Algebra II: Strand 6. Exponentials and Logarithms; Topic 2. Applications; Topic Notes
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STRAND 6. EXPONENTIALS AND LOGARITHMS
TOPIC 6.2: APPLICATIONS
Topic Notes
Mathematical focus
This topic focuses upon mathematical modeling and applications that use exponential and
logarithmic functions.
Topic overview
This topic contains 3 tasks:
6.2.1: E. coli Growth Rate
6.2.2: Bacteria Growth
6.2.3: Exponential Decay
The tasks in Topic 6.2 focus upon exploring bacteria growth as a model of population
growth in general and also on exponential decay as a model for cooling or heating
processes.
TExES standards focus
TExES Standard II.004 Patterns and algebra. The teacher uses patterns to model and
solve problems and formulate conjectures. The beginning teacher:
(A) Recognizes and extends patterns and relationships in data presented in tables,
sequences, or graphs.
(B) Uses methods of recursion and iteration to model and solve problems.
TExES Standard II.008 Patterns and algebra. The teacher understands exponential
and logarithmic functions, analyzes their algebraic and graphical properties, and
uses them to model and solve problems. The beginning teacher:
(A) Recognizes and translates among various representations (e.g., written,
numerical, tabular, graphical, algebraic) of exponential and logarithmic functions.
(D) Uses a variety of representations and techniques (e.g., numberical methods,
tables, graphs, analytic techniques, graphing calculators) to solve equations,
inequalities, and systems involving exponential and logarithmic functions.
(E) Models and solves problems involving exponential growth and decay.
(H) Uses the exponential function to model situations and solve problems in
which the rate of change of a quantity is proportional to the current amount of the
quantity [i.e., f !(x) = kf (x) ].
TExES Standard V.018 Mathematical processes and perspectives. The teacher
understands mathematical reasoning and problem solving. The beginning
teacher:
(F) Evaluates how well a mathematical model represents a real-world situation.
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
Algebra II: Strand 6. Exponentials and Logarithms; Topic 2. Applications; Topic Notes
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TExES Standard V.019 Mathematical processes and perspectives. The teacher
understands mathematical connections both within and outside of mathematics
and
`how to communicate mathematical ideas and concepts.
(B) Understands how mathematics is used to model and solve problems in other
disciplines (e.g., art, music, science, social science, business).
TEKS, TAKS focus
TEKS 2A.1 Foundations for functions. The student uses properties and attributes of
functions and applies functions to problem situations.
TEKS 2A.11 Exponential and logarithmic functions. The student formulates
equations and inequalities based on exponential and logarithmic functions, uses a
variety of methods to solve them, and analyzes the solutions in terms of the
situation. The student is expected to:
(D) determine solutions of exponential and logarithmic equations using graphs,
tables, and algebraic methods; and
(F) analyze a situation modeled by an exponential function, formulate an equation
or inequality, and solve the problem.
TAKS Objective 1: The student will describe functional relationships in a variety of ways.
TAKS Objective 2: The student will demonstrate an understanding of the properties and
attributes of functions.
TAKS Objective 3: The student will demonstrate an understanding of linear functions.
TAKS Objective 5: The student will demonstrate an understanding of quadratic and
other nonlinear functions.
TAKS Objective 9: The student will demonstrate an understanding of percents,
proportional relationships, probability, and statistics in application problems.
TAKS Objective 10: The student will demonstrate an understanding of the mathematical
processes and tools used in problem solving.
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
3
Algebra II: Strand 6. Exponentials and Logarithms; Topic 2. Applications; Topic Notes
Materials
Task 1
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Graphing Calculator
Graph Paper
Large Grid Paper
One meter stick
Thermometers or data collection device with a temperature
probe
Pint-size plastic containers or 8-oz Styrofoam cups
Hot water
Stop watches
Transparencies
Transparencies pens
Task 2
*
Task 3
*
*
*
*
*
*
*
*
*
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Procedure
Participants work in groups of 3 or 4. Some background questions (provided on instructor
transparencies) provide a setting for the situations in which participants will investigate
the use of exponential mathematical models.
Summary
The central focus of these tasks is to give participants experience with various growth
situations that can be modeled with exponential models. They see that repeated addition
results in a linear model, while repeated multiplication results in an exponential model.
Extensions
Use Census Bureau data to investigate human population growth patterns.
Assessments
After completing Task 6.2.1, participants may be asked to complete the following set of
questions for homework or as a group assessment:
The equation below represents E. coli growth data under realistic conditions, where y
represents the Klett Value and x represents the elapsed time in minutes.
y=
215.03
(1+ 248.52e )
!.03x
1. Using a graphing calculator, sketch a graph of the Klett Value versus Elapsed Time.
Describe the plot.
2. Describe the bacteria growth, based on the plot. Compare to the previous Klett value
plots.
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
Algebra II: Strand 6. Exponentials and Logarithms; Topic 2. Applications; Topic Notes
4
Transitions to the classroom
Which of the tasks are appropriate only for teachers? Which for students? How would we
modify a teacher-task to make it student appropriate?
Teacher use only
Task 1
Task 2
Task 3
Modify for students
*
Ready for students
*
*
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.