## Reflection: Developing a Conceptual Understanding Introduction to Trigonometry - Section 2: Discussion Phase

As my students were filling out their tables, I heard many remark, “I bet there’s a pattern here!”  I loved hearing this! And as we began to discuss the trends in their columns, a couple students remarked, “I remember this – isn’t opp/hyp the same thing as sine?”  So at this point, we did name the columns.  I asked if they were familiar with SOHCAHTOA and they all were, which probably helped to make the assigning of the trig function names to the columns go very smoothly.

After discussing general trends in the columns, I asked each group what value they calculated for the ratio of opp/hyp for 30o.  As each group reported their value, I began to see surprise register on a lot of faces.  During the measuring phase, the students were very aware of the fact that their circles had different radii and that each group had very different data. However, all the groups reported that they had gotten .5 (or very close to this) for the sine of 30o.  This led to one of those very long silent pregnant pauses (or textbook disequilibrium, as I have recently learned) when I could tell everyone was thinking.  After a few moments of silence, I asked, “Why, for 30 degrees, are all of your ratios for opposite to hypotenuse equal to one half?”  And then it seemed to click – several students blurted out, “The side opposite the 30o is half the hypotenuse!”  Next I asked everyone for their results for the tangent of 45o, and very quickly it was clear that everyone’s answer was very close to one.  Again I asked, “Why?” and, this time more quickly, students responded that, for 45o the triangle an isosceles right triangle – the opposite and the adjacent sides are the same!

I went around the room and asked for their trig values for 10o.  Again everyone’s values were quite close, and I asked, “Why?”  After another long moment of silence, a young man shouted out, “I get it.  The triangles are similar!  Angle-Angle Similarity Postulate!”  This was a very cool moment (yes, I’m a math nerd…) because the pieces all seemed to fall in place for the students.

A Great Discussion
Developing a Conceptual Understanding: A Great Discussion

# 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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