss Cotangent, Secant, and Cosecant when they come up later in the year. They are all actual lengths in the unit circle!
| 17c_exploring_trig_functions.docx |
If you're teaching Algebra 2, after this throw in a little Cosine Ferris Wheel from MARS and you've got yourself a decent start to a trig unit.
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