Georgia: Advanced Finite Mathematics Math Standards
49 standards · 9 domains
AFM.AQR
- AFM.AQR.5.1 Find the union, intersection, difference, complement, and Cartesian product of sets, and classify sets as equal, subsets, and power sets.
- AFM.AQR.5.2 Justify whether the union of subsets of a set is a partition of that set.
- AFM.AQR.5.3 Given a relation on two sets, determine whether the relation is a function and find its inverse relation, if it exists.
- AFM.AQR.5.4 Determine the equivalence classes given an equivalence relation on a set; determine whether the union of equivalence classes of a set is a partition of that set.
- AFM.AQR.5.5 Prove set relations, including DeMorgan’s Laws and equivalence relations.
- AFM.AQR.5.6 Prove statements in Boolean algebra.
- AFM.AQR.5.7 Simplify Boolean algebra expressions using Karnaugh maps (K-maps).
- AFM.AQR.6.1 Use the addition rule to count the number of outcomes in a disjoint set of sample spaces. Use the principle of inclusion-exclusion to count the number of outcomes in the union of sample spaces.
- AFM.AQR.6.2 Apply the axioms of probability to determine the probability of dependent and independent events, including use of the multiplication rule for independent events.
- AFM.AQR.6.3 Find expected value.
- AFM.AQR.6.4 Apply Bayes’ Theorem to determine conditional probability.
- AFM.AQR.6.5 Calculate the number of permutations of a set with n elements. Calculate the number of permutations of r elements taken from a set of n elements.
- AFM.AQR.6.6 Calculate the number of subsets of size r that can be chosen from a set of n elements.
- AFM.AQR.6.7 Calculate the number of combinations with repetitions of r elements from a set of n elements
- AFM.AQR.6.8 Prove combinatorial identities.
- AFM.AQR.6.9 Apply a combinatorial argument to prove the binomial theorem.
- AFM.AQR.6.10 Use the pigeonhole principle to prove statements about counting.
- AFM.AQR.7.1 Identify simple graphs, complete graphs, complete bipartite graphs, and trees. Identify graphs that have Euler and Hamiltonian cycles.
- AFM.AQR.7.2 Construct the complement and the line graph of a graph.
- AFM.AQR.7.3 Use the adjacency matrix of a graph to determine the number of walks of length n in a graph.
- AFM.AQR.7.4 Prove statements about graph properties.
- AFM.AQR.7.5 Prove that every connected graph has a minimal spanning tree.
- AFM.AQR.7.6 Use Kruskal’s algorithm and Prim’s algorithm to determine the minimal spanning tree of a weighted graph.
ABSTRACT & QUANTITATIVE REASONING – SET THEORY
- AFM.AQRa.5 Use set theory to describe relationships and equivalence when solving contextual, mathematical problems used to explain real-life phenomena.
ABSTRACT & QUANTITATIVE REASONING – PROBABILITIES AND COMBINATORICS
- AFM.AQRb.6 Calculate and solve combinatorics problems to make sense of a real-life, contextual problem.
ABSTRACT & QUANTITATIVE REASONING – GRAPH THEORY
- AFM.AQRc.7 Apply graph theory to solve contextual, mathematical problems and to explain real-life phenomena.
AFM.LR
- AFM.LR.2.1 Use a counterexample to disprove a statement.
- AFM.LR.2.2 Prove statements directly from definitions and previously proved statements.
- AFM.LR.2.3 Prove statements indirectly by proving the contrapositive of the statement.
- AFM.LR.2.4 Apply the method of reductio ad absurdum (proof by contradiction) to prove statements.
- AFM.LR.2.5 Use the method of mathematical induction to prove statements involving the positive integers.
- AFM.LR.3.1 Construct truth tables that represent conditional, biconditional, and quantified statements; use truth tables to determine whether the statement is true or false and use Venn diagrams to illustrate the relationship represented by these truth tables.
- AFM.LR.3.2 Represent logic operations such as AND, OR, NOT, NOR, and XOR (exclusive OR) using logical symbolism, determine whether statements involving these operations are true or false, and interpret such symbols into English.
- AFM.LR.3.3 Apply modus ponens and modus tollens to analyze logical arguments to determine whether it is valid, invalid, a tautology, or a contradiction.
- AFM.LR.3.4 Write the negation, converse, contrapositive, and inverse of a conditional statement and find the truth of each.
- AFM.LR.3.5 Represent the dichotomy between “true” and “false” with 1s and 0s. Use 1s and 0s to calculate whether a statement is true or false by constructing Boolean logic circuits.
- AFM.LR.3.6 Convert binary and hexadecimal numbers into decimal, and convert from binary to hexadecimal, and vice versa. Add binary integers and use 2’s complement to subtract binary integers.
LOGICAL REASONING – METHODS OF PROOF
- AFM.LRa.2 Apply methods of proof to prove or disprove mathematical statements; explain reasoning and justify thinking through mathematical induction when formulating mathematical arguments.
LOGICAL REASONING – LOGICAL SYMBOLISM AND BINARY
- AFM.LRb.3 Interpret, represent, and communicate logical arguments to explain reasoning and justify thinking when solving problems and to explain real-life phenomena.
MATHEMATICAL MODELING
- AFM.MM.1.1 Explain contextual, mathematical problems using a mathematical model.
- AFM.MM.1.2 Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.
- AFM.MM.1.3 Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.
- AFM.MM.1.4 Use various mathematical representations and structures with this information to represent and solve real-life problems.
NUMERICAL REASONING – NUMBER THEORY
- AFM.NR.4.1 Apply the divides relation to positive integers and calculate one integer modulo another integer.
- AFM.NR.4.2 Find the inverse of an integer for a certain modulus.
- AFM.NR.4.3 Calculate the floor and the ceiling of a real number.
- AFM.NR.4.4 Prove statements involving properties of numbers.
- AFM.NR.4.5 Prove statements involving the floor and ceiling functions.
- AFM.NR.4.6 Prove the Fundamental Theorem of Arithmetic, the Euclidean algorithm, and Fermat’s Little Theorem.