I asked if we had similar numbers for the ratios and, for the most part, they said we did. I looked at them with a furrowed brow and said "But my triangles were not the same as yours. Why would we have the same ratios?" I was surprised (in a really awesome way) at how quickly someone said that all of our triangles for a particular angle were similar (reflecting on this, I should have given more wait time and had them talk about this in their groups). We explored that a little further to emphasize that each side length is being multiplied by the scale factor and that the scale factor divided by the scale factor is 1, hence the ratios will be the same.
Then I explained that they had created a trig table which could be used to find the side length or acute angle of a right triangle. I showed them a trig table for all whole numbers from 1° to 90° and told them that, back before there were computers, people had accurately created triangles with all of these angles and calculated the ratios.
Side note: Up until yesterday, I was going to have them use the trig table for all their trig work in cycle 1. But I changed my mind yesterday afternoon. I use trig tables in grade 10 applied very successfully. Many of those students have trouble with any solution that requires multiple steps, so giving them the trig table removes one layer of abstraction - they don't need to think about which trig ratio they are finding, just that it's opposite/hypotenuse, for example. However, my grade 10 academic students should all be able to work through that and I decided it might be doing them a disservice by waiting to introduce the names of the ratios.
Back to today's class. I then told them that all of these values had been programmed into their calculators <insert speech about needing a good scientific calculators>. Although it might have been fun to have student give the ratios names, we went with tradition instead.
We went over some basics to make sure everyone knew how to use their calculators and that none was in radian mode.
I really emphasized that they should use the brackets (parentheses to my American readers) on their calculators to help keep their answers as exact as possible. I showed them that for part (c), above, using 0.88 resulted in the answer being off by a full degree. Then we moved on to actually using trig.
They were doing really well up to this point, but got stuck on the next one.
I was so happy that no one said "cross multiply!". We looked back at what we had done to get rid of the fraction when there was a number in the denominator, and followed the same process. They got stuck again. Many wanted to subtract cos(40) from both sides. I wrote it on the whiteboard x * 0.6 = 30 (where 0.6 was my bad approximation of cos 40) and they still wanted to subtract. But when I rewrote that as 0.6x = 30 the light bulb went on and they said "We need to divide!". They don't seem to be looking for shortcuts and seem to be really buying in to understanding why we are doing what we are doing. I like that.
Here is today's homework set. We have a PD day tomorrow and Monday is Thanksgiving so my next blog post should be on Tuesday.
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