Introducing Sine, Cosine, and Tangent

Today's class started off a bit weird as a student got sick in class. We were working out in the hallway before finally switching to another classroom.

I started students by comparing their work from the night before. Most students had completed the work successfully and were getting comfortable with the concept of opposite and adjacent sides. I had students continue to the third page of work in Are Relationships Predictable? While there were some questions, students were getting more comfortable with using the Pythagorean theorem to find the length of either a missing leg or the missing hypotenuse length. They also were comfortable calculating the side ratios for the given angle.

The next page started naming the ratios. I told students that the ratios that they had been calculating actually had names. I went through the names and wrote the corresponding side ratios underneath the name. I also drew a right triangle on the board and used two different colors to indicate which sides were adjacent and opposite based upon the angle being used. Some students readily connected that they were still calculating the same ratios, we were just shortening the reference to the ratios by naming them.

The question came up about how to round or record an answer such as square root of 48. I told students that for standardized tests, such as the SAT or ACT, they would be expected to simplify the radical. We briefly revisited how to factor and simplify radicals. Most students said they didn't remember this, until I mentioned factor trees, at which point they all remembered simplifying radicals. I'll practice and revisit this periodically as the class gets closer to taking the PSAT in April.

One thing I was pleased with was that some students started to see the relationship that the sine of one angle was the same as the cosine of the other angle. We didn't discuss this but it is a noticing that can be built upon and used to discuss why this relationship exists.

Students continued to work through these problems. The remaining problems were assigned as homework for those that didn't finish the work in class.

The geometry team had a meeting this morning to discuss where we were and where we were going. One part of the discussion centered around covering the Pythagorean theorem. Now, my class has been using the Pythagorean theorem, albeit, with some assistance. But, this is a topic they should be familiar with and one that I believe can be used with some assistance rather than re-teaching the Pythagorean theorem. A colleague said they were spending a couple of days going through the Pythagorean theorem and its converse.

I know that there is a constant complaint that students never remember what they learned the prior year. This is a huge issue of frustration. Yet, teachers don't hold students accountable for their prior years of learning. Instead, they turn around and re-teach topics. It is no wonder that students don't retain information because what they have learned is that the "important" topics will be re-taught to them and they don't have to worry about retaining any learning.

I don't believe this situation will change until we teachers learn to hold students accountable for their learning and put the burden of retaining information on students shoulders. Re-teaching topics over and over again simply promotes the very learning behaviors that cause so much frustration. Stop re-teaching and hold students accountable for their prior learning. You may be surprised at how well students respond and rally to the challenge.