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# Introduction to Mathematical Modeling

1. COURSE DESCRIPTION. Mathematical modeling uses graphical, numerical, symbolic, and verbal
techniques to describe and explore real-world data and phenomena. Emphasis is on the use of elementary
functions to investigate and analyze applied problems and questions , on the use of appropriate supporting
technology, and on the effective communication of quantitative concepts and results. THIS COURSE IS
NOT AN APPROPRIATE PREREQUISITE FOR PRECALCULUS OR CALCULUS. Students who must
take precalculus must understand the implications of taking MATH 1101 (See the instructor immediately if
you have any questions).

2. PREREQUISITE. Knowledge of high school algebra II , or equivalent. This includes algebraic
expressions, first degree equations and inequalities, exponents, radicals, solving and graphing linear
equations, factoring quadratic expressions, and other topics.

3. COURSE OBJECTIVES.

Algebra. Students will demonstrate the ability to:

a. Graph points.
b. Graph linear, piecewise linear, exponential, logarithmic, and quadratic equations and
functions. and identify horizontal asymptotes.
c. Determine the equation of a line given two points or one point and the slope.
d. Determine the absolute value of a quantity.
e. Solve and estimate solutions to linear, quadratic, exponential, and logarithmic equations,
including use of the properties of exponents and common and natural logarithms.
f. Solve linear systems of two equations by substitution and elimination, including systems
that have a unique solution, no solution, or many solutions.
g. Simplify expressions using the laws of exponents and logarithms.
h. Calculate average rate of change of any function.
i. Perform arithmetic calculations to answer questions regarding two-variable data
presented in tabular, graphical, or equation form.
j. Express and compare very large and very small numbers using scientific notation and
orders of magnitude.
k. Factor quadratic expressions .
l. Complete the square of quadratic expressions.
m. Express the square root of negative numbers in terms of the imaginary unit, i.
n. Given conversion factors, convert units of measure.
o. Use the quadratic formula to solve quadratic equations

Functions.
Students will demonstrate:

a. The understanding of the definitions of function, domain, range, independent and
dependent variables, and input and output.
b. The ability to determine if tables, graphs, and equations represent functions.
c. The ability to determine the domain and range of functions as mathematical abstractions
or in a physical context.
d. The ability to determine from the graph of a function the values of the independent
variable for which the function increases, decreases, or remains constant.

Linear and piecewise linear functions. Students will demonstrate the ability to:

a. Determine when two real-world variables are related by a linear or piecewise linear
function.
b. Calculate, and interpret average rate of change as slope.
c. Model the behavior of two real-world variables that are directly proportional or are
related by a linear or piecewise linear function using tables, graphs, equations.
d. Evaluate linear and piecewise linear functions.
e. Use a linear function to approximate the value of a non-linear function.
f. Interpret the intersection of the graphs of linear functions as equilibrium points.

Exponential Functions. Students will demonstrate the ability to:

a. Determine when two real-world variables are related by an exponential function.
b. Model the behavior of two real-world variables that are related by an exponential
function using tables, graphs, equations, or combinations thereof including such
applications as population growth and decay, radioactive decay, simple and compound
interest, inflation, the Malthusian dilemma, musical pitch, and the Rule of 70.
c. Change the base of an exponential function to determine rate of growth/decay,
growth/decay factor, and effective and nominal interest rate .
d. Express continuous growth/decay in terms of the number e.
e. Evaluate exponential functions.
f. Determine the exponential equation model from the table or graphical model.
g. Compare linear to exponential growth.

Logarithmic Functions. Students will demonstrate:

a. The ability to determine when two real-world variables are related by a logarithmic
function.
b. The ability to model the behavior of two real-world variables that are related by a
logarithmic function using tables, graphs, equations, or combinations thereof including
such applications as pH and the decibel system.
c. The understanding of the natural logarithm.
d. The ability to graph logarithmic functions.

Quadratic Functions.
Students will demonstrate the ability to:

a. Estimate horizontal intercepts of quadratic functions from their graphs.
b. Determine the horizontal intercepts of quadratic functions in factored form.
c. Determine the vertex, axis of symmetry, and horizontal and vertical intercepts of
quadratic functions in either the a-b-c or a-h-k forms.
d. Convert quadratic functions from the a-b-c form to the a-h-k form and vice versa.
e. Determine when two real-world variables are related by a quadratic function by
calculating the average rate of change of the average rates of change.
f. Model the behavior of two real-world variables that are related by a quadratic function
using tables, graphs, equations, or combinations thereof including such applications as
maximum area for fixed perimeter, minimum perimeter for fixed area, free fall ,
maximum profit, and break-even analysis.

4. COURSE COVERAGE. We will cover the following sections from the text:

Chapter 1 Making Sense of Data and Function (1.1 - 1.5)
Chapter 2 Rates of Change and Linear Functions (2.1 - 2.8)
Chapter 3 When Lines Meet: Linear Systems (3.1 - 3.2, 3.4)
Chapter 4 The Laws of Exponents and Logarithms: Measuring the Universe (4.1 - 4.7)
Chapter 5 Growth and Decay: An Introduction to Exponential Functions (5.1 - 5.6)
Chapter 6 Logarithmic Links: Logarithmic and Exponential Functions (6.1 - 6.5)
Chapter 8 Quadratic and Other Polynomial Functions (8.1 - 8.4)

IMPORTANT NOTE: Georgia State University and its faculty are not responsible for outcomes due to
individual technical issues, nor scheduled WileyPlus downtimes. It is expected that the students will be
responsible for completing their work in a timely fashion as to alleviate any pressures these scheduled
downtimes occur. All students will be notified of these downtimes by WileyPlus through the
announcements page of the course.

5. COURSE EVALUATION. Your course grade will be determined as follows:

a. Three midterm tests (closed book/notes, 15% each): 45%
b. WileyPlus online homework and quizzes : 20%
c. Excel Projects: 10%
d. Final Exam (closed book/notes, cumulative): 25%
e. Bonus points: For example, each of the Pre-QL and Post-QL questions carry 0.3 bonus point.

Example of Course Grade Computation:
Test Grades: T1 = 88, T2 = 72, T3 = 68; WileyPlus online homework and quizzes = 85;
Excel Project average = 90; Final Exam = 76; Bonus: 2.7.
Then the overall course score is 0.15*(88+72+68) + 0.20*85 + 0.10*90 + 0.25*76 + 2.7 = 81.9

6. GRADING SCALE. We will use the following grading scale (the letter grade A + is NOT available):

For example, the above score of 81.9 would make a letter grade of B for the course .

7. Makeup Policy: Your final exam grade will replace your missed exam grade. No make-up exams will
be given unless in some extreme situations. Absence from the final exam will result in a grade of F for the
course unless arrangements are made PRIOR (at least one week before the final exam) to its administration.

8. CALCULATOR Policy . You are recommended to have a scientific calculator or a graphing calculator .
If you are not strong in mathematics, I strongly recommend you obtain a graphing calculator. You are not
allowed to share calculator with any other party in your class during any in class quiz or exam, unless
permitted by your instructor.

9. Academic assistance at GSU:

1. Attend academic assistance session, MW 12:00-1:15pm, GCB 401

2. Visit the Math Assistance Complex (MAC), Kell Hall 122 (phone: 404-413-6462).

3. Visit the Counseling Center for Learning assistance, Test anxiety classes, and Student support
services (phone: 404-413-1641)

4. African American Student Services (phone: 404-413-1530)

5. A private tutor list is available at Math Assistance Complex and Math Department

# Arithmetic and Problem Solving

Brief description: A deep examination of topics in mathematics that are
relevant for elementary
school teaching. Problem solving . Number systems:
whole numbers, integers, rational numbers (fractions) and real numbers (dec-
imals) and the relationships between these systems. Understanding multipli-
cation and division, including why standard computational algorithms work.
Properties of arithmetic. Applications of elementary mathematics.

Course Objectives: To strengthen and deepen knowledge and understand-
ing of arithmetic, how it is used to solve a wide variety of problems, and
how it leads to algebra. In particular, to strengthen the understanding of
and the ability to explain why various procedures from arithmetic work.
To strengthen the ability to communicate clearly about mathematics, both
orally and in writing. To promote the exploration and explanation of mathe-
matical phenomena. To show that many problems can be solved in a variety
of ways.

Topical Outline:

Problem solving: Polya’s principles. Writing explanations.

Numbers: The natural numbers, the whole numbers, the rational numbers
(fractions), and the real numbers (decimals). The decimal system and place
value. Representing decimals with bundled objects. Representing decimals
on a number line. Comparing sizes of decimals. Finding decimals in between
decimals. Rounding decimals. The meaning of fractions. The importance
of the whole associated with a fraction. Improper fractions . Equivalent
fractions. Simplest form of a fraction . Fractions as numbers on number
lines. Comparing sizes of fractions: by giving them common denominators,
by converting to decimals , and by cross-multiplying. Using other reasoning to
compare sizes of fractions. Solving fraction problems with the aid of pictures.
Percent. Benchmark percentages and their common fraction equivalents.
Solving percentage problems with the aid of pictures. Solving percentage
problems numerically.

Addition and subtraction : Interpretations of addition and subtraction. The
relationship between addition and subtraction. Explaining why the standard
algorithms for adding and subtracting whole numbers and decimals work.
Using regrouping in situations other than base 10, for example in calculating
elapsed time by replacing 1 hour with 60 minutes. Adding and subtracting
fractions. Explaining why we add and subtract fractions the way we do.
The importance of the whole when adding and subtracting fractions, espe-
cially in story problems. Recognizing and writing story problems for fraction
addition and subtraction. Recognizing story problems that are not solved
by fraction addition or subtraction. Mixed numbers. Understanding when
percentages should and should not be added. Calculating percent increase
and decrease with the aid of pictures. Calculating percent increase and de-
crease numerically. Percent of versus percent increase or decrease. The
commutative and associative properties of addition and their use in mental
arithmetic. Using properties of addition to aid the learning of basic addition
facts. Other (mental) methods for adding and subtracting : rounding and
compensating, subtracting by adding on. Writing equations that correspond
to a mental method of calculation (to demonstrate the connection between
mental arithmetic and algebra).

Multiplication: The meaning of multiplication. Ways of showing multiplica-
tive structure: with groups, with arrays, and with tree diagrams. Using the
meaning of multiplication to explain why various problems can be solved by
multiplying. Explaining why multiplication by 10 is easy in the decimal sys-
tem. Why the commutative and associative properties of multiplication and
the distributive properties make sense and how to illustrate them with arrays,
areas of rectangles, and volumes of boxes. Using properties of arithmetic in
solving arithmetic problems mentally. Writing equations that correspond
to a mental method of calculation (to demonstrate the connection between
mental arithmetic and algebra). Using properties of arithmetic to aid in the
learning of basic multiplication facts. The distributive property and FOIL.
Using multiplication to estimate how many. The partial products multiplica-
tion algorithm. Using pictures and the distributive property to explain why
the standard and partial products procedures for multiplying whole numbers
are valid. Explaining why non-standard strategies for multiplying can be
correct or incorrect, such as explaining why 23 × 23 ≠ 20 × 20 + 3 × 3
and explaining why 32 × 28 = 30 × 30 − 2 × 2. The meaning of multi-
plication for fractions. Recognizing and writing story problems for fraction
multiplication. Recognizing story problems that are not problems for frac-
tion multiplication. Explaining why the procedure for multiplying fractions
works. Powers. (Optional: scientific notation.) Multiplication of decimals:
explaining why the procedure for the placement of the decimal point is valid.
Multiplication of negative numbers. Understanding that multiplication does
not always “make bigger.”

Division: The meaning of division (two interpretations, with or without re-
mainder). Understanding when the answer to a story problem solved by
whole number division is best expressed as a decimal, as a mixed number,
or as a whole number with a remainder. Why dividing by zero is unde-
fined. The scaffold method of division. Explaining why the scaffold and
standard longhand procedure for dividing whole numbers works. Explaining
why some non-standard methods of division are valid. The relationship be-
tween fractions and division: explaining why . Calculating decimal
representations of fractions. Explaining the relationship among remainder,
mixed number, and decimal answers to division problems .

Text: Mathematics for Elementary Teachers , first edition, by Sybilla Beck-
mann, Addison-Wesley, 2005

Chapter 1 Problem Solving, Chapter 2 Numbers, Chapter 3 Fractions, Chap-
ter 4 Addition and Subtraction, Chapter 5 Multiplication, Chapter 6 Multi-
plication of Fractions, Decimals, and Negative Numbers, Chapter 7 Division,
through section 7.3 only.

Math 7001: for graduate credit, students must complete an additional
course project. The project could consist of several essays, or a longer paper,
in which the student discusses some aspect of the course material in depth,
or in which the student relates the course material to their future teaching
(e.g., with a collection of lesson plans or with a discussion of some lesson
plans). However, other creative ideas could also be acceptable. For example,
students might think of a creative way to tie their course project for math
7001 to something they will be doing for one of their other courses.

# Lesson Plan for Ellipse and the Mars Orbital Path

 Lesson Plan Title Ellipse and the Mars Orbital Path Grade Level Algebra II , 8th to 12th grade Concept/ Topic to Teach The characteristic of the ellipse formula will be presented. Transforming the elliptical quadratic formula to the standard elliptical equation will be presented by completing the square . Content Standards 17.0 Given a quadratic equation of the form ax 2 + by2 + cx + dy + e = 0, students can use the method for completing the square to put the equation into standard form and can recognize whether the graph of the equation is a circle, ellipse, parabola, or hyperbola. Students can then graph the equation General Goals Students will learn the geometric characteristic and features of the ellipse equation. Some of the characteristics will be the elliptical focus points, eccentricity, and extremities. Specific Objectives Students will learn that when the eccentricity (e) is between 0 and one, it is an ellipse. If e = 0, it is a circle. The student will learn to convert an elliptical quadratic formula to a more visual equation by completing the square. The student will see the minor differences of the equation from a circle equation. The student will also learn about the other parameters of the elliptical polynomial equation. For a real life application, the student will go through an exercise of determining the Mars’ elliptical orbit. Required Materials Algebra Textbook . Pictures of Elliptical shapes . Internet access for student research. Anticipatory Set The teacher will give a string that is longer than the width of the focus points. Students are to pair up and use the string and a writing instrument to draw an elliptical shape. Students will write down objects have the elliptical shapes Step-by-Step Procedures Teacher will review the quadratic equation. The teacher will review how to represent it with polynomials by completing the square. The teacher will review the quadratic equation of a circle and parabola. The teacher will show a picture of ellipse. The teacher will squeeze a circle to an elliptical shape. The teacher will give the quadratic equation (using the polynomial format) of an ellipse and its characteristics: eccentricity, focus, and extremities. Plan for Guided Practice The teacher will go through an example of translating a quadratic formula into its polynomial format. The teacher will point out the characteristics of the elliptical polynomial quadratic equation The teacher will calculate the eccentricity , Plan for Independent Practice The students will break into groups of three to four and work on textbook problems . They will work on exercises to convert elliptical quadratic equations to its polynomial format. They will identify the focus, extremities of the minor axis, and vertices. They will draw the elliptical shape. They will work as a group to determine the elliptical polynomial equation on the orbital path of Mars. They should seek assistance from their peers first. If they need help outside of their group, the teacher can assist or an advanced student will be asked to help. If they finish early, they can start on their homework assignments. Assessment (based on objectives) The student will be given equations, eccentricity numbers, and geometric figures in a test. The student will be asked to determine which are associated to an ellipse. Quadratic formulas will be given. The student will be asked to translate it to the quadratic polynomial equation of an ellipse. The student will be asked what the eccentricity, the foci, and other elliptical parameters are. The students will be asked to derive the orbital path of Mars given different orbital parameters that differ from its real parameters.. Adaptations (ELL students or special populations) Students will enter in their journal the definition(s) of eccentricity, axis, axes, focal point, focus, and other words they don’t know. They are to write down their questions. The teacher or their peers will answer the questions . They will write down the answers in their journal . They are advise to use the free tutoring classes or come in early or after school for sessions with the teacher Extensions (for gifted students) Students will go on the Internet and research Kepler ’s discovery that the Mars orbital path was elliptical with the Sun being at one of the focus. Students will investigate the orbits of Halley’s comet which has a more elongated elliptical orbit. The students will determine when we will see Halley’s comet again. The students can also determine the intercontinental orbit used for a space vehicle to travel from Earth to Mars.
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