Modeling Data with Exponential Functions


Goals
(1) Students will explore exponential models and share helpful webpages and online videos.
(2) Students will apply exponential regression to model a set of data using graphing calculators.
(3) Students will create their own exponential model and work out an example problem.
(4) Students will evaluate and analyze the reasonableness of their peers' work.

Abstract
Real-world data often follows the mathematical pattern of one of the functions that we study in this course. If we use one of those functions to match up with observed data and draw a curve that closely fits that data, we can use it to make predictions about future and past values in the data set.

Problem Statement
Statistics taken by researchers are valuable for more than just taking a count of how many people are in a city, or the median household income of a state, or the price of hogs at a rural market. If we chart that data on a graph and see if the data follows the pattern of a linear, parabolic, or exponential curve, we can create an equation to make predictions about what WASN'T counted, or HAS NOT YET been counted. Making future predictions is called EXTRAPOLATION, and making within the domain of a data set is called INTERPOLATION.

You are going to use extrapolation and exponential regression models to predict the future population of a country.

Instructor Suggestions
(1) Set the stage by discussing the "Problem Statement" (see above) with the class.
(2) Distribute the "Population Explosion" activity sheet (see attachment) and allow the students to individually read the first part of the activity.
(3) Have students explore YouTube, KhanAcademy, or another site with tutorial videos to find helpful examples to share with classmates on the Functions group.
(4) After students have a grasp on the process of creating and using an exponential growth model, have them create and create a map on GoogleMaps with 4 countries to study, paste sticky notes with their reasoning, and share with the Functions group.
(5) Students will research population levels by year of each country and bookmark their source.
(4) After students complete the "Population Explosion" activity, they should upload images or share video of themselves working out the populations of each country in 2050 and a graph of their data and model.
(5) As students' posts come in and are shared with the Functions group, students will comment on each other's work calculate the residuals of one of their classmate's models, determining which country's population best follows the model.

Materials
"Population Explosion" Activity Sheet, student Google accounts for GoogleMap, student Diigo account for sharing and viewing links and images, 4 days of time with a laptop cart or computer lab, graphing calculators

Time
Introduction of problem statement (5 minutes), Student tutorial exploration (1 day in lab), GoogleMap creation (20 minutes in lab), Researching population levels by year (40 minutes in lab) Exponential regression in the calculators and extrapolation (40 minutes), Working out calculations in MS Paint, SMART Notebook, or similar drawing program (1 day), Calculating residuals and critiquing work (1 day)

NCTM Standards Addressed
Data Analysis and Probability, Algebra, Communication, Connection, Representation

Curriculum Integration
This activity will be used in a course studying functions and modeling of data. At my school, it could potentially fit in AP Statistics, or the "Functions, Statistics, and Trigonometry" class.

Further Investigation
This lesson could be extended to a historical/sociological discussion about why some countries have higher growth rates than others, or analyzing education/socioeconomic factors along with growth data.

Variations/Comments
If less lab time is available the instructor could provide a list of country choices and a chart of past populations for each country to cut down one day. Calculations could also be written by hand and scanned in by students or the instructor in lieu of using a drawing program. A group GoogleMap could be created to see all countries being modeled together.

References/Resources
www.diigo.com - Sharing videos, worked out examples, map of places, and analyzing peers' work
maps.google.com - creating map of countries modeled

Video Tutorials

EXAMPLE STUDENT WORK:

Student Diigo page
Slideshow of Diigo bookmarks - could be used by the teacher to quickly "flip" through the student's bookmarks
Student Google map
Google Map with sticky note annotations
Student work with peer annotations and teacher feedback