Exploring Exponential Growth

North Carolina Council of Teachers of Mathematics 43rd Annual State Conference

Christine Belledinbelledin@ncssm.edu

The North Carolina School of Science and Mathematics

Durham, NC

GOALS FOR THE SESSION

• We will show how to use data about grain production and population growth in Uganda to compare linear and exponential growth.

• We will show how students can understand the meaning of the constants in an exponential function by relating them back to our context.

WHERE THIS IDEA COMES FROM…

Reverend Thomas Robert Malthus(1766-1834)

British cleric and scholar

Known for theories about population growth and change.

MALTHUSIAN THEORY

FACTS ABOUT HUNGER

• Total number of children that die each year from hunger:

• Percent of world population considered to be starving:

• Number of people who will die from hunger today:

• Number of people who will die of hunger this year:

1.5 million

33%

20,866

7,615,360

Grain production for

Uganda

in

1000’s of tons

Year   Grains

1998   2085

1999   2178

2000   2112

2001   2309

2002   2368

2003   2508

2004   2274

2005   2459

2006   2667

2007   2631

BELOW IS GRAPH OF THE DATA

We would like to build a linear model for the data set.

2 4 6 8 10 12 140

500

1000

1500

2000

2500

3000

Grains

Years Since 1995

Gra

in P

roduced in 1

000's

of

tons

LINEAR FUNCTION

2 4 6 8 10 12 140

500

1000

1500

2000

2500

3000

f(x) = 61.2545454545455 x + 1899.69090909091

Grains

Y=61.255x+1899.7

USING OUR LINEAR MODEL

• Interpret the slope and intercept in context.

• Make predictions about future food production.

• Later compare growth of food production to population growth.

POPULATION GROWTH FOR UGANDA

To the right is a table of Uganda’s

population in millions in the years from

1995 to 2009.

Year   Population

1995   20.71996   21.21997   21.91998   22.51999   23.22000   24.02001   24.72002   25.52003   26.32004   27.22005   28.22006   29.22007   30.32008   31.42009   32.4

CREATE A SCATTER PLOT OF THE DATA

0 2 4 6 8 10 12 14 160

5

10

15

20

25

30

35

Population of Uganda

Years Since 1995

Popuati

on in m

illions

CONSIDER VARIOUS MODELS

• Linear• Quadratic• Exponential

FROM PREVIOUS WORK

We know • Linear growth is governed by

constant differences.• Exponential growth is governed by

constant ratios.

Let’s use this knowledge to find a model for population...

ANOTHER OPTION: RE-EXPRESSING THE DATAWe can re-express the data using inverse functions.

If we think the appropriate model is an exponential

function, let’s use the logarithm to “straighten” the data.

Consider the ordered pairs (time, ln(population)). Look

at the graph of this re-expressed data.

COMPARING GROWTH

Can we find ways to compare growth of food production to population?

EX. 2: FOOD PRODUCTION VS. POPULATION GROWTH

1. The population of a country is initially 2 million people and is increasing at 4% per year. The country's annual food supply is initially adequate for 4 million people and is increasing at a constant rate adequate for an additional 0.5 million people per year.

a. Based on these assumptions, in approximately what year will this country rst experience shortages of food?

Taken from Illustrative Mathematics

FOOD SUPPLY VS. POPULATION CONTINUED…

b. If the country doubled its initial food supply and maintained a constant rate of increase in the supply adequate for an additional 0.5 million people per year, would shortages still occur? In approximately which year?

c. If the country doubled the rate at which its food supply increases, in addition to doubling its initial food supply, would shortages still occur?

WHY ARE THESE PROBLEMS SO POWERFUL?

• Students see that mathematics can help us understand important real-life issues

• Students have the chance to create mathematical models.

• We can help students make sense of the constants in the models. (Interpret constants in context.)

• Students build tools to help them distinguish between different types of growth based on mathematical principles.

CCSS CONTENT STANDARDS

HSF-LE.A.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.

HSF-LE.A.1a Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

HSF-LE.A.1b Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

HSF-LE.A.1c Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

HSF-LE.A.2 Construct linear and exponential functions, including arithmetic and geometric

sequences, given a graph, a description of a relationship, or two input-output pairs (include

leading these from a table).

.

MORE CCSS CONTENT STANDARDS

S.ID.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

b. Informally assess the fit of a function by plotting and analyzing residuals.

Represent data on two quantitative variables on a scatterplot, and describe how the variables are related.

c. Fit a linear function for a scatter plot that suggests a linear association.

S.ID.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

CCSS MATHEMATICAL PRACTICES

1. Make sense of problems and persevere in solving them

2. Reason abstractly and quantitatively

3. Construct viable arguments and critique the reasoning of others

4. Model with mathematics

5. Use appropriate tools strategically

6. Attend to precision

7. Look for and make use of structure

8. Look for and express regularity in repeated reasoning

RESOURCES FOR TEACHERS

• NCSSM Algebra 2 and Advanced Functions websites

www.dlt.ncssm.edu/AFM

http://www.dlt.ncssm.edu/algebra/

See Linear Data and Exponential Functions

• Link to NEW Recursion Materials

http://www.dlt.ncssm.edu/stem/content/lesson-1-introduction-recursion

• NCSSM CCSS Webinar: Session 1: Using Recursion to Explore Real-World Problems

http://www.dlt.ncssm.edu/stem/using-recursion-explore-real-world-problems

MORE RESOURCES

• Illustrative Mathematics

http://www.illustrativemathematics.org/standards/hs

Tasks that illustrate part F-LE.A.1.a

F-LE Equal Differences over Equal Intervals 1

F-LE Equal Differences over Equal Intervals 2

F-LE Equal Factors over Equal Intervals

• The Essential Exponential by Al Bartlett

http://www.albartlett.org/books/essential_exponential.html

LINKS TO DATA AND INFORMATION

Gapminder

http://www.gapminder.org/

World Hunger Map Link

http://www.wfp.org/hunger/downloadmap

Link to Data for Uganda

http://faostat.fao.org/site/609/DesktopDefault.aspx?PageID=609#ancor

My Contact Information:

Christine Belledin – NC School of Science and Mathematicsbelledin@ncssm.edu

For copies of the presentation and other materials, please visit

http://courses.ncssm.edu/math/talks/conferences/ after Monday, November 4.

Thank you for attending!