To begin today's class, I draw the attention of my students back to the "Fascinating Chart" that was part of the Similar Triangles Project in Unit 1. Many students have referred continuously to this document over the course of the semester, and with each reference, they're able to make more sense of it.
Look at the first slide of today's Prezi to see the questions I pose to students at the start of class. The purpose of this opener is to briefly review the understandings we've gained over the last few weeks about how sine and cosine behave on the unit circle, and to bring back to mind how strangely tangent behaves in comparison.
I give students a few minutes to find the handout and to either think about these questions on their own or to discuss them with tablemates, then I lead a whole-class discussion about what's happening in each of these columns. I want to see if students can use their new knowledge of the graphs of sine and cosine to "see" the shapes of these graphs in the sequences of values in this chart. I'll point out to students how "quickly" sine goes from 0 to 0.5, and then how it slowly moves through the "0.8's and 0.9's" between 55 and 90 degrees. I'll try to get students to really visualize what this looks like on the graph of the sine function.
Then, I'll ask them to visualize what the graph of a tangent function must look like. How fast does it grow? I want students to see that for half the graph (from 0 to 45 degrees), tangent moves gradually from 0 to 1, then for the other half (from 45 to 85), it accelerates from 1 to more than 11. Students have remarked on the strangeness of this column in comparison to the others over the course of the semester. Most of them also continue to seek a good explanation as to why tangent is "undefined" at 90 degrees. Today, I tell them, we're going to see how the graph of the tangent function helps us make sense of this third - and so far, most strange - periodic function.
Definitions of Tangent: "Turn and Sketch"
In an adaptation of the Turn-and-Talk strategy, I ask my students to "Turn-and-Sketch" some visual definitions of tangent. Specifically, I ask for both the right triangle definition and the unit circle definition. I find that it's useful to have students draw something as they discuss these definitions, because we're doing a lot of visualization today.
After students spend a few minutes sketching and discussing the word tangent in pairs, I ask for volunteers to share their work on the chalkboard. Depending on what they draw, I direct the conversation toward the similarities and differences between sine & cosine and tangent. I repeat the question from the end of today's opener: why is tangent undefined at 90 degrees?
Looking at Graphs
I have posted a few key graphs in today's Prezi. If I decide I want a little more control over the mini-lesson, I may project my Desmos browser window and manipulate the graphs as necessary.
We take a look at the graphs of sine_and_cosine again. I tell the class that the main thing I'm interested is where these two graphs intersect. Where is it? How is this expressed in the fascinating chart? Most importantly for today's purposes, what is the value of tangent when sine and cosine are equal? And why does this happen?
After discussing these questions as a class, we move to a graph of y = tan(x) and y = cot(x). Once again, I say that I'm interested in the intersection points of these two graphs. I ask the same questions as before. Some students need a reminded of the definition of cotangent, so I refer back to the right triangle definition: it's just the multiplicative inverse of tangent, or adjacent/opposite. So why do these graphs intersect where they do? See today's explaining the math resource for more notes on this conversation.
Domain and Range
On Slide #11 of today's Prezi, I ask students to identify the domain and range of the sine, cosine, and tangent functions. This is not the first time I have asked these questions, but it's the most formally I've stated them. I expect that at this point, these questions will make more sense than ever. My hope is that the work we've done with the graphs of y = sin(x) and y = cos(x) should make it clear that both of these can take any real number as an input, and that the output will always be between -1 and 1. With tangent, on the other hand, the domain has some constraints. For example, we've known for a while that tan(x) is undefined if x = 90. So how can we state the domain of tangent? On the other hand, there are no limits to the range of y = tan(x). Why is this?
This leads to our first real discussion of asymptotes.
Features of the Tangent Function & Handout
On Slide #12 are three more questions about the features of the tangent function. As students begin to discuss these questions, I distribute today's handout, Graphing Tangent Functions. The questions are included on this handout. Just like students were asked to do on their Check in Quiz yesterday, I am going to annotate this graph. The annotations will serve both as answers to these questions and notes that students can use as they continue their work.
Guided Note Taking
To begin the Graphing Tangent Functions handout, students answer some questions from the end of today's whole class discussion. I show them the steps for graphing a tangent function and I demonstrate these steps while asking students to take notes on this process. As we work, I give students an introduction to the ideas of asymptotes and inflection points. I don't worry about how deeply they understand these ideas - I'm just excited to give them a little glimpse of these big ideas.
Students then apply what they know as they practice graphing and annotating a few tangent functions. As students work, they have access to laptops and the link to a simple Desmos applet that allows them to change the a, b, c parameters on y = a * tan(bx) + c. In addition to being useful, this applet is visually stunning - and like tangent - a bit weird in its behavior.
Students also have the option of using TI-83 graphing calculators, which I encourage, but don't spend explicit time on during today's lesson.
Included here are two links to Desmos graphs: the one I've described above with just tangent and three sliders, and another that adds sine and cosine to the same graph. It's interesting to watch all three graphs change at once as each parameter is changed. Depending on my students' grasp of these ideas, I may simply show them the second applet, or may suggest the idea and ask them to try to make it.
Problem Set #15 is a Unit 4 Exam Review assignment. Students will have the next class period to work on it and prepare for the exam. I hand it out today so they can get started tonight.
Today's exit slip is another Index Card Essay. Today the prompt is simply:
What is an asymptote?
It's delightfully instructive to see how students answer this question.