Standards - Mathematics

Construct a sampling distribution for a random event or random sample.

Use the binomial theorem to construct the sampling distribution for the number of successes in a binary event or the number of positive responses to a yes/no question in a random sample.

Use the normal approximation of a proportion from a random event or sample when conditions are met.

Use the central limit theorem to construct a normal sampling distribution for the sample mean when conditions are met.

Find the long-term probability of a given range of outcomes from a random event or random sample.

Perform inference procedures based on the results of samples and experiments.

Use a point estimator and margin of error to construct a confidence interval for a proportion or mean.

Interpret a confidence interval in context and use it to make strategic decisions.

Perform a significance test for null and alternative hypotheses.

Interpret the significance level of a test in the context of error probabilities, and use the results to make strategic decisions.

Critique the validity of reported conclusions from statistical studies in terms of bias and random error probabilities.

Conduct a randomized study on a topic of student interest (sample or experiment) and draw conclusions based upon the results.

Represent logic statements in words, with symbols, and in truth tables, including conditional, biconditional, converse, inverse, contrapositive, and quantified statements.

Represent logic operations such as and, or, not, nor, and x or (exclusive or) in words, with symbols, and in truth tables.

Use truth tables to solve application-based logic problems and determine the truth value of simple and compound statements including negations and implications.

Determine whether statements are equivalent and construct equivalent statements.

Determine whether a logical argument is valid or invalid, using laws of logic such as the law of syllogism and the law of detachment.

Determine whether a logical argument is a tautology or a contradiction.

Prove a statement indirectly by proving the contrapositive of the statement.

Use multiple representations and methods for counting objects and developing more efficient counting techniques. Note: Representations and methods may include tree diagrams, lists, manipulatives, overcounting methods, recursive patterns, and explicit formulas.

Develop and use the Fundamental Counting Principle for counting independent and dependent events.

Use various counting models (including tree diagrams and lists) to identify the distinguishing factors of a context in which the Fundamental Counting Principle can be applied.

Using application-based problems, develop formulas for permutations, combinations, and combinations with repetition and compare student-derived formulas to standard representations of the formulas.

Using application-based problems, develop formulas for permutations, combinations, and combinations with repetition and compare student-derived formulas to standard representations of the formulas.

Using application-based problems, 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.

Using application-based problems, calculate the number of subsets of size r that can be chosen from a set of n elements, explaining this number as the number of combinations n choose r.““

Using application-based problems, calculate the number of combinations with repetitions of r elements from a set of n elements as (n + r - 1) choose r.““

Use various counting techniques to determine probabilities of events.

Use the Pigeonhole Principle to solve counting problems.

Find patterns in application problems involving series and sequences, and develop recursive and explicit formulas as models to understand and describe sequential change.

Determine characteristics of sequences, including the Fibonacci Sequence, the triangular numbers, and pentagonal numbers.

Use the recursive process and difference equations to create fractals, population growth models, sequences, and series.

Use mathematical induction to prove statements involving the positive integers.

Use mathematical induction to prove statements involving the positive integers.

Use vertex and edge graphs to model mathematical situations involving networks.