**Lesson Plan Linearity in Algebra**

**LEVEL**: 10-12

This lesson is designed for students in grade levels 10 to 12 who have mastered basic math concepts or can use a calculator to solve basic operations.

**OVERVIEW**: Understand the concept of linear and exponential relationships and relates this knowledge to human population growth over time.

**PURPOSE**: The purpose of this lesson is to help students learn about linear and exponential relationships. Students will learn how to graph both linear and exponential information.

**MATERIALS:** Graph paper, pencils, rulers, calculators, computer or smartboard.

**OBJECTIVE(**s): Students will learn how to:

1. Solve a real life math problem involving multiple and sequential steps in order to answer a question.

2. Graph the results of their problem solving to give a visual representation of the results.

3. Explain the difference between a linear and an exponential relationship.

4. Apply this knowledge to a study of world population growth by making a graph of world population data from 1650 to 2000 (projected).

**ACTIVITIES AND PROCEDURES:**

1. Present the following problem to your students: Imagine you are four years old. A rich aunt wants to provide for your future. She has offered to do one of two things.

**Option 1:** she would give you $1000 a year until you are twenty- one (seventeen years from now); or Option 2: she would give you $1 this year, $2 next year, and so on, doubling the amount each year until you were 21.

Which would you choose? Why? Which way would you have the most money when you were twenty-one?

2. Students would develop grid first then graph information. Students would be paired if computations are too difficult-The stronger with weaker students. If some students cannot complete the graph, allow them access to a completed graph to study.

3. Study the graph and answer the following questions:

A. How much money would you have when you were 21 if you chose option 1? How much would you have if you chose option 2?

B. If you only received money for ten years, which option would give you the most money?

C. How many years would it be before you had the same amount of money with both options?

D. Why did the money in option 2 increase so rapidly after the fourteenth year?

E. Which line do you think would look most like the world’s population growth from 1650 to 2000? Why?

F. Look at the graph. Option 1 represents a simple, direct relationship and is called a linear relationship. Option 2 shows an exponential relationship in which for every year the amount doubles. Some exponential relationships increase even more than this. Which option is linear? Which option is exponential?

4. The estimated world population from 1650 to 2000 is listed in the chart below. Make your own graph of this information. This line graph will show how fast the world’s population is growing. Do you think that a line showing this population growth would look more like the linear or the exponential line from the last exercise? Why?

YEAR WORLD POPULATION (in millions, estimated)

1650 500

1700 600

1750 700

1800 900

1850 1300

1900 1700

1950 2500

1976 4000

2000 7000

Which type of relationship does your graph represent–linear or exponential?

There are concerns that as world population increases there will be shortages of food, water, and the quality of life will be threatened worldwide. What do you think? Write a problem, graph and explain your views.

As part of the lesson your students could write the equations they would use to calculate exponential growth.

Scenarios with bacteria are also great when teaching about exponential growth.

Question: How will you assess the students?

Grace,

This comment is in regards to Assignment 7-1 Math Anxiety.

Students who are weaker in math might be reluctant to pair with students who are stronger in math. Prior to the lesson or in the beginning of the school year you might need to engage your students in team building activities so that your students are comfortable working with each other.

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