Arithmetic Concepts
132 concepts · Grades 3-5, 6-8, 9-12, K-2 · 246 prerequisite connections
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Concepts flow left to right, from foundational to advanced. Hover to highlight connections. Click any concept to learn more.
Connected Families
Arithmetic concepts have 96 connections to other families.
All Arithmetic Concepts
Fractions
A number representing a part of a whole, written as one integer over another non-zero integer.
Equivalent Fractions
Two or more fractions that look different but represent exactly the same amount or value.
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 comparison of two quantities that shows their relative sizes, written as $a:b$ or $\frac{a}{b}$.
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
Arranging three or more fractions from least to greatest (or greatest to least).
Mixed Numbers
A number consisting of a whole number and a proper fraction combined, such as $2\frac{3}{4}$.
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
Multiplying two fractions by multiplying the numerators together and the denominators together.
Dividing Fractions
Dividing by a fraction by multiplying by its reciprocal (inverting the divisor and multiplying).
Fraction of a Number
Finding a fractional part of a whole number by multiplying the fraction by that number.
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
Adding, subtracting, multiplying, and dividing numbers that contain decimal points.
Percent of a Number
Calculating a given percentage of a quantity by converting the percent to a decimal (or fraction) and multiplying.
Percent Change
The ratio of the change in a quantity to the original value, expressed as a percentage.
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.
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 equal groups fit into a total amount.
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.
Square Roots
The square root of a number $a$ is the non-negative value $b$ such that $b \times b = a$; it is the inverse of squaring and is written $\sqrt{a}$. For example, $\sqrt{25} = 5$ because $5^2 = 25$.
Absolute Value
The distance of a number from zero on the number line, always non-negative; written $|x|$.
Addition as Combining
Understanding addition as joining or combining two or more quantities to form a larger whole amount.
Subtraction as Difference
Understanding subtraction as finding the gap or difference between two quantities.
Multiplication as Scaling
Understanding multiplication as stretching or shrinking a quantity by a factor—scaling up or down from the original.
Multiplication as Area
Understanding multiplication as finding the area of a rectangle with given side lengths.
Division as Sharing
Understanding division as distributing a quantity equally among a number of groups or recipients.
Division as Inverse
Understanding division as the inverse of multiplication—recovering the missing factor in a product.
Inverse Operations
Pairs of operations that undo each other: addition/subtraction and multiplication/division are inverse pairs.
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
Multiplication distributes over addition: $a(b + c) = ab + ac$, linking two operations together.
Identity Elements
Special numbers that leave any other number unchanged under a given operation: 0 for addition, 1 for multiplication.
Operation Closure
When an operation on elements of a set always produces an element in the same set.
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
Understanding roots as undoing exponentiation—finding what was raised to a power.
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
Thinking about multiplicative relationships between quantities that scale together.
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 where quantities change at a constant rate, graphing as a straight line.
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 of the form $y = kx$ (where $k \neq 0$) that always passes through the origin; when one quantity doubles, the other doubles, and the ratio $\frac{y}{x}$ remains constant.
Inverse Variation
A relationship where $y = \frac{k}{x}$: as one quantity doubles, the other halves—their product stays constant.
Constraints
Conditions or limitations that restrict which values a variable or solution can take in a problem.
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
The process of simplifying a fraction or expression by removing (dividing out) common factors that appear in both the numerator and denominator, leaving 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.
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.
Length Measurement
Measuring how long something is using standard units (inches, centimeters, feet, meters) or non-standard units (paper clips, hand spans), by comparing the object's length to repeated copies of the chosen unit laid end to end.
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 pattern where each term changes by a consistent rule, such as adding the same number each time.
Skip Counting
Counting forward by a number other than 1, jumping by equal intervals such as 2s, 5s, or 10s.
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
A step-by-step algorithm for dividing a multi-digit number (dividend) by another number (divisor), producing 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 context into mathematical relationships and solving them.
Counting
Determining the total number of objects in a set by assigning one number to each object.
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
A number system using ten symbols (0-9) where each place represents a power of ten.
Zero
The number representing the absence of quantity; the additive identity and placeholder in positional notation.
Magnitude
The size or absolute value of a quantity, considering only how large it is and ignoring direction or sign.
Ordering Numbers
Arranging a collection of numbers from least to greatest (ascending) or greatest to least (descending).
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
A relationship where one quantity increases as another decreases, with constant product.
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
Discrete quantities come in separate, countable units; continuous quantities can take any value.
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 largest positive integer that divides evenly into two or more given numbers with no remainder.
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 of $x$, written $\sqrt[3]{x}$, is the number that when multiplied by itself three times equals $x$. Unlike square roots, cube roots are defined for negative numbers.
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.