Classical Algebra for Honours Math

Created by: University of Waterloo, powered by Möbius

Description

The Classical Algebra for Honours Math Content Pack is a Möbius course developed by the University of Waterloo that you can use as a customizable starting point to a complete classical algebra course in Möbius. This Content Pack covers a typical first-year classical algebra course centered on proof—the defining property of mathematics. This Content Pack provides an introduction to the language of mathematics and proof techniques through a study of the basic algebraic systems of mathematics: integers, integers modulo n, rational numbers, real numbers, complex numbers, and polynomials. This customizable resource contains 14 units of sectioned lessons and assignments enhanced with Möbius capabilities including algorithmic questions, in-lesson questions with unlimited practice, adaptive questions, interactive narratives, videos, immediate feedback, and end-of-section assignments.

How does Möbius take the University of Waterloo’s content to the next level?

  • Lessons contain interactive elements like Interactive Narratives, HTML objects, Math Apps, or Geogebra applications to help solidify difficult STEM concepts.

  • Learn how Möbius’ unique STEM question types throughout this content provide the best STEM learning experience.

  • Work with over 50 configurable assessment properties when modifying existing or building your own assessments.

Course Structure

  • All content is organized into 14 units for easy navigation
  • Course materials provide a solid foundation for creating your course offering which include 76 lessons that promote Active Learning by including illustrative visualizations, interactive elements, and example problems with immediate feedback
  • Different forms of assessment materials are distributed across 11 assignments to evaluate student comprehension through a variety of different question types
  • Pull from a vast selection of over 530 questions that you can use to create your own lessons and assignments or supplement existing ones
14 units
76 lessons
11 assignments
530+ questions

Unit 0: Getting Started

Unit 1: Introduction to the Language of Mathematics

  • 1.1 The Language

  • 1.2 Sets

  • 1.3 Mathematical Statements and Negation

  • 1.4 Quantifiers and Quantified Statements

  • 1.4.1 Universal and Existential Quantifiers

  • 1.4.2 Quantifiers and Language

  • 1.4.3 Negation of Quantifiers

  • 1.5 Nested Quantifiers

  • 1.5.1 Two Quantifiers

  • 1.5.2 An Arbitrary Number of Quantifiers

  • 1.5.3 Negation of Nested Quantifiers

  • 1.6 Practice Problems

Unit 2: Logical Analysis of Mathematical Statements

  • 2.1 Truth Tables and Negation

  • 2.2 Conjunction and Disjunction

  • 2.3 Logical Operators and Algebra

  • 2.4 Implication

  • 2.5 Converse and Contrapositive

  • 2.6 If and Only If

  • 2.7 Practice Problems

Unit 3: Proving mathematical Statements

  • 3.0 Introduction

  • 3.1 Proving Universally Quantified Statements

  • 3.2 Proving Existentially Quantified Statements

  • 3.3 Proving Implications

  • 3.4 Divisibility of Integers

  • 3.4.1 Transitivity of Divisibility

  • 3.4.2 Divisibility of Integer Combinations

  • 3.5 Proof by Contrapositive

  • 3.6 Proof by Contradiction

  • 3.6.1 Proving Uniqueness

  • 3.7 Proving If and Only If Statements

  • 3.8 Practice Problems

Unit 4: Mathematical Induction

  • 4.1 Notation for Summations, Products and Recurrences

  • 4.2 Proof by Induction

  • 4.3 The Binomial Theorem

  • 4.4 Proof by Strong Induction

  • 4.5 Practice Problems

Unit 5: Sets

  • 5.1 Introduction

  • 5.2 Set-builder Notation

  • 5.3 Set Operations

  • 5.4 Subsets of a Set

  • 5.5 Subsets, Set Equality and Implications

Unit 6: The Greatest Common Divisor

  • 6.1 The Division Algorithm

  • 6.2 The Greatest Common Divisor

  • 6.3 Certificate of Correctness and Bézout's Lemma

  • 6.4 The Extended Euclidean Algorithm

  • 6.5 Further Properties of the Greatest Common Divisor

  • 6.6 Prime Numbers

  • 6.7 The Unique Factorization Theorem

  • 6.8 Prime Factorizations and the Greatest Common Divisor

  • 6.9 Practice Problems

Unit 7: Linear Diophantine Equations

  • 7.1 Linear Diophantine Equations in Two Variables

  • 7.2 Finding All Solutions in Two Variables

  • 7.3 Practice Problems

Unit 8: Congruence and Modular Arithmetic

  • 8.1 Congruence

  • 8.2 Elementary Properties of Congruence

  • 8.3 Congruence and Remainders

  • 8.4 Linear Congruences

  • 8.5 Non-Linear Congruences

  • 8.6 Congruence Classes and Modular Arithmetic

  • 8.7 Fermat's Little Theorem

  • 8.8 The Chinese Remainder Theorem

  • 8.9 Splitting a Modulus

  • 8.10 Practice Problems

Unit 9: The RSA Public-Key Encryption Scheme

  • 9.1 Public-Key Cryptography

  • 9.2 Implementing the RSA Scheme

  • 9.3 Proving that the RSA Scheme Works

  • 9.4 Practice Problems

Unit 10: Complex Numbers

  • 10.1 Standard Form

  • 10.2 Conjugate and Modulus

  • 10.3 The Complex Plane and Polar Form

  • 10.4 De Moivre's Theorem

  • 10.5 Complex n-th Roots

  • 10.6 Square Roots and the Quadratic Formula

  • 10.7 Practice Problems

Unit 11: Polynomials

  • 11.1 Introduction

  • 11.2 Arithmetic with Polynomials

  • 11.3 Roots of Complex Polynomials and the Fundamental Theorem of Algebra

  • 11.4 Real Polynomials and the Conjugate Roots Theorem

  • 11.5 Integer Polynomials and the Rational Roots Theorem

  • 11.6 More Examples for Roots and Factoring

  • 11.7 Practice Problems

Unit 12: Additional Material

  • 12.1 Prime Numbers and the Riemann Hypothesis

Quizzes

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