Arithmetic

Use digits, zero, and the number line to represent whole numbers and distances from zero. Place numbers accurately and identify predecessors, successors, and intervals.

Read, write, compose, and decompose whole numbers using ones, tens, hundreds, and larger powers of ten. Use expanded form and place-value models to explain each digit's value.

Compare and order whole numbers using place value and inequality symbols. Round to a chosen place and judge when an exact or rounded value is appropriate.

Interpret addition as combining, increasing, and measuring total distance. Build addition facts and solve practical problems with number lines, objects, and written equations.

Interpret subtraction as taking away, finding a difference, and finding a missing addend. Use addition to check subtraction and handle regrouping with place-value models.

Add and subtract multi-digit whole numbers using standard written algorithms. Record regrouping clearly and verify results through estimation and inverse operations.

Interpret multiplication as equal groups, arrays, scaling, and repeated addition. Develop fluent multiplication facts from patterns rather than isolated memorization.

Interpret division as sharing equally, forming groups, and finding a missing factor. Represent exact quotients and quotients with remainders in context.

Use the commutative, associative, distributive, identity, and inverse properties to rewrite calculations efficiently. Recognize which properties do not apply to subtraction or division.

Multiply multi-digit whole numbers using partial products and the standard algorithm. Connect each written step to place value and the distributive property.

Divide whole numbers using partial quotients and long division. Interpret remainders, estimate quotient digits, and check answers by multiplication.

Trace arithmetic from tally marks and early numeral systems through zero, Hindu-Arabic notation, written algorithms, mechanical calculators, and electronic computation. Connect each development to changes in trade, science, education, and everyday calculation.

Represent whole numbers in binary and other positional bases, convert between bases, and perform simple calculations outside base ten. This reveals which features belong to place value generally and which are specific to decimal notation.

Use decomposition, compensation, doubling, halving, compatible numbers, and benchmark values to calculate mentally. Estimate before calculating and use bounds to detect implausible results.

Identify factors, multiples, factor pairs, and common multiples. Use arrays and multiplication facts to solve grouping and scheduling problems.

Distinguish prime and composite numbers and find prime factorizations using factor trees or repeated division. Apply the fundamental theorem of arithmetic to compare factorizations.

Find greatest common divisors and least common multiples from lists, prime factorizations, and the Euclidean algorithm. Apply them to simplifying quantities and coordinating repeating cycles.

Apply divisibility tests and interpret remainders in division problems. Connect quotient-remainder form to repeating patterns and checks on whole-number calculations.

Calculate with congruences and clock-style number systems. Use modular arithmetic for calendars, check digits, cyclic schedules, and parity arguments.

Represent fractions as parts of a whole, measurements, quotients, ratios, and points on the number line. Work with proper fractions, improper fractions, and mixed numbers.

Generate equivalent fractions by scaling numerator and denominator together. Reduce fractions to simplest form using common factors and greatest common divisors.

Compare and order fractions using common denominators, cross-products, benchmarks, and number-line reasoning. Explain comparisons without relying only on a procedure.

Add and subtract fractions with like or unlike denominators. Use least common multiples, mixed numbers, estimation, and visual models to support the calculations.

Interpret fraction multiplication as scaling and as finding a fraction of a quantity. Multiply fractions and mixed numbers, simplify efficiently, and estimate the size of products.

Interpret fraction division through measurement, sharing, and missing-factor questions. Divide using reciprocals and verify the result by multiplication.

Read, write, compare, and round decimal numbers using place value to the right of the decimal point. Connect tenths, hundredths, and thousandths to fractions and measurements.

Add, subtract, multiply, and divide decimals while tracking place value. Estimate first, place decimal points logically, and check results with inverse operations.

Convert terminating and repeating decimals to fractions and convert fractions to decimal form through division. Recognize how denominator factors determine whether a decimal terminates or repeats.

Compare two quantities through part-to-part and part-to-whole ratios. Simplify ratios, create equivalent ratios, and use ratio tables in mixtures, scales, and sharing problems.

Calculate unit rates and compare quantities with different units. Apply rates to prices, speed, fuel use, productivity, and other everyday measurements.

Recognize proportional relationships and solve them using scaling, unit rates, tables, equations, and cross-products. Distinguish proportional situations from relationships with fixed additions or other patterns.

Interpret percent as a rate per hundred and convert among percentages, fractions, and decimals. Calculate a percentage, find the whole or rate, and distinguish percent change from percentage-point change.

Calculate arithmetic means and weighted means, including missing-value problems. Judge when an average fairly summarizes quantities and how weights alter the result.

Represent gains, losses, elevations, temperatures, and opposing directions with positive and negative numbers. Compare signed numbers and use absolute value as distance from zero.

Add, subtract, multiply, and divide signed numbers using number-line and pattern-based reasoning. Apply sign rules to balances, changes, and multi-step calculations.

Interpret whole-number exponents as repeated multiplication and apply the product, quotient, and power rules. Handle zero and negative exponents through consistent numerical patterns.

Recognize perfect squares and cubes and calculate square and cube roots. Approximate nonperfect roots and see why some values cannot be written as fractions.

Classify numbers as natural, whole, integer, rational, irrational, or real. Use the nesting of these sets and determine when an operation stays within a given set.

Evaluate numerical expressions using grouping symbols and the conventional order of operations. Rewrite expressions with fraction bars, exponents, and nested parentheses without changing their value.

Write very large and very small quantities as a number times a power of ten. Compare values and perform the four operations in scientific notation.

Distinguish exact values from measured or rounded values. Work with significant figures, absolute and relative error, tolerances, and upper and lower bounds.

Convert measurements using conversion factors and dimensional analysis. Calculate accurately with length, mass, capacity, time, temperature, and compound units.

Calculate unit prices, discounts, sales tax, tips, markups, currency conversions, and shared costs. Build itemized calculations that make each assumption and rounding decision visible.

Calculate simple and compound interest, repeated percentage growth, loan costs, and savings balances. Compare advertised rates using consistent time periods and realistic payment assumptions.

Use calculators without surrendering estimation or judgment, including memory, parentheses, exponent, and fraction functions. Detect entry errors by checking magnitude, units, and inverse calculations.

Build spreadsheet formulas for totals, percentages, repeated growth, unit conversions, and scenario comparisons. Use cell references, formatting, and independent checks to create a reliable arithmetic worksheet.

Recognize additive, multiplicative, alternating, and recursive numerical patterns. Describe rules, extend sequences, and distinguish a convincing pattern from a proven claim.

Support arithmetic claims with definitions, cases, counterexamples, parity arguments, and divisibility reasoning. Write short explanations showing why a method works rather than presenting only an answer.

Take a real quantitative problem from question to result: identify known and unknown quantities, choose operations, preserve units, calculate, validate, and communicate the answer. Complete cases such as comparing purchases, planning materials, or checking a bill.

Connect arithmetic mastery to algebra, geometry, statistics, number theory, computing, finance, and skilled trades. Identify next-step courses, useful skill assessments, and proof-of-skill artifacts such as checked worksheets, budgets, and calculation reports.