Standards - Mathematics

Design and carry out an investigation to determine whether there appears to be an association between two categorical variables, and write a persuasive argument based on the results of the investigation.

Distinguish between quantitative and categorical data and between the techniques that may be used for analyzing data of these two types.

Analyze the possible association between two categorical variables.

Summarize categorical data for two categories in two-way frequency tables and represent using segmented bar graphs.

Interpret relative frequencies in the context of categorical data (including joint, marginal, and conditional relative frequencies).

Identify possible associations and trends in categorical data.

Generate a two-way categorical table in order to find and evaluate solutions to real-world problems.

Aggregate data from several groups to find an overall association between two categorical variables.

Recognize and explore situations where the association between two categorical variables is reversed when a third variable is considered (Simpson’s Paradox).

Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (or and not“).“

Explain whether two events, A and B, are independent, using two-way tables or tree diagrams.

Compute the conditional probability of event A given event B, using two-way tables or tree diagrams.

Recognize and describe the concepts of conditional probability and independence in everyday situations and explain them using everyday language.

Explain why the conditional probability of A given B is the fraction of B’s outcomes that also belong to A, and interpret the answer in context.

Identify numbers written in the form $a + bi$, where $a$ and $b$ are real numbers and $i^2 = -1$, as complex numbers.

Identify numbers written in the form $a + bi$, where $a$ and $b$ are real numbers and $i^2 = -1$, as complex numbers.

Use matrices to represent and manipulate data.

Multiply matrices by scalars to produce new matrices.

Add, subtract, and multiply matrices of appropriate dimensions.

Describe the roles that zero and identity matrices play in matrix addition and multiplication, recognizing that they are similar to the roles of 0 and 1 in the real numbers.

Find the additive and multiplicative inverses of square matrices, using technology as appropriate.

Explain the role of the determinant in determining if a square matrix has a multiplicative inverse.

Factor polynomials using common factoring techniques, and use the factored form of a polynomial to reveal the zeros of the function it defines.

Prove polynomial identities and use them to describe numerical relationships.

Explain why extraneous solutions to an equation may arise and how to check to be sure that a candidate solution satisfies an equation. Extend to radical equations.

For exponential models, express as a logarithm the solution to $ab^{ct} = d$, where $a$, $c$, and $d$ are real numbers and the base $b$ is 2 or 10; evaluate the logarithm using technology to solve an exponential equation.

Create equations and inequalities in one variable and use them to solve problems. Extend to equations arising from polynomial, trigonometric (sine and cosine), logarithmic, radical, and general piecewise functions.

Solve quadratic equations with real coefficients that have complex solutions.

Solve simple equations involving exponential, radical, logarithmic, and trigonometric functions using inverse functions.

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales and use them to make predictions. Extend to polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions.

Explain why the x-coordinates of the points where the graphs of the equations $y = f(x)$ and $y = g(x)$ intersect are the solutions of the equation $f(x) = g(x)$.

Explain why the x-coordinates of the points where the graphs of the equations $y = f(x)$ and $y = g(x)$ intersect are the solutions of the equation $f(x) = g(x)$.

Functions can be described by using a variety of representations: map\ping diagrams, function notation (e.g., $f(x) = x^2$), recursive definitions, tables, and graphs.

Identify the effect on the graph of replacing $f(x)$ by $f(x) + k$, $k cdot f(x)$, $f(k cdot x)$, and $f(x + k)$ for specific values of $k$ (both positive and negative); find the value of $k$ given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Extend to polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions.

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Note: Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximums and minimums; symmetries (including even and odd); end behavior; and periodicity. Extend to polynomial, trigonometric (sine and cosine), logarithmic, reciprocal, radical, and general piecewise functions.