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.
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.
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.
Generate expressions in equivalent forms based on context and explain how the quantities are related. [Grade 7, 7]
Develop and apply properties of integer exponents to generate equivalent numerical and algebraic expressions. [Grade 8, 3]
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. [Grade 8, 4]
Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used, expressing answers in scientific notation. [Grade 8, 6]
Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities. [Grade 8, 6a]
Interpret scientific notation that has been generated by technology. [Grade 8, 6b]
Solve multi-step real-world and mathematical problems involving rational numbers (integers, signed fractions, and decimals), converting between forms as needed. Assess the reasonableness of answers using mental computation and estimation strategies. [Grade 7, 8]
Use variables to represent quantities in a real-world or mathematical problem and construct algebraic expressions, equations, and inequalities to solve problems by reasoning about the quantities.
Solve word problems leading to equations of the form $px + q = r$ and $p(x + q) = r$, where $p$, $q$, and $r$ are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.
Solve word problems leading to inequalities of the form $px + q > r$ or $px + q < r$, where $p$, $q$, and $r$ are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. [Grade 7, 9, and linear portion of Algebra I with Probability, 11]
Create equations in two variables to represent relationships between quantities in context; graph equations on coordinate axes with labels and scales and use them to make predictions. Limit to contexts arising from linear functions. [Algebra I with Probability, 12 partial]
Represent constraints by equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Limit to contexts arising from linear. [Algebra I with Probability, 13 partial]
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. [Grade 8, 11]
Identify the effect on the graph of replacing $f(x)$ by $f(x) + k,$ $k \cdot f(x), f (kx),$ 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 explain the effects on the graph using technology, where appropriate. Limit to linear functions. [Algebra I with Probability, 23]
Construct a function to model the linear relationship between two variables.
Interpret the rate of change (slope) and initial value of the linear function from a description of a relationship from two points in a table or graph. [Grade 8, 16]
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 $f(x) = g(x)$. Limit to linear equations. [Algebra I with Probability, 19]
Find approximate solutions by graphing the functions, making tables of values, or finding successive approximations, using technology where appropriate.
Note: Include cases where $f(x)$ is linear and $g(x)$ is constant or linear. [Algebra I with Probability, 19 edited]
