Starting Right Triangle Trigonometry

Today we started the unit on right triangle trigonometry. I began by asking students what trigonometry meant to them. Students identified the prefix tri and said it had something to do with three. I broke the word down tri | gonometry and wrote 3 under the tri. Some students connected gon with polygons and said it had to do with a figure or sides. I separated the word further tri | gon | ometry and wrote sides under the gon. Another student said the metry had to do with measure. I wrote measurement under ometry. I then said that we were using triangles as a way to measure things and would expand on how similar triangles can be used.

With that, we started the investigation using Are Relationships Predictable? package. Of course, this investigation required students to draw a 30^o, 60^o, 90^o triangle, measure its sides, and then calculate ratios. As usual, some students still struggled with using a protractor to draw the appropriate angles and others had difficulty measuring sides lengths.

Once students had triangles drawn, I had a lot of questions, not unexpectedly, about which side was an opposite side and which was an adjacent side. We went through these using an example triangle I drew on the board. Many students were okay at this point, but I found a few that had inverted the ratio, calculating hypotenuse / adjacent rather than adjacent / hypotenuse, for example. I also encountered a couple of students that were either mixing angles and side measurements together or working exclusively with angle measurements as they tried to calculate their ratios.

We did reach a point as a class where students could see that the ratios, within given measurement error, were the same, regardless of the size of the triangle drawn. We then proceeded to the next piece where two triangles, each with a missing side length, were given. Students needed to calculate the missing side length and then calculate out given side ratios.

Students seemed baffled about how to find the missing side length. I reminded the class that these were right triangles, which prompted several to use the Pythagorean theorem to find the missing distance. As other students caught onto this idea, I walked around to check their work. The given side lengths were for one leg and the hypotenuse. What I saw was students calculate the missing side length as if it were the hypotenuse. As a result, they calculated a leg length that was longer than the hypotenuse. None of the students recognized this error.

I then had to remind the class that the hypotenuse was the longest side in a right triangle. If they calculated the missing side length and it was longer than the hypotenuse they did something wrong.

One student had calculated the missing side length as 4 when the hypotenuse was 6 and the other leg was 3. I asked her to recheck her calculations. At this point, class was ending. I asked the class to complete this second page. I also pointed out that the side length they calculated should be greater than 4 and less than 6.

I'll see how they did with this piece and proceed on with some work with right triangles and naming the ratios that the class has been calculating.