Representing Trigonometric Functions
This lesson unit is intended to help you assess how well students are able to:
- Model a periodic situation, the height of a person on a Ferris wheel, using trigonometric functions.
- Interpret the constants a, b, c in the formula h = a + b cos ct in terms of the physical situation, where h is the height of the person above the ground and t is the elapsed time.
It is helpful if students have met graphs of the sine and cosine functions before this lesson unit.
The unit is structured in the following way:
- Before the lesson, students attempt the assessment task individually. You then review their solutions and formulate questions for students to answer in order for them to improve their work.
- In the lesson, students engage in pairs or threes on a related card-matching task. Throughout their work they justify and explain their decisions to peers. In a whole-class discussion, students explain and extend their solutions and methods.
- Finally, students work alone on a task similar to the assessment task.
- Each student will need a copy of Ferris Wheel and Ferris Wheel (revisited), a scientific calculator (not a graphing calculator), a mini-whiteboard, a pen, and an eraser.
- Each small group of students will need a copy of Card Set A: Graphs, Card Set B: Functions, Card Set C: Descriptions of the wheels, a large sheet of paper, a glue stick, and a pair of scissors. Some teachers cut the cards before the lesson; others ask students to do this for themselves.
- If you decide to split the lesson over two teaching sessions you will also need some paper clips.
20 minutes before the lesson, a 100-minute lesson (or two 50-minute lessons), and 20 minutes in a follow-up lesson. Timings are approximate. Exact timings will depend on the needs of your class.
This lesson involves a range of mathematical practices from the standards, with emphasis on:
- MP1: Make sense of problems and persevere in solving them
- MP2: Reason abstractly and quantitatively
- MP3: Construct viable arguments and critique the reasoning of others
- MP4: Model with mathematics
- MP5: Use appropriate tools strategically
- MP6: Attend to precision
- MP7: Look for and make use of structure
- MP8: Look for and express regularity in repeated reasoning
Mathematical Content Standards
This lesson asks students to select and apply mathematical content from across the grades, including the content standards:
- H.A-CED: Create equations that describe numbers or relationships
- H.A-SSE: Interpret the structure of expressions
- H.F-BF: Build a function that models a relationship between two quantities
- H.F-BF: Build new functions from existing functions
- H.F-TF: Model periodic phenomena with trigonometric functions