Arithmetic
Free lessons and practice worksheets for arithmetic — worked examples, common mistakes, and step-by-step solutions.
- 01Arithmetic4 min read
Addition Properties
Addition properties are fundamental mathematical rules that govern how numbers combine in addition operations. The three main properties are commutative (order doesn't matter), associative (grouping doesn't matter), and identity (adding zero changes nothing). These properties form the foundation for arithmetic fluency and algebraic thinking in elementary mathematics, appearing in CCSS.1.OA and CCSS.2.OA standards.
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Addition
Addition is the fundamental arithmetic operation that combines two or more numbers to produce their total sum. The process follows consistent rules regardless of whether adding single digits like 3 + 4 = 7 or larger numbers like 127 + 358 = 485. Addition appears in CCSS standards from kindergarten through grade 2, building from simple counting strategies to multi-digit algorithms with regrouping.
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Decimal Arithmetic
Decimal arithmetic involves performing addition, subtraction, multiplication, and division operations with decimal numbers. The fundamental principle requires aligning decimal points for addition and subtraction, while multiplication and division follow specific rules for decimal placement. These operations form the foundation for working with money, measurements, and precise calculations in mathematics.
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Decimal Word Problems
Decimal word problems combine decimal arithmetic with real-world contexts such as shopping, cooking, and measurement. These problems require identifying which mathematical operation to use based on the situation described, then performing calculations with numbers that include decimal points. Common scenarios involve calculating change from purchases, finding totals for multiple items, or determining unit prices.
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Factors, GCF & LCM
Factors are whole numbers that divide evenly into another number without leaving a remainder. The Greatest Common Factor (GCF) represents the largest number that divides into two or more numbers, while the Least Common Multiple (LCM) is the smallest number that both original numbers divide into evenly. These concepts appear in CCSS Grade 4 standards for finding factor pairs and identifying prime and composite numbers.
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Intro to Multiplication
Multiplication represents repeated addition of equal groups, where 4 × 3 means adding 3 four times to get 12. This operation appears in CCSS 3.OA standards as students transition from counting individual objects to working with equal groups. Arrays, equal groups, and skip counting provide visual foundations for understanding multiplication before memorizing facts.
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Long Division
Long division is a systematic method for dividing large numbers by breaking the process into smaller, manageable steps. The algorithm involves repeatedly dividing, multiplying, subtracting, and bringing down digits until the entire dividend is processed. This method works with any divisor and produces exact quotients with remainders when necessary.
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Modular Arithmetic
Modular arithmetic deals with remainders after division, focusing on what's left over when one integer is divided by another. The notation a mod n represents the remainder when a is divided by n, always yielding a value between 0 and n-1. Two numbers are congruent modulo n if they leave the same remainder when divided by n, written as a ≡ b (mod n).
- 09Arithmetic3 min read
Multiplication & Division in Daily Life
Multiplication and division represent two fundamental operations that solve opposite problems in everyday situations. Multiplication determines the total when combining equal groups, such as finding the cost of 8 notebooks at $3 each. Division splits quantities into equal parts or determines how many groups can be formed, like sharing 24 cookies among 6 people.
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Multiplication Properties
Multiplication properties are mathematical rules that describe how numbers behave when multiplied together. These properties include the commutative property (order doesn't matter), associative property (grouping doesn't matter), identity property (multiplying by 1), and distributive property (multiplying over addition). The zero property states that any number multiplied by 0 equals 0.
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Multiplication
Multiplication combines equal groups into a single total, making it a faster alternative to repeated addition. The operation uses two factors to produce a product, such as 6 × 4 = 24. This fundamental arithmetic operation appears throughout mathematics, from basic counting to advanced algebra and calculus.
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Order of Operations
Order of operations is a set of rules that determines which mathematical operations to perform first in an expression containing multiple operations. The standard convention follows PEMDAS: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right). Without these rules, the expression 3 + 4 × 2 could equal either 11 or 14, creating mathematical ambiguity.
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Percentages
A percentage represents a part of 100, expressing fractions and ratios in a standardized form. The word "percent" comes from the Latin "per centum," meaning "by the hundred." Converting between percentages, decimals, and fractions forms the foundation for solving proportion problems across mathematics.
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Rounding & Estimation
Rounding transforms precise numbers into simpler, approximate values by replacing digits with zeros based on their position. The process follows a consistent rule: examine the digit immediately to the right of the target place value, then round up if it's 5 or greater, or round down if it's less than 5. For example, 347 rounded to the nearest hundred becomes 300 because the tens digit (4) is less than 5.
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Subtraction
Subtraction finds the difference between two numbers by taking away a smaller quantity from a larger one. The operation answers questions like "How many are left?" or "How much more is needed?" This fundamental arithmetic skill appears in CCSS.1.OA and CCSS.2.NBT standards, beginning with simple take-away problems and progressing to multi-digit calculations requiring regrouping.