# Representing Trigonometric Functions

## Mathematical Goals

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.

## Introduction

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.

## Materials required

- 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.

## Time needed

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.

## Lesson Type

## Mathematical Practices

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