Lesson 3

Exponential Growth

Class:    Math II

Time: One 50-minute class

Overview:

This lesson will help students model the real world situation using the knowledge of inverse variation or exponential functions.

Objectives:

The student:

1. will learn how to fit an inverse-variation function to real world data by trial and error.

2. will use residuals to judge how well a function fits data

3. will use residuals to improve the fit of a function

NCTM Standards:

Understand patterns, relations and functions.

Use Mathematical models to represent and understand quantitative relationships.

• identify essential quantitative relationships in a situation and determine the class or classes of functions that might model the relationships;

• use symbolic expressions, including iterative and recursive forms, to represent relationships arising from various contexts.

• draw reasonable conclusions about a situation being modeled.

GPS Standards:

• ### MA1P5. Students will represent mathematics in multiple ways.

Prerequisite Skills:

Students should be able to graph exponential functions. Students should be able to use function transformations (dilations and translations) and fit an exponential function by trial and error.

Fathom Pre-requisites: Students should be able to

• Create attributes defined by formulas

• Make a scatter plot of two attributes

• Graph a least-squares line

• Make a residual plot

Fathom Skills: Students will learn how to

• Use a residual plot to recognize a better function model.

Materials Needed

Activity Worksheet: Worksheets to be distributed at the beginning of the class.

Computers with Fathom 2 along with survey feature

A sheet of notebook paper

Activity: During the activity student will work in pairs and the teacher will begin the activity by distributing the worksheet and demonstrating the beginning of the activity by taking a sheet of paper and folding it into half and asking students how many folds they see and what fraction of the total area does each part represent. Teacher may draw a table on the board or over head display to show how to record the number of folds and rectangles. And directing the students to work in pairs, teacher will go around the class room assisting and scaffolding students in completing the task.

Assessment:

Students will be assessed on the basis of classroom discussion, participation, ease of technology use, completion of task and involvement in the activity.

Evaluation:

Student worksheet will be evaluated based on the following rubric.

Rubric for Modeling the non-linear data

 Question Points Completion of task The student completes the task (4-5 pts) The student miss two to three  questions(2-3 pts) The student does not complete the task.  (0-1 pt) Mathematical Concepts Explanation shows complete understanding of the mathematical concepts used to solve the problem(s).        (4-5 pts) Explanation shows substantial understanding of the mathematical concepts used to solve the problem(s). (2-3pts) Explanation shows very limited understanding of the underlying concepts needed to solve the problem(s) OR is not written. (0-1pts) Explanation/ Justification of reasoning Explanation is detailed and clearly justifies. (4-5 pts) Explanation is clear but does not justify. (2-3 pts) Provides little or no explanation (0-1pt) Type of graph chosen Graph fits the data well and makes it easy to interpret. (4-5 pts) Graph is adequate and does not distort the data, but interpretation of the data is somewhat difficult. (2-3 pts) Graph seriously distorts the data making interpretation almost impossible (0-1 pt)