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. [Grade 8, 8]
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. [Grade 8, 9]
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. [Grade 8, 10]
Apply and extend knowledge of operations of whole numbers, fractions, and decimals to add, subtract, multiply, and divide rational numbers including integers, signed fractions, and decimals.
Identify and explain situations where the sum of opposite quantities is 0 and opposite quantities are defined as additive inverses.
Interpret the sum of two or more rational numbers, by using a number line and in real-world contexts.
Explain subtraction of rational numbers as addition of additive inverses.
Use a number line to demonstrate that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.
Extend strategies of multiplication to rational numbers to develop rules for multiplying signed numbers, showing that the properties of the operations are preserved.
Divide integers and explain that division by zero is undefined. Interpret the quotient of integers (with a non-zero divisor) as a rational number.
Convert a rational number to a decimal using long division, explaining that the decimal form of a rational number terminates or eventually repeats. [Grade 7, 4]
Solve real-world and mathematical problems involving the four operations of rational numbers, including complex fractions. Apply properties of operations as strategies where applicable. [Grade 7, 5]
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 in a pattern or terminates.
Convert a decimal expansion that repeats in a pattern into a rational number. [Grade 8, 1]
Locate rational approximations of irrational numbers on a number line, compare their sizes, and estimate the values of irrational numbers. [Grade 8, 2]
Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. [Grade 7, 6]
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]
Express and compare very large or very small numbers in scientific notation. [Grade 8, 5]
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]
