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.
Given a scatter plot that suggests a linear association, informally draw a line to fit the data, and assess the model fit by judging the closeness of the data points to the line.
Use a linear model of a real-world situation to solve problems and make predictions.
Describe the rate of change and y-intercept in the context of a problem using a linear model of a real-world situation.
Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects, using relative frequencies calculated for rows or columns to describe possible associations between the two variables.
Verify experimentally the properties of rigid motions (rotations, reflections, and translations): lines are taken to lines, and line segments are taken to line segments of the same length; angles are taken to angles of the same measure; and parallel lines are taken to parallel lines.
Given a pair of two-dimensional figures, determine if a series of rigid motions maps one figure onto the other, recognizing that if such a sequence exists the figures are congruent; describe the transformation sequence that verifies a congruence relationship.
Use coordinates to describe the effect of transformations (dilations, translations, rotations, and reflections) on two- dimensional figures.
Given a pair of two-dimensional figures, determine if a series of dilations and rigid motions maps one figure onto the other, recognizing that if such a sequence exists the figures are similar; describe the transformation sequence that exhibits the similarity between them.
Analyze and apply properties of parallel lines cut by a transversal to determine missing angle measures.
Use informal arguments to establish that the sum of the interior angles of a triangle is 180 degrees.
Informally justify the Pythagorean Theorem and its converse.
Apply the Pythagorean Theorem to find the distance between two points in a coordinate plane.
Apply the Pythagorean Theorem to determine unknown side lengths of right triangles, including real-world applications.
Informally derive the formulas for the volume of cones and spheres by experimentally comparing the volumes of cones and spheres with the same radius and height to a cylinder with the same dimensions.
Use formulas to calculate the volumes of three-dimensional figures (cylinders, cones, and spheres) to solve real-world problems.
Calculate unit rates of length, area, and other quantities measured in like or different units that include ratios or fractions. [Grade 7, 1]
Represent a relationship between two quantities and determine whether the two quantities are related proportionally.
Use equivalent ratios displayed in a table or in a graph of the relationship in the coordinate plane to determine whether a relationship between two quantities is proportional.
Identify the constant of proportionality (unit rate) and express the proportional relationship using multiple representations including tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
Explain in context the meaning of a point (x,y) on the graph of a proportional relationship, with special attention to the points (0,0) and (1, r) where r is the unit rate. [Grade 7, 2]
Solve multi-step percent problems in context using proportional reasoning, including simple interest, tax, gratuities, commissions, fees, markups and markdowns, percent increase, and percent decrease. [Grade 7, 3]
Determine whether a relationship between two variables is proportional or non-proportional. [Grade 8, 7]
Graph proportional relationships.
