# Lesson 19

Beyond Circles

### Lesson Narrative

The goal of this lesson is for students to develop their skills modeling periodic data. The context in this lesson is the amount of the Moon visible in the night sky for each night in January 2018. The minimum, maximum, and midline values here are relatively unambiguous but it is not possible to estimate the period accurately without more data. In addition, regardless of the period chosen, a trigonometric model only approximates the data.

In this lesson, students engage in aspects of mathematical modeling (MP4). The data is provided and students are prompted to think about the midline, amplitude, and period before choosing their model. The horizontal translation is not directly addressed, however, and the period turns out to be the most interesting part of the story (since an accurate estimate requires more data). As students work, they will need to adjust their model based on its fit to the data (MP2). In addition, discussions about which model “best fits” the data provide an opportunity to make sense of and critique the reasoning of others (MP3).

### Learning Goals

Teacher Facing

• Create a trigonometric function to model data, and use the model to make predictions.

### Student Facing

• Let's use trigonometric functions to model data.

### Required Preparation

Acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if each student has their own device. (Desmos is available under Math Tools.)

### Student Facing

• I can create a model of data that is approximately periodic and use the model to make predictions.