Prefix, Number and Name of Course: MAT 114 Functions and
Modeling
Credit Hours: 3
In Class Instructional
Hours: 3 Labs: 0 Field
Work: 0
Catalog
Description:
Prerequisite: 3 years high school mathematics or equivalent.
Describe and explore
real-world functions, data and phenomena through graphic, numeric, symbolic,
and verbal representations. Use elementary functions (linear, polynomial,
power, and exponential) to investigate and analyze applied problems (supported
by the use of appropriate technology)
Reasons
for Addition or Revision: Recent
recommendations from the Committee on the Undergraduate Program in Mathematics
(CUPM), a committee of the Mathematical Association of America, emphasize using
functions and modeling to develop quantitative literacy in students. Their
summary statement advocates offering an introductory level mathematics course
that provide students with
Éa college level academic experience
that emphasizes the use of algebra and functions in problem solving and
modeling, provides a foundation in quantitative literacy, supplies the algebra
and other mathematics needed in partner disciplines, and helps meet
quantitative needs in, and outside of, academia. Students address problems
presented as real world situations by creating and interpreting mathematical
models. Solutions to the problems are formulated, validated, and analyzed using
mental, paper and pencil, algebraic, and technology-based techniques as
appropriate. (CUPM/CRAFTY, 2007)
This new
course is built around those recommendations with an emphasis on using linear,
polynomial, power, and exponential functions to model real-world phenomena and
on the effective communication of conclusions about the world from those
models. The course targets students who
have successfully completed three years of high school mathematics (the current
NYS graduation requirement) and will prepare them for other college level
mathematics-based courses including partner disciplinesÕ statistics courses.
Furthermore, this course will prepare students for MAT 124–Precalculus,
which the department is currently revising to align its topics with those
developed in MAT 114.
|
Student Learning Outcomes: Students will: |
Content
Reference |
Assessment: |
|
1. demonstrate comprehension of the function concept
through multiple representations (verbal, numeric, graphic, and symbolic) and
functional fluency by utilizing the appropriate representations to explore
and describe functional relationships. |
I – V |
1. Group work and classroom activities, individual
assignments, quizzes, exams, projects |
|
2. identify and describe patterns in linear, polynomial,
power, and exponential functions and how they are evident in the various
representations including tables of data, graphs, and equations. |
I – V |
2. Group work and classroom activities, individual
assignments, quizzes, exams, projects |
|
3. identify appropriate input values (domain) and output
values (range) natural to a given context, determine inputs for which the
function values increase, decrease or remain constant, find inputs resulting
in a maximum or a minimum output value, and identify inputs which result in
outputs that are less than or greater than a given value using algebra and
technology. |
I – V |
3. Group work and classroom activities, individual assignments, quizzes, exams, projects |
|
4. determine, in contexts involving more than one
function, appropriate input values for which the functions are equal,
appropriate interval(s) of values for which one function is greater than the
others, and significant points and values relevant to the specific context. |
I – V |
4. Group work and classroom activities, individual
assignments, quizzes, exams, projects |
|
5. use patterns to find exact or approximate equations
that model the data, manually and using technology, to solve for relevant
values, discuss trends and long-term behavior, and explain rates of change in
linear, polynomial, power, and exponential functions. |
I – V |
5. Group work and classroom activities, individual
assignments, quizzes, exams, projects |
|
6. analyze and describe results of investigations of applied
problems and data using appropriate mathematical language and notation with
linear, polynomial, power, and exponential functions on domains natural to
the application. |
I – V |
6. Group work and classroom activities, individual
assignments, quizzes, exams, projects |
|
7. identify real-world phenomena, and describe
characteristics of those phenomena, that can be approximately modeled by
linear, polynomial, power, and
exponential functions |
I – V |
7. Group work and classroom activities, individual
assignments, quizzes, exams, projects |
Course
Content: Note: The content listed below
cannot be separated from the pedagogy implied in the above objectives. That is,
the content stems from studentsÕ active exploration of data and models in
context with technology serving as an integral part of their learning.
Functions
introduced through applications are the main focus of the course.
I.
Functions and Models
A. Function concept
i. Numeric and symbolic representation: definition, notation,
evaluation, domain, range, dependent/independent variable, linear, nonlinear,
ii. Graphic representation: vertical line test; intervals where
positive, negative, increasing, decreasing; concavity, rate of change, maximum,
minimum, intercepts, zeros
iii. Verbal representation: develop numeric, symbolic, and
graphic representations from verbal descriptions (i.e., word problems); develop
verbal representation (i.e., construct meaning) from graphic, symbolic, and
numeric representation
B. Mathematical models
i. Concept, extract mathematics from real-world context to
develop models
ii. Scatterplots, approximate models
iii. Distinguishing appropriate models, limitations,
compare/contrast characteristics
iv. Recursive representation of arithmetic, geometric sequences
and related models
II.
Linear Models
A. Review of linear functions (as required, i.e. just in time
teaching)
i. Direct variation, slope as rate of change, intercepts,
point-slope form, slope-intercept form, standard form
ii. Solutions to equations and inequalities- algebraically,
graphically, and via tables
B. Linear models and applications
i. Develop linear models of phenomena from business and
economics, and the physical,
life,
and social sciences
ii. Models of direct variation, constant rate of change
(successive differences)
iii. Interpret linear models, assess linearity, interpolation,
extrapolation
iv. Linear regression (least square, median-median on
calculator), residuals, residual
plots, correlation coefficient,
correlation vs. causality
v.
Solutions to linear systems of
equations and inequalities graphically and
algebraically (matrix solutions
optional), optimization
III.
Polynomial Models
A. Review of quadratic functions (as required, i.e. just in
time teaching)
i. Standard form, vertex form, axis of symmetry
ii. Solving quadratic equations, factoring, completing the
square, zeros (algebraically,
graphically,
table), quadratic formula
B.
Polynomial models and applications
i. Develop polynomial models of phenomena from business and
economics, and the physical, life, and social sciences
ii. Models of direct variation with power of x, assess
quadratic models– second-order differences, linear rate of change, cubic
models– third-order differences, etc.
iii. Polynomial regression (calculator),
interpret models- significant values, limitations
IV.
Power Models
A. Review of power functions (as required, i.e. just in time
teaching)
i. Exponents, scientific notation, order of magnitude
ii. Roots and radical exponents
B. Power models and applications
i.
Develop power models of phenomena
from business and economics, and the
physical, life, and social sciences
ii.
Models of inverse variation with
power of x, rational exponents and
radicals,
extraneous roots
iii. Power regression (calculator),
interpret models- significant values, limitations
V.
Exponential Models
A. Review of exponential functions (as required, i.e. just in
time teaching)
i. General form (symbolic and graphic), growth, decay
ii. Properties including domain, range, intercept
B. Exponential models and applications
i.
Develop exponential models of
phenomena from business and economics, and the
physical, life, and social sciences
ii. Models of constant proportional rate of change (successive
quotients), long term
implications
iii. Exponential regression (calculator), interpret models-
significant values, limitations
Resources:
Classic
Scholarship in the Field:
Cohen, D.
(Ed.), Crossroads in Mathematics:
Standards for Introductory College
Mathematics
Before
Calculus. MAA, 1995. (http://www.imacc.org/standards/)
Demana, F., Waits, B., Clemens,
S., and Greene, M., Intermediate Algebra:
A Graphing
Approach. Reading, MA, Addison Wesley, 1994.
Keedy, M. and Bittinger, M.,
College Algebra: A Functions Approach.
Reading, MA.,
Addison-Wesley,
1978.
Koshy, T., College Algebra and
Trigonometry. New York, NY, McGraw-Hill, 1986.
Larson, R., Hostetler, R., and
Neptune, C., Intermediate Algebra: Graphs and Functions.
Lexington,
MA, D.C. Heath, 1994.
McKeague, C., Intermediate Algebra
with Trigonometry. New York, NY, Academic Press, 1983.
Nancil, C., College Algebra and
Trigonometry. New York, NY, MacMillan, 1983.
Sobel, M., and Lerner, N., Algebra
and Trigonometry. Englewood Cliffs, NJ, Prentice Hall, 1987.
Swokowski,
E., Fundamentals of Algebra and Trigonometry. Boston, MA,
Prindle, Weber, and Schmidt,
1987.
Current
Scholarship in the Field:
Angel, A., Semmler, R., and Runde,
D., Intermediate Algebra for College Students.
Upper
Saddle River, NJ: Prentice Hall, 2003.
Bittinger, M., Beecher, J.,
Ellenbogen, D., and Penna, J., Fundamentals of College Algebra:
Graphs
and Models. Reading, MA, Addison-Wesley, 2001.
Bittinger, M., Ellenbogan, D., and
Johnson, B., Elementary and Intermediate Algebra:
Concepts
and Applications. Boston, MA, Pearson Education, Inc., 2006.
Burgis, K. and Morford, J.,
Investigating College Algebra and Trigonometry with Technology.
Key
College, 2006.
Crauder, B., Evans B., and Noell
A., Functions and Change: A Modeling Approach to College
Algebra.
Houghton Mifflin, 2007.
Committee on the Undergraduate
Program in Mathematics Curriculum Guide 2004, MAA
Larson, R.,
Hostetler, R., and Edwards, B., Algebra and Trigonometry, A Graphing Approach.
Houghton Mifflin, 2008.
Small, D., An Urgent Call to Improve Traditional
College Algebra Programs. MAA, 2007
Yoshiwara, K.
and Yoshiwara, B., Modeling, Functions, and Graphs: Algebra for College
Students. Thomson, 2007.
Periodicals:
American Mathematics Monthly
College Mathematics Journal
Math Horizons
Mathematics Magazine
Mathematics Teacher
Electronic or Audiovisual
Resources:
MAAÕs Committee on the Undergraduate Program in Mathematics (CUPM) Illustrative Resources: http://www.maa.org/cupm/ill_ref/part2/A.html
College Algebra Reform Papers: http://www.oswego.edu/nsf-precalc/CollegeAlgebraReform