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   20851999   21782000   21122001   23092002   23682003   25082004   22742005   24592006   26672007   2631

BELOW IS GRAPH OF THE DATAWe 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 1995Gra

in P

rodu

ced

in 1

000'

s of

ton

s

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 UGANDATo the right is a table of Uganda’s population in millions in the years from1995 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

Popu

atio

n in

mill

ions

CONSIDER VARIOUS MODELS

• Linear• Quadratic• Exponential

FROM PREVIOUS WORKWe 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 STANDARDSHSF-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

geometricsequences, given a graph, a description of a relationship, or two input-output

pairs (include leading these from a table). .

MORE CCSS CONTENT STANDARDSS.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 PRACTICES1. Make sense of problems and persevere in solving

them2. Reason abstractly and quantitatively3. Construct viable arguments and critique the

reasoning of others4. Model with mathematics5. Use appropriate tools strategically6. Attend to precision7. Look for and make use of structure8. Look for and express regularity in repeated

reasoning

RESOURCES FOR TEACHERS• NCSSM Algebra 2 and Advanced Functions websites

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

See Linear Data and Exponential Functions

• Link to NEW Recursion Materialshttp://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 Mathematicshttp://www.illustrativemathematics.org/standards/hsTasks that illustrate part F-LE.A.1.aF-LE Equal Differences over Equal Intervals 1F-LE Equal Differences over Equal Intervals 2F-LE Equal Factors over Equal Intervals

• The Essential Exponential by Al Bartletthttp://www.albartlett.org/books/essential_exponential.html

LINKS TO DATA AND INFORMATIONGapminder http://www.gapminder.org/World Hunger Map Linkhttp://www.wfp.org/hunger/downloadmapLink to Data for Uganda http://faostat.fao.org/site/609/DesktopDefault.aspx?PageID=609#ancorMy Contact Information:Christine Belledin – NC School of Science and Mathematicsbelledin@ncssm.eduFor copies of the presentation and other materials, please visit http://courses.ncssm.edu/math/talks/conferences/ after Monday, November 4.

Thank you for attending!