Arithmetic Concepts
134 concepts · Grades 3-5, 6-8, 9-12, K-2 · 250 prerequisite connections
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Arithmetic concepts have 99 connections to other families.
All Arithmetic Concepts
Fractions
A fraction is a number of the form $\frac{a}{b}$ where $a$ (the numerator) counts how many equal parts you have and $b$ (the denominator, which must not be zero) tells how many equal parts the whole is divided into.
Equivalent Fractions
Two fractions $\frac{a}{b}$ and $\frac{c}{d}$ are equivalent if they represent the same value, which happens exactly when $a \times d = b \times c$ (cross-multiplication gives equal products).
Decimals
Numbers written with a decimal point where each position to the right represents tenths, hundredths, thousandths, etc.
Percentages
A way of expressing a quantity as a fraction of 100, written with the symbol % to mean 'per hundred.'
Ratios
A ratio compares two or more quantities by showing how many times one contains the other, written as $a:b$ or $\frac{a}{b}$. Unlike fractions, ratios can compare parts to parts, not just parts to wholes.
Rates
A rate is a ratio that compares two quantities measured in different units, expressing how much of one quantity corresponds to a given amount of another. It is often written as 'per' one unit of the second quantity, such as miles per hour or dollars per pound.
Fraction on a Number Line
Locating and representing a fraction as a precise point on a number line by dividing the unit interval into equal parts.
Comparing Fractions
Determining which of two fractions is greater, less, or equal using common denominators, benchmarks, or cross-multiplication.
Ordering Fractions
Ordering fractions means arranging a set of fractions from least to greatest (or greatest to least) by converting them to a common denominator or to decimals so their sizes can be directly compared.
Mixed Numbers
A mixed number combines a whole number and a proper fraction, such as $3\frac{1}{4}$, representing the sum of the whole part and fractional part: $3 + \frac{1}{4} = \frac{13}{4}$.
Improper Fractions
A fraction where the numerator is greater than or equal to the denominator, representing a value of one or more.
Mixed-Improper Conversion
The process of converting between mixed-number form and improper-fraction form, which represent the same value.
Adding Fractions with Like Denominators
Adding fractions that share the same denominator by adding the numerators and keeping the denominator.
Subtracting Fractions with Like Denominators
Subtracting fractions that share the same denominator by subtracting the numerators and keeping the denominator.
Adding Fractions with Unlike Denominators
Adding fractions with different denominators by first rewriting them with a common denominator (usually the LCD), then adding numerators.
Subtracting Fractions with Unlike Denominators
Subtracting fractions with different denominators by first rewriting them with a common denominator, then subtracting numerators.
Multiplying Fractions
To multiply fractions, multiply the numerators together and the denominators together: $\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$. Simplify the result by cancelling common factors.
Dividing Fractions
Dividing by a fraction means multiplying by its reciprocal: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$. This works because division asks 'how many groups of this size fit?'
Fraction of a Number
Finding a fraction of a number means multiplying that number by the fraction: $\frac{a}{b}$ of $n$ equals $\frac{a}{b} \times n = \frac{a \times n}{b}$. It answers 'what is this part of the whole amount?'
Decimal-Fraction Conversion
Converting between fraction form and decimal form of a number: divide numerator by denominator for fraction-to-decimal, and use place value to go the other way.
Decimal Operations
Decimal operations — addition, subtraction, multiplication, and division — follow the same rules as whole-number arithmetic but require careful attention to decimal point placement and alignment.
Percent of a Number
Calculating a given percentage of a quantity by converting the percent to a decimal (or fraction) and multiplying.
Percent Change
Percent change measures how much a quantity has increased or decreased relative to its original value, calculated as $\frac{\text{new} - \text{original}}{\text{original}} \times 100\%$.
Percent Applications
Using percentages to solve real-world problems involving tax, tip, discount, markup, and simple interest.
Adding Fractions
Adding fractions combines parts of a whole by rewriting both with a common denominator and then adding the numerators.
Fraction Line Plots
A fraction line plot displays fractional data by placing marks above a number line scaled in fractional units (halves, quarters, eighths, etc.).
Addition
The arithmetic operation of combining two or more numbers into a single total, representing joining or accumulating quantities.
Subtraction
Finding the difference between two numbers by removing one quantity from another, or measuring the gap between them.
Multiplication
Finding the total when a quantity is repeated a given number of times; the result of repeated addition of equal groups.
Division
Splitting a quantity into equal parts, or finding how many times one number fits into another. Division answers two questions: 'How many in each group?' and 'How many groups?'
Order of Operations
The agreed-upon sequence for evaluating expressions: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).
Exponents
An operation representing repeated multiplication: $a^n$ means $a$ multiplied by itself $n$ times. For example, $2^3 = 2 \times 2 \times 2 = 8$. Exponents extend to zero, negative, and fractional powers.
Square Roots
The non-negative number $b$ such that $b^2 = a$, written $\sqrt{a} = b$ — the inverse of squaring.
Absolute Value
The distance of a number from zero on the number line, always non-negative; written $|x|$. For any real number, absolute value strips away the sign and returns only the magnitude.
Addition as Combining
Understanding addition as the act of joining or combining two or more quantities to form a larger whole amount. This model helps students see addition as a physical action rather than an abstract rule.
Subtraction as Difference
Understanding subtraction as finding the gap or difference between two quantities, rather than just 'taking away.' This comparison model asks 'how many more?' or 'how far apart?'
Multiplication as Scaling
Understanding multiplication as resizing or scaling a quantity by a factor. Multiplying by 2 doubles, by 0.5 halves, and by 1 leaves unchanged — it stretches or shrinks the original number.
Multiplication as Area
Understanding multiplication as calculating the area of a rectangle: length times width gives the number of unit squares that fit inside. This visual model connects arithmetic to geometry.
Division as Sharing
Understanding division as distributing a total equally among a given number of groups. This 'fair sharing' model asks: if I share equally, how many does each group get?
Division as Inverse
Understanding division as the reverse of multiplication: if $a \times b = c$, then $c \div b = a$. This inverse relationship lets you undo multiplication to find missing factors.
Inverse Operations
Operations that undo each other: addition undoes subtraction, multiplication undoes division, and vice versa. Applying an operation followed by its inverse returns you to the starting value.
Commutativity
A property where swapping the order of two operands does not change the result: $a \ \star\ b = b\ \star\ a$.
Associativity
A property where changing the grouping of operands does not change the result: $(a \star b) \star c = a \star (b \star c)$.
Distributive Property
The rule that multiplying a sum equals the sum of individual products: $a(b+c) = ab + ac$. It links multiplication and addition, allowing you to break apart or combine terms.
Identity Elements
Special numbers that leave any other number unchanged under a given operation: 0 for addition, 1 for multiplication.
Operation Closure
A set is closed under an operation if performing that operation on members of the set always produces a result that is also in the set. For example, integers are closed under addition.
Operation Hierarchy
The layered relationship between arithmetic operations, where each is built from the previous: multiplication from addition, exponentiation from multiplication.
Repeated Operations
Applying the same operation multiple times in succession, where the repetition is often compressed into a higher-level operation: repeated addition becomes multiplication ($n \cdot a$), and repeated multiplication becomes exponentiation ($a^n$).
Square vs Cube Intuition
Understanding $x^2$ as the area of a square with side $x$ (2D), and $x^3$ as the volume of a cube (3D).
Roots as Inverse Growth
Roots reverse the process of exponentiation: the $n$th root of $a$ finds the number that, raised to the $n$th power, produces $a$. For example, $\sqrt[3]{8} = 2$ because $2^3 = 8$.
Unit Rate
A rate expressed as a quantity per single unit of another quantity, such as miles per hour or cost per item.
Proportional Reasoning
The ability to recognize and work with multiplicative relationships between quantities. If one quantity doubles, a proportional quantity also doubles — the ratio stays constant.
Constant of Proportionality
The constant ratio $k$ between two proportional quantities: if $y = kx$, then $k$ is the constant of proportionality.
Linear Relationship
A relationship between two variables where the rate of change is constant, producing a straight line when graphed. Expressed as $y = mx + b$ where $m$ is the slope.
Nonlinear Relationship
A relationship between two quantities where the rate of change is not constant—the graph is curved, not a straight line.
Direct Variation
A proportional relationship $y = kx$ that always passes through the origin — when one quantity doubles, so does the other.
Inverse Variation
A relationship where $y = \frac{k}{x}$: as one quantity doubles, the other halves—their product stays constant.
Constraints
Conditions or restrictions that limit which values are allowed in a problem. Constraints narrow the set of possible solutions, such as 'x must be positive' or 'the total cannot exceed 100.'
Balance Principle
The rule that any operation applied to one side of an equation must also be applied to the other side to preserve equality.
Equality as Relationship
Understanding $=$ not as 'the answer is' but as expressing that two expressions represent the same value.
Inequality Intuition
Understanding that $<$ and $>$ describe ordering relationships—one quantity is strictly smaller or larger than the other.
Bounds
The upper and lower limits within which a quantity must lie; often expressed as $a \leq x \leq b$.
Monotonicity
A function or sequence that consistently moves in one direction only—always increasing or always decreasing throughout its domain.
Symmetry in Operations
When exchanging or swapping operands or roles in an operation produces the same result or a symmetrically related one.
Invariants
Quantities or properties that remain unchanged during a process, operation, or transformation—values that stay the same no matter how the system is rearranged or acted upon.
Cancellation
Cancellation is the process of removing a common factor from the numerator and denominator of a fraction, or from both sides of an equation, to simplify. It works because dividing both parts by the same nonzero number leaves an equivalent but simpler form.
Equivalence
When two expressions, numbers, or objects represent the same value or are interchangeable in every relevant context.
Telling Time
Reading analog and digital clocks to determine the current time in hours, half hours, quarter hours, and five-minute intervals.
Elapsed Time
Calculating the amount of time that passes between a start time and an end time, using hours and minutes in base-60 arithmetic rather than base-10.
Money Counting
Identifying coins and bills by their value and adding them together to find a total amount of money.
Making Change
Calculating how much money is returned to a buyer when they pay more than the purchase price, using subtraction with dollars and cents or the counting-up strategy.
Length Measurement
Measuring how long something is using standard units (cm, m, in, ft) by finding the difference between start and end marks.
Weight Measurement
Measuring how heavy something is using standard units such as grams, kilograms, ounces, and pounds, by comparing an object's weight against known reference amounts on a balance or scale.
Simple Patterns
A repeating pattern is a sequence of elements (colors, shapes, numbers, or sounds) that repeats in a predictable cycle.
Growing Patterns
A growing pattern is a sequence where each term increases by following a consistent rule, such as adding the same number each time (2, 5, 8, 11, ...) or multiplying by a constant factor (3, 6, 12, 24, ...). Recognizing the rule lets you predict any term in the sequence.
Skip Counting
Counting forward by a number other than 1, jumping by equal intervals such as 2s, 5s, or 10s to produce the multiples of that number.
Picture Graphs
A way of displaying data using pictures or icons, where each picture represents one unit (or a set number of units), and the total for each category is found by counting or multiplying the number of pictures by the scale value.
Bar Graphs
A chart that uses rectangular bars of different heights or lengths to represent and compare quantities, where each bar's length is proportional to the value it represents and categories are shown on one axis.
Tally Charts
A method of recording and organizing data by drawing tally marks grouped in sets of five, where four vertical lines are crossed by a fifth diagonal line.
Multi-Digit Addition and Subtraction
Adding and subtracting numbers with three or more digits using the standard algorithm, which involves regrouping (carrying) in addition and borrowing in subtraction.
Multi-Digit Multiplication
Multiplying numbers with two or more digits using the standard algorithm, partial products, or the area (box) model.
Long Division
Long division is a step-by-step method for dividing large numbers by breaking the problem into a series of easier steps: divide, multiply, subtract, bring down, and repeat. It produces a quotient and possibly a remainder.
Adding and Subtracting Decimals
Adding and subtracting numbers with decimal points by aligning the decimal points vertically so that digits with the same place value line up.
Multiplying Decimals
Multiplying numbers that contain decimal points by first multiplying as if they were whole numbers, then placing the decimal point in the product based on the total number of decimal places in both factors.
Dividing Decimals
Dividing numbers that contain decimal points, typically by converting the divisor to a whole number (multiplying both divisor and dividend by a power of 10) and then performing long division.
Decimal Place Value
The value assigned to each digit's position to the right of the decimal point: the first position is tenths ($\frac{1}{10}$), the second is hundredths ($\frac{1}{100}$), the third is thousandths ($\frac{1}{1000}$), and so on.
Integer Operations
Adding, subtracting, multiplying, and dividing integers—numbers that include positive values, negative values, and zero.
Operations with Rational Numbers
Extending addition, subtraction, multiplication, and division to the full set of rational numbers—including fractions, decimals, mixed numbers, and their negative counterparts.
Word Problems
Word problems require translating a real-world scenario described in natural language into mathematical relationships, identifying the unknown quantity, setting up equations or expressions, and solving them to answer the question.
Unknown Factor Problems
An unknown factor problem asks you to find a missing number in a multiplication equation, such as $? \times 6 = 48$ or $8 \times ? = 56$.
Counting
Determining the total number of objects in a set by assigning exactly one number to each object in sequence, where the last number spoken equals the total count (the cardinality of the set).
Number Sense
An intuitive understanding of numbers, their relative size, and how they relate to each other and to real quantities.
Place Value
The value a digit represents based on its position in a number; the same digit means different amounts in different places.
More and Less
Comparing two quantities to determine which is greater, which is smaller, or whether they are equal.
Equal
Having exactly the same value or amount; the relationship expressed by the symbol $=$ between two expressions.
Integers
The set of whole numbers extended in both directions: positive whole numbers, their negatives, and zero.
Rational Numbers
Numbers that can be expressed as a ratio of two integers ($\frac{a}{b}$ where $b \neq 0$).
Irrational Numbers
An irrational number is a real number that cannot be expressed as a ratio of two integers $\frac{p}{q}$; its decimal expansion goes on forever without repeating any fixed block of digits.
Real Numbers
The complete set of all rational and irrational numbers, filling every point on the continuous number line.
Complex Numbers
Numbers of the form $a + bi$ where $a, b$ are real and $i = \sqrt{-1}$; they extend the real numbers to solve $x^2 = -1$.
Quantity
An amount or number of something that can be measured or counted; a quantity combines a number with a unit.
Number as Measure
Using numbers to represent the size or amount of a real-world quantity, always paired with a unit of measurement.
Base-Ten System
The positional numeral system using ten as its base, where each digit's value depends on its position, with each place worth ten times the place to its right.
Zero
The number representing the absence of quantity; the additive identity and placeholder in positional notation.
Magnitude
Magnitude measures the size or length of a quantity — for a vector (a, b), it is sqrt(a^2 + b^2). For a single number, magnitude is its absolute value: how far it is from zero, ignoring sign or direction.
Ordering Numbers
Ordering numbers is the process of arranging numbers in sequence from smallest to largest (ascending order) or largest to smallest (descending order). To order numbers, compare them using place value, common denominators, or convert to the same form (e.g. all decimals).
Comparison
Determining how two quantities relate in terms of size or value, using the symbols $<$, $>$, or $=$.
Unit Fraction
A fraction with numerator 1, like $\frac{1}{3}$ or $\frac{1}{8}$, representing exactly one equal part of a whole.
Decimal Representation
Writing fractions as digits to the right of a decimal point, using place values of tenths, hundredths, thousandths, etc.
Percent as Ratio
A ratio comparing a quantity to 100, written with the % symbol; 'per cent' literally means 'per hundred'.
Scaling
Changing the size of a quantity by multiplying by a factor, making it proportionally larger (factor $> 1$) or smaller (factor $< 1$).
Proportionality
A relationship where two quantities maintain a constant ratio: doubling one always doubles the other, giving $y = kx$.
Inverse Quantity
The reciprocal or multiplicative inverse of a quantity, where multiplying a number by its inverse yields one. Inverse quantities appear whenever two measurements are inversely related, so that doubling one halves the other.
Estimation
Finding a quick approximate answer by rounding to convenient values and computing mentally—no exact calculation needed.
Number Line
A straight line where each point represents a number, with equal spacing giving a visual model of all real numbers. The number line extends infinitely in both directions, with negative numbers to the left of zero and positive numbers to the right, providing a geometric representation of order and distance.
Density of Numbers
The property that between any two distinct real numbers, there are infinitely many other real numbers—no two are 'adjacent'.
Infinity Intuition
The concept of endlessness or unboundedness—a process that goes on forever with no final stopping point.
Finite vs Infinite
Finite describes a quantity or set with a definite end; infinite describes something that goes on forever without bound.
Discrete vs Continuous
The distinction between quantities that take separate, distinct values (discrete, like number of students) and quantities that can take any value in a range (continuous, like height or temperature).
Parity (Even/Odd)
The classification of integers as even (evenly divisible by 2, with no remainder) or odd (not divisible by 2).
Divisibility Intuition
Understanding when one whole number divides evenly into another, leaving no remainder—the foundation of factor and multiple relationships.
Factors
Whole numbers that divide evenly into a given number with no remainder—the 'building blocks' that multiply together to make it.
Multiples
Numbers obtained by multiplying a given number by positive integers: the skip-counting sequence $n, 2n, 3n, 4n, \ldots$
Prime Numbers
Integers greater than 1 whose only positive divisors are 1 and themselves—they cannot be factored further.
Composite Numbers
Integers greater than 1 that can be expressed as a product of two smaller positive integers; they are the opposite of primes.
Greatest Common Factor
The greatest common factor (GCF) of two or more numbers is the largest positive integer that divides each of them evenly, with no remainder. It is also called the greatest common divisor (GCD).
Least Common Multiple
The smallest positive integer that is divisible by each of two or more given numbers—where their multiples first coincide.
Numerical Structure
The underlying patterns, relationships, and algebraic properties—like commutativity and distributivity—that organize numbers into coherent systems.
Exponent Rules
A set of laws governing how exponents behave under multiplication, division, and raising to a power: product rule ($a^m \cdot a^n = a^{m+n}$), quotient rule ($a^m / a^n = a^{m-n}$), power rule ($(a^m)^n = a^{mn}$), zero exponent ($a^0 = 1$ for $a \neq 0$), and negative exponent ($a^{-n} = \frac{1}{a^n}$).
Scientific Notation
A way of writing very large or very small numbers as $a \times 10^n$, where $1 \leq |a| < 10$ and $n$ is an integer.
Scientific Notation Operations
Performing addition, subtraction, multiplication, and division on numbers expressed in scientific notation.
Cube Roots
The cube root $\sqrt[3]{x}$ is the number that, when cubed, gives $x$ — defined for all real numbers, including negatives.
Prime Factorization
Writing a whole number as a product of prime numbers; every composite number has exactly one such representation (up to order).
Negative Numbers
Negative numbers are numbers less than zero, used to represent direction, deficit, or values below a reference point.