LESSON PLAN TOPIC: Exponential vs. Linear Equations                     TITLE: World Population Review

AUTHOR: Cathy Brown

GRADE: 9-12						          Integrated Discipline(s):  Mathematics/Science/Social Studies

8.4 LISTENING

8.4.1 By the end of the eighth grade, students will identify information gained and complete tasks through listening.

12.1 NUMERATION/NUMBER SENSE

12.1.2 By the end of twelfth grade, students will express the equivalent forms of numbers using exponents, radicals, scientific notation,

            absolute values, fractions, decimals, and percents.

12.2 COMPUTATION/ESTIMATION

12.2.1 By the end of twelfth grade, students will solve theoretical and applied problems using numbers in equivalent forms, radicals, exponents,

            scientific notation, absolute values, fractions, decimals, and percents, ratios and proportions, order of operations, and properties of real numbers.

12.2.2 By the end of twelfth grade, students will justify solutions to mathematical problems.

12.2.3 By the end of twelfth grade, students will perform estimations and computations of real numbers mentally, with paper and pencil, and with technology.

12.5 DATA ANALYSIS, PROBABILITY, AND STATISTICAL CONCEPTS

12.5.1 By the end of twelfth grade, students will select a sampling technique to gather data, analyze the resulting data and make inferences.

12.5.2 By the end of twelfth grade, students will write equations and make predictions from sets of data.

OBJECTIVES:  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).

5.   Explain some of the reasons for the growth in the world's population.

• Participation in class discussion will be part of overall class participation grade.
• Student work for today will be handed in to assess their grasp of the concept
• Points are given for right answers, showing work, neatness

MATERIALS:

• Graph paper, pencils, rulers, calculators, blackboard & chalk

PROCEDURES:  The concept of exponential (vs. linear) relationships is a difficult concept for many students to understand.  This lesson helps students understand the difference between the two and relates this knowledge to human population growth over time.  PURPOSE:  The purpose of this lesson is to help students learn about an exponential relationship and how it relates to human population growth and the current global population crisis.  Students will learn how to graph both exponential and linear information.

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?

Vocabulary

None--

Activities

1.  Each student will have a graph paper and a ruler.  I will walk through this problem doing the steps on the board as they follow on their papers.
Put money on the left, vertical margin, using units of $5,000.
Put years on the horizontal margin, starting with year one to seventeen years. 
Demonstrate on the board where to put the information on the graph and how to connect the lines, and you will do this as a class.
Find the year along the line at the bottom of the graph.  Then, find the amount of money for that year along the left side of the page.  Match up these two amounts and place a dot.  
When you have placed all your dots, draw a straight, solid line to represent option 1, $1000 per year, and a curved, dotted line to represent option 2, $1 the first year and double that amount every year.
(Allow calculators.  Option to give students an empty, labeled graph if this is new and difficult for them.)
					
2.   Each student will study the graph and answer the following questions (this will be a group/class exercise)
     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. Which option is linear?  Which option is exponential?
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. 

3.   The estimated world population from 1650 to 2000 is listed in the chart below (This will be a hand out class problem).  Make your own graph of this information, putting population figures (in millions) on the left vertical margin, and years on the horizontal margin. 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?
				
Find the year along the line at the bottom of the graph.  Then find the correct population for that year along the left side of the page.  With your pencil and ruler, draw one dot for each pair of information.  When you have placed all of the dots on the graph, connect them with one curved line.


     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?

4.   To understand why world population is now growing so fast, we will discuss some issues.  This activity (also a class hand out) will help you understand one of them.  Read the four "family histories" below and answer the questions.  It might be useful to draw a "family tree" for each one to help you with the math.

     Family A:  A has one child.  If that child has one child, how many grandchildren does A have?  If the
     grandchild has one child, how many great grand-children does A have?

     Family B:  B has two children and each of them has two children.  How many grandchildren does B have?  If each grandchild has two children, how many great-grandchildren does B have?

     Family C:  C has three children and each of them has three children.  How many grandchildren does C have?  If each grandchild has three children, how many great-grandchildren does C have?

Family D: D has four children and each of them has four children. How many grandchildren does D have? If each grandchild has four children, how many great-grandchildren does D have?

• Clear, concise instructions

• Use of visuals

• Modeling of expectations

• Repeating of instructions

• Model a variety of problem solving strategies.

• Assist students to develop criteria for choosing a strategy

• Apply reading comprehension strategies to problem solving (i.e. deciding what the main idea is and what information is extraneous to the problem).

• Use simpler problems of the same type to model methods.

• Encourage students to represent problem in pictorial or other visual form before attempting solution.

• Focus students' understanding of algebraic symbols as representing real numbers or operations.

• Permit extended time for problem solving, including pre-solution stage trial and error.

• Teach prediction and estimation skills.

• Consider the use of pictorial flow charts to plan strategies before setting up equations.

• Other modifications as needed by IEP in consultation with the special education teacher

The number of children "multiply" each generation.  For family B there are twice as many children each generation and for family D there are four times as many. 

	Few families really have the same number of children each generation.  But these examples help explain one reason why the world's population has 

	grown rapidly in the last 100 years.  

Another reason is that in most areas of the world, people are living longer.  Up until 125 years ago, the world's population was increasing slowly.  Although the 

	number of births multiplied, many babies did not live and large numbers of children and adults died from diseases.  Over the past 150 years diet, nutrition, 

	and health care have improved.  Scientists have discovered cures for many diseases.  As a result, the death rate has been declining rapidly.  With more 

	people being born and living longer, the result has been a big jump in the world's population.
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?  Discuss &/or debate.

Academy Curricular Exchange, Columbia Education Center, Mathematics, Margaret V. Smith, Reg. II Observation & Assessment Center; Salt Lake City, UT at http://712educators.about.com/gi/dynamic/offsite.htm?site=http%3A%2F%2Fofcn.org%2Fcyber.serv%2Facademy%2Face%2Fmath%2Fhigh.html

Ideas for helping students with special needs in mathematics at http://www.bced.gov.bc.ca/specialed/landbdif/42.htm