Tomorrow I will be “introducing” graphs of trigonometric functions to my IB Math Standard Level 1 class, so I’ve got Trig on the brain. Yes, yes, they already learned it in Algebra 2/Trig Honors, but I find that it’s best to build from the ground up when it comes to Trigonometry. Sometimes students catch something that did not make sense to them the first time around. I should probably sleep, but I know this post won’t happen if I don’t get it done tonight…
From the start of the school year, my IB Math SL 1 students have learned:
- 30-60-90 and 45-45-90 triangles and proportions
- right triangle trigonometry
- the unit circle (my best template of all time is here)
- laws of sines & cosines
- arc length & sector area
Check out this YouTube video on the order in which I teach the Unit Circle: https://youtu.be/MaORCIxMeVw
Part 1: Class Discussion
I begin this epic day by asking students to graph their height on a Ferris wheel ride over time. As you can imagine, students’ Ferris wheel sketches take them underground, back in time, or on very “angular” adventures. Some students begin their rides in the middle of the sky. (I usually walk around and say, “Yup. Nope. Yup. Yup. Nope,” as students draft their guesses… because this encourages students to want to get the “right” answer. You definitely have to know your audience, though.)
These sketches launch us into a great discussion on: where Ferris wheel rides begin (close to but not ON the ground), and how we might picture them as nice rounded rides, but in all actuality we stop periodically to pick up new people and drop off others. All of the “stop n’ go” aside, we jump into group Ferris wheel scenarios where we’re assuming the ride is continuous.
Part 2: Group Discussion
- I provide each group with one of 4 scenarios involving different Ferris Wheels.
- The scenarios have different radii, periods, central axes, starting locations, and lengths of rotation.
- Students work individually and then with their group to develop a consensus on labeling their axes and sketching a graph of their assigned scenario.
- 1 person in the group is responsible for transcribing the scenario’s key information and graph onto a presentation-worthy graph paper.
Part 3: Carousel Walk
- I post the graphs around the room in order from 1-4. *Because I generally have 8 groups in my room, there are 2 different sets of students rotating through 2 different sets of posters, 1 – 4.
- Students in Group 1 begin at Poster 2. Students in Group 2 begin at Poster 3, etc.
- They have 1-2 minutes to discuss the commonalities and differences between their scenarios and the graph in front of them.
- I call time and students rotate to discuss the next poster.
- We repeat this until students have considered and compared ALL 4 Ferris wheel scenarios.
At the end of this carousel walk, we have a summarizing discussion in which students often bring up vocabulary like period, amplitude, etc., which demonstrates their prior knowledge. It also helps students to quantify their key vocabulary (amplitude, period, phase shift, vertical shift/principal axis, etc.) with real-world applications of this terminology. ADDED BONUS: it gives them a context when they’re required to answer Ferris-wheel type IB Questions. 🙂
Part 4: Graphing Sine, Cosine, and Tangent (Basic Notes)
I ask students to take out their unit circles so we can use the sine, cosine, and tangent values at each of the quadrantal angles to create our trigonometric function graphs. Here is the link to my Template for Graphing Trigonometric Functions. I could do more with this using Geogebra, but I need to find a good link. Message me if you have one that works well for you!
Part 5: Transformation Exploration
Usually by this point the bell has rung, and students are rushing off to lunch or practice. HOWEVER, like a Boy Scout, I am ALWAYS prepared. heh. And our discussion of basic trigonometric graphs often flows naturally into a discussion of transformed trigonometric functions. Of course, we finish this activity the following class period.
In the next week or so, I will post a file in which I lead students through graphing transformed sine and cosine graphs with their graphing calculators. They make a prediction for the effect of each constant on the graph, and then test out their predictions.
Because this is the first time in the year that we have worked with transformations of functions, I dedicate a significant amount of time to solidifying these concepts. Students work with transformations ALL YEAR LONG, so the foundation is super important.
Part …200?: Ferris Wheel Follow-Up
When we get to the “writing equations of trigonometric functions” stage of our learning, I actually scan the Ferris wheel group graphs (#1 – 4) the students made on day 1 of this unit.
- I upload them to padlet.
- I project Graph #1 on padlet from my computer.
- Students work in pairs (using the padlet link I provide) to post their guess for the sine or cosine equation of the graph provided. (I alternate between instructing: “Write a transformed sine equation for this graph.” and “Write a transformed cosine equation for this graph.”)
While we don’t necessarily get to all 4 graphs in the allotted time, our discussion brings everything back to the key vocabulary from Day 1, reinforces the context for our sine and cosine graphs, and students develop a greater understanding of the connections between trigonometric graphs and their equations.
I will be adding a Part B to this post later – we still need to address my “frame” approach to graphing trigonometric functions. I picked it up from another teacher in the county and it has worked wonders!
So that’s all for now, folks. Happy Graphing!