**Core 120**

**Mathematical
Ideas**

**Revised: 11/07**

**Prerequisite: CORE 098 Mathematical Skills**

All students enrolled in this course should have taken CORE 098 Mathematical Skills and attained a minimum grade of “C” in that course, or they should have been exempted from the requirement of taking Core 098 by the mathematics department.

**Course Description**: In order to fully participate in
society today, a person must have knowledge of the contributions of
mathematics. Mathematics has become an indispensable tool for analysis,
quantitative description, decision-making, and the efficient management of both
private and public institutions. Consequently, a familiarity with essential
concepts of mathematics is necessary for one to function intelligently as both a
private individual and a responsible citizen. As such, this course is divided
into four units, each covering an aspect of mathematics that is conceptually
significant and highly relevant. The first unit deals with issues of fairness
and strategy in voting and elections. In the second, students learn about
collecting, organizing, interpreting, and presenting statistical data. The
third unit involves the use of mathematics to solve problems related to
organizing and managing complex activities, and a final unit on symmetry and
fractal geometry establishes connections between mathematics and art and
highlights some applications. On some occasions, units on other suitable topics
may replace those denoted here.

**Course Objectives: **

In this course, all students will gain experience with and increase proficiency in working with mathematics on a conceptual level, and develop an appreciation for the utility of mathematics. Specifically, students will

1) Construct and analyze good examples and counterexamples.

2) Communicate using precise, technical terminology.

3) Look at the same problem in multiple ways.

4) Prove statements.

5) Construct and apply algorithms.

6) Create structures and systems that model problems and information.

7) Develop and apply abstractions of concrete ideas.

8) Appreciate the relevance and significance of math in the world around them, especially in unexpected areas.

**Teaching Procedures:**

Classes are mainly composed of lecture/discussion and group work. Homework is discussed in class regularly. In addition, we frequently use computer demonstrations in class to illustrate mathematical ideas and techniques.

**Assessment:**

**A) Exams**

This course consists of four
units. At the end of each unit, there will be a one-period in-class test. The
last such test will be given during the final exam period. There will be no
comprehensive final exam. Each test will cover the material examined during the
preceding unit. Questions will not only be computational in nature, but rather
they will also require students to offer written explanations of and logical
arguments based on the various concepts from the unit. **Each exam will be
worth 20% of the final grade. **

**B) Reflection on the Objectives**

On the last day of class, students
will write a reflection on the course objectives. The eight objectives relate
to critical thinking skills that are transferable throughout a liberal arts
education. As a result, it is necessary that students understand these skills
and their power. In order to ensure this, on the last day of class, the
instructor will give a written exam in which students will be required to
reflect on the course objectives. The instructor will choose four of the eight
objectives and ask students to carefully explain what these objectives mean and
provide an example of how they were achieved in the course. **This reflection
will be worth 10% of the final grade.**

**C) Class Participation and
Attendance**

Persistence and commitment are
necessary if one intends to achieve her or his full potential, and they are
necessary attributes for success in mathematics. Students are expected to
prepare for, attend, and participate in every class. If a student must miss a
class, she or he should inform the instructor why she or he will not attend by
calling or emailing the instructor prior to the class. Unexcused absences will
negatively affect the final grade. **Class participation and attendance is
worth 10% of the final grade.**

**Textbook:** P. Tannenbaum, *Excursions in Modern
Mathematics*, 6^{th} edition, Pearson Prentice Hall, 2007. ISBN:
0-13-187363-6

**Make-Up Exams:**

Make-up exams will only be given as a result of extreme circumstances, such as sudden accidents, illnesses, and court appearances. If you are unable to attend an exam, then you should inform the instructor prior to the exam time. The instructor determines whether or not you should be given a make-up exam.

If the instructor decides to give a student a make-up exam, then the exam is scheduled at the instructor’s discretion. Make-up exams will not be the same as the original exam, and in general they will be more challenging than the original exam. This is not a punishment, but rather a matter of fairness. No two exams have the same difficulty level, and as a matter of fairness, the more challenging exam should go to the person who is being granted an exception from following the ordinary course schedule.

** **

**Special Requirements:**

You will be allowed to use a calculator on exams. Only a basic scientific calculator is necessary for this course. While you may use a graphing calculator, there is certainly no advantage to having one.

**Outline of Material Covered (tentative):**

I.
**The Mathematics of Social Change**

This deals with mathematical applications in social science. How do groups make decisions? How are elections decided? How can power be measured? When there are competing interests among members of a group, how are conflicts resolved in a fair and equitable way?

1. Voting. The Paradoxes of Democracy

1.1 Preference Ballots and Preference Schedules

1.2 The Plurality Method

1.3 The Borda Count Method

1.4 The Plurality-with-Elimination Method

1.5 The Method of Pairwise Comparisions

1.6 Breaking Ties

1.7 Ranking

2. Weighted Voting Systems. The Power Game

2.1 Weighted Voting Systems

2.2 The Banzhaf Power Index

II.
**Statistics**

In one way or another, statistics affects all of our lives. Government policy, insurance rates, our health, and our diet are all governed by statistical laws. This section deals with some of the basic elements of statistics. How are statistical data collected? How are they summarized so that they say something intelligible? How are they interpreted? What are the patterns of statistical data?

1. Collecting Data. Censuses, Surveys, and Clinical Studies

1.1 Sizing up the Population

1.2 Census versus Survey

1.3 Case Study 1. Biased Samples: The 1936 Literary Digest Poll

1.4 Case Study 2. Quota Sampling: The 1948 Presidential Election

1.5 Random Sampling

1.6 Case Study 3. Stratified Sampling: Modern Public Opinion Polls

1.7 Case Study 4. Counting the Uncountable: The U.S. Census

1.8 Clinical Studies

1.9 Case Study 5. Controlled Experiments: The 1954 Salk Vaccine Fields Trials

2. Descriptive Statistics: Graphing and Summarizing Data

2.1 Graphical Representation of Data

2.2 Variables: Quantitative and Qualitative. Continuous and Discrete

2.3 Numerical Summaries of Data

2.4 Measures of Spread

III.
**Management Science**

This deals with methods for solving problems involving the organization and management of complex activities – that is, activities involving either a large number of steps and/or a large number of variables (building a skyscraper, putting a person on the moon, organizing a banquet, scheduling classrooms at a big university, etc.). Efficiency is the name of the game in all these problems. Some limited or precious resource (time, money, raw materials) must be managed in such a way that waste is minimized. We deal with problems of this type (consciously or unconsciously) every day of our lives.

1. Euler Circuits. The Circuit Comes to Town

1.1 Assorted Routing Problems

1.2 Graphs

1.3 Euler’s Theorems

1.4 Fleury’s Algorithm

1.5 Graph Models

1.6 Eulerizing Graphs

2. The Traveling Salesman Problem. Hamilton Joins the Circuit

2.1 Hamilton Circuits

2.2 The Traveling Salesman Problem

2.3 Solving the Traveling Salesman Problem

2.4 Efficiency Considerations

2.5 The Nearest Neighbor Algorithm

2.6 The Cheapest Link Algorithm

IV.
**Growth and Symmetry**

This deals with nontraditional geometric ideas. What do sunflowers and seashells have in common? How do populations grow? What are the symmetries of a pattern? What is the geometry of natural (as opposed to artificial) shapes? What kind of geometry lies hidden in a cloud?

1. Symmetry of Motion. Mirror, Mirror off the Wall

1.1 Symmetry of Motion

1.2 Rigid Motions

1.3 Reflections

1.4 Rotations

1.5 Translations

1.6 Glide Reflections

1.7 Symmetry of Motion Revisited

1.8 Classifying Patterns

2. Symmetry of Scale and Fractals. Fractally Speaking

2.1 The Koch Snowflake

2.2 The Sierpinski Gasket

2.3 The Chaos Game

2.4 Symmetry of Scale in Art and Literature

2.5 The Mandelbrot Set

2.6 Fractals

2.7 Fractal Geometry, a New Frontier