Grade 5

Students add and subtract multi-digit whole numbers through commonly used large place values by aligning place values, regrouping across one or more zeros, and using place-value representations to explain and check the standard algorithms. They add and subtract decimals through hundredths, estimate differences using rounding or compatible numbers, and assess whether exact results are reasonable. They solve one-step addition problems involving whole numbers, decimals, and appropriately represented fractions, as well as multi-step addition and subtraction problems using equations, models, and contextual reasoning.

Read and interpret tables, scaled bar graphs, line graphs, line plots with halves and fourths, and first-quadrant coordinate graphs to retrieve, compare, total, and analyze data. Calculate and interpret the mean, median, mode, and range of small, ungrouped data sets, including whole numbers and familiar decimals or fractions. Represent, compare, and investigate simple probabilities as fractions from 0 to 1, identifying equally likely outcomes and relating experimental results to theoretical probability.

Read, write, identify, and expand nonnegative decimals through the thousandths place, including tenths, hundredths, thousandths, zero placeholders, equivalent trailing-zero forms, and conversions between decimals and fractions with denominators of 10, 100, or 1,000. Compare, order, and round decimals through the thousandths place, and add or subtract them by aligning place values, inserting equivalent zeros, and regrouping as needed. Solve contextual problems involving decimals through the thousandths place by selecting appropriate addition, subtraction, multiplication, or division, representing quantities with equations or models, and checking results through estimation and place-value reasoning.

Students recall division facts with products through 100 and divide two-digit and multi-digit whole-number dividends by one-digit divisors using place value, partial quotients, area models, or the standard algorithm. They divide typically up to four-digit dividends by two-digit divisors, estimate quotients using rounding and compatible numbers, and verify results with multiplication while expressing remainders as less than the divisor. Students solve division word problems involving equal groups or group size, represent relationships with equations or models, and interpret remainders as leftovers, ignored quantities, or additional groups, with limited contexts involving a unit fraction divided by a whole number.

Apply divisibility rules for 2, 3, 4, 5, 6, 9, and 10 to multi-digit whole numbers to identify factors and multiples. Identify factors and multiples, and classify whole numbers through 100 as prime or composite, recognizing that 1 is neither. Find common factors and common multiples of familiar whole numbers and determine the greatest common factor and least common multiple for pairs of numbers no greater than 100.

Students identify, represent, locate, and interpret proper, improper, and mixed numbers using area models, sets, number lines, and benchmark fractions, and recognize, generate, convert, and simplify equivalent fractions. Students compare and order positive fractions and mixed numbers with common or unlike denominators using equivalent fractions, common numerators or denominators, benchmarks, and number-line reasoning. Students add and subtract fractions and mixed numbers with like denominators, including regrouping, and solve contextual problems involving unlike-denominator addition and subtraction, finding a fraction of a quantity, and division situations with a whole number and a unit fraction.

Classify triangles by angle measures as acute, right, or obtuse and by side lengths as equilateral, isosceles, or scalene, and classify quadrilaterals hierarchically using properties such as parallel or perpendicular sides, congruent side lengths, and right angles, including trapezoids, parallelograms, rectangles, rhombuses, squares, and kites. Identify, measure, and draw angles from 0° to 180° with a protractor, correctly distinguishing acute, right, obtuse, and straight angles regardless of ray length or orientation. Identify lines of symmetry, parallel lines, and perpendicular lines, and plot and interpret points represented by nonnegative whole-number ordered pairs in the first quadrant of the coordinate plane.

Convert among metric length units (kilometers, meters, centimeters, and millimeters), capacity units (kiloliters, liters, and milliliters), and mass units (kilograms, grams, and milligrams), as well as customary length (inches, feet, yards, miles), capacity (fluid ounces, cups, pints, quarts, gallons), and weight (ounces, pounds, tons), using the given whole-number, decimal, or familiar fractional relationships. Estimate and solve area and perimeter problems for rectangles, composite rectilinear figures, and other polygons, including partitioning figures, inferring missing side lengths, using square units, and distinguishing surface coverage from distance around a figure. Solve elapsed-time and multistep measurement word problems involving hours and minutes, unit conversions, whole numbers, fractions, and decimals while selecting appropriate operations and preserving units.

Recall multiplication facts through 12, identify patterns when multiplying by 0, 1, 10, 100, and 1,000, and multiply by multiples of 10 using place-value relationships. Use place-value decomposition, partial products, area models, and the standard algorithm to multiply whole numbers, including 2-digit by 2-digit, 3-digit by 2-digit, and larger cases such as 4-digit by 2-digit, with accurate regrouping and alignment. Estimate products and solve multiplication problems involving multi-digit whole numbers, decimals, and fractions by interpreting equal groups, arrays, area, or scaling, selecting appropriate operations and checking the magnitude and units of results.

Students identify, extend, and compare numerical and arithmetic patterns in lists and tables, including whole-number patterns and familiar decimal sequences generated by repeated addition, subtraction, multiplication, or division, and represent corresponding pairs on coordinate grids to predict missing or later values and explain term-to-term or input-based rules. Students generate and complete input-output tables, distinguish inputs from outputs, and infer or apply numerical rules involving a fixed operation or two straightforward operations. Students evaluate grouped numerical expressions with parentheses using whole-number addition, subtraction, multiplication, and division, applying grouping and multiplication or division before addition or subtraction, and use variables to represent unknowns in solving and checking one-step equations and contextual problems with appropriate units.

Read, write, and represent whole numbers from 0 through 999,999,999 in standard, word, and expanded forms, identifying each digit’s value and recognizing powers of 10 through $10^3$. Compare and order multi-digit whole numbers using place value, and round them to any specified place, including the nearest 10, 100, and 1,000, to estimate sums and products with reasonable accuracy. Multiply and divide whole numbers by 10, 100, and 1,000 using place-value shifts, including interpreting zeros as placeholders and recognizing that quotients may be terminating decimals.