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Every rule and concept in Pre-Algebra, with its definition and a link to the lesson that teaches it. Search by term, by idea, or by lesson title.

Glossary (226)

“At Least One” via the Complement: The complement of an event is everything except that event, and $P(\text{not }A) = 1 - P(A)$ .; Data, Statistics & Probability › Going Deeper: Advanced Statistics & Probability
A Circle Fact: The Inscribed Angle (Preview): An inscribed angle has its vertex on the circle; a central angle has its vertex at the center.; Geometry Basics › Going Deeper: Advanced Geometry
A letter is a placeholder for a number: A variable like $x$ just stands for a number you don't know yet.; Core Ideas in Plain Terms › Variables and Solving Equations
A Plan for Word Problems: Fractions › Word Problems with Fractions
A Plan of Attack: Geometry Basics › Real-World Geometry Problems
Absolute Value as Distance Between Two Numbers: You already know $|x|$ measures distance from $0$ .; Rational Numbers › Going Deeper: Advanced Rational-Number Ideas
Absolute-Value Equations and Inequalities (Preview): Because $|x|$ measures distance from $0$ , an equation like $|x| = 5$ has two answers: $x = 5$ or $x = -5$ (both are $5$ units away).; Integers › Going Deeper: Advanced Integer Ideas
Add the Opposite: To subtract an integer, add its opposite.; Integers › Subtracting Integers
Angle Chasing: The Exterior Angle Rule: Because a triangle's angles sum to $180^\circ$ and a straight line also measures $180^\circ$ , an exterior angle (formed by extending one side) equals the sum of the two remote interior angles: Here the exterior angle at $B$ equals $x + y$ .; Geometry Basics › Going Deeper: Advanced Geometry
Angles: Measured in degrees.; Course Review › Geometry Basics
Area from Vertices: The Shoelace Idea: You can find the area of a triangle straight from its corner coordinates $(x_1,y_1)$ , $(x_2,y_2)$ , $(x_3,y_3)$ — no height needed: \[ Area= $\tfrac{1}{2}$ | x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2) |.; The Coordinate Plane › Going Deeper: Advanced Coordinate Geometry
Arithmetic in Scientific Notation: You can multiply and add numbers written in scientific notation without first expanding them.; Exponents & Roots › Going Deeper: Advanced Exponent & Root Ideas
Breaking a Number into Primes: The prime factorization of a number writes it as a product of prime numbers only.; Factors & Multiples › Prime Factorization
Building a Table: A table of values is a neat list of ordered pairs.; The Coordinate Plane › Tables of Values
Classifying Triangles: By sides:; Geometry Basics › Triangles
Closure and Properties of the Integers: A set is closed under an operation if combining any two members always lands you back inside the set.; Integers › Going Deeper: Advanced Integer Ideas
Closure: Staying Inside the Family: A set is closed under an operation if combining two members always gives another member.; Rational Numbers › Going Deeper: Advanced Rational-Number Ideas
Combined Rates: Work Problems & Average Speed: When two workers (or pipes, or machines) act together, add their rates, not their times.; Ratios, Rates & Proportions › Going Deeper: Advanced Ratio & Proportion Ideas
Common Denominator First, Then Sign Rules: To add or subtract fractions, first get a common denominator.; Rational Numbers › Adding and Subtracting Fractions
Comparing Powers with Different Bases: When two powers have different bases and different exponents, the exponent rules cannot combine them — so you compare by evaluating (or estimating) each one.; Exponents & Roots › Going Deeper: Advanced Exponent & Root Ideas
Complementary and Supplementary Angles: Two angles are complementary if their measures add to $90^\circ$ .; Geometry Basics › Basic Vocabulary: Points, Lines, and Angles
Complex Fractions (a Fraction of Fractions): A complex fraction is a fraction whose numerator, denominator, or both are themselves fractions, like $\dfrac{\ \frac{2}{3}\ }{\ \frac{4}{5}\ }$ .; Fractions › Going Deeper: Advanced Fraction Ideas
Compound Interest: A Preview of $A = P\left(1 + \frac{r}{n}\right)^{nt}$: Simple interest ( $I = Prt$ ) is always figured on the original principal.; Percents › Going Deeper: Advanced Percent Ideas
Continued Fractions: A continued fraction builds a number out of a whole part plus $1$ over another whole part plus $1$ over $\ldots$ , stacking reciprocals downward.; Fractions › Going Deeper: Advanced Fraction Ideas
Converting: Percent means “per $100$ .” Percent $\to$ decimal: divide by $100$ (move point $2$ left): $45\%=0.45$ .; Course Review › Percents
Counting Divisors from the Prime Factorization: Once a number is written in prime-power form, you can count how many factors it has without listing a single one.; Factors & Multiples › Going Deeper: Advanced Number Theory
Cylinders: For a cylinder with radius $r$ and height $h$ : \[ V = π r^2 h.; Geometry Basics › Surface Area and Volume of Prisms and Cylinders
Deciding Which to Use: The hardest part of a word problem is choosing the right tool.; Factors & Multiples › Word Problems: GCF or LCM?
Decimal $\to$ Fraction: Use the Place Value: Write the decimal's digits over the place value of the last digit, then simplify.; Decimals › Converting Between Fractions & Decimals
Decimals Are Dense: Always Room in Between: Between any two different decimals there is always another decimal — in fact infinitely many.; Decimals › Going Deeper: Advanced Decimal Ideas
Definition: The absolute value of a number is its distance from $0$ on the number line.; Integers › Absolute Value
Deriving the Surface Area of a Cylinder: Imagine unwrapping a can.; Geometry Basics › Going Deeper: Advanced Geometry
Direct vs. Inverse Variation: When two quantities stay in a constant ratio, we say $y$ varies directly with $x$ : $ $y = kx \qquad\text{equivalently}\qquad \frac{y}{x} = k \;(\text{constant}).$ $Here$ k$ is the constant of proportionality.; Ratios, Rates & Proportions › Going Deeper: Advanced Ratio & Proportion Ideas
Distance from Zero: The absolute value of a rational number is its distance from $0$ on the number line, so it is never negative.; Rational Numbers › Absolute Value of Rational Numbers
Distributive Property & Like Terms: Distributive: $a(b+c)=ab+ac$ (multiply the outside by each inside term).; Course Review › Introduction to Algebra
Dividing by a Decimal — Move the Point: When the divisor is a decimal, slide its point to the right until it becomes a whole number, and slide the dividend's point the same number of places (add zeros if needed).; Decimals › Dividing Decimals
Dividing by a Whole Number: When the divisor is a whole number, place the decimal point in the quotient directly above the point in the dividend, then divide normally.; Decimals › Dividing Decimals
Dividing the Plane: The two axes split the plane into four regions called quadrants.; The Coordinate Plane › The Four Quadrants
Divisibility Rules: Course Review › Factors & Multiples
Divisibility tricks: A number is divisible by $3$ exactly when its digit sum is, and by $9$ when its digit sum is divisible by $9$ .; Core Ideas in Plain Terms › Going Deeper: Exponents and Number Sense
Equivalent, Simplify & Compare: Multiply or divide top and bottom by the same number for an equivalent fraction.; Course Review › Fractions
Estimating Roots More Precisely: Trapping a root between two integers is a first step; you can sharpen the estimate by testing a decimal in the middle.; Exponents & Roots › Going Deeper: Advanced Exponent & Root Ideas
Every Fraction Becomes One or the Other: Convert a fraction to a decimal by dividing the numerator by the denominator.; Rational Numbers › Terminating vs. Repeating Decimals
Expected Value: The expected value is the long-run average outcome of a random situation.; Data, Statistics & Probability › Going Deeper: Advanced Statistics & Probability
Exponent Rules: Course Review › Exponents & Roots
Exponent shortcuts: The product rule for exponents, $x^a \cdot x^b = x^{a+b}$ : three 2s multiplied together, times four more 2s, is seven 2s — $2^3 \cdot 2^4 = 2^7 = 128$ .; Core Ideas in Plain Terms › Going Deeper: Exponents and Number Sense
Exponents & Perfect Squares: In $b^n$ , $b$ is the base and $n$ the exponent: $b^n=\underbrace{b\times b\times\cdots\times b}_{n\text{ times}}$ .; Course Review › Whole Numbers & Operations
Factoring: The Distributive Property in Reverse: The distributive property $a(b + c) = ab + ac$ can be run backward.; Introduction to Algebra › Going Deeper: Toward Real Algebra
Fast Mental-Math & Estimation Tricks: A few reliable shortcuts:; Whole Numbers & Operations › Going Deeper: Advanced Whole-Number Ideas
First Make the Denominators Match: To add or subtract fractions with different denominators, rewrite them over a common denominator.; Fractions › Adding & Subtracting: UNLIKE Denominators (LCD)
Four Properties to Know: Introduction to Algebra › The Properties of Operations
Fraction $\leftrightarrow$ Decimal: Fraction $\to$ decimal: divide numerator by denominator.; Course Review › Decimals
Fraction $\to$ Decimal: Divide: To turn a fraction into a decimal, divide the numerator by the denominator.; Decimals › Converting Between Fractions & Decimals
From Expression to Function: A function is a rule that takes an input and produces exactly one output.; Introduction to Algebra › Going Deeper: Toward Real Algebra
From Rule to Graph: A rule like $y = x+2$ or $y = 2x$ describes how $y$ depends on $x$ .; The Coordinate Plane › Graphing a Linear Relationship
GCF & LCM: GCF (greatest common factor): the largest factor two numbers share — multiply the primes they have in common.; Course Review › Factors & Multiples
Get the Variable onto One Side: When the variable appears on both sides, like $5x - 3 = 2x + 9$ , first gather all the variable terms onto one side and all the plain numbers onto the other.; Equations & Inequalities › Going Deeper: Advanced Equations & Inequalities
Heron's Formula: Area From Three Sides: You can find a triangle's area from its side lengths alone—no height needed.; Geometry Basics › Going Deeper: Advanced Geometry
Independent vs. Dependent Events and Conditional Probability: Two events are independent if one happening does not change the other's probability (like two separate coin flips).; Data, Statistics & Probability › Going Deeper: Advanced Statistics & Probability
Inequalities: Solve like equations, but the sign shows a range of solutions ( $<,\le,>,\ge$ ).; Course Review › Equations & Inequalities
Inequalities and the Sign-Flip Rule: An inequality uses $<$ , $>$ , $\le$ , or $\ge$ instead of an equals sign, and describes a whole range of values.; Introduction to Algebra › Going Deeper: Toward Real Algebra
Key Words: Certain words signal each operation:; Introduction to Algebra › Writing Expressions from Word Phrases
Lattice Points and Taxicab Distance: A lattice point is a point whose coordinates are both integers, like $(3,-2)$ — exactly where grid lines cross.; The Coordinate Plane › Going Deeper: Advanced Coordinate Geometry
Line Up the Decimal Points: Write the numbers in a column so the decimal points sit directly above one another (this automatically lines up tenths with tenths, hundredths with hundredths).; Decimals › Adding & Subtracting Decimals
Line Up the Points, Compare Left to Right: To compare two decimals, stack them so the decimal points line up and add trailing zeros so both have the same number of places.; Decimals › Comparing & Ordering Decimals
Line Up the Points, Then Use Sign Rules: To add or subtract decimals, line up the decimal points and work as usual.; Rational Numbers › Adding and Subtracting Decimals
Look at the Next Digit: To round to a given place, find that place and look at the single digit just to its right:; Decimals › Rounding Decimals
Lowest Terms: A fraction is in lowest terms (fully simplified) when the numerator and denominator share no common factor except $1$ .; Fractions › Simplifying Fractions to Lowest Terms
Maps and Grids: Coordinates appear everywhere: on maps, game boards, and city grids.; The Coordinate Plane › Real-World Coordinates and Patterns
Mean Proportional & Continued Proportions: In the proportion $\frac{a}{b} = \frac{b}{c}$ , the repeated middle term $b$ is called the mean proportional (or geometric mean) of $a$ and $c$ .; Ratios, Rates & Proportions › Going Deeper: Advanced Ratio & Proportion Ideas
Measures of Center & Spread: Course Review › Data, Statistics & Probability
Measuring Spread: MAD and Standard Deviation: Range uses only two values.; Data, Statistics & Probability › Going Deeper: Advanced Statistics & Probability
Mixed Numbers: Mixed $\to$ improper: $a\tfrac{b}{c}=\dfrac{a\cdot c+b}{c}$ .; Course Review › Fractions
Mixture and Concentration Problems: A concentration is a percent: the amount of pure substance divided by the total amount of mixture.; Percents › Going Deeper: Advanced Percent Ideas
Modeling Multi-Step Word Problems: Real problems become algebra in three moves: (1) let a letter stand for the unknown quantity; (2) translate each sentence into an expression or equation; (3) solve, then check the answer against the words.; Introduction to Algebra › Going Deeper: Toward Real Algebra
Modular Arithmetic & Clock Arithmetic: Often we care only about the remainder of a division, not the quotient.; Whole Numbers & Operations › Going Deeper: Advanced Whole-Number Ideas
Modular Arithmetic with Negatives: When we divide, the remainder is what is left over.; Integers › Going Deeper: Advanced Integer Ideas
More Than One Answer: An inequality compares two expressions that are not necessarily equal.; Equations & Inequalities › What an Inequality Is
Multiply by the Reciprocal: To divide by a fraction, multiply by its reciprocal (flip the second fraction).; Fractions › Dividing Fractions & Mixed Numbers
Multiply Every Term Inside: To expand, multiply the number outside the parentheses by each term inside, keeping track of signs: \[ a(b + c) = ab + ac a(b - c) = ab - ac.; Introduction to Algebra › Using the Distributive Property to Expand
Multiply, Then Count the Places: Ignore the decimal points at first and multiply the numbers as whole numbers.; Decimals › Multiplying Decimals
Negatives are just "below zero": Numbers live on a line: zero in the middle, positives to the right, negatives to the left.; Core Ideas in Plain Terms › Negative Numbers and the Number Line
Not Every Number Is Rational: $\sqrt{2}$: Some numbers can never be written as a fraction of integers; we call them irrational.; Rational Numbers › Going Deeper: Advanced Rational-Number Ideas
Number Bases: Counting in Binary: Our usual system is base ten: each place is a power of $10$ ( $\ldots, 1000, 100, 10, 1$ ).; Whole Numbers & Operations › Going Deeper: Advanced Whole-Number Ideas
Number Line & Absolute Value: Integers are the whole numbers and their opposites: $\ldots,-3,-2,-1,0,1,2,3,\ldots$ On a number line, numbers increase to the right.; Course Review › Integers
One Move to Solve: A one-step equation needs just a single inverse operation to get the variable alone.; Equations & Inequalities › One-Step Equations
Only Like Terms Combine: Like terms have the exact same variable part (same letter, same power).; Introduction to Algebra › Combining Like Terms
Order of Operations (PEMDAS): Course Review › Whole Numbers & Operations
Parts of a Circle: Geometry Basics › Circles
PEMDAS Still Rules: Follow the usual order: Parentheses, Exponents, Multiply/Divide (left to right), Add/Subtract (left to right).; Integers › Order of Operations with Integers
PEMDAS, Now with Fractions and Signs: Use the same order every time: Parentheses, Exponents, Multiply/Divide (left to right), Add/Subtract (left to right).; Rational Numbers › Order of Operations with Rational Numbers
Percent Error: How Far Off a Measurement Is: In science, percent error compares a measured (or estimated) value to the true (accepted) value, relative to that true value: \[ percent error = measured - actual actual × 100%.; Percents › Going Deeper: Advanced Percent Ideas
Percent Of, Percent Change & Money: Course Review › Percents
Percent vs. Percentage Points: A percentage point is an absolute gap between two percents; a percent change is relative to the starting percent.; Percents › Going Deeper: Advanced Percent Ideas
Perfect, Abundant, and Deficient Numbers: Compare a number to the sum of its proper divisors (all its divisors except itself):; Factors & Multiples › Going Deeper: Advanced Number Theory
Perimeter & Area: Course Review › Geometry Basics
Permutations and Combinations: When the Counting Principle is used to arrange or choose from one group, two special cases appear.; Data, Statistics & Probability › Going Deeper: Advanced Statistics & Probability
Place Value & Rounding: Each place is worth ten times the one to its right: ones, tens, hundreds, thousands, ten-thousands, … Digits are grouped into periods of three (ones, thousands, millions) by commas.; Course Review › Whole Numbers & Operations
Place Value & The Four Operations: Places after the point: tenths, hundredths, thousandths, …; Course Review › Decimals
Places to the Right of the Point: A decimal point separates the whole-number part from the fractional part.; Decimals › Decimal Place Value; Reading & Writing Decimals
Placing and Ordering: Rational numbers fill in the gaps between the integers.; Rational Numbers › Rational Numbers on the Number Line
Plotting a Point: To plot the point $(x,y)$ :; The Coordinate Plane › Plotting Points and Naming Coordinates
Points on the Axes: If a point sits on an axis, it is not in any quadrant.; The Coordinate Plane › The Four Quadrants
Points, Axes & Quadrants: The plane is formed by a horizontal $x$ -axis and vertical $y$ -axis meeting at the origin $(0,0)$ .; Course Review › The Coordinate Plane
Preview: a Logarithm Undoes an Exponent: A square root answers “what number squared gives this?” A logarithm answers the matching question for exponents: “what exponent do I put on the base to get this number?” Just as squaring and square-rooting are opposites, exponents and logarithms are opp…; Exponents & Roots › Going Deeper: Advanced Exponent & Root Ideas
Primes, Factors & Prime Factorization: A factor divides a number evenly; a multiple is what you get counting by that number.; Course Review › Factors & Multiples
Probability & Counting: Probability of an event: $P(E)=\dfrac{\text{favorable outcomes}}{\text{total outcomes}}$ , a value from $0$ (impossible) to $1$ (certain).; Course Review › Data, Statistics & Probability
Proportions & Cross-Multiplication: A proportion states two ratios are equal: $\dfrac{a}{b}=\dfrac{c}{d}$ .; Course Review › Ratios, Rates & Proportions
Pythagorean Theorem: For a right triangle with legs $a,b$ and hypotenuse $c$ (the side opposite the right angle):; Course Review › Geometry Basics
Quartiles, IQR, and Outliers: The median splits ordered data in half.; Data, Statistics & Probability › Going Deeper: Advanced Statistics & Probability
Quick Tests for Even Division: These shortcuts tell you if a number divides evenly without doing the long division.; Factors & Multiples › Divisibility Rules
Ratios, Rates & Unit Rates: A ratio compares two quantities: $a$ to $b$ , $a{:}b$ , or $\tfrac{a}{b}$ .; Course Review › Ratios, Rates & Proportions
Reading a proof: Do not read a proof the way you read a rule.; Introduction to Algebra › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Equations & Inequalities › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; The Coordinate Plane › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Geometry Basics › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Data, Statistics & Probability › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Whole Numbers & Operations › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Integers › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Factors & Multiples › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Fractions › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Decimals › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Ratios, Rates & Proportions › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Percents › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Rational Numbers › Why It Works
Reading a proof: Do not read a proof the way you read a rule.; Exponents & Roots › Why It Works
Rectangular Prisms: For a box with length $l$ , width $w$ , and height $h$ : \[ V = l × w × h SA = 2(lw + lh + wh).; Geometry Basics › Surface Area and Volume of Prisms and Cylinders
Reverse Percent: Finding the Original After a Change: If a price already includes a markup or discount, you cannot just take that percent off the final number to get back.; Percents › Going Deeper: Advanced Percent Ideas
Rounding Error and Bounds: When you round, the true value lives in a band around your rounded answer.; Decimals › Going Deeper: Advanced Decimal Ideas
Same Bottom, Add the Tops: When two fractions have the same denominator, add or subtract the numerators and keep the denominator.; Fractions › Adding & Subtracting: LIKE Denominators
Same Row or Same Column: Two points lie on the same horizontal line when they share the same $y$ -coordinate.; The Coordinate Plane › Distance on a Horizontal or Vertical Line
Same Sign Rules as Multiplying: Division follows the exact same sign rules as multiplication.; Integers › Dividing Integers
Scaling Area & Volume: When a figure is enlarged by a scale factor $k$ , its lengths multiply by $k$ — but its area multiplies by $k^2$ and its volume by $k^3$ : $ $\text{length} \times k, \qquad \text{area} \times k^2, \qquad \text{volume} \times k^3.$ $ This is why doubling a pho…; Ratios, Rates & Proportions › Going Deeper: Advanced Ratio & Proportion Ideas
Scientific Notation: Write as $a\times10^n$ where $1\le a<10$ and $n$ is an integer.; Course Review › Exponents & Roots
Scientific Notation: A Bridge for Tiny and Huge Decimals: Scientific notation writes a number as a single nonzero digit, a decimal part, and a power of ten: $a \times 10^n$ with $1 \le a < 10$ .; Decimals › Going Deeper: Advanced Decimal Ideas
Show the Answer on a Number Line: Because an inequality has many solutions, we picture them on a number line.; Equations & Inequalities › Graphing Inequality Solutions
Sign Rule Is the Same; Method Depends on Form: For both multiplication and division: same signs give a positive, different signs give a negative.; Rational Numbers › Multiplying and Dividing Rational Numbers
Sign Rules: Adding: same signs $\to$ add and keep the sign; different signs $\to$ subtract and keep the sign of the larger absolute value.; Course Review › Integers
Sign Rules for Products: Multiply the absolute values, then decide the sign:; Integers › Multiplying Integers
Similar Triangles and Proportional Reasoning: Two triangles are similar if they have the same shape but not necessarily the same size: their corresponding angles are equal, and their corresponding sides are in the same ratio (scale factor).; Geometry Basics › Going Deeper: Advanced Geometry
Slope–Intercept Form: $y=mx+b$ (Preview): Every non-vertical line can be written as \[ y=mx+b, \] where $m$ is the slope and $b$ is the $y$ -intercept — the $y$ -value where the line crosses the $y$ -axis (that is, where $x=0$ ).; The Coordinate Plane › Going Deeper: Advanced Coordinate Geometry
Slope: Rate of Change: The slope of a line measures how steeply it rises.; The Coordinate Plane › Going Deeper: Advanced Coordinate Geometry
Solve Like an Equation, with One Warning: You solve inequalities almost exactly like equations: use inverse operations and do the same thing to both sides.; Equations & Inequalities › Solving Inequalities
Solving Equations: An equation is a balance.; Course Review › Equations & Inequalities
Solving Literal Equations (Rearranging Formulas): A literal equation has several letters, and we solve for one of them in terms of the others.; Introduction to Algebra › Going Deeper: Toward Real Algebra
Square Roots: $\sqrt{a}$ asks “what number squared gives $a$ ?” Squaring and square-rooting undo each other: $\sqrt{49}=7$ since $7^2=49$ .; Course Review › Exponents & Roots
Straight Across: To multiply fractions, multiply the numerators and multiply the denominators.; Fractions › Multiplying Fractions & Mixed Numbers
Substitute, Then Compute: To evaluate an expression, replace each variable with its given value and follow the order of operations (PEMDAS).; Introduction to Algebra › Evaluating Expressions by Substitution
Successive Percent Changes Multiply (They Don't Add): When one percent change follows another, you multiply their factors; you do not add the percents.; Percents › Going Deeper: Advanced Percent Ideas
Sums of Consecutive Integers: The sum of the integers from $1$ to $n$ has a famous closed form (Gauss's trick: pair the first with the last): \[ 1 + 2 + 3 + … + n = $\frac{n(n+1)}{2}$ .; Integers › Going Deeper: Advanced Integer Ideas
Tables & Rules: A rule like $y=2x+1$ pairs each $x$ with a $y$ .; Course Review › The Coordinate Plane
Terminating vs. Repeating: When you divide, one of two things happens:; Decimals › Converting Between Fractions & Decimals
Terminating vs. Repeating: Write the fraction in lowest terms and look at the denominator:; Course Review › Rational Numbers
The $10x - x$ Trick for Repeating Decimals: A repeating decimal is an infinite sum, but a little algebra collapses it to a fraction.; Decimals › Going Deeper: Advanced Decimal Ideas
The Angle Sum Rule: The three angles of any triangle always add up to $180^\circ$ : \[ ∠ A + ∠ B + ∠ C = 180^°.; Geometry Basics › Triangles
The Area Formulas: Area is the amount of surface a flat shape covers, measured in square units.; Geometry Basics › Area of Rectangles, Squares, Parallelograms, Triangles, and Trapezoids
The Big Idea: A variable is a letter (like $x$ , $n$ , or $y$ ) that stands for a number we do not know yet, or a number that can change.; Introduction to Algebra › Variables and What They Represent
The Big Idea: An equation is a math sentence that says two things are equal.; Equations & Inequalities › What an Equation Is
The Big Idea: The coordinate plane is made from two number lines that cross at right angles.; The Coordinate Plane › The Coordinate Plane
The Big Idea: The integers are the whole numbers together with their negatives and zero: \[ …, -3, -2, -1, 0, 1, 2, 3, … \] Integers do not include fractions or decimals.; Integers › What Integers Are
The Big Idea: A rational number is any number that can be written as a fraction $\dfrac{a}{b}$ , where $a$ and $b$ are integers and $b \neq 0$ .; Rational Numbers › What a Rational Number Is
The Building Blocks: Geometry Basics › Basic Vocabulary: Points, Lines, and Angles
The Distance Formula: For two points not on the same horizontal or vertical line, the straight-line distance still comes from a right triangle.; The Coordinate Plane › Going Deeper: Advanced Coordinate Geometry
The Division Algorithm (the exact rule behind remainders): Every long-division problem is really a statement about whole numbers.; Whole Numbers & Operations › Going Deeper: Advanced Whole-Number Ideas
The Euclidean Algorithm: For large numbers, factoring can be slow.; Factors & Multiples › Going Deeper: Advanced Number Theory
The Four Operations: Course Review › Fractions
The Full Strategy: To fully simplify an expression: first use the distributive property to remove any parentheses, then combine all like terms.; Introduction to Algebra › Simplifying: Distribute, Then Combine
The GCF–LCM Product Theorem: For any two whole numbers $a$ and $b$ , the greatest common factor and least common multiple are locked together by \[ GCF(a,b) × LCM(a,b) = a × b.; Factors & Multiples › Going Deeper: Advanced Number Theory
The Golden Ratio: The golden ratio appears when a length is split so that the whole is to the larger part as the larger part is to the smaller part.; Ratios, Rates & Proportions › Going Deeper: Advanced Ratio & Proportion Ideas
The Harmonic Mean: The harmonic mean of two positive numbers $a$ and $b$ averages their reciprocals instead of the numbers themselves: \[ H = 2 $\frac{1}{a}$ + $\frac{1}{b}$ = $\frac{2ab}{a+b}$ .; Fractions › Going Deeper: Advanced Fraction Ideas
The Largest Shared Factor: The greatest common factor of two numbers is the biggest number that divides both.; Factors & Multiples › Greatest Common Factor (GCF)
The Mediant: Another In-Between Number: There is a second trick for landing between two positive fractions $\dfrac{a}{b}$ and $\dfrac{c}{d}$ : add the tops and add the bottoms to form the mediant $\dfrac{a+c}{b+d}$ .; Rational Numbers › Going Deeper: Advanced Rational-Number Ideas
The Midpoint Formula: The midpoint of the segment joining $(x_1,y_1)$ and $(x_2,y_2)$ is the point exactly halfway between them.; The Coordinate Plane › Going Deeper: Advanced Coordinate Geometry
The One Big Rule: When you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign.; Equations & Inequalities › Flipping the Inequality Sign
The Parts of a Fraction: A fraction names part of a whole (or part of a group).; Fractions › Fraction Vocabulary & Equivalent Fractions
The Rationals Are Dense: Between any two different rational numbers there is always another rational number — in fact, infinitely many.; Rational Numbers › Going Deeper: Advanced Rational-Number Ideas
The Rule: A measure of center is one number that summarizes a whole list of numbers.; Data, Statistics & Probability › Mean, Median, and Mode (Measures of Center)
The Rule: In our number system, the place of a digit tells you its value.; Whole Numbers & Operations › Place Value & Reading Large Numbers
The Rule: On the number line, the number farther to the right is greater.; Integers › Comparing and Ordering Integers
The Rule: A ratio compares two quantities by division — it tells you how much of one thing there is for each amount of another.; Ratios, Rates & Proportions › What a Ratio Is
The Rule: The word percent comes from the Latin per centum, meaning “per hundred.” A percent is just a special fraction whose denominator is always $100$ .; Percents › What Percent Means (Per Hundred)
The Rule: An exponent is a short way to write repeated multiplication.; Exponents & Roots › Exponent Notation: Base & Exponent
The Rule for Right Triangles: In a right triangle, the two shorter sides (the legs, $a$ and $b$ ) and the longest side (the hypotenuse, $c$ , opposite the right angle) are related by \[ a^2 + b^2 = c^2.; Geometry Basics › The Pythagorean Theorem
The Smallest Shared Multiple: The least common multiple of two numbers is the smallest number that both divide into — the first multiple they share.; Factors & Multiples › Least Common Multiple (LCM)
The Sum of All Divisors: You can also add up all the divisors straight from the prime factorization.; Factors & Multiples › Going Deeper: Advanced Number Theory
The Weighted Mean: A plain mean treats every value equally.; Data, Statistics & Probability › Going Deeper: Advanced Statistics & Probability
Three ways to write the same amount: A fraction like $\tfrac34$ , the decimal $0.75$ , and the percent $75\%$ are the same number in different clothes.; Core Ideas in Plain Terms › Fractions, Decimals, and Percents
Three-Term Ratios $a:b:c$: A ratio can compare three or more quantities at once, such as $2:3:4$ .; Ratios, Rates & Proportions › Going Deeper: Advanced Ratio & Proportion Ideas
Tidy Up, Then Solve: Sometimes a side has more than one term with the same variable.; Equations & Inequalities › Combining Like Terms First
Transformations as Coordinate Rules: Moving or flipping a shape changes each point by a simple rule:; The Coordinate Plane › Going Deeper: Advanced Coordinate Geometry
Translate, Then Compute: Decide which operation the situation calls for, line up units, and keep money to two decimal places (cents).; Decimals › Word Problems with Decimals (Money & Measurement)
Translate, Then Compute: Real situations mix signs and fractions: temperatures, money owed, elevation, and recipes.; Rational Numbers › Word Problems with Rational Numbers
Translate, then undo in reverse: Turn the words into an equation phrase by phrase ("three more than twice a number" $\to 2n + 3$ ).; Core Ideas in Plain Terms › Problem-Solving Playbook
Translating Phrases: Course Review › Introduction to Algebra
Translating Situations: Real situations often use negatives: temperatures below zero, money owed (debt), and elevation below sea level.; Integers › Word Problems with Integers
Triangular Numbers & the Sum $1+2+\cdots+n$: The triangular numbers count dots stacked in a growing triangle: $1, 3, 6, 10, 15, \ldots$ , where the $n$ th one is the sum $1 + 2 + 3 + \cdots + n$ .; Whole Numbers & Operations › Going Deeper: Advanced Whole-Number Ideas
Turn Words into Math: To solve a word problem, let a variable stand for the unknown, translate the words into an equation, then solve.; Equations & Inequalities › Writing Equations from Word Problems
Turning a Pattern into a Formula: Many number patterns grow by the same amount each step.; Introduction to Algebra › Going Deeper: Toward Real Algebra
Two Cases: Same signs: add the absolute values and keep the common sign.; Integers › Adding Integers
Two Conditions at Once: A compound inequality traps the variable between two bounds, like $-1 < 2x + 3 \le 9$ .; Equations & Inequalities › Going Deeper: Advanced Equations & Inequalities
Two Different Things: An expression is a combination of numbers, variables, and operations with no equals sign, such as $3x + 7$ .; Introduction to Algebra › Expressions vs. Equations
Two Equations, Two Unknowns: A system is two equations that must both be true at the same time, usually with two variables like $x$ and $y$ .; Equations & Inequalities › Going Deeper: Advanced Equations & Inequalities
Two Kinds of Whole Numbers: Factors & Multiples › Prime and Composite Numbers
Two Names for the Same Amount: An improper fraction has a numerator greater than or equal to its denominator (like $\tfrac{11}{4}$ ).; Fractions › Improper Fractions & Mixed Numbers
Two Ways to Compare: To decide which of two fractions is larger:; Fractions › Comparing & Ordering Fractions
Types of Angles: Geometry Basics › Basic Vocabulary: Points, Lines, and Angles
Undo in Reverse Order: A two-step equation has two operations, like $2x + 3 = 11$ .; Equations & Inequalities › Two-Step Equations
Undo with the Opposite: Every operation has an inverse (an opposite) that undoes it:; Equations & Inequalities › Inverse Operations and Keeping Balance
Unit Fractions and Egyptian Fractions: A unit fraction has a numerator of $1$ , like $\dfrac{1}{2}$ or $\dfrac{1}{7}$ .; Fractions › Going Deeper: Advanced Fraction Ideas
Variables, Terms & Evaluating: A variable is a letter standing for an unknown number.; Course Review › Introduction to Algebra
Vocabulary of an Expression: Consider $\ 5x + 2y - 8$ .; Introduction to Algebra › Expressions vs. Equations
Volume & Surface Area: Course Review › Geometry Basics
Weighted Averages & Mixtures: A plain average treats every item equally, but a weighted average counts some values more heavily according to a ratio of “weights” (amounts).; Ratios, Rates & Proportions › Going Deeper: Advanced Ratio & Proportion Ideas
What a Factor Is: A factor of a number divides into it evenly, leaving no remainder.; Factors & Multiples › Factors of a Number
What a Multiple Is: A multiple of a number is what you get by multiplying it by $1, 2, 3, 4, \dots$ In other words, it is any number in that number's “times table.” Multiples go on forever, while factors are a short, finite list.; Factors & Multiples › Multiples of a Number
What Perimeter Means: The perimeter is the total distance around a shape: just add up the lengths of all the sides.; Geometry Basics › Perimeter of Polygons
What They Are: A rational number can be written as a fraction $\dfrac{a}{b}$ of integers with $b\ne0$ .; Course Review › Rational Numbers
When Is Each One Most Useful?: Data, Statistics & Probability › Mean, Median, and Mode (Measures of Center)
When the Variable Disappears: Sometimes, while solving, the variable cancels out completely and you are left with a statement of pure numbers.; Equations & Inequalities › Going Deeper: Advanced Equations & Inequalities
Wholes with Wholes, Fractions with Fractions: Add (or subtract) the whole-number parts and the fraction parts separately.; Fractions › Adding & Subtracting Mixed Numbers
Why $(-)(-) = (+)$ : A Real Proof: The rule “a negative times a negative is positive” is not arbitrary—it is forced by the distributive property, $a(b+c) = ab + ac$ .; Integers › Going Deeper: Advanced Integer Ideas
Why Cross-Multiplication Works — and the Mediant: When you compare $\dfrac{a}{b}$ and $\dfrac{c}{d}$ (with positive $b$ and $d$ ), multiplying both by $bd$ does not change which is larger, since $bd>0$ .; Fractions › Going Deeper: Advanced Fraction Ideas
Why Negative Exponents Mean “One Over”: The quotient rule not only explains $a^{0}=1$ ; it also tells us what a negative exponent must mean.; Exponents & Roots › Going Deeper: Advanced Exponent & Root Ideas
Why Only $2$ and $5$ Make a Decimal Terminate: A decimal terminates exactly when it can be written with a denominator that is a power of ten.; Decimals › Going Deeper: Advanced Decimal Ideas
Why the Divisibility Rules Work: The quick tests for divisibility are not magic — each one comes from place value.; Whole Numbers & Operations › Going Deeper: Advanced Whole-Number Ideas
Why the Pythagorean Theorem Is True (A Proof): Take four copies of a right triangle with legs $a$ , $b$ and hypotenuse $c$ , and arrange them inside a big square whose side is $a + b$ .; Geometry Basics › Going Deeper: Advanced Geometry