Students add and subtract multi-digit whole numbers through 1,000,000 using place-value reasoning, partial sums, and the standard algorithms, including regrouping across zeros. They estimate sums and differences, check results for reasonableness, and solve missing-value equations using the inverse relationship between addition and subtraction. They model and solve one-step, comparison, and multi-step word problems involving whole-number quantities with equations, bar models, or number lines.
Organize and represent categorical, discrete, and measured data in frequency tables, pictographs, scaled bar graphs, line graphs, and line plots, using accurate labels, keys, intervals, and scales, including halves, fourths, and eighths on line plots. Interpret these displays to compare values and frequencies, find totals and differences, identify changes over ordered sequences, and calculate the median, mode, and range for small whole-number data sets. Describe, compare, and express simple probabilities as fractions, conduct repeated chance experiments, and distinguish theoretical probability from experimental results in terms of impossible, unlikely, equally likely, likely, and certain events.
Students represent, read, write, and expand nonnegative decimals through hundredths, identifying the ones, tenths, and hundredths places and connecting decimals with fractions, models, and equivalent forms.
Students compare, order, and locate decimals through hundredths and add or subtract them by aligning place values and regrouping across ones, tenths, and hundredths.
Students apply these skills to solve measurement and money problems involving tenths and hundredths, including interpreting quantities, finding totals, differences, costs, change, and unknown amounts.
Interpret division as equal grouping and equal sharing, recall related multiplication–division facts with products up to 100, and solve whole-number missing-factor problems. Divide two-, three-, and four-digit whole numbers by one-digit divisors using place-value models, partial quotients, or the standard algorithm, writing exact quotients or quotients with remainders less than the divisor. Estimate quotients using rounding or compatible numbers, check results with multiplication, and solve division situations by representing equations, interpreting remainders appropriately, and determining whether answers are reasonable.
Identify factors, multiples, and factor pairs for whole numbers through 100 using multiplication, division, arrays, and skip-counting, including common factors and common multiples. Classify numbers through 100 as prime, composite, or neither, recognizing that primes have exactly two positive factors, composites have more than two, and 1 is neither. Recognize even and odd numbers and apply divisibility rules for 2, 3, 5, 9, and 10 to determine factors and multiples.
Students interpret, represent, and locate positive fractions, proper fractions, improper fractions, and mixed numbers using area, set, and number-line models, including unit fractions and fractions greater than one. They recognize, generate, and simplify equivalent fractions; decompose fractions into sums of like-denominator unit fractions; compare and order fractions using common numerators or denominators, benchmarks, equivalent fractions, and number lines. They add and subtract fractions and mixed numbers with like denominators, multiply fractions by whole numbers, and solve related word problems, including regrouping and expressing results as mixed numbers when appropriate.
Students identify points, lines, line segments, rays, and parallel, intersecting, and perpendicular lines; classify, measure, and draw acute, right, and obtuse angles from 0° to 180° using a protractor; and use angle addition and subtraction to find unknown whole-number angle measures.
Students classify triangles by angle measures and side lengths, classify quadrilaterals as trapezoids, parallelograms, rectangles, rhombi, or squares using their attributes and overlapping relationships, and identify the properties of parallelograms, rectangles, and squares and congruent figures.
Students identify and draw lines of symmetry, and recognize translations, rotations, and reflections as transformations that preserve a figure’s size and shape while changing its position or orientation.
Measure and compare length, mass, and capacity using appropriate customary and metric units, read rulers and scales, and convert within systems, including inches, feet, yards, miles; millimeters, centimeters, meters, kilometers; grams and kilograms; and liters and milliliters. Convert and interpret time in seconds, minutes, hours, days, and weeks, read analog and digital clocks to the minute, measure angles with a protractor to the nearest degree, and recognize benchmark angles of 90°, 180°, and 360°. Find and compare the area of rectangles and composite rectangles, determine missing dimensions from area, find polygon perimeters, distinguish area from perimeter, and solve measurement problems involving whole-number and familiar mixed-unit quantities through addition, subtraction, multiplication, division, and elapsed-time reasoning.
Students interpret multiplication as equal groups, arrays, rectangular area, and multiplicative comparisons, connecting factors and products through equations, diagrams, repeated addition, and the commutative and associative properties. They recall multiplication facts through 12×12, recognize whole-number patterns, apply the distributive property, multiply numbers up to four digits by one-digit factors and two two-digit factors, and multiply by multiples of 10 using place-value reasoning, partial products, area models, and the standard algorithm. They estimate whole-number products by rounding or using compatible numbers and solve one- and two-step multiplication and multiplicative comparison problems with unknown factors, quantities, or products.
Students identify, describe, generate, extend, and analyze repeating, growing, and numerical patterns involving whole-number rules such as adding, subtracting, multiplying, or dividing by a fixed number, using sequences, diagrams, number lines, and tables to find missing or later terms and infer features such as increase, repetition, alternation, or oddness and evenness. They complete and interpret input-output tables, determine and verify consistent one- or two-step rules, and evaluate whole-number expressions with addition, subtraction, multiplication, division, and parentheses while finding missing values in equivalent expressions. They represent equality as a balance and use symbols to solve one-step whole-number equations and simple one- or two-step word problems involving addition, subtraction, multiplication, and division, checking solutions with inverse operations and substitution.
Understand, read, write, and represent whole numbers from 0 through 1,000,000 in standard, expanded, and word forms, identifying each digit’s place and value through the millions place and using zeros as placeholders. Compare and order whole numbers, identify whole numbers between benchmarks, and locate them on scaled number lines using place-value reasoning and the symbols <, >, and =. Recognize ten times as much and one tenth as much through base-ten relationships, and round whole numbers through the millions place, including to the nearest 10, 100, and 1,000.