Right off the bat, let me say that I like trig. Or at least I like all the triangles and the easy applications for finding all sorts of lengths. Students in my classes generally like that too. What they find difficult, however, is translation of trigonometric functions. I mean, who looks at y = a*sin([p/(2pi][x-h])+k and says "ooooh I want to do that"? My students initially look at that and panic. You think I'm kidding.
This year I hope to change that--I'm stressing transformations for every family of functions we look at rather than limiting it to the chapter dealing with transformations of functions. Those of you with access to Desmos think that's easy. Without student devices I can tell you it's not. I'm also introducing trigonometry by embedding it in the unit circle. If all of the triangles we look at are always brought back to the circle, then all of a sudden the Law of Cosines isn't such a big deal. Shifting functions up, down, left, and right is less scary (I hope).
Anyway, that's all to say we starting looking at trigonometry today by playing with hula hoops.
This year I hope to change that--I'm stressing transformations for every family of functions we look at rather than limiting it to the chapter dealing with transformations of functions. Those of you with access to Desmos think that's easy. Without student devices I can tell you it's not. I'm also introducing trigonometry by embedding it in the unit circle. If all of the triangles we look at are always brought back to the circle, then all of a sudden the Law of Cosines isn't such a big deal. Shifting functions up, down, left, and right is less scary (I hope).
Anyway, that's all to say we starting looking at trigonometry today by playing with hula hoops.
We started the class talking about what students remembered about trigonometry--just to get the proverbial wheels turning. They started off slow but then started picking up steam. Then I threw a stick into the spokes: we're going to start talking about trigonometry by looking at circles. We watched the following video:
Me: If you were the person recording this video, what would the graph of your experience look like? Guess. Try to sketch it.
Student1: How do we graph that?
Student2: What are the variables?
Me: So you have to identify your independent and dependent variables? (I probably gave too much away there).
Student3: Time!
Student4: Height?
Student2: Are we assuming constant speed?
Me: Sure, I think we can make that assumption.
Students gave a lot of triangle-looking mountains as graphs. Brilliant.
Me: We're going to refine our ideas by making our own Ferris Wheel out of a hula hoop. *cue student rejoicing*
Student1: How do we graph that?
Student2: What are the variables?
Me: So you have to identify your independent and dependent variables? (I probably gave too much away there).
Student3: Time!
Student4: Height?
Student2: Are we assuming constant speed?
Me: Sure, I think we can make that assumption.
Students gave a lot of triangle-looking mountains as graphs. Brilliant.
Me: We're going to refine our ideas by making our own Ferris Wheel out of a hula hoop. *cue student rejoicing*
| I had 7 hula hoops so I had 7 groups. Easy. Students got their materials and started to play with the hula hoops. Really? We've been doing experiments all year and today you decide to revert to 3rd grade now? I felt like a broken record going around the room: "let's start working", "do you have a tape measure?", "what point are you measuring on your hula hoop?". In less than 5 minutes kids were mostly settled down and working well. | ||
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We ran out of time to formalize what I wanted to talk about with them. Finishing the questions on the lab/worksheet was homework. The next day we started class by having students show their graphs. Some student had graphs looking kind of like a bell curve and others had graphs looking like a sine curve. They quickly came to the conclusion that this difference was due to where people started measuring from--either halfway up or from the bottom.
Me: So if trigonometry is about triangles, where are the triangles in our circle?
Enter Geogebra and interactive websites:
Me: So if trigonometry is about triangles, where are the triangles in our circle?
Enter Geogebra and interactive websites:
| unit_circle_and_the_sine_curve.ggb |
These led to some great conversations about the geometric nature of sine, cosine, and tangent (not to mention cosecant, secant, and cotangent). Sine is the vertical distance of a point on the circle above the x-axis. Cosine is the horizontal distance of a point on the circle from the y-axis. Tangent is the length of the tangent through the point on the circle to the x-axis.
Then we talk about ratios, SohCahToa, and solving triangles using neat stuff like:
Then we talk about ratios, SohCahToa, and solving triangles using neat stuff like:
Introducing trig this way adds extra day to the crazy end of the year I-can't-believe-we-had-more-than-10-snow-days blitz to try to cover all the standards. What I hope is that it deepens student understanding and cuts down on the number of days we need to review these concepts.
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