Module 1:  Chapter 2

• Read and use standard symbols and terminology for describing and denoting finite and infinite sets, inclusion in a set, set equality, cardinality.
• Be able to read, interpret and properly represent sets using set-builder notation.

• Read and use standard symbols and terminology for describing and denoting subsets, proper subset, complement of a set.
• Use Venn diagrams to represent sets and their relationships.

• Read and use standard symbols and terminology for describing and denoting set union, intersection, difference, equality of sets and cartesian products.
• Describe sets in words.
• Shade Venn diagrams representing various set operations.
• Know and apply deMorgan's Laws, and be able to explain those laws in words and using Venn diagrams.

• Know the cardinal number formula and be able to explain it in words or with Venn diagrams.
• Analyze data from surveys, tables or reports when given sufficient information to reason out the cardinalities of all 8 regions into which 3 sets generally divide the universal set.

(alternatively extension labeled 'Infinite Sets and their Cardinalities')

• Be able, where possible, to define one-to-one correspondences between finite and infinite sets; where this is not possible show why.
• Know and apply the meaning of aleph-null.

Module 2:  Chapter 1

• Distinguish between inductive and deductive arguments.
• Solve problems using inductive and deductive reasoning.

• Apply inductive reasoning to solve problems involving number patterns and figurate numbers.

• Use a variety of problem-solving strategies to solve problems.

• Read, construct and interpret pie graphs, histograms, point graphs and line graphs.
• Use reasonable estimates to quickly approximate the results of arithmetic calculations.

Module 3:  Chapter 3

• Know the definition of the word 'statement' and be able to distinguish between statements and non-statements.
• Symbolize compound statements formed by conjunctions, disjunctions and negations.
• Identify and use universal and existential quantifiers.
• Negate statements formed from universal ('all do' or 'all do not') or existential ('some do' or 'some do not') quantifiers.
• Know the definitions of real numbers, integers, whole numbers, counting numbers, rational numbers, and irrational numbers, and the relationships among these sets.

• Know the rules for determining the truth values of negations, conjunctions and disjunctions.
• Use truth tables, in which each column can be evaluated by reference to at most two other columns, to determine the truth values of compound statements formed by negations, conjunctions and disjunctions of simple statements.
• Validate deMorgan's Laws using truth tables.
• Symbolize compound verbal statements in terms of simple statements, and evaluate the truth values of those statements.

• Know and be able to explain the reasons for the rule used to evaluate the truth value of a conditional.
• Translate a verbal or symbolic conditional into a disjunction, and negate it.
• Use truth tables, in which each column can be evaluated by reference to at most two other columns, to determine the truth values of compound statements formed by negations, conjunctions, conditionals and disjunctions of simple statements.
• Given a circuit translate it as a compound statement, and vice versa.
• Evaluate a compound statement by evaluating the corresponding circuit.

• Know and apply the definitions of the converse, inverse and contrapositive of a conditional.
• Evaluate the relative truth values of a statement and its converse, inverse and contrapositive.
• Know, interpret and apply alternative verbal forms of the conditional.
• Use the converse, inverse and contrapositive to write equivalent forms of conditional statements.
• Know and apply the definition of the biconditional.
• Know and apply the definition of 'consistent' and 'contrary'.

• Use Euler diagrams to represent conditionals, negations and quantified statements.
• Use Euler diagrams to represent and evaluate arguments.

• Given an argument involving compound statements, write the argument as a conditional, evaluate the conditional and determine the validity of the argument.
• Given a set of premises, use modus ponens, modus tollens, disjunctive syllogism and/or transitivity, as appropriate, to draw a valid conclusion.  (It is not required that you know the terms ' modus ponens', 'modus tollens', or 'disjunctive syllogism'.  You should know the term 'transitivity' and recognize when you are using it.  It's not required but it's a good idea to understand the meaning of 'direct argument' (vernacular for 'modus ponens') and 'argument by contradiction' (vernacular for 'modus tollens').) 

Module 4:  Chapter 4

• Given a table defining the symbols for Egyptian, Roman or Chinese numeration systems, be able to write a given Hindu-Arabic number in the notation of that system, and be able to write a number in any of those systems in Hindu-Arabic form.
• Perform addition or subtraction using Egyptian symbols (as opposed, for example, to translating the Egyptian symbols to Hindu-Arabic notation and doing the arithmetic in the usual fashion before translating back to Egyptian symbols).
• Perform multiplication using the Egyptian algorithm.

• Given a table defining the symbols for Babylonian, Mayan or Greek numeration systems, be able to write a given Hindu-Arabic number in the notation of that system, and be able to write a number in any of those systems in Hindu-Arabic form.

• Do arithmetic in the Hindu-Arabic system by first expanding each number in powers of 10.
• Interpret numbers represented on an abacus.
• Apply the Russian Peasant Method for multiplication.
• Apply the above to solve problems.

• Convert a base-10 number to another specified base (including hexadecimal).
• Convert a number in a specified base (including hexadecimal) to a base-10 number.
• Convert between binary and hexadecimal representations of a number.

(alternatively listed as 'Extension:  Modular Systems')

• Perform 'clock arithmetic' on a clock with a given number of hours.
• Perform modular arithmetic for a given modulus.

Module 5:  Chapter 5

• Know and apply the definitions of prime and composite numbers.
• Apply divisibility tests for divisibility by 2, 3, 4, 5, 6, 8, 9, 10 and 12 to a given number.
• Find the prime factorization of a given number.
• Find all the natural-number factors of a given number.
• Use the Sieve of Erasthones to find the primes up to number n, where n is a specified natural number.

• Explain how we can prove that there are infinitely many primes.
• Check the Mersenne numbers M_n = 2^n - 1 for values of n up through 10 and identify corresponding Mersenne primes.
• Generate Fermat numbers 2^(2^n) + 1 for values of n up to 3, and identify those which are prime.

• Identify a given number as perfect, deficient or abundant.
• List the perfect numbers less than 100.
• Verify Goldbach's Conjecture for a given number by showing how it is the sum of two primes.
• Verify whether two given numbers are amicable.
• Identify twin primes up to 100.
• State Fermat's Last Theorem and explain how it is related to the Pythagorean Theorem.

• Find the greatest common factor or least common multiple of two or more given numbers based on their prime factorizations.
• Find the greatest common factor or least common multiple of two or more given using division by primes.
• Find the greatest common factor of two numbers using the Euclidean Algorithm

• Know and identify the pattern of the Fibonacci sequence.
• Know and apply the recursion formula for the Fibonacci sequence.
• Be able to prove and/or verify stated characteristic of the Fibonacci sequence.
• Show how the successive ratios of the Fibonacci sequence approach the golden ratio.

Module 6:  Chapter 7

• Know the properties of equality and use them to solve linear equations.
• Write the linear equation that models a given linear phenomenon.
• Apply the above to solve problems, especially problems involving geometrical quantities, mixtures, interest and rates.

• Write and reduce to lowest terms the ratio of two given quantities.
• Properly apply the word 'per' to the ratio of two quantities.
• Write and solve statements of proportion.
• Know the forms of the statements of direct variation, inverse variation, and variation with a power.
• Apply direct variation to solve problems.
• Write equations for, apply and solve problems related to joint or combined variation.