DISCRETE MATHEMATICS 2019-2020

SECTION MQN11

COURSE SYLLABUS

Discrete Mathematics Course Description: In Discrete Mathematics we will use the principles and properties of natural numbers, whole numbers and integers to explore set theory, number theory and graph theory. We begin by reviewing logical thinking and revisiting proofs. Students then apply their ability to reason and think critically to develop algorithms that provide solutions to real questions such as

1. What is the most efficient route to plow all the streets in this neighborhood after a snowstorm (or pick up all the trash)?

2. What is the best way to schedule eight committee meetings without any conflicts, given that some people are on more than one committee?

3. How can we schedule all the tasks on this large project (like a construction project or a new product launch) so that the entire project is finished in the least amount of time?

4. Will there be enough phone numbers available to accommodate all the phones, faxes, and mobile phones in this area?

5. What is the optimal medicine dosage for a patient, in order to maintain the right amount of medicine in the body while it is naturally metabolized?

6. How can we model and analyze a changing population, or the changing amount of money in an investment program?

Students will be provided with a deep understanding of how math works in the real world and to prepare them to solve problems that requires them to present results that go beyond the information given.

Number Theory Course Description: “Mathematics is the queen of sciences and number theory is the queen of mathematics. She often condescends to render service to astronomy and other natural sciences, but in all relations she is entitled to the first rank.”–Carl Friedrich Gauss (1777-1855)

The theory of numbers is concerned, at least in its elementary aspects, with properties of the integers and more particularly with the positive integers 1, 2, 3, … (also known as the natural numbers). The origin of this misnomer dates back to the early Greeks for whom the word number meant positive integer, and nothing else. The goal of this course is to learn a good amount of interesting mathematics, made interesting because we will work out plenty of examples and applications. In addition, in the months of May and June we will focus on solving word problems with extensive practice time built into the classroom period.

A few reasons to study number theory: 1. In some ways the most basic piece of mathematics, for you can build everything else from natural numbers. From there you can get to calculus, topology, etc. “God made the integers, all the rest is the work of man.” –Leopold Kronecker (1823-1891) 2. It includes some of the most elegant mathematics! Number theory uses techniques from algebra, analysis, geometry and topology, logic and computer science, and often drives development in these fields. 3. Has some great applications – eg., RSA public key cryptography, construction of expander graphs, coding theory, etc. 4. It’s a great place to learn how to read and write proofs. 5. It’s a rich source of conjecture which are easy to state and VERY hard to prove. Essential Questions:

• What is discrete mathematics? • How can knowing the properties of numbers help me do calculations in math? • What is number theory?

Instructor: Mr. Goncalves Classroom: Room 401 Schedule: Monday, Tuesday, Thursday, Friday 9th period Contact Information:

• Phone: Office (212) 772-1220 extension 4011 • E-Mail: ggoncalves@erhsnyc.net

Suggested Materials: 1. Karl J. Smith. Nature of Mathematics. 12th ed. Brooks/Cole 2012, Cengage Learning.

2. Burton, David M. Elementary Number Theory. 6th ed. Boston: Allyn and Bacon, 1976. McGraw-Hill.

3. Briggs, William L. Ants, Bikes, & Clocks: Problem Solving for Undergraduates. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2005. Print.

4. Calculator – TI-83 or TI-84 required Each student must have his/her own graphing calculator 5. Pencil and eraser Ink is not allowed on any assignment or test. Exams and Quizzes Exams and quizzes are an important part of our coursework. Exams allow you to assess yourself and allow me to evaluate how the class is progressing. Old exams and quizzes are also excellent study tools for future exams. Be prepared for exams and quizzes; there will be no opportunity for a retake or revision. Be aware of when exams/quizzes are scheduled; you will be expected to take them even if you are absent that week. If you are absent on a quiz/exam day, it is your responsibility to speak with me to set up a day after school to make up the quiz/exam. All exams and quizzes must be made up within 5 days upon return to school, otherwise the quiz/exam will be scored as a zero. To schedule a make up exam you must email your teacher with possible dates and times you are available for the make up. Your teacher will email you a confirmation of your exam make up date and time. Ideally, the make up exam should be scheduled before your return to school. All assessments will be opened notebook. That is, you can use your notebook to assist you during examinations. J Classwork Classwork must be done in class, which includes staying on task, following directions, participation, completing a task, and attendance. Participation is the key element in making our classroom environment constructive and exciting. We will apply the skills we learn to group assignments, individual work and various tasks. To receive a 100% in this category students must complete assigned work in class, stay on task, follow directions, bring all necessary materials to class, and be an active learner. As our classroom is a place of inquiry, discussion and collaboration, we expect students to respect one another and all questions that arise. Absences and lateness with negatively affect your classwork grade. Homework You will be assigned homework weekly. Homework is extremely important in the study of mathematics. You need to be able to complete problems on your own at home in order to master the topic. Late homework will not be accepted except for an absence. If a student is absent, it is the responsibility of the student to show missed homework the same day he/she returns to school (If a student is absent Monday, returns to school on Tuesday, he/she is responsible for showing missed homework on Tuesday). This means that the student can show missed homework the day they return to school or the following day and receive full credit.

ERHS Academic Honesty Requirement: ERHS Academic Honesty Requirements apply to all assigned material in this class. This includes homework, quizzes, tests, projects, extra credit work, and anything else assigned. Failure to do your own work, or providing work to others, will result in a zero for the assignment and a referral to the Principal. Grading Policy: Criteria for computing grades: Weight Quizzes 70% Homework/ Projects 20% Classwork/Attendance/Engagement 10%

Course Outline – Discrete Mathematics: Subject to change Chapter 1 The Foundations: Logic and Proofs

1.1 Propositional Logic 1.2 Applications of Propositional Logic 1.3 Propositional Equivalences 1.4 Predicates and Quantifiers 1.5 Nested Quantifiers 1.6 Rules of Inference 1.7 Introduction to Proofs 1.8 Proof Methods and Strategy **Field trip to Escape the Room NYC

Chapter 2 Basic Structures: Sets, Functions, Sequences, Sums, Matrices and Cryptography

2.1 Sets 2.2 Set Operations 2.3 Functions 2.4 Sequences and Summations 2.5 Cardinality of Sets 2.6 Matrices *Cryptography using Matrices *Linear Programming *Fractal Geometry with BeetleBlocks **Trip to Washington DC to visit the Spy Museum and N.S.A

Chapter 10 Graphs

10.1 Graphs and Graph Models 10.2 Graph Terminology 10.5 Euler and Hamilton Paths 10.6 Shortest-Path Problems **Pacman in the park 10.7 Planar Graphs 10.8 Graph Coloring

Topology and Fractals If time permits

Definition Examples

Introduction to Coding If time permits

SCRATCH

*Handouts will be provided **Field trips will depend on monetary and schedule availability

Course Outline – Number Theory: Subject to change Chapter 1: Preliminaries 1.1 Mathematical Induction 1.2 The Binomial Theorem Chapter 2: Divisibility Theory in the Integers 2.1 Early Number Theory 2.2 The Division Algorithm 2.3 The Greatest Common Divisor 2.4 The Euclidean Algorithm 2.5 The Diophantine Equation ax + by = c Chapter 3: Primes and Their Distribution 3.1 The Fundamental Theorem of Arithmetic 3.2 The Sieve of Eratosthenes* 3.3 The Goldbach Conjecture* Chapter 4: The Theory of Congruences 4.1 Carl Friedrich Gauss 4.2 Basic Properties of Congruence 4.3 Binary and Decimal Representations of Integers 4.4 Linear Congruences and the Chinese Remainder Theorem* Chapter 5: Fermat’s Theorem 5.1 Pierre de Fermat 5.2 Fermat’s Little Theorem and Pseudoprimes Chapter 10: Introduction to Cryptography 10.1 From Caesar Cipher to Pubic Key Cryptography Additional Topics Selected from Chapters 11-15 of the textbook, or other sources. Possible choices include: • Perfect Numbers • Mersenne Primes and Amicable Numbers • Fermat Numbers • Fermat’s Last Theorem • Sum of Two Squares • Fibonacci • Finite and Infinite Continued Fractions • Pell’s Equation • Problem Solving • Group Theory • Binary, Octal, and Hexadecimal numbers *If time permits