Developing a connection between trigonometric ratios and slope

Today, we continued working through the final problems on the Relationships with Meaning packet. I started by asking a student to share their work on problem 6, the first homework problem from last class. I projected the work on the board and asked students to comment on the work. Typically I want students to say whether or not they agree or disagree, to ask questions, or to comment on things they notice.

For the most part, students were not commenting. I finally asked students to look at the diagram that was sketched. Finally, a couple of students recognized that one of the leg lengths was longer than the hypotenuse. Several others jumped in about this not being possible. We had a good discussion about making calculations and then using other ways to check for the reasonableness of the calculation. While it is sometimes painful to display incorrect work, it also exposed a mistake that many students had made and how they could use this to learn and do better next time.

With that correction, I had students re-work their answers to this problem. They also took advantage to fix their results for the next problem. The results were again displayed and this time, the student presenting realized that she had reversed the sine and cosine ratios. She jumped back up and corrected her mistake.

The final problem in this set involved the equation sin(B) = 1 / √2. Invariably, students freeze when facing values like this. I pointed out that √2 is a value written with two squiggly lines, but so is 37. We're just more used to seeing a value like 37 rather than √2. With that, students tackled the final problem. Some got stuck on how to use the Pythagorean theorem when one of the values was √2. They started to realize that √2 x √2 = √4. They also realized that √4 = 2. With this they could now tackle the trigonometric ratios.

As a final piece, I had students look at the sin(B) = 1 / √2 result. I again reinforced that the PSAT and ACT would not provide this as a possible answer and that they needed to simplify the expression so that there was no radical in the denominator. Many students recognized that multiplying the numerator and denominator by √2 would achieve the desired result. We now had
sin(B) = 1 / √2 = √2  / 2.

I had students proceed to the next three statements and determine whether or not these were true or false. With the aid of the trig table, students were able to correctly determine which statements were true.

We then moved to the final page of the packet. The first set of problems asked students to draw slope defining triangles. I was concerned about the wording and asked students what a slope defining triangle would look like. A few students suggested that the triangle should be a right triangle. I felt students could proceed ahead with the work and asked them to draw their triangles and calculate their slopes.

The first issue I saw as I walked around the room was that students wanted to use the entire line length drawn. This meant that they had triangle vertices that did not correspond to grid line intersections and their slopes were not correct. The next issue was that some students still did not know how to calculate slope. The third issue I ran into was that students had trouble identifying whether the slope was positive or negative.

I spent a good deal of time walking around and helping correct misunderstandings and confusion. The one-on-one work seemed to help and students started to show a better understanding of why slopes were positive or negative and how the slopes were calculated.

The final three problems involved finding missing side lengths and two of the problems involved square roots. I was pleased when many of the students tackled these problems and were actually doing them correctly. Some students still had a little issue with the hypotenuse length of √116. With a little help they realized that squaring √116 produces a value of 116. The other positive in the class's work with these three problems was that students were resisting the urge to grab a calculator to turn the square root into an approximate decimal value.

I concluded the class by asking them to think about their slope calculations and to make connections of their calculations with sine, cosine, and tangent. I want to see how many students can identify the connection at this point.

We'll continue next class with working on the next investigation packet.