I include Warm ups with a Rubric as part of my daily routine. My goal is to allow students to work on Math Practice 3 each day. Grouping students into homogeneous pairs provides an opportunity for appropriately differentiated math conversations. The Video Narrative explains this lesson’s Warm Up-Graph of Tangent which asks students to hypothesize about the shape of a tangent graph.
I also use this time to correct and record the previous day's Homework.
I found this amazing applet linking the graphs of sine, cosine, and tangent to the unit circle. The sine and cosine versions were obvious. It took me a minute to realize how the graph of tangent was created. But when I did figure it out, it was obvious how brilliant it was. This is how I decided to build the conceptual portion of this lesson. I found addition information in the CK12 curriculum whose link it located in the resource below. Please look through this resource if this concept of particular method of building the tangent graph off of the unit circle is unfamiliar.
The lesson begins by asking the students to recall the value of tangent on a unit circle. This knowledge is fundamental to the understanding of the structure of the graph. Next, we connect similar triangles located on the unit circle to the value of tangent at any angle (Math Practice 7). I then link to the applet showing the entire graph of tangent using this concept of similar triangles. I pause the applet at points to show the similar triangles as they are not drawn.
Next, we draw a sample graph in their notes and talk about its major features. The first thing to talk about is period. I allow the students some time to discuss with each other and then we share as a class. I also ask how this period compares to the period of sine or cosine (Math Practice 2).
The asymptotes are pretty obvious. It is important to show them how to represent all the asymptotes using ±π.
Next, we talk about amplitude. Obviously, tangent doesn’t have amplitude since is goes to infinity in both directions. Instead, we could have a vertical stretch or shrink. Tthe students notes that tangent hits 1 or -1 at the half way mark from the intersection to the asymptotes
Domain and range should also be discussed. Finally, I ask the students to graph this using degrees rather than radians, noting the change in the period and asymptotes.
Now that students have a basic understanding of the graph of a tangent, we practice transforming them. In problems like these where the students already have the skills, I have them attempt the transformations first and then we discuss them as a class. When we look at them as a class, I model efficient ways to graph these functions. I start with locating the asymptotes and then showing the students how the x-intercepts are centered between each intercept. I also have them graph one point on either side of the intercept to give shape to the curve. On the parent graph of tangent, the point equidistant from the x-intercept and the asymptote is located up one unit. If there is a vertical stretch or shrink, this point will move up or down accordingly. I really like to show the students these patterns rather than give them numerical locations (Math Practice 8).
Each type of transformation is presented by itself and then there are several examples using multiple transformations. If time is running short, I may have them describe the transformations verbally. It is also useful to have the students identify horizontal translations as well as asymptotes in terms of degrees as well as radians.
The first 8 problems in this Homework ask students to graph transformations of tangent functions. The next three problems give the students a graph and ask them to write an equation to represent the graph. The final problem is a critical thinking task about multiple representations given a horizontal shift on a tangent function (Math Practice 8).
I use an exit ticket each day as a quick formative assessment to judge the success of the lesson.
This Exit Ticket assesses students' ability to transform a tangent function.