# Approximating Angles

Amidst the many things I do with Right Triangles, I added a bit last year to try an add an idea of what trends in trig ratios mean for angle values. Eventually we use the inverse buttons on the calculator, but it helps to add some process to the magic. Yes, the calculator can tell you, but how is the calculator coming to its conclusion?

First, have them crank out a trig table, 0-90, increments of 5:

A few will mention the patterns. I have introduced the Unit Circle at this point, so I've been hinting that trig is a pattern lovers dream come true. All the patterns! A lot find it interesting that sin and cos have the same values but in a different order.

Next I demonstrate how you'd use the table given a few dimensions of a right triangle. A good moment to discuss the relativity of the words "opposite" and "adjacent."

Random new addition, I gave them a challenge. Using a person (160-190cm) as the leg of a triangle, and with only a tape measure, can you plot out a triangle with a 20º, 50º, and 65º angle with the floor? I set them loose in the hallway to experiment and then had them determine how well they did.

I have them in groups of 5-6 all the time, so to ensure there was enough to do, they had to split their table into two teams.

Some interesting results. A lot of groups were able to get into the neighborhood pretty well. Some were consistently off (see: the Y2 45 40 43 and R2 51 57 48 squads up there). I overhead a lot of interesting strategies. Many correctly guessing that creating a 20º angle would require quite the triangle.

Now were they mindful of the table while doing this? Not really. Most used their own ideas of what 20º, 50º, and 65º looked like and used the table to verify the experiment. One intrepid group immediately sat on the floor and worked backwards from the table to determine everything in advance (W1 on the purple card). Another was similarly calculating, although didn't think about the work backwards part (G1 on the purple card).

For some added practice they tackled this the next day (PDF link):

A few weeks ago we used the tape measures for a linear speed challenge. Fun to see the "today's going to be GOOD" reactions when they saw them set out again.

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