## Reflection: Student Ownership Introduction to Trigonometry - Section 2: Discussion Phase

In the first year that I used this activity, soon after attaching the names sincos, and tan to their ratios, a young man remarked, “Wait a minute.  I’ve heard all this stuff about sine waves.  What’s wave-like about this?”  This was a great question – one that I had not anticipated – and was one that let me extend the lesson far beyond what I had first imagined.

I asked the students to enter  on their graphing calculators.  Then I had them go to the “WINDOW” feature of the calculator.  I am a huge fan of using this feature; I think that, when students are asked to think about and adjust their windows, they are really thinking deeply about the function, how the function behaves and why, and this goes a long way toward preparing students for future math courses.  (For this reason, I absolutely hate the ZOOM FIT feature on the TI calculators, and never introduce my students to it!)

With their windows in mind, I asked the students about our minimum and maximum x values – where do our angle values begin and end? We set our Xmin and Xmax values to go from 0 to 90.  Then I asked about our values for sine – “What is the range of our sine values?  What should our Ymin and Ymax be?”  We had discussed trends previously (that sine increased from 0 to 1, while cosine decreased from 1 to 0) so the students decided our Ymin and Ymax should be 0 and 1.

When the students pressed GRAPH, I was surprised to see that they got quite excited by the curve.  (This made me realize that they had really never seen much beyond a line and a parabola.)  However the excitement wore off quite soon, and the same young man remarked, “It still doesn’t look much like a wave.”  So I asked the students to return to their windows, change their Xmax to 180, and graph the function again.  When the excitement wore off, I asked them to explain to me what we had just accomplished by changing our Xmax to 180.  After a moment or two of thought, they realized that, by changing from 90 to 180, we had just included obtuse angles.  Then I asked them to again change their Xmax, this time to 360.  And this time they got the full “wave” effect!  I asked about the x-values for  –  “Wait a minute. Can we have an angle that is bigger than 180o?”  My students had done a project in which they had investigated convex polygons, so they remembered measuring angles larger than 180o and were not bothered by the addition of these angles to their curve.

Finally I asked what we had just accomplished by going from 0 to 360 degrees.  There was a lengthy silence, and finally a young lady (not even one of the top students!) answered, “We went around a whole circle.”  I answered, “Awesome! And what would happen if we changed our X-max to 720 or 1080?” The students responded that we would just keep traveling around the circle.  I suspect that, in that moment at least, my freshmen Geometry students “got” trigonometry a lot better than some of my Precalculus students!

Student Ownership: An Unanticipated Extension

# Introduction to Trigonometry

Unit 7: Right Triangle Trigonometry
Lesson 1 of 3

## Big Idea: Students learn about right triangle trigonometry by creating similar triangles and producing their own trig tables.

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Standards:
Subject(s):
Math, Geometry, Trigonometry, Right Triangle Trigonometry, trig, 10th grade
75 minutes

### Beth Menzie

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