Grade 6

Students identify variables, constants, coefficients, and terms; translate between words and algebraic expressions; evaluate expressions with whole numbers, integers, fractions, and finite decimals using grouping symbols, whole-number exponents, and order of operations; and generate equivalent forms by combining like terms, applying properties, expanding, and factoring simple expressions. Students represent equations with balance models, tape diagrams, and one-step or simple multi-step contexts; write and solve addition, subtraction, multiplication, division, and fractional-coefficient equations; and check solutions by substitution while preserving equality. Students interpret, write, solve, and graph simple one-variable inequalities with whole numbers and familiar rational values, and use input-output tables, rules, ordered pairs, and coordinate graphs to describe dependent and independent variables and distinguish proportional from non-proportional patterns.

Students distinguish statistical questions, collect categorical and numerical data consistently, and organize observations in tables or frequency tables. They choose, create, and interpret bar graphs, dot plots, line plots, line graphs, histograms, circle graphs, and informal box plots, using accurate scales to identify frequencies, trends, clusters, gaps, center, and spread. They calculate and apply mean, median, mode, range, interquartile range, and informal mean absolute deviation to compare distributions and support descriptive conclusions while considering variability, outliers, misleading displays, and sample representativeness.

Students read, write, expand, compare, order, locate, round, and convert finite decimals through the thousandths place, connecting them to fractions with denominators of 10, 100, or 1,000, mixed numbers, and measurement. They add, subtract, multiply, and divide decimals and whole numbers, including multiplication or division by 10, 100, and 1,000, while applying order of operations and using place-value reasoning, compatible numbers, and estimation to check results. Students solve multistep money, measurement, and other word problems involving positive terminating decimals, interpret remainders and units, and judge whether answers are reasonable.

Students represent, locate, compare, order, simplify, and convert positive fractions, improper fractions, and mixed numbers, including generating equivalent forms and finding least common denominators. They add and subtract fractions and mixed numbers with like and unlike denominators, multiply fractions and mixed numbers, and divide fractions using models and reciprocals, including finding a fraction of a quantity. Students use benchmark fractions, estimation, area and number-line models, and operation reasoning to solve and check single-step and multistep fraction problems in context.

Students identify and classify points, lines, rays, segments, angles, triangles, quadrilaterals, polygons, and solids using defining properties such as side lengths, angle measures, parallelism, perpendicularity, symmetry, and relationships among special quadrilaterals. They measure and construct angles, use complementary, supplementary, triangle, quadrilateral, and full-turn angle relationships to find unknown measures, and reason about congruence, informal similarity, dilations, and lines and rotational symmetry. Students represent and perform translations, reflections, and rotations on coordinate planes, construct nets of prisms and pyramids, and identify informal cross-sections of common solids.

Students represent, interpret, compare, and order integers on number lines, including zero, opposites, absolute value, and positive or negative quantities in contexts such as temperature, elevation, gains, losses, and direction. Students read and plot integer ordered pairs, locate points in all four quadrants or on the axes, identify quadrants, graph polygons, and recognize or create symmetry through reflections across the $x$- and $y$-axes. Students find horizontal and vertical distances and side lengths using absolute differences of coordinates and use coordinates to solve perimeter and area problems involving simple rectangles and polygons, without applying diagonal distance formulas.

Students analyze and decompose routine and non-routine problems by identifying relevant, irrelevant, and missing information; representing relationships with diagrams, tables, graphs, equations, expressions, and number lines; and working with whole numbers, integers, fractions, decimals, ratios, rates, percentages, and familiar measurements. They select efficient operations, tools, and strategies; work backward or use logical elimination; estimate before and after calculating; check precision, rounding, units, inverse operations, and algebraic solutions by substitution; and interpret results in context. They communicate complete justifications, compare and critique strategies and solutions, create related problems and statistical questions, recognize and generalize patterns, organize cases, and form, test, support, or disprove conjectures using examples and counterexamples.

Students estimate and measure length, mass, capacity, and time using appropriate customary and metric units, record measurements to specified precision, calculate elapsed time, and convert among common units, including inches, feet, yards, miles; ounces, pounds, tons; cups, pints, quarts, gallons; millimeters, centimeters, meters, kilometers; milligrams, grams, kilograms; milliliters, liters, kiloliters; and seconds, minutes, hours, days, and weeks. Students calculate perimeter and area of polygons, rectangles, parallelograms, triangles, and composite figures; determine missing side lengths and dimensions; and solve scale-drawing and measurement problems involving whole numbers, fractions, and decimals. Students find and interpret the volume and surface area of rectangular prisms and familiar solids using unit cubes, nets, formulas, and decomposition, while solving multi-step contextual problems and applying conversion factors with consistent linear, square, cubic, and compound units.

Determine theoretical probabilities for simple events in finite, equally likely sample spaces, including cards, colored objects, fair dice, and spinners, by listing outcomes with organized lists, tables, or tree diagrams and expressing probabilities as fractions, decimals, or percentages from 0 to 1. Find complementary probabilities, compare likelihoods, identify fair and unfair games, and use probability to make decisions or predict frequencies through multiplication by the number of trials. Conduct experiments and simulations to calculate relative frequencies, explain short-run variation and increasing stability with more trials, compare experimental and theoretical probabilities, and solve simple compound-event problems without conditional probability or advanced probability models.

Students interpret, write, represent, compare, simplify, and generate equivalent ratios using words, colon notation, fractions, tables, double number lines, and tape diagrams, and solve part-to-part, part-to-whole, missing-value, scaling, and other ratio problems through multiplicative reasoning. Students calculate, interpret, compare, and use unit rates, including price per item and speed, convert rates between compatible units, and solve problems involving whole numbers, fractions, and decimals while attending to units and order. Students interpret percent as per hundred, relate fractions, decimals, and percents, represent and estimate percents with hundred grids and benchmarks, find a percent of a quantity or the whole from a familiar percent, and solve simple discount, tax, and tip problems.

Students read, write, expand, compare, order, round, and identify the place value of whole numbers through billions, and use estimation to assess sums, differences, products, and quotients. They accurately add, subtract, multiply, and divide multi-digit whole numbers, including division by one- and two-digit divisors, using standard algorithms, partial products or quotients, powers of ten, remainders, divisibility rules, and multi-step problem-solving strategies. They apply operation properties and order of operations, evaluate powers and parentheses, represent squares and cubes, identify factors, multiples, primes, composites, and factor pairs, and determine prime factorizations, common factors and multiples, greatest common factors, and least common multiples.