Grade 8

Students identify variables, coefficients, constants, terms, and like terms, and translate between algebraic expressions and precise verbal descriptions involving sums, differences, products, quotients, grouping, and whole-number exponents. They evaluate expressions by substituting integer or rational values and applying order of operations, and recognize equivalent expressions using the commutative, associative, and distributive properties. They expand expressions with the distributive property, combine like terms, factor common numerical factors, and simplify multi-step expressions containing parentheses, variables, constants, and integer or rational coefficients.

Classify data as categorical or numerical and as discrete or continuous, select appropriate censuses, surveys, observations, or experiments, and evaluate samples and survey questions for representativeness and bias. Calculate and interpret mean, median, mode, and range, explain how outliers affect measures of center, construct and interpret histograms, and identify misleading scales, intervals, labels, and visual features in data displays. Construct scatter plots of paired numerical data and describe positive, negative, or nonexistent associations, including linear or nonlinear patterns, clusters, gaps, and outliers, without inferring causation or using regression or correlation coefficients.

Students model equations with balance diagrams, interpreting the equal sign relationally and writing one-variable linear equations and inequalities from contexts involving quantities, rates, totals, comparisons, and constraints. They solve one-step and two-step linear equations with integer and rational coefficients using equality-preserving inverse operations, including variables in different positions, and verify solutions by substitution while recognizing when an equation has one, no, or infinitely many solutions. They solve one-step inequalities, reversing the inequality when multiplying or dividing by a negative, and represent rational and compound solution sets symbolically and on number lines with appropriate open or closed endpoints and shaded regions.

Students convert among fractions, terminating decimals, and percents, simplify equivalent forms, and compare or order positive and negative rational numbers, including zero, using signs, common units, and number-line reasoning. Students add, subtract, multiply, and divide numerical fractions, mixed numbers, decimals, and signed fractional quantities, interpreting operations, simplifying results, estimating, and recognizing that division by zero is undefined. Students evaluate numerical expressions involving rational numbers by applying grouping symbols, whole-number exponents, and the order of multiplication or division before addition or subtraction, while preserving equivalent values and signs.

Use angle relationships for intersecting lines, perpendicular lines, and parallel lines cut by a transversal to determine unknown measures with vertical, corresponding, alternate, linear-pair, and supplementary relationships. Apply triangle angle sums, exterior-angle and isosceles relationships, and polygon angle-sum formulas to determine unknown angles, including angles in regular and irregular convex polygons. Classify triangles by angle and side lengths, apply the Pythagorean theorem to find unknown side lengths, and use its converse to determine whether a triangle is right.

Students represent, compare, and order integers on a number line, identify opposites and additive inverses, and interpret absolute value as nonnegative distance from zero for integers and rational numbers. They add, subtract, multiply, and divide integers using sign rules, including the roles of zero and undefined division by zero, and represent non-exact quotients as rational numbers. They evaluate and solve multi-step numerical problems using grouping symbols, nonnegative integer exponents, and the conventional order of operations while interpreting signed quantities and checking results for reasonableness.

Students identify independent and dependent variables and distinguish linear relationships by their constant rate of change, including positive, negative, and zero rates and proportional and nonproportional relationships. They represent and interpret linear relationships in tables, graphs, equations, and contexts, determining rate of change from $\Delta y/\Delta x$, identifying the initial value, and connecting these features to $y=mx+b$. They graph linear equations, write equations from tables and graphs using whole-number or simple fractional slopes, and compare relationships by matching inputs and outputs and interpreting their rates of change and initial values, including when the relationships intersect.

Students determine radius and diameter using $d=2r$ and $r=d/2$, and calculate circle circumference with $C=2\pi r=\pi d$ and area with $A=\pi r^2$, reporting linear or square units and exact or appropriately approximated values. They find the area of polygonal composite figures by decomposing or subtracting familiar regions and convert metric units of length, area, and volume across millimeters, centimeters, meters, and kilometers using factors of $10$, $10^2$, and $10^3$, respectively. They calculate prism and cylinder volumes and surface areas using $V=Bh$, $V=\pi r^2h$, $SA=2B+Ph$, and $SA=2\pi r^2+2\pi rh$, and solve composite-solid problems by adding component measures or subtracting missing sections while excluding shared internal surfaces from exterior area.

Interpret and evaluate powers with whole-number exponents, including integer and negative bases, zero exponents, parentheses, and the product, quotient, and power-of-a-power rules, without using negative or fractional exponents. Recognize perfect squares and determine their principal square roots from 0 through 225, and estimate nonperfect square roots between consecutive integers whose squares bound them. Write large and small nonzero numbers in scientific notation and convert between scientific and standard form, using $a \times 10^n$ with $1 \le a < 10$ and interpreting positive and negative integer exponents through place value.

Students calculate discounts, markups, taxes, tips, percent increases and decreases, and simple interest using percent equations, decimal multipliers, and $I=Prt$, distinguishing the original amount, rate, change, and final total. They compare financial options by translating rates, unit costs, fixed fees, and transaction charges to a common basis using tables, equations, graphs, and unit rates. They determine and compare theoretical and experimental probabilities and analyze compound events with organized lists and tree diagrams, adding probabilities of mutually exclusive outcomes, multiplying probabilities of successive outcomes, and accounting for dependent events.

Students write, simplify, generate, and compare ratios in colon, word, and fraction forms using positive whole-number and rational quantities, preserving the order and multiplicative relationship. They interpret rates and calculate and compare unit rates, including rates involving fractions and decimals, across tables, graphs, equations, and real-world contexts with appropriate units. Students recognize proportional relationships in tables, graphs, and equations, determine and interpret the constant of proportionality, solve proportions using equivalent ratios, and apply proportional reasoning to scale drawings while maintaining consistent units.

Plot points in all four quadrants, identify points on the axes and origin, and determine horizontal and vertical distances using absolute differences of coordinates. Apply coordinate rules for translations, reflections across the axes, and quarter- or half-turn rotations about the origin, including sequences of transformations in a specified order, to predict image coordinates and analyze preserved lengths, angles, and shape. Identify congruent figures through rigid motions and equal corresponding measurements, and identify similar figures through proportional corresponding side lengths, congruent angles, and a single scale factor, including dilation.