Exploring Exponential Growth
North Carolina Council of Teachers of Mathematics 43rd
Annual State Conference
Christine Belledin
belledin@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:
1.5 million
• Percent of world population considered to
be starving:
33%
• Number of people who will die from hunger
today:
20,866
• Number of people who will die of hunger
this year:
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.
Grains
Grain Produced in 1000's of tons
3000
2500
2000
1500
1000
500
0
0
2
4
6
8
Years Since 1995
10
12
14
LINEAR FUNCTION
Grains
3000
Y=61.255x+1899.7
2500
2000
y = 61.255x + 1899.7
1500
1000
500
0
0
2
4
6
8
10
12
14
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
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
Population
20.7
21.2
21.9
22.5
23.2
24.0
24.7
25.5
26.3
27.2
28.2
29.2
30.3
31.4
32.4
CREATE A SCATTER PLOT OF THE DATA
Population of Uganda
35
Popuation in millions
30
25
20
15
10
5
0
0
2
4
6
8
10
Years Since 1995
12
14
16
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 DATA
We 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.
2.
3.
4.
5.
6.
7.
8.
Make sense of problems and persevere in solving them
Reason abstractly and quantitatively
Construct viable arguments and critique the reasoning of
others
Model with mathematics
Use appropriate tools strategically
Attend to precision
Look for and make use of structure
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 Mathematics
belledin@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!