## Determining Functions Using Regression: What's the Function?

- Lesson

This activity allows students to look for functions within a given set of data. After analyzing the data, students should be able to determine what type of function best represents the data.

Before participating in this lesson, students should understand the relationships between various functions and their graphs. Their knowledge should cover linear, quadratic, higher-degree-polynomial, exponential, and trigonometric functions. They should also be able to use a graphing calculator to edit data, create scatter plots, and draw functions over those plots.

Instructions for using TI-83 and TI-84 calculators for regression are available here.

**Structure**

Each experiment in this section stands alone. Consider selecting the activities that apply functions that students have already seen.

Options for presenting these activities:

- Use a single experiment and have students perform the activity in pairs or groups. Then, share results as a class.
- Provide each group with the materials for each activity.
- Have different groups perform different activities. Groups can make presentations on their experiment, their graph of the data, and the function determined.
- Set up activity stations. Give students ten minutes to perform the experiment, and maybe start the calculations. After the ten minutes, rotate the groups to the next station until all students have done each activity.

**Activity 1: Counting M&M's** (Exponential Decay)

Be sure that students have at least 35 M&M's so that they can do the experiment repeatedly. Students will pour the candies onto a plate, and remove the ones that have the “'M”' face up, and putour the rest of the candies back into the cup. Doing this repeatedly yields an exponential decay function. *Remind the student not to eat the candy, and that it has been used by other students in other classes.*

**Activity 2: Inflating a Balloon with Breath **(Power Function)

Students measure the circumference of a balloon while counting the breaths used to fill it up. Encourage students to do some practice breaths before starting to measure the balloon. Get team members involved in determining how big a breath should be, and in helping the student blowing up the balloon to practice being consistent.

(This lesson is adapted from *Navigations through Mathematical Connections 9-12*.)

**Activity 3: Height of Cups** (Linear)

Students measure the height of the stack of cups as they put on or take off cups. An alternative procedure is to stack cups on the floor, measuring the distance between the top of the stack and the surface of a desk.

(This lesson is adapted from *Navigating through Algebra 6-8*.)

**Activity 4: Volume of a Box** (Cubic Function)

Students find the volume of a box made from a 20 cm x 20 cm grid with squares of equal size cut out from each corner of the grid. Different-sized corners yield boxes of different volumes. Students graph the length against the volume.

- You could extend this activity by asking students to find the length, width, and surface area of each box in addition to its volume. The problem would then include linear, quadratic, and cubic models.
- The post office uses the concept of girth to limit the size of boxes sent in the mail (length + width ≤ 108 inches). Ask students to find the box with the greatest volume for the least surface area that meets postal regulations.

**Activity 5: Weather Patterns** (Sinusoidal or Periodic)

Students gather data about local temperatures over a year. To minimize the data but still give meaningful results, students can use the temperature on the 1st and 15th of each month.

- This information reduces the domain to 24 values but still yields a fairly smooth curve.
- You can find the average monthly temperature for many cities in almanacs on the Internet. Alternatively, give students this table of weather data for Washington, DC. The temperatures in this table reflect the
*average*high temperatures for the dates within one week of the listed date. For example, the temperature listed for Jan 15 reflects the average high temperature for all days from Jan 8 through Jan 22. - Alternatively, all of the temperature data for Washington, DC, in 2011 can be found in the Washington Temperatures spreadsheet (Excel).
- This activity works best when the calculator is in radian mode.

**References**

A TI-83 or TI-84 calculator is required for these experiments. The remaining materials are listed by experiment:

- Activity Sheets
- Counting M&M's Activity Sheet (Exponential Decay)
- M&Ms
- paper plate
- cup or other container

- Inflating a Balloon Activity Sheet(Power Function)
- uninflated balloon (which when inflated is as nearly spherical as possible)
- tape measure with centimeters

- Stacking Cups Activity Sheet(Linear Function)
- paper or plastic cups
- ruler with centimeters

- Volume of a Box Activity Sheet(Cubic Function)
- grid paper cut into 20 cm x 20 cm squares (enough paper for each group to do 3–6 experiments)

- Weather Patterns Activity Sheet(Sinusoidal, Periodic Function)
- Internet access to look up local temperature averages OR table of weather data for Washington DC

**Assessment Options**

- Use the temperature data in Activity 5 to predict temperatures before and after the period covered by the data. Then, check to see how close your predictions come to the actual temperature values.

**Extensions**

- Extend Activity 4: Volume of a Box by asking students to find the length plus width (girth). and the surface area of each box in addition to its volume.
- Extend Activity 5: Weather Patterns by asking students to find temperature data for different cities. Then, discuss why the period for the graphs is always the same, but the amplitude varies. Ask students, How does this relate to the context of the problem?
- Give students a function type or a graph. Have them suggest a context that would yield that graph.

**Questions for Students**

- Explain how you chose a function type.
- Which, if any, other function types might be used to model the data?
- How did you derive the equation?
- What restrictions are placed on the domain and range?
- How could you use the model to predict future values?
- For which experiments did you get a curve that exactly describes the data? Which were more of a “best function of fit”? Why didn’t some of the data plots match a curve perfectly?

**Teacher Reflection**

- Extend Activity 4: Volume of a Box by asking students to find the length plus width. and the surface area of each box in addition to its volume.
- Extend Activity 5: Weather Patterns by asking students to find temperature data for different cities. Then, discuss why the period for the graphs is always the same, but the amplitude varies. Ask students, How does this relate to the context of the problem?
- Give students a function type or a graph. Have them suggest a context that would yield that graph.

### Determining Functions Using Regression

### Using a Calculator for Finding the Equation of a Function

To determine the function of best fit for a set of data, students should recognize which category of function bests fit the data and know how to use technology to obtain a function. This lesson teaches these skills and prepares students for the subsequent lesson(s), in which they will collect their own data.

Instructions on using TI83 and TI84 calculators for regression is available here.

### Learning Objectives

By the end of this lesson,students will be able to:

- Analyze data to determine the type of function that most closely describes the data.
- Demonstrate an understanding of how modifying parameters changes the graphs of functions by writing equations for those functions.

### NCTM Standards and Expectations

- Understand and compare the properties of classes of functions, including exponential, polynomial, rational, logarithmic, and periodic functions.

- Use symbolic algebra to represent and explain mathematical relationships.

- Judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried out by technology.

- Draw reasonable conclusions about a situation being modeled.