Connections in Mathematics

Instructor:  Dr. Stephen C. Smith (ssmith@cvgs.k12.va.us)

Course Description:
This course provides students with introductory experiences in symbolic logic, binary and other bases, probability, conditional probability, set theory, voting schemes and apportionment theory, topics of personal finance and investment, and the calculus necessary to participate in the Senior Science Scenario project during the final weeks of the year.  Both EXCEL and a variety of website-based applications will be used throughout the year.  Emphasis is placed on conceptual understanding, solving real world applications, and fostering mathematical reasoning and communication.

Course Materials:
The text for this course will be “The Nature of Mathematics”, 10th ed., by Smith (no relation).  We will also use "Calculus" by Larson, Hostetler, and Edwards, 5th ed.  You will be responsible for any text while it is in your possession.  

Methodology
The beginning of most class periods will be used to answer questions on the material that is due for that day.  The rest of the class period will consist of a variety of activities which will include lecture, individual and group problem solving, and exploration of questions and concepts.  It is strongly advised that you prepare for each class by working assigned homework problems and by reading and taking notes on the text to be covered in the next class. 

Study Aids:
There are many reference books and web sites widely available that can serve as study aids for this course.  However, it is unlikely that any materials beyond those provided in class will be necessary.  If you feel at any time that you require additional assistance, please discuss this with me at the beginning or end of the next class.

Participation:
You should plan to be actively involved in class.  This means being attentive and participating in class discussions and activities.  

Absences (consult the Student Handbook for additional information):
When you miss any amount of class time, for any reason, it is your responsibility to contact a student colleague in the class to obtain the information you missed.  

Foreseeable absences for any reason need to be discussed with the instructor in advance.  Failure to do so will result in an unexcused absence.

If a student is absent (excused) for only one class meeting, upon return he/she is expected to have completed the work which was due on the day of absence.  If a test was missed, the student is expected to take the test on the day of return.  If a student misses two or more consecutive class meetings, he/she should talk to the instructor to devise a plan to catch up.  

Work missed because of an unexcused absence cannot be made up.  If a test is missed because of an unexcused absence, then that test grade will be lowered by 10 points for each day late.  

Tardiness (consult the Student Handbook for additional information):
You are expected to be in our class, ready to learn, by our starting time.  Given my responsibilities as the Director of the Governor’s School, I might not be in the room; that does not relieve you of your responsibility to be in the class, ready to learn, by the beginning of class.  I will permit one unexcused tardy without any grade penalty.  After that, I may lower your grade for each unexcused tardy.

Honor Code:
Students are required to pledge all work that they turn in for a grade.  The complete pledge will be written out by hand and signed prior to completing the work.  The pledge is as follows:
"I have not given, received, or observed any unauthorized assistance on this assignment."

Grading
The grading scale is a standard 11/10/10/10 point scale.

Percentage and Grade Equivalent:
89.5-100 A
79.5-89.4 B
69.5-79.4 C
59.5-69.4 D
Earning less than 60 points will result in a failing grade for the course.
 
Course Description (First Semester):
During the first semester students will work with introductory experiences in symbolic logic, binary and other bases, voting methods, apportionment schemes and paradoxes, probability, conditional probability, set theory, and non-routine problem solving.  Emphasis is placed on conceptual understanding, solving real world applications, and fostering mathematical reasoning and communication.

Specific Course Content and Objectives (First Semester):
the student will be able to: 

  • translate sentences to symbolic form,
  • construct truth tables,
  • state the converse, inverse and contrapositive of statements,
  • determine the validity of an argument,
  • design a simple circuit (or a gate) as a logic application,
  • understand and use basic set theory concepts, including intersections, unions, complements, distributive and De Morgan’s laws, and cardinality,
  • recall and be able to compare and contrast voting methods, voting dilemmas, apportionment methods and paradoxes
  • convert numbers in the decimal, binary, octal, and hexadecimal systems,
  • recall the processes necessary and properly solve problems related to the Fundamental Counting Principle, permutations, and combinations,
  • compute probabilities and evaluate expected values,
  • compute conditional probabilities,
  • identify and solve problems using an Euler Circuit
  • identify a Hamiltonian Cycle
  • identify and construct a minimal spanning tree
  • solve Traveling Salesman problems
 
The First Semester Grade Will be Determined as Follows (1,000 points):
  1. Tests (5 @ 120 pts = 600 pts)
  2. Test Corrections (5 @ 20 pts = 100 pts) Specific requirements must be met…
  3. Final Exam (1 @ 200 pts) This is a comprehensive semester final.
  4. Projects (3 @ 30 pts = 90 pts)  These are completed individually or in teams.
  5. Class Participation (10 pts): Asking or answering questions well, putting problems on the board, and generally being attentive and engaged
It is your responsibility to keep track of the points you have earned and the assignments you have completed.  Progress reports will report progress for the entire semester thus far.  To reiterate, all grades will be cumulative from the beginning of the semester!
 
Tentative Course Schedule for First Semester:
We will follow the sequence of topics below, although adjustments will be made depending on how quickly we are able to move as a group.
 
2.1: Symbolic Logic
2.2: Truth Tables and Conditionals
2.3: Operators and Laws of Logic
2.4: Logical Proof
2.6: Logic Circuits 
16.1: Voting Methods
16.2 Voting Dilemmas
16.3: Apportionment
16.4: Apportionment Paradox
Project #1
3.4: Binary + Octal + hexadecimal
10.1: Sets, subsets, Venn diagrams
10.2: Sets—combined operations, DeMorgan’s Laws
10.3: Permutations
10.4: Combinations
10.5: Complex Counting
Project #2
11.1: Probability
11.2: Math Expectation
11.3 Probability Models 
11.4: Calculated Probabilities
Project #3
15.1: Euler Circuits and Hamiltonian Cycles
15.2: Trees and Minimum Spanning Trees
Special Topics
REVIEW FOR EXAM EXAMS
 
Course Description (Second Semester)
This course provides students with experiences in topics of personal finance and investment and the use of EXCEL to facilitate calculations such as those used in an amortization schedule.  Students will also learn the calculus basics necessary to participate in the Senior Science Scenario project during the sixth six weeks.  The use of EXCEL and the free website www.Wolframalpha.com will be explored. 

Throughout the course emphasis is placed on conceptual understanding, solving real world applications, and fostering mathematical reasoning and communication.

Specific Course Content and Objectives (Second Semester): the student will be able to:

  • Identify how payroll related taxes affect income; topics include
       Gross, taxable and adjusted income
       Social Security Tax/Medicare, Federal, State, Local payroll taxes
       Tax brackets
  • Identify and estimate major household expenses including:
       Housing, insurance, transportation, and miscellaneous living expenses
       College expenses, home purchasing, having a baby
  • Explore credit card benefits and fees, debit card information, etc.
  • Compute origination fees, discounts, repayments, and total cost of loans
       Compute simple interest
       Compute compound interest (including discrete and continuous compounding)
       Compute APR for add-on interest loans
  • Define the terms “stocks,” “bonds,” “mutual funds,” and “annuities"
       Compute the effective (annual) yield
       Compute average annual returns
       Compute the present and future value (given certain assumptions) of annuities
  • Create an amortization schedule for a home purchase
  • Use Microsoft EXCEL to numerically investigate data
  • Use Wolframalpha.com to investigate various mathematical questions
  • Define, compare, and contrast the average and instantaneous rates of change
  • Perform elementary differentiation
  • Apply differential calculus to solve optimization problems
  • Use Wolframalpha.com to perform elementary integration
  • Solve ordinary differential equations algebraically
  • Solve ordinary differential equations numerically using Euler's Method
  • Complete the Senior Science Scenario Project
The Second Semester Grade Will be Determined as Follows (1,000 points): 
  1. Tests (3 @ 120 pts = 360  pts)
  2. Test Corrections (3 @ 20 pts = 60 pts) Specific requirements must be met…
  3. Final Exam (1 @  200 pts) This is a comprehensive semester final.
  4. Finance Project (1 @ 80 pts = 80 pts)  This is completed independently.
  5. Smaller Projects (3 @ 50 pts = 150 pts)  These are completed independently or in teams.
  6. Class Participation: Asking or answering questions well, putting problems on the board, and generally being attentive and engaged is expected.  Failing to do so can result in a deduction of points from the total points earned.
  7. S-Cubed grade (1@150 pts)

Tentative Course Schedule for Second Semester:
We will follow the sequence of topics below, although adjustments will be made depending on how quickly we are able to move as a group.

Personal Finance (Project 1)
9.1: Simple and Compound Interest
9.2: Installment Loans and Credit Cards
Presentations of Projects
Introduction to Investments
Investments Continued
9.5: Annuities
9.6: Amortization
Smaller Project 1: Buying a Home
2.1: Tangent Lines, Limits, and Derivatives
2.2: Basic Differentiation Rules and Rate of Change
Using Wolframalpha.com
3.1: Finding Global Extrema
Smaller Project 2: Calculus
3.7: Applied Max and Min
3.10: Optimization
4.1: Antiderivatives
Special Topics