Standards - Mathematics

Define the real number system as composed of rational and irrational numbers.

Explain that every number has a decimal expansion; for rational numbers, the decimal expansion repeats or terminates.

Convert a decimal expansion that repeats into a rational number.

Locate rational approximations of irrational numbers on a number line, compare their sizes, and estimate the values of the irrational numbers.

Develop and apply properties of integer exponents to generate equivalent numerical and algebraic expressions.

Use square root and cube root symbols to represent solutions to equations.

Evaluate square roots of perfect squares (less than or equal to 225) and cube roots of perfect cubes (less than or equal to 1000).

Explain that the square root of a non-perfect square is irrational.

Estimate and compare very large or very small numbers in scientific notation.

Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used.

Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities.

Interpret scientific notation that has been generated by technology.

Determine whether a relationship between two variables is proportional or non-proportional.

Graph proportional relationships.

Interpret the unit rate of a proportional relationship, describing the constant of proportionality as the slope of the graph which goes through the origin and has the equation y = mx where m is the slope.

Interpret y = mx + b as defining a linear equation whose graph is a line with m as the slope and b as the y-intercept.

Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in a coordinate plane.

Given two distinct points in a coordinate plane, find the slope of the line containing the two points and explain why it will be the same for any two distinct points on the line.

Graph linear relationships, interpreting the slope as the rate of change of the graph and the y-intercept as the initial value.

Given that the slopes for two different sets of points are equal, demonstrate that the linear equations that include those two sets of points may have different y-intercepts.

Compare proportional and non-proportional linear relationships represented in different ways (algebraically, graphically, numerically in tables, or by verbal descriptions) to solve real-world problems.

Solve multi-step linear equations in one variable, including rational number coefficients, and equations that require using the distributive property and combining like terms.

Determine whether linear equations in one variable have one solution, no solution, or infinitely many solutions of the form x = a, a = a, or a = b (where a and b are different numbers).

Represent and solve real-world and mathematical problems with equations and interpret each solution in the context of the problem.

Solve systems of two linear equations in two variables by graphing and substitution.

Explain that the solution(s) of systems of two linear equations in two variables corresponds to points of intersection on their graphs because points of intersection satisfy both equations simultaneously.

Interpret and justify the results of systems of two linear equations in two variables (one solution, no solution, or infinitely many solutions) when applied to real-world and mathematical problems.

Determine whether a relation is a function, defining a function as a rule that assigns to each input (independent value) exactly one output (dependent value), and given a graph, table, mapping, or set of ordered pairs.

Evaluate functions defined by a rule or an equation, given values for the independent variable.

Compare properties of functions represented algebraically, graphically, numerically in tables, or by verbal descriptions.

Distinguish between linear and non-linear functions.

Construct a function to model a linear relationship between two variables.

Interpret the rate of change (slope) and initial value of the linear function from a description of a relationship or from two points in a table or graph.

Analyze the relationship (increasing or decreasing, linear or non-linear) between two quantities represented in a graph.

Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities, describing patterns in terms of positive, negative, or no association, linear and non-linear association, clustering, and outliers.