I have designed a 4 to 5 day instructional unit for a GPS Math IV course. Each of these lessons was designed for a 90-minute class period. However, the format of these lessons and the amount of time spent on individual activities can be modified to fit the needs of different groups of students. This unit is designed to introduce students to the characteristics of the graphs of tangent, cotangent, cosecant, and secant. Students will also learn how to graph transformations of these functions. Day 1: Students use their knowledge of the Unit Circle and the aid of a graphing calculator to graph the tangent function by hand and determine the characteristics of the graph. These characteristics include: domain, range, asymptote(s), x-intercept(s), y-intercept(s), maximum, and minimum. Day 2: Using reasoning similar to Day 1, students graph the functions of cotangent, cosecant, and secant by hang and determine the characteristics of the graphs. Students also use the Shodor Data Flyer applet to explore the effects of the parameters a, b, c, and d in f(x) = a f(bx – c) + d, where f(x) is one of the trig functions. The lesson I have provided for Day 2 covers a good bit of material and could easily be split into two lessons. Again, this decision would depend on the group of students being taught. Day 3: Students are placed in groups to practice graphing transformations of these functions. Each group presents a problem. Day 4: Students are given an assessment. This assessment could be used for a test or quiz grade. For example, if you wanted to combine this unit with another four or five day unit, this assessment could serve as a quiz grade.
Day 1: Graph of Tangent
Standards addressed:
MM4A3. Students will investigate and use the graphs of the six trigonometric functions.
Understand and apply the six basic trigonometric functions as functions of real numbers.
Determine the characteristics of the graphs of the six basic trigonometric functions.
Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift.
Materials: Pencil, Unit Circle (completed), trigonometric functions chart, characteristics of trigonometric graphs chart, coordinate plane worksheet, and a graphing calculator.
Objectives:
Students will discover the connection between the coordinates of the Unit Circle and the graph of Tangent.
Students will discover the characteristics of the graph of tangent.
Introduction: I will begin by reviewing cos
and sin
Small Group Investigation: Students will work in pairs to complete the tangent portions of the following charts. During this investigation, students should use their unit circle and graphing calculator.
Trigonometric Functions Chart & Characteristics of Trig Graphs. In the previous unit, students filled out the sine and cosine sections.
Whole Class Discussion: I will have the tangent portions of the charts on the board (blank), along with a coordinate plane. I will ask students to come up to the board to fill in sections of the charts. I will also ask a student to come up to the board to graph the tangent function. I will then bring the class together so the students can discuss the solutions to the two charts. After we have discussed the characteristics of the tangent function, students will graph the tangent function on the coordinate plane worksheet I have provided.
Discussion Questions:
What do the asymptotes represent?
What is the behavior of the graph from (-pi/2) to (pi/2)? (I.e. the behavior between two asymptotes)
How can we be sure the function is increasing over that entire interval?
Is the graph supposed to touch the asymptotes?
Closure: So far, we have discussed the characteristics of the graphs of sine, cosine, and tangent. I am going to have students recall the definitions of cot θ, csc θ, and sec θ. Students may start out defining these as hypotenuse over opposite, etc. Questions I can ask the students include: How are these functions related to sine, cosine, and tangent? I.e. reciprocals. How do you define these reciprocals in terms of sine, cosine, and tangent? For homework, students are going to investigate the graphs of cotangent, cosecant, and secant.
Homework: Fill out the cotangent, cosecant, and secant portions of the two charts.
Day 2: Graphs of Cotangent, Cosecant, & Secant / Graphing Transformations
Standards addressed:
MM4A3. Students will investigate and use the graphs of the six trigonometric functions.
Understand and apply the six basic trigonometric functions as functions of real numbers.
Determine the characteristics of the graphs of the six basic trigonometric functions.
Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift.
Materials: Pencil, Unit Circle (completed), trigonometric functions chart, characteristics of trigonometric graphs chart, coordinate plane worksheet, exploring parameters learning task, a graphing calculator, and access to an I-pad or computer.
Objectives:
Students will discover the connection between the coordinates of Unit Circle and the graphs of cosecant, secant, and cotangent.
Students will discover the characteristics of the graphs of cosecant, secant, and cotangent.
Students will compare the graphs of sine, cosine, and tangent with their reciprocals.
Students will be able to graph transformations of the tangent, cotangent, cosecant, and secant functions.
Whole Class Discussion: We will start off the period by discussing the solutions to the cotangent, cosecant, and secant portions of the charts. Students were asked to complete these portions for homework. I will have these portions of the charts on the board (blank), along with three coordinate planes. I will ask students to come up to the board to fill in sections of the charts. I will also ask students to come up to the board to graph these functions. After we have discussed the characteristics of these functions, students will graph these functions on the coordinate plane worksheet I have provided.
Discussion Questions:
What do you notice about the values of the tangent and cotangent values in the chart?
How is this different from the graph of tangent?
What are the values of sin θ at -2π, -π, 0, π, and 2π?
What are the values of csc θ at -2π, -π, 0, π, and 2π?
I would like students to conclude that these values are x-incepts for sine and asymptotes for cosecant.
What are the values of cos θ at -3π/2, -π/2, π/2, and 3π/2?
What are the values of sec θ at -3π/2, -π/2, π/2, and 3π/2?
Exploring Parameters Learning Task: Students will work with their partner to complete the Exploring the Effects of the Parameters learning task.
After all students have completed numbers 1 through 4, I will bring the class together to discuss each of the parameters and the effects they have on each of the functions. Students will work on #5 for the remainder of the class period.
Homework: Finish #5 on the learning task. We will discuss the solutions first thing tomorrow.
Day 3: Additional Practice Graphing Transformations
Standards addressed:
MM4A3. Students will investigate and use the graphs of the six trigonometric functions.
Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift.
Materials: Pencil or pen (three colors), completed Unit Circle, trigonometric functions/graphs charts graphing calculator, markers, chart paper.
Introduction: We will begin the period by discussing the solutions to #5 on the Exploring Parameter’s learning task.
Small Group Work: Since we just covered graphing transformations of tangent, cotangent, cosecant, and secant yesterday, the students have not had a lot of time to practice transformations. The point of today’s class is to allow students to practice graphing these functions and clear up any misconceptions they have. Today, they have their teacher and fellow students as resources. Students also need to be familiar with graphing transformations for their assessment tomorrow. As soon as we are done going over the homework, I will quickly separate the students into groups and pass out the Graphing Transformations worksheet. The students will go through and solve each of the problems as a group. I will assign each group a problem to put on chart paper to present to the class. I will tell each group to make sure they really understand the problem they are presenting and to be prepared to answer questions about it
Whole Class Discussion: Once all the groups have completed their charts, I will bring the class together. I will have groups come up to the board and present their graphs. I wanted every group to get through all the graphs that were going to be presented, so they could compare theirs with the group presenting. I will ask the group presenting to explain step by step how they graphed their transformation.
Closure and Extension: During the last 10 minutes of the class, I am going to discuss the quiz that the students are taking tomorrow. “Make sure you know”:
How to graph all six trigonometric functions.
The domain, range, and period of the 6 trigonometric functions
How to graph transformations involving tangent, cotangent, cosecant, and secant.
Go ahead and start thinking about why these things work the way they do.
During this time, I will assign them a few homework problems (attached below) as well. If we finish with the group discussion early, they can go ahead and start working on these in class.
Homework:
y = - tan(x + π)
y = 2 sec(3x)
y = 2 csc(x + 2π)
y = cot (2x) + 2
Day 4: Assessment
Standards addressed:
MM4A3. Students will investigate and use the graphs of the six trigonometric functions.
Determine the characteristics of the graphs of the six basic trigonometric functions.
Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift.
Materials: scratch paper, pen or pencil (at least two colors).
Introduction: I will begin the class period by going over the homework from the night before. We will discuss specific questions about the homework and general questions about the assessment. Since there were only four homework questions, I am planning on having us discuss each question. This way we can address any misconceptions the students may have about the graphs of tangent, cotangent, cosecant, and secant before they begin the quiz.
Assessment
Remaining Class Time: When students finish their assessment, they can pick up a worksheet on inverses to go ahead and start on.
The worksheet serves as a review for inverses since we will be working with them tomorrow. This also provides something for students to do while they wait for the rest to finish. Any remaining class time can be spent working on this worksheet.
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