You will want to have copies of the “calculating tangent values” handout ready as soon as you’re done with this discussion. The goal of this part of the lesson is for students to recognize that while sine, cosine, and tangent functions are all trigonometric ratios, sine and cosine values can be read directly from the unit circle, while tangent values come from the ratio of sine to cosine. I begin this class by asking my students to reflect silently for a moment about the similarities and differences between the sine, cosine and tangent functions. (MP2) I give the further directive that when they’re ready to share their ideas, I will know because they will put all writing materials down and be looking quietly at me. Even my seniors have fun with this because they like the idea of all staring at the teacher at the same time…trying to get a response out of me. I keep it deadpan until the last kid is ready, then feign surprise to see so many eyes all focused on me. I sometimes make a goofy comment about my features, like “Am I glowing now?” or “I must be having a really bad hair day today.” Other times I just ask for volunteers or draw popsicle sticks (see my strategies folder – “Calling on kids”).
When everyone is ready I have students share their ideas by either posting them on the board individually or by having a scribe post for the class. I like to have students get up and moving because it seems to keep them more engaged, even if it’s just to write on the board, but it takes more time, so I have to weigh the value for each class and lesson. I’ve included two versions of the format I use for recording this kind of information, a venn diagram and a compare/contrast table. You can hand a copy of one of these out before you ask the students to reflect if you want to give them more guidance, which can be especially helpful for students with language or learning differences.
I always hope that at least a few students will remember how we generated the sine and cosine graph using the idea of unit circle, but don’t be too frustrated if you only get responses about the trigonometric ratios they learned in geometry. I just take whatever the students give me and then guide them toward the unit circle and graphing sine and cosine functions. Your students may more easily see that both sine and cosine use the hypotenuse while tangent doesn’t. You can ask what the value of the hypotenuse is in a unit circle and move from there to the tangent being sine/cosine.
There is a video narrative in my resources that further discusses the pedagogy for this section.
To close this lesson, I ask students to share and compare their summaries with their table with the goal of coming to a consensus about the effects of different transformations on the graph of the tangent function. (MP3) I then ask one member from each table to share the consensus of his/her table with the class. We discuss any discrepancies to reach a class consensus, with direction from me as needed. The final piece of this lesson is for each student to update her/his summary to reflect the class consensus and then complete the Challenge problem sheet as homework.