Exponential Models
UNIT OVERVIEW In many
quantitative problems, key variables are related in patterns that
are described well by linear models. But there are many other important
situations in which
variables are related by nonlinear patterns. Among the most important nonlinear
patterns of
change are those that can be modeled well by rules of the form
. Since the independent
or manipulated variable occurs in such rules as an exponent, those relations are
called
exponential models. Exponential models often are useful in solving problems
involving
change in populations, pollution, temperature, bank savings, drugs in the
bloodstream, and
radioactive materials. In fact, it is the widespread usefulness of exponential
models that has
led us to introduce them in the algebra and functions strand ahead of the more
familiar, but
less applicable, power and polynomial models. These models will be addressed in
Course 2.
Another very strong reason for introducing exponential models at this relatively
early
stage of the curriculum is the fact that the difference equation for exponential
growth, which
is NEXT= NOW ×b, is a natural counterpoint to the difference equation for linear
change,
which is NEXT =NOW +b. Capitalizing on these connections and comparisons as you
and
your students work through this unit will help students begin to develop
recursive (or sequential)
thinking.
Unit 6 Objectives
■ To recognize and give examples of situations in which exponential models are
likely to match the patterns of change that are observed or expected. This
modelrecognition skill should apply to information given in data tables,
graphs, or verbal descriptions of related changing variables.
■ To find exponential rules to match patterns of change in exponential model
situations.
This should include rules in the “y= . . .” and “NOWNEXT” forms.
■ To use exponential rules and graphing calculators or computer software to
produce tables and graphs to answer questions about exponential change of
variables
■ To interpret an exponential function rule in order to sketch or predict the
shape of its graph and the pattern of change in tables of values
■ To describe major similarities and differences between linear and exponential
patterns of change
Variety of Models
and Situations
This unit covers a rich sample of the problems that are
modeled by exponential growth and
decay and extensive analysis of the mathematical properties of those models.
However, not
all students need to master (or even encounter) all of those applications and
properties in their
first exposure. For successful progress in the algebra strand, all students
should study the core
material in Lessons 1, 2, 4, and 5. More ambitious and interested students will
find important
applications and mathematical insights in the material of Lesson 3 on compound
growth , but
coverage is not essential to progress through the future units.
The in dividual lessons are designed to engage students in explo ration of a rich
variety of
situations in which variables change exponentially over time. Following the
instructional
model, it is important that the contextspecific investigations of each lesson
be summarized
and analyzed in class discussions that articulate the key mathematical ideas
embedded in each
context so that students come away from the investigations with some broad
generalizable
ideas, not simply memories of specific problems.
In Lesson 1, “Exponential Growth,” students investigate and model exponential
growth.
This introductory lesson asks students to describe and draw conclusions about
several different
situations in which exponential models are the best fit to patterns of change.
The goal is
to get students sensitive to the existence of patterns different from linear ,
constant additive
rates of change and to begin developing skill in writing rules of the form NEXT
=NOW× b
and based on analysis
of the given situation.
Lesson 2, “Exponential Decay,” presents a variety of problem situations in which
some quantity
is decreasing by a constant factor over each unit of time. The students will see
how this
behavior translates into exponential rules with 0 <b <1 and the related table and
graph patterns.
In Lesson 3, “Compound Growth,” two fundamental types of exponential growth are
introduced. These are growth in populations and in investments, where the growth
rates are
commonly given as percentages.
Lesson 4, “Modeling Exponential Patterns in Data,” involves students in finding
exponential
models for patterns in experimental data, where the exponential pattern is not
exact.
Students do some data collection and study the pattern in the data. Then they
analyze several
other sets of data from contexts where exponential patterns of change might be
expected, and
they use the regression capabilities of calculators or software to find
reasonable models.
See Teaching Masters 161a–161c for Maintenance tasks that students can work
after Lesson 1.
New Approach to
Exponential Models
The focus on exponential functions in this unit represents
a new approach to this topic in
algebra. For many years, elementary algebra courses in high school have included
practice in
evaluating exponential expressions like 5^{2} or 2^{3} and in
using properties of exponents to simplify
exponential expressions like or
. More recently, algebra curricula have begun
to emphasize the patterns of change that are implied by those variable
exponential expressions.
Using graphing calculators as tools, students have less need for the rules of
formal symbol
manipulation when solving practical applications of exponential relations. For
this reason,
this early algebra unit develops only a few simple properties of exponents.
Those properties
appear near the end of the unit, at a point when students’ rich prior encounters
with exponential
change and expressions should make using the familiar formal rules quite
natural.
Knowledge of formal properties for exponents and ability to use them in
reasoning by
symbol manipulations will be developed further at several subsequent points in
the curriculum.
Exponents occur again in the “Power Models” unit of Course 2. At that point the
familiar
list of rules for operating with symbolic exponential expressions (including
negative and
fractional exponents) is examined. In that unit and in other geometry and
probability units,
students will encounter a variety of examples illustrating exponential patterns
of change.
Extensive symbolic manipulation skills will be learned in Courses 3 and 4 of the
Contemporary
Mathematics in Context (CMIC) curriculum. Practice of skills is available in the
Teaching Resources and the Reference and Practice books for Courses 13.
Comparing Linear and
Exponential Models
Exponential models have several key properties that make
them useful for describing and reasoning
about common patterns of change. While linear models match patterns of change at
a
constant additive rate with graphs that are straight lines, exponential models
match patterns
of change at a constant multiplicative rate with graphs that are curves. For
instance, consider
the following comparisons of two basic linear and exponential models:
Linear Model 
Exponential Model 
Rules:
NEXT= NOW ×2 
NEXT= NOW ×2 
Tables: 




Graphs:
To match patterns of decrease by a constant multiplicative
factor, the rules for exponential
change would be or
NEXT= NOW× b with values of b between 0 and 1.
(Exponential decay is the focus of Lesson 2.) The graphs of exponential decay
decrease in a
pattern that is asymptotic to the xaxis as the following example shows:
Linear Model 
Exponential Model 
Rules: y= 16 – 2x; NEXT
=NOW –
2 
NEXT= NOW× 0.5 
Tables: 




Graphs:
Exponential models of growth or decay are useful when the
change in a quantity over time
occurs at a pace that is proportional to the size of the quantity. For instance,
the population of
China is nearly 1.4 billion and the population of the United States is only
about 0.29 billion.
Both countries have population growth of about 1% per year, but that 1 percent
converts into
14 million people per year in China and only 2.9 million people in the United
States.
Similarly, two bank deposits of $1,000 and $5,000 earning the same 4% interest
will increase
in one year by $40 and $200, respectively.