# math.wiki

> Free lessons for all of K-12 math, from counting to calculus: 1467 lessons in 18 courses, laid out as one map where every topic links to what it needs first and what it leads to. Each lesson opens with a question, then explains why the idea exists, defines it, works examples and ends with exercises and answers. Licence: CC BY-NC 4.0.

- Full text manifest: https://math.wiki/llms-full.txt
- Map and search: https://math.wiki/
- For teachers: https://math.wiki/teachers/
- For homeschools: https://math.wiki/homeschool/
- Sitemap: https://math.wiki/sitemap.xml

## Courses

- [Kindergarten](https://math.wiki/course/kindergarten/): Elementary, 36 lessons in 10 chapters
- [Grade 1](https://math.wiki/course/grade-1/): Elementary, 30 lessons in 11 chapters
- [Grade 2](https://math.wiki/course/grade-2/): Elementary, 45 lessons in 11 chapters
- [Grade 3](https://math.wiki/course/grade-3/): Elementary, 47 lessons in 12 chapters
- [Grade 4](https://math.wiki/course/grade-4/): Elementary, 58 lessons in 12 chapters
- [Grade 5](https://math.wiki/course/grade-5/): Elementary, 46 lessons in 12 chapters
- [Math 6](https://math.wiki/course/math-6/): Middle school, 98 lessons in 13 chapters
- [Math 7](https://math.wiki/course/math-7/): Middle school, 89 lessons in 13 chapters
- [Math 8](https://math.wiki/course/math-8/): Middle school, 89 lessons in 13 chapters
- [Algebra 1](https://math.wiki/course/algebra-1/): High school, 112 lessons in 13 chapters
- [Geometry](https://math.wiki/course/geometry/): High school, 128 lessons in 12 chapters
- [Algebra 2](https://math.wiki/course/algebra-2/): High school, 121 lessons in 13 chapters
- [Precalculus](https://math.wiki/course/precalculus/): High school, 115 lessons in 12 chapters
- [Statistics](https://math.wiki/course/statistics/): High school, 95 lessons in 11 chapters
- [Calculus 1](https://math.wiki/course/calculus-1/): College, 71 lessons in 10 chapters
- [Calculus 2](https://math.wiki/course/calculus-2/): College, 35 lessons in 7 chapters
- [Linear Algebra](https://math.wiki/course/linear-algebra/): College, 39 lessons in 6 chapters
- [Discrete Mathematics and Proof](https://math.wiki/course/discrete-math/): College, 76 lessons in 10 chapters

## Printable worksheets

Printable PDF worksheets, quizzes and chapter tests with answer keys, plus a one-page handout per lesson. Chapter 1 of every course is free to download; the rest come with a plan (see https://math.wiki/pricing/). Every worksheet has its own page at https://math.wiki/worksheets/<course>/<lesson>/ listing its problems.

- [Free math worksheets with answer keys](https://math.wiki/worksheets/free/): every free sheet on one page
- [Middle school math worksheets with answer keys](https://math.wiki/worksheets/middle-school/): Grades 6, 7 and 8
- [High school math worksheets with answer keys](https://math.wiki/worksheets/high-school/): Algebra 1, Geometry, Algebra 2, Precalculus
- [Elementary math worksheets with answer keys](https://math.wiki/worksheets/elementary/): Grades 4 and 5
- [Kindergarten worksheets with answer keys](https://math.wiki/worksheets/kindergarten/): 36 handouts, 36 worksheets, 0 quizzes and 10 chapter tests
- [Grade 1 worksheets with answer keys](https://math.wiki/worksheets/grade-1/): 30 handouts, 30 worksheets, 0 quizzes and 11 chapter tests
- [Grade 2 worksheets with answer keys](https://math.wiki/worksheets/grade-2/): 45 handouts, 45 worksheets, 0 quizzes and 11 chapter tests
- [Grade 3 worksheets with answer keys](https://math.wiki/worksheets/grade-3/): 47 handouts, 47 worksheets, 0 quizzes and 12 chapter tests
- [Grade 4 worksheets with answer keys](https://math.wiki/worksheets/grade-4/): 58 handouts, 58 worksheets, 22 quizzes and 12 chapter tests
- [Grade 5 worksheets with answer keys](https://math.wiki/worksheets/grade-5/): 44 handouts, 44 worksheets, 19 quizzes and 12 chapter tests
- [Math 6 worksheets with answer keys](https://math.wiki/worksheets/math-6/): 98 handouts, 98 worksheets, 37 quizzes and 13 chapter tests
- [Math 7 worksheets with answer keys](https://math.wiki/worksheets/math-7/): 89 handouts, 89 worksheets, 34 quizzes and 13 chapter tests
- [Math 8 worksheets with answer keys](https://math.wiki/worksheets/math-8/): 89 handouts, 89 worksheets, 37 quizzes and 13 chapter tests
- [Algebra 1 worksheets with answer keys](https://math.wiki/worksheets/algebra-1/): 112 handouts, 112 worksheets, 42 quizzes and 13 chapter tests
- [Geometry worksheets with answer keys](https://math.wiki/worksheets/geometry/): 128 handouts, 128 worksheets, 47 quizzes and 12 chapter tests
- [Algebra 2 worksheets with answer keys](https://math.wiki/worksheets/algebra-2/): 121 handouts, 121 worksheets, 47 quizzes and 13 chapter tests
- [Precalculus worksheets with answer keys](https://math.wiki/worksheets/precalculus/): 115 handouts, 115 worksheets, 46 quizzes and 12 chapter tests
- [Statistics worksheets with answer keys](https://math.wiki/worksheets/statistics/): 95 handouts, 95 worksheets, 37 quizzes and 11 chapter tests
- [Calculus 1 worksheets with answer keys](https://math.wiki/worksheets/calculus-1/): 71 handouts, 71 worksheets, 29 quizzes and 10 chapter tests
- [Calculus 2 worksheets with answer keys](https://math.wiki/worksheets/calculus-2/): 35 handouts, 35 worksheets, 15 quizzes and 7 chapter tests
- [Linear Algebra worksheets with answer keys](https://math.wiki/worksheets/linear-algebra/): 39 handouts, 39 worksheets, 16 quizzes and 6 chapter tests

### Worksheets by topic

- [Algebra Worksheets](https://math.wiki/worksheets/topic/algebra/): 132 printable PDF worksheets with answer keys
- [Geometry Worksheets](https://math.wiki/worksheets/topic/geometry/): 196 printable PDF worksheets with answer keys
- [Fractions and Decimals Worksheets](https://math.wiki/worksheets/topic/fractions/): 81 printable PDF worksheets with answer keys
- [Calculus Worksheets](https://math.wiki/worksheets/topic/calculus/): 107 printable PDF worksheets with answer keys
- [Statistics Worksheets](https://math.wiki/worksheets/topic/statistics/): 159 printable PDF worksheets with answer keys
- [Trigonometry Worksheets](https://math.wiki/worksheets/topic/trigonometry/): 79 printable PDF worksheets with answer keys
- [Number Sense and Integers Worksheets](https://math.wiki/worksheets/topic/integers/): 187 printable PDF worksheets with answer keys
- [Functions Worksheets](https://math.wiki/worksheets/topic/functions/): 115 printable PDF worksheets with answer keys
- [Probability Worksheets](https://math.wiki/worksheets/topic/probability/): 49 printable PDF worksheets with answer keys
- [Measurement Worksheets](https://math.wiki/worksheets/topic/measurement/): 83 printable PDF worksheets with answer keys
- [Linear Algebra Worksheets](https://math.wiki/worksheets/topic/linear-algebra/): 49 printable PDF worksheets with answer keys

## Topics

- [Algebra](https://math.wiki/topic/algebra/): 130 lessons in 9 subtopics. Working with unknowns. Letters stand for numbers, and the rules of arithmetic still hold.
- [Calculus](https://math.wiki/topic/calculus/): 123 lessons in 8 subtopics. Change and accumulation: derivatives and integrals.
- [Fractions, ratio and percent](https://math.wiki/topic/fractions-ratio-percent/): 101 lessons in 3 subtopics. Parts of a whole, comparisons by multiplying, and per-hundred.
- [Functions](https://math.wiki/topic/functions/): 128 lessons in 13 subtopics. Rules that turn one number into exactly one other, and the graphs they draw.
- [Geometry](https://math.wiki/topic/geometry/): 209 lessons in 11 subtopics. Shapes, space, and the reasoning that proves facts about them.
- [Linear algebra](https://math.wiki/topic/linear-algebra/): 50 lessons in 6 subtopics. Vectors, matrices and the linear transformations they describe: solving many equations at once, and what a matrix does to space.
- [Logic, proof and discrete math](https://math.wiki/topic/logic-and-proof/): 101 lessons in 6 subtopics. How mathematical truth is argued, plus sets, counting structures and a first look at linear algebra.
- [Measurement](https://math.wiki/topic/measurement/): 93 lessons in 12 subtopics. Putting numbers on the world: length, area, volume, angle, time.
- [Number](https://math.wiki/topic/number/): 211 lessons in 9 subtopics. What numbers are, and what you do with them: counting, place value, the four operations, negatives, fractions as numbers, and the real line.
- [Probability](https://math.wiki/topic/probability/): 53 lessons in 4 subtopics. Measuring how likely something is.
- [Statistics](https://math.wiki/topic/statistics/): 163 lessons in 7 subtopics. Collecting, describing, and drawing careful conclusions from data.
- [Trigonometry](https://math.wiki/topic/trigonometry/): 93 lessons in 7 subtopics. Angles and triangles, then the waves they turn into.

## Curriculum Standards

Math curriculum standards mapped to lessons and worksheets on https://math.wiki:

- [Curriculum standards directory](https://math.wiki/standards/): Common Core and AP overview
- [Common Core math standards (CCSS)](https://math.wiki/standards/ccss/): 506 standards, Kindergarten to High School
- [AP math curriculum topics](https://math.wiki/standards/ap/): 215 topics across AP Calculus, AP Precalculus and AP Statistics

## Algebra

- [Absolute value equations](https://math.wiki/equations-with-absolute-value/): Distance from zero gives two cases; solve both and check.
- [Absolute value inequalities](https://math.wiki/absolute-value-less-than/): Close to zero means trapped between two numbers: an and.
- [Adding and subtracting polynomials](https://math.wiki/add-polynomials/): Combine like terms across both.
- [Adding and subtracting rational expressions](https://math.wiki/add-subtract-rational-expressions/): Rename over the common denominator, then combine the numerators.
- [Algebra](https://math.wiki/algebra/): Working with unknowns. Letters stand for numbers, and the rules of arithmetic still hold.
- [Arithmetic sequences and the explicit formula](https://math.wiki/explicit-formula-arithmetic/): The nth term straight from n, which is a linear rule.
- [Arithmetic series and partial sums](https://math.wiki/partial-sum/): The total of the first n terms.
- [Combining like terms](https://math.wiki/combine-like-terms-negatives/): Subtraction is adding the opposite, and the sign travels with the term.
- [Completing the square](https://math.wiki/complete-the-square-leading-one/): Add and subtract the missing number, then rewrite.
- [Completing the square, leading coefficient not one](https://math.wiki/complete-the-square-leading-not-one/): Factor the coefficient out first.
- [Compound inequalities: and, or](https://math.wiki/and-inequalities/): Both conditions hold; the solution is the overlap.
- [Degree, terms and standard form](https://math.wiki/degree-of-a-polynomial/): The highest power, and the degree of each term.
- [Deriving the quadratic formula](https://math.wiki/derive-the-quadratic-formula/): Complete the square on the general quadratic and the formula falls out.
- [Direct variation](https://math.wiki/direct-variation/): A line through the origin: y is a constant times x.
- [Distribute, then combine](https://math.wiki/distribute-and-combine/): Open the brackets first, then collect what matches.
- [Equations from proportions and percents](https://math.wiki/proportion-equations/): Cross multiply, then solve the linear equation that results.
- [Equations of the form p(x + q) = r, from words](https://math.wiki/equations-of-the-form-p-x-plus-q/): Write equations like 3(x + 4) = 27 from a word problem and solve them by dividing first or by distributing first.
- [Equations with variables on both sides](https://math.wiki/collect-variables-one-side/): Subtract the smaller variable term from both sides, then solve.
- [Equivalent expressions: the distributive property and like terms](https://math.wiki/equivalent-expressions-with-whole-numbers/): Two expressions are equivalent when they agree for every value; make them by distributing, factoring out a common factor and combining like terms.
- [Evaluating expressions and formulas](https://math.wiki/substitute-one-variable/): Put the number where the letter is, then compute.
- [Expression, equation and inequality](https://math.wiki/expression-vs-equation/): An expression is a number you have not worked out yet; an equation and an inequality are claims that can be true or false.
- [Expressions](https://math.wiki/expressions/): Number sentences with letters in them, and the rules for rewriting them.
- [Factored form](https://math.wiki/factored-form-of-a-quadratic/): Written as a product, the roots are visible; three forms, three things made easy.
- [Factoring by grouping](https://math.wiki/factor-by-grouping/): Pair the terms, factor each pair, then factor the shared bracket.
- [Factoring completely and choosing a strategy](https://math.wiki/factor-completely/): Take out the common factor first, then factor what is left, and check nothing more can go.
- [Factoring out the greatest common factor](https://math.wiki/factor-out-a-monomial/): Pull the shared factor to the front and write what is left in brackets.
- [Factoring special patterns](https://math.wiki/recognize-difference-of-squares/): a squared minus b squared factors into a sum times a difference.
- [Factoring trinomials](https://math.wiki/factor-trinomials/): Reverse the expansion: find two binomials that multiply to the trinomial.
- [Factoring when the leading coefficient is not one](https://math.wiki/factor-trinomial-leading-not-one/): Split the middle term, then factor by grouping.
- [Factoring when the leading coefficient is one](https://math.wiki/factor-trinomial-leading-one/): Find two numbers that multiply to the constant and add to the middle.
- [Fitting a quadratic to data](https://math.wiki/quadratic-modeling/): Three points determine a parabola.
- [Geometric sequences and the explicit formula](https://math.wiki/explicit-formula-geometric/): The nth term straight from n, which is exponential.
- [Geometric series](https://math.wiki/geometric-series-applications/): Repayments, doubling, and the total of a shrinking sequence.
- [Graphing a line from a table of values](https://math.wiki/graph-from-a-table/): Pick inputs, compute outputs, plot, join.
- [Graphing a line from its equation](https://math.wiki/graph-from-slope-intercept/): Plot the intercept, then step by the slope.
- [Graphing a linear inequality in two variables](https://math.wiki/graph-a-linear-inequality/): Draw the boundary line, dashed or solid, then shade the side that works.
- [Graphing from standard form](https://math.wiki/graph-a-quadratic-from-standard-form/): Find the axis, the vertex and the intercepts, then sketch.
- [Inequalities](https://math.wiki/inequalities/): Statements about greater and less, whose solutions are whole stretches of the line.
- [Inequalities on the number line](https://math.wiki/graph-simple-inequality/): Open circle for strict, closed for inclusive, then shade the side that works.
- [Infinite geometric series](https://math.wiki/infinite-geometric-sum/): First term over one minus the ratio.
- [Interpreting the parts of an expression](https://math.wiki/interpret-parts-of-an-expression/): Reading terms, factors and coefficients as quantities, and seeing a complicated expression as built from simpler pieces.
- [Interval and set-builder notation](https://math.wiki/interval-notation/): Brackets and parentheses as a compact way to name a stretch of the line.
- [Inverse operations and the properties of equality](https://math.wiki/properties-of-equality/): Do the same thing to both sides and the equation still holds. This is the license for every step.
- [Inverse, joint and combined variation](https://math.wiki/inverse-variation/): When one quantity goes up the other goes down by the same factor; the product stays fixed.
- [Linear equations](https://math.wiki/linear-equations/): Finding the number that makes a statement true, when the unknown is not raised to a power.
- [Linear programming](https://math.wiki/linear-programming/): Maximize or minimize a quantity over a shaded region; the best value sits at a corner.
- [Linear relationships](https://math.wiki/linear-relationships/): Straight lines: constant rate, and every way of writing one.
- [Literal equations: solving for one variable](https://math.wiki/solve-for-a-variable/): Treat every other letter as a number and isolate the one you want.
- [Maximum, minimum and projectile problems](https://math.wiki/maximize-or-minimize/): The vertex answers how much and when.
- [Modeling with linear relationships](https://math.wiki/linear-modeling/): Choose the variables, fit the line, read a prediction off it, and say where the model stops making sense.
- [Monomials and the exponent rules](https://math.wiki/multiply-monomials/): Multiply the numbers, add the exponents.
- [Multi-step equations](https://math.wiki/multi-step-combine-first/): Collect like terms on each side, then solve as a two-step equation.
- [Multi-step inequalities](https://math.wiki/multi-step-inequalities/): Distribute, collect, solve, flip if needed.
- [Multiplying and dividing rational expressions](https://math.wiki/multiply-divide-rational-expressions/): Factor, flip for division, cancel, multiply.
- [Multiplying polynomials](https://math.wiki/multiply-by-a-monomial/): Distribute the single term into every term.
- [Multiplying two binomials](https://math.wiki/multiply-two-binomials/): Every term in the first times every term in the second.
- [No solution, one solution, or every number](https://math.wiki/no-solution-and-identity/): When the letters vanish, what is left tells you which of the three you have, and why.
- [Number and shape patterns from a rule](https://math.wiki/patterns-from-a-rule/): A pattern is built from its rule, then studied for features the rule does not say, such as every term being odd.
- [One-step equations](https://math.wiki/one-step-add-subtract/): Undo an addition by subtracting, and the other way around.
- [One-step inequalities and the sign flip](https://math.wiki/one-step-inequalities/): Same moves as an equation, with one exception coming.
- [Operations with radicals](https://math.wiki/multiply-and-divide-radicals/): Roots multiply and divide as one root.
- [Parallel and perpendicular lines](https://math.wiki/parallel-slopes/): Same slope, different intercept, and why they never meet.
- [Point-slope form and the line through two points](https://math.wiki/write-from-point-and-slope/): Drop the point and slope into the form.
- [Polynomial and rational inequalities with sign charts](https://math.wiki/polynomial-and-rational-inequalities/): Solving polynomial and rational inequalities by finding zeros and undefined points and testing the sign on each interval between them.
- [Polynomial long division](https://math.wiki/polynomial-long-division/): Same procedure as with numbers, one power at a time.
- [Polynomials](https://math.wiki/polynomials/): Sums of powers of a variable, and how they add, multiply and factor.
- [Proving polynomial identities](https://math.wiki/prove-polynomial-identities/): Proving that two polynomial expressions are equal for every value, and using identities such as the one behind Pythagorean triples.
- [Quadratic expressions and parabolas](https://math.wiki/quadratic-expressions-and-parabolas/): What ax squared plus bx plus c looks like on a graph, and why it curves.
- [Quadratic inequalities](https://math.wiki/quadratic-inequalities/): Find the roots, then test which stretches of the line satisfy the inequality.
- [Quadratics](https://math.wiki/quadratics/): Expressions and equations where the variable is squared, and the parabolas they draw.
- [Radicals and rational expressions](https://math.wiki/radicals-and-rational-expressions/): Roots and fractions with variables in them.
- [Rate of change](https://math.wiki/rate-of-change/): How much the output moves for each step of the input.
- [Rationalizing a denominator](https://math.wiki/rationalize-a-denominator/): Multiply top and bottom to clear the root below the line, using the conjugate when needed.
- [Recursive formulas](https://math.wiki/recursive-formula/): Define each term from the one before, plus a starting value.
- [Remainder and factor theorems](https://math.wiki/remainder-and-factor-theorems/): Plugging in a number tells you the remainder, and a zero remainder means a factor.
- [Repeating patterns](https://math.wiki/repeating-patterns/): A repeating pattern has a core, like red-blue or circle-square-square, that repeats, so the next piece can be predicted.
- [Rewriting an expression to show what it means](https://math.wiki/rewrite-an-expression-to-show-its-meaning/): Rewriting an expression in a different form shows something new about the situation, as a + 0.05a = 1.05a shows a 5 percent increase.
- [Roots, zeros and x-intercepts](https://math.wiki/roots-zeros-and-x-intercepts/): Three names for the same thing, seen in the equation, the function and the graph.
- [Sequences and series](https://math.wiki/sequences-and-series/): Lists of numbers with a rule, and what happens when you add them up.
- [Sequences, terms and notation](https://math.wiki/sequence-basics/): An ordered list with a rule; the nth term and its index.
- [Sigma notation](https://math.wiki/read-sigma-notation/): The index, the limits, and the rule being summed.
- [Simplifying a rational expression and where it is undefined](https://math.wiki/rational-expression-domain/): Any value that makes the denominator zero is excluded, and it stays excluded after canceling.
- [Simplifying a square root](https://math.wiki/simplify-square-roots/): Pull out perfect square factors.
- [Simplifying radicals](https://math.wiki/simplify-radicals/): Pull perfect squares out from under the root.
- [Simplifying rational expressions](https://math.wiki/rational-expressions/): Fractions with polynomials: factor, cancel, note what is not allowed.
- [Slope](https://math.wiki/slope/): Rise over run: how steep a line is, and its sign.
- [Slope from a graph or a table](https://math.wiki/slope-from-a-graph/): Count the rise, count the run, divide.
- [Slope triangles: why a line has one slope everywhere](https://math.wiki/slope-triangles/): Any two right triangles drawn along the same line are scaled copies, so rise over run is the same wherever you measure it.
- [Slope-intercept form](https://math.wiki/read-slope-and-intercept/): In y = mx + b, m is the slope and b is where it starts.
- [Slopes of perpendicular lines](https://math.wiki/perpendicular-slopes/): Slopes multiply to negative one: flip and change the sign.
- [Solving a rational equation](https://math.wiki/solve-rational-equation/): Multiply through by the common denominator, then solve what is left.
- [Solving linear inequalities](https://math.wiki/solve-linear-inequalities/): Same moves as an equation, except the sign flips when you multiply or divide by a negative, and why.
- [Solving quadratics by factoring](https://math.wiki/zero-product-property/): If a product is zero, at least one factor is zero, and this is why factoring solves equations.
- [Solving quadratics with square roots](https://math.wiki/square-root-property/): If something squared equals a number, that something is plus or minus the root.
- [Solving radical equations](https://math.wiki/solve-one-radical/): Isolate the root, square both sides, solve.
- [Solving systems by elimination](https://math.wiki/elimination-by-adding/): When the coefficients are already opposites, add the equations.
- [Solving systems by graphing](https://math.wiki/what-a-system-solution-is/): A pair of numbers that makes both equations true at once.
- [Solving systems by substitution](https://math.wiki/substitution-when-isolated/): Drop the expression for y into the other equation.
- [Special products](https://math.wiki/polynomial-special-products/): The square of a binomial and the sum-times-difference, recognized on sight.
- [Standard form of a line](https://math.wiki/standard-form-meaning/): Ax + By = C, and when this form is the convenient one.
- [Synthetic division](https://math.wiki/synthetic-division/): A shorthand for dividing by x minus a number.
- [Systems in three variables](https://math.wiki/three-variable-systems/): Eliminate one variable to get a two-variable system, then finish.
- [Systems in word problems](https://math.wiki/mixture-problems/): Two amounts and two totals: quantity and value.
- [Systems of equations](https://math.wiki/systems-of-equations/): Two conditions at once, and the point that satisfies both.
- [Systems of linear inequalities](https://math.wiki/systems-of-inequalities/): Shade each region; the overlap is the answer.
- [Systems with a curve](https://math.wiki/nonlinear-systems/): A line and a parabola can meet twice, once, or not at all.
- [Terms, coefficients and constants](https://math.wiki/terms-coefficients-constants/): Naming the parts of an expression: the pieces added together, the number multiplying a letter, and the plain numbers.
- [The discriminant](https://math.wiki/discriminant-and-number-of-roots/): Positive gives two, zero gives one, negative gives none that are real.
- [The discriminant and the number of real solutions](https://math.wiki/discriminant-and-the-graph/): Two crossings, a touch, or a miss.
- [The distributive property with variables](https://math.wiki/distribute-over-a-sum/): Multiply the outside number into every term inside.
- [The fundamental theorem of algebra](https://math.wiki/fundamental-theorem-of-algebra/): A polynomial of degree n has exactly n roots, counting repeats and complex ones.
- [The overlap of two shaded regions](https://math.wiki/overlap-of-inequality-regions/): Where both shadings agree is the solution set.
- [The quadratic formula](https://math.wiki/quadratic-formula/): One formula that solves every quadratic, and where it comes from.
- [The rational root theorem](https://math.wiki/rational-root-theorem/): The only possible fraction roots come from the first and last coefficients.
- [The remainder and factor theorems](https://math.wiki/factor-theorem/): A zero remainder means you have found a factor, and a root.
- [The shape of a parabola](https://math.wiki/parabola-shape/): Why squaring makes a symmetric curve, and which way it opens.
- [The slope formula and the four kinds of slope](https://math.wiki/slope-from-two-points/): Difference in y over difference in x, and why the order does not matter as long as it matches.
- [The vertex and the axis of symmetry](https://math.wiki/vertex-of-a-parabola/): The turning point, the highest or lowest value the expression takes.
- [Translating between words and expressions](https://math.wiki/translate-words-to-expressions/): Turn five more than twice a number into an expression, and watch the order.
- [Two-step equations](https://math.wiki/two-step-with-negatives/): Signed coefficients and constants.
- [Two-step inequalities](https://math.wiki/two-step-inequalities/): Undo two operations, watching for the flip.
- [Using a letter for the unknown number](https://math.wiki/letter-for-an-unknown/): A letter stands for the number we do not know yet in an equation that models a story.
- [Using the quadratic formula](https://math.wiki/apply-the-quadratic-formula/): Identify a, b and c, substitute carefully, simplify the radical.
- [Variables](https://math.wiki/variables/): A letter that stands for a number we do not know yet, or for any number at all.
- [Vertex form: reading, converting and graphing](https://math.wiki/vertex-form-meaning/): In a(x - h) squared plus k, the vertex is at h, k, and the sign tricks people.
- [What a letter stands for](https://math.wiki/variable-meaning/): A letter can hold one unknown number, or stand for any number at all. Which one it is depends on the sentence around it.
- [What a solution is, and checking one](https://math.wiki/what-a-solution-is/): A number that makes the sentence true. Check by putting it back in.
- [What makes an expression quadratic](https://math.wiki/what-makes-a-quadratic/): Degree two: the variable is squared and no higher.
- [Writing a parallel or perpendicular line through a point](https://math.wiki/write-parallel-or-perpendicular-line/): Take the slope you need, then use point-slope.
- [Writing an inequality for a condition](https://math.wiki/write-an-inequality-for-a-condition/): Turn a condition such as "at least 12" or "less than 5 dollars" into an inequality and see that it has infinitely many solutions.
- [Writing and interpreting constraints](https://math.wiki/model-constraints-with-inequalities/): Turning the limits in a situation into equations and inequalities, and deciding which solutions are viable in context.

## Calculus

- [Absolute and conditional convergence](https://math.wiki/absolute-and-conditional-convergence/): Whether a series still converges once every term is made positive.
- [Accumulation and net change](https://math.wiki/accumulation-functions/): The integral of a rate is the total amount accumulated.
- [Alternating series and their error bound](https://math.wiki/alternating-series-error-bound/): The error after stopping is no bigger than the first term left out.
- [Antiderivatives and indefinite integrals](https://math.wiki/antiderivative-of-a-power/): Reversing the power rule by raising the exponent and dividing.
- [Antiderivatives of trig, exponential and log functions](https://math.wiki/antiderivatives-of-trig-functions/): Reading the trig derivative rules backwards.
- [Applications of derivatives](https://math.wiki/applications-of-derivatives/): What the derivative tells you about a function and the world.
- [Applications of integrals](https://math.wiki/applications-of-integrals/): Area, volume, distance, and total change.
- [Approximating area with Riemann sums](https://math.wiki/area-under-a-curve-idea/): Approximating a curved region by stacking simple rectangles inside it.
- [Arc length](https://math.wiki/arc-length-formula-derivation/): Adding up tiny straight hypotenuses along the curve and taking a limit.
- [Arc length and speed on parametric and polar curves](https://math.wiki/speed-of-a-parametric-curve/): The rate at which arc length is swept out as the parameter advances.
- [Area between curves](https://math.wiki/area-with-curves-that-cross/): Splitting the region at each intersection so the top curve stays on top.
- [Area in polar coordinates](https://math.wiki/polar-area-between-curves/): Subtracting the inner sector area from the outer one over a shared range of angles.
- [Average value of a function](https://math.wiki/average-value-geometric-meaning/): The height of the rectangle with the same area as the region under the curve.
- [Calculus](https://math.wiki/calculus/): Change and accumulation: derivatives and integrals.
- [Choosing a convergence test](https://math.wiki/choosing-a-convergence-test/): Reading the shape of the terms to decide which test will settle the question.
- [Choosing an integration technique](https://math.wiki/choosing-an-integration-technique/): Reading the shape of an integrand to decide which method to try.
- [Combining differentiation rules](https://math.wiki/choosing-differentiation-rules/): Deciding which rules apply and in what order for a layered expression.
- [Comparing growth rates at infinity](https://math.wiki/growth-rates-at-infinity/): Exponentials outgrow powers and powers outgrow logarithms in the long run.
- [Completing the square inside an integral](https://math.wiki/completing-the-square-in-an-integral/): Reshaping a quadratic denominator so an inverse trig or log form appears.
- [Concavity and inflection points](https://math.wiki/concavity-from-the-second-derivative/): A positive second derivative bends the graph up and a negative one bends it down.
- [Continuity](https://math.wiki/continuity/): No holes, no jumps, no gaps: the limit equals the value.
- [Continuity at a point](https://math.wiki/continuity-at-a-point/): A function is continuous at a point when the limit there exists and equals the value.
- [Continuity on an interval](https://math.wiki/continuity-on-an-interval/): A function continuous at every point of an interval, with one sided continuity at the ends.
- [Convergence of sequences](https://math.wiki/sequence-limit-laws/): Sums, products and quotients of convergent sequences behave as expected.
- [Convergence tests](https://math.wiki/convergence-tests/): Comparison, ratio, integral, alternating: tools for deciding whether a series settles.
- [Critical points and local versus absolute extrema](https://math.wiki/critical-points/): Inputs where the derivative is zero or does not exist.
- [Curve sketching and connecting f, f' and f''](https://math.wiki/sketch-a-curve-from-sign-charts/): Assembling intercepts, asymptotes, rises, falls and bends into one picture.
- [Definition of the derivative](https://math.wiki/derivative-definition/): The limit of the slope of a secant as the two points merge.
- [Derivatives](https://math.wiki/derivatives/): The rate of change at an instant, and the rules for finding it.
- [Derivatives of exponential functions](https://math.wiki/derivative-of-exponential-base-a/): A base other than e brings a factor of the natural log of that base.
- [Derivatives of inverse functions](https://math.wiki/inverse-function-derivative-formula/): The slope of the inverse is the reciprocal of the slope of the original at the partner point.
- [Derivatives of inverse trigonometric functions](https://math.wiki/derivative-of-inverse-trig-functions/): Arcsine, arccosine and arctangent have algebraic derivatives with no trig left in them.
- [Derivatives of logarithms and logarithmic differentiation](https://math.wiki/derivative-of-natural-log-derivation/): Differentiating the exponential relationship implicitly gives the result.
- [Derivatives of parametric curves](https://math.wiki/tangent-lines-to-parametric-curves/): Finding the slope at a parameter value and writing the line through that point.
- [Derivatives of the trigonometric functions](https://math.wiki/derivative-of-the-other-trig-functions/): Each one follows from sine and cosine through the quotient rule.
- [Derivatives of trig, exponential and log functions](https://math.wiki/derivatives-of-trig-exp-log/): Sine to cosine, e to itself, log to one over x, and where each comes from.
- [Differentiability: corners, cusps and piecewise functions](https://math.wiki/corners-cusps-and-vertical-tangents/): The three shapes where a continuous graph still has no single tangent slope.
- [Differential equations](https://math.wiki/differential-equations/): Equations about a rate, and finding the function that obeys them.
- [Differential equations and their solutions](https://math.wiki/verifying-a-solution-to-a-differential-equation/): Substituting a candidate function and its derivatives to check the equation holds.
- [Discontinuities and how to remove them](https://math.wiki/jump-discontinuity/): A break where the left and right limits both exist but disagree.
- [Dividing before integrating](https://math.wiki/long-division-before-integrating/): Splitting a top heavy rational integrand into a polynomial plus a proper remainder.
- [Euler's method](https://math.wiki/eulers-method-error/): Smaller steps reduce the error, and concavity decides which way the estimate leans.
- [Exponential growth and decay models](https://math.wiki/proportional-rate-of-change/): This single condition forces exponential behavior.
- [Extrema on a closed interval](https://math.wiki/closed-interval-method/): Comparing values at the critical points and at both endpoints to find the largest and smallest.
- [Finding a derivative from the definition](https://math.wiki/derivative-from-first-principles/): Evaluating the limit of the difference quotient by hand for a given function.
- [Graphing an accumulation function](https://math.wiki/extrema-of-an-accumulation-function/): The accumulated amount peaks where the rate crosses from positive to negative.
- [Implicit differentiation](https://math.wiki/implicit-differentiation-idea/): Treating y as an unnamed function of x so the chain rule applies to every y term.
- [Improper integrals](https://math.wiki/improper-integral-infinite-limit/): Integrating out to infinity by taking a limit of ordinary integrals.
- [Improper integrals with an infinite integrand](https://math.wiki/improper-integral-unbounded-integrand/): Handling an integral where the function blows up at an endpoint.
- [Increasing and decreasing from the derivative](https://math.wiki/sign-charts-for-derivatives/): Splitting the line at critical points and testing the sign of the derivative in each piece.
- [Infinite limits and vertical asymptotes](https://math.wiki/vertical-asymptotes-from-limits/): A vertical asymptote is a place where a one sided limit runs off to infinity.
- [Infinite series and partial sums](https://math.wiki/partial-sums-of-a-series/): The running totals whose limit defines the sum of a series.
- [Integrals](https://math.wiki/integrals/): Adding up infinitely many infinitely thin pieces, and the surprise that undoes the derivative.
- [Integration by partial fractions](https://math.wiki/partial-fractions-repeated-and-quadratic-factors/): Splitting a rational function into simple fractions, from distinct linear factors to repeated and irreducible quadratic ones, and integrating each piece.
- [Integration by parts](https://math.wiki/integration-by-parts-derivation/): Integrating both sides of the product rule and rearranging.
- [L'Hopital's rule](https://math.wiki/lhopital-zero-over-zero/): Replacing the ratio of two vanishing functions with the ratio of their derivatives.
- [Limits](https://math.wiki/limits/): What a function is heading toward, even where it never arrives.
- [Limits at infinity and horizontal asymptotes](https://math.wiki/horizontal-asymptotes-from-limits/): A horizontal asymptote is the value a function levels off toward far out on the axis.
- [Limits by algebra: factoring and rationalizing](https://math.wiki/limits-by-factoring/): Removing a common factor that makes both top and bottom zero so substitution works.
- [Linear approximation and differentials](https://math.wiki/error-estimation-with-differentials/): Turning a small measurement error into an estimated error in a computed quantity.
- [Logistic growth](https://math.wiki/logistic-carrying-capacity/): The ceiling a logistic population approaches but never passes.
- [Making a piecewise function continuous](https://math.wiki/continuity-of-a-piecewise-function/): Choosing a parameter so the two pieces meet at the boundary.
- [Maxima, minima and critical points](https://math.wiki/extrema/): Where the derivative is zero or undefined, and the tests that tell peak from valley.
- [Newton's method](https://math.wiki/newtons-method/): Using tangent lines over and over to close in on a root.
- [One sided limits and when a limit exists](https://math.wiki/two-sided-limit-exists-criterion/): A limit exists exactly when the left and right limits agree.
- [Optimization](https://math.wiki/optimization-on-a-closed-interval/): Checking the ends of the sensible domain as well as the critical points.
- [Parametric, polar and vector valued functions](https://math.wiki/parametric-polar-and-vector-calculus/): Doing calculus on curves described by a parameter, by an angle, or by a position vector.
- [Position, displacement and distance from velocity](https://math.wiki/total-distance-versus-displacement/): Integrating velocity gives displacement while integrating speed gives distance traveled.
- [Position, velocity and acceleration](https://math.wiki/particle-at-rest-and-direction-change/): Reading zeros and sign changes of velocity as stops and reversals.
- [Power series and the interval of convergence](https://math.wiki/endpoint-testing-for-power-series/): Checking each end separately, since the ratio test says nothing there.
- [Properties of definite integrals](https://math.wiki/additivity-over-intervals/): An integral over a long interval is the sum of the integrals over its parts.
- [Related rates](https://math.wiki/related-rates-with-geometry-formulas/): Linking changing lengths to changing areas or volumes.
- [Representing functions as power series](https://math.wiki/integrating-a-series-term-by-term/): Term by term integration gives series for functions that resist antidifferentiation.
- [Second and higher derivatives](https://math.wiki/second-derivative/): The rate at which the first derivative itself is changing.
- [Separable differential equations](https://math.wiki/separable-with-an-initial-condition/): Solving for the constant after integrating, then choosing the branch through the starting point.
- [Sequences and series in calculus](https://math.wiki/series/): Infinite sums: when they settle, and how functions become polynomials.
- [Signed area and integrals from geometry](https://math.wiki/signed-area/): Regions below the axis count as negative in a definite integral.
- [Slope fields](https://math.wiki/sketching-a-slope-field/): Drawing a short segment at each grid point with the slope the equation gives there.
- [Slope of a polar curve](https://math.wiki/slope-of-a-polar-curve/): Differentiating the x and y forms with respect to the angle and dividing.
- [Surface area of a solid of revolution](https://math.wiki/surface-area-of-revolution/): Adding the side areas of thin bands swept out by an arc.
- [Tangent lines and normal lines](https://math.wiki/equation-of-a-tangent-line/): Using the derivative as the slope and the point of contact to build a line.
- [Taylor and Maclaurin series](https://math.wiki/taylor-series/): Rebuild a function from its derivatives at one point.
- [Taylor polynomials](https://math.wiki/taylor-coefficients-from-derivatives/): Matching derivatives at the center forces each coefficient to be a derivative over a factorial.
- [The binomial series](https://math.wiki/binomial-series/): A power series for a binomial raised to any real exponent.
- [The chain rule](https://math.wiki/chain-rule-in-leibniz-notation/): Writing the rule as a product of rates makes the canceling pattern visible.
- [The common Maclaurin series](https://math.wiki/maclaurin-series-for-e-to-the-x/): Every derivative is the same at zero, so the coefficients are reciprocal factorials.
- [The definite integral](https://math.wiki/definite-integral/): The limit of Riemann sums: exact area, with sign.
- [The definite integral as a limit of Riemann sums](https://math.wiki/definite-integral-as-a-limit/): The exact accumulated amount defined as the limit of finer and finer approximations.
- [The derivative as a function](https://math.wiki/sketching-the-derivative-graph/): Turning the slopes of one graph into the heights of another.
- [The derivative as a rate of change](https://math.wiki/interpreting-a-derivative-in-context/): Saying what a rate means for the situation, with its units and its sign.
- [The derivative at a point](https://math.wiki/derivative-at-a-point/): A single number giving the slope of the curve at one place.
- [The direct and limit comparison tests](https://math.wiki/limit-comparison-test/): Two positive series whose terms stay in a fixed ratio behave the same way.
- [The disk method](https://math.wiki/disk-method/): Slicing a solid of revolution into thin circles perpendicular to the axis.
- [The epsilon delta definition of a limit](https://math.wiki/formal-limit-definition/): The precise statement that for every tolerance there is a window of inputs that meets it.
- [The extreme value theorem](https://math.wiki/extreme-value-theorem/): A continuous function on a closed interval reaches a highest and a lowest value.
- [The first and second derivative tests](https://math.wiki/first-derivative-test/): A sign change of the derivative at a critical point tells whether it is a peak or a valley.
- [The first fundamental theorem of calculus](https://math.wiki/derivative-of-an-integral-with-variable-limits/): The chain rule attaches to the first fundamental theorem when the upper limit moves faster.
- [The fundamental theorem of calculus](https://math.wiki/fundamental-theorem-of-calculus/): Area under a rate is total change; integrals and derivatives undo each other.
- [The geometric series test](https://math.wiki/geometric-series-test/): A geometric series converges exactly when the common ratio is smaller than one in size.
- [The integral test and p series](https://math.wiki/p-series-test/): The sum of one over n to a power converges exactly when the power is bigger than one.
- [The intermediate value theorem](https://math.wiki/bisection-method/): Halving an interval again and again to trap a root as closely as wanted.
- [The limit laws](https://math.wiki/sum-and-difference-limit-laws/): The limit of a sum is the sum of the limits when both limits exist.
- [The mean value theorem](https://math.wiki/rolles-theorem/): A smooth curve with equal endpoint heights must level off somewhere in between.
- [The net change theorem in context](https://math.wiki/accumulation-in-context/): Totaling water, cars or charge from a rate given in the units of the situation.
- [The nth term test for divergence](https://math.wiki/nth-term-test/): If the terms do not shrink to zero the series cannot converge.
- [The power rule and basic differentiation rules](https://math.wiki/derivative-of-a-polynomial/): Applying the power rule to each term of a polynomial.
- [The product rule](https://math.wiki/product-rule-derivation/): Adding and subtracting a middle term splits the change of a product into two pieces.
- [The quotient rule](https://math.wiki/quotient-rule-derivation/): The quotient rule follows from the product rule applied to a reciprocal.
- [The ratio and root tests](https://math.wiki/ratio-test/): Comparing each term with the one before it to judge convergence.
- [The second fundamental theorem of calculus](https://math.wiki/evaluating-a-definite-integral/): Finding an antiderivative and subtracting its values at the two ends.
- [The shell method](https://math.wiki/shell-method/): Building a solid from thin cylindrical shells parallel to the axis.
- [The squeeze theorem and the limit of sine x over x](https://math.wiki/limit-of-sin-x-over-x/): Near zero, sine of x and x itself are so close that their ratio approaches one.
- [The Taylor remainder and the Lagrange error bound](https://math.wiki/lagrange-error-bound/): A ceiling on the remainder built from the next derivative and the distance from the center.
- [The trapezoidal rule and Simpson's rule](https://math.wiki/trapezoidal-rule-accuracy/): Trapezoids sit above a concave up curve and below a concave down one.
- [The washer method](https://math.wiki/washer-method/): Subtracting an inner circle from an outer one when the solid has a hole.
- [Trigonometric integrals](https://math.wiki/trigonometric-integrals/): Using identities to rewrite powers of sine, cosine, secant and tangent before integrating.
- [Trigonometric substitution](https://math.wiki/trigonometric-substitution/): Replacing a variable with a trig function to clear a square root.
- [u-substitution](https://math.wiki/u-substitution-indefinite/): Replacing the inner function and its differential, then changing back at the end.
- [Using series to evaluate limits and integrals](https://math.wiki/approximating-an-integral-with-a-series/): Integrating a series term by term to estimate a definite integral.
- [Vector valued functions and motion in the plane](https://math.wiki/speed-and-distance-in-planar-motion/): Speed is the length of the velocity vector and distance is the integral of that speed.
- [Volumes with known cross sections](https://math.wiki/volumes-with-known-cross-sections/): Stacking squares, triangles or semicircles built on a base region.
- [What a limit is: graphs and tables](https://math.wiki/limits-from-a-graph/): Finding what value a graph approaches near a point by looking at the curve from both sides.
- [Work as an integral](https://math.wiki/work-as-an-integral/): Adding force times distance over many tiny displacements.
- [Writing a differential equation from a description](https://math.wiki/modeling-with-differential-equations/): Turning a sentence about how fast something changes into an equation.

## Fractions, ratio and percent

- [A fraction as division](https://math.wiki/fraction-as-division/): The bar in a fraction means divide, so three fourths is three things shared among four.
- [A fraction as part of a whole](https://math.wiki/fraction-as-part-of-whole/): Split a whole into equal parts; the fraction says how many parts you have.
- [A fraction of a fraction of an amount](https://math.wiki/fraction-of-a-fraction-of-a-quantity/): Two cuts in a row, where the second one applies to what the first cut left.
- [A fraction of a set](https://math.wiki/fraction-of-a-set/): A fraction that counts objects in a group instead of pieces of one object.
- [A fraction of a whole number](https://math.wiki/fraction-of-a-whole-number/): Splitting a number into equal groups and taking some of those groups.
- [A fraction times a fraction](https://math.wiki/fraction-times-fraction/): Taking a part of a part, which multiplies the tops together and the bottoms together.
- [A fraction times a whole number](https://math.wiki/fraction-times-whole-number/): Taking several copies of a fraction, which is the same as taking a fraction of that number.
- [Adding and subtracting fractions](https://math.wiki/add-subtract-fractions-unlike/): Rename to same-size pieces first.
- [Adding and subtracting fractions with like denominators](https://math.wiki/add-fractions-like-denominators/): Counting unit fractions that are already the same size, so only the top numbers add.
- [Adding and subtracting fractions with unlike denominators](https://math.wiki/add-unlike-any-denominators/): Renaming both fractions in shared pieces before the counts can be added.
- [Adding and subtracting mixed numbers with like denominators](https://math.wiki/add-and-subtract-mixed-numbers-like-denominators/): Mixed numbers with the same denominator are added and subtracted whole part and fraction part, trading a whole for fractions when needed.
- [Adding mixed numbers](https://math.wiki/add-mixed-numbers/): Adding the whole parts and the fraction parts, or rewriting both as improper fractions.
- [Adding tenths and hundredths as fractions](https://math.wiki/add-tenths-and-hundredths/): Tenths are rewritten as hundredths so a number of tenths and a number of hundredths can be added.
- [Area and length models of a fraction](https://math.wiki/fraction-area-model/): A shape cut into equal pieces with some of them shaded, used to picture a fraction.
- [Benchmark fractions](https://math.wiki/benchmark-fractions/): Familiar fractions such as one half and one fourth, used as reference points for judging others.
- [Benchmark percents and mental percent](https://math.wiki/benchmark-percents/): Ten, twenty five, fifty and one hundred percent, used as reference points.
- [Changing the units of a rate](https://math.wiki/convert-a-rate-to-other-units/): Rewriting feet per second as miles per hour by multiplying by carefully chosen ones.
- [Common denominators and the least common denominator](https://math.wiki/rename-two-fractions-to-a-common-denominator/): Rewriting both fractions so their pieces match, before comparing, adding or dividing.
- [Comparing against a benchmark](https://math.wiki/compare-using-benchmarks/): Deciding which side of one half each fraction falls on and comparing from there.
- [Comparing by finding a common denominator](https://math.wiki/compare-using-a-common-denominator/): Renaming both fractions in matching pieces so the counts can be compared directly.
- [Comparing fractions](https://math.wiki/compare-fractions/): Same denominator, same numerator, or a benchmark like one half.
- [Comparing fractions with the same denominator or numerator](https://math.wiki/compare-fractions-same-denominator/): When the pieces are the same size, the fraction holding more pieces is the larger one.
- [Converting among fractions, decimals and percents](https://math.wiki/fraction-to-percent/): Finding how many hundredths a fraction comes to.
- [Counting by unit fractions](https://math.wiki/count-by-unit-fractions/): Skip counting one third, two thirds, three thirds and on past the whole.
- [Deciding which of two ratios is larger](https://math.wiki/compare-two-ratios/): Putting two comparisons on the same footing before saying which one is stronger.
- [Decomposing a fraction into unit fractions](https://math.wiki/decompose-fraction-into-unit-fractions/): Writing a fraction as a sum of copies of its own unit fraction.
- [Discounts, sales tax, tips and commission](https://math.wiki/discount-and-sale-price/): An amount taken off a price, and the price that is left behind.
- [Dividing by multiplying by the reciprocal](https://math.wiki/divide-fractions-invert-and-multiply/): Turning a division into a multiplication by flipping the second fraction over.
- [Dividing fractions](https://math.wiki/divide-fractions/): How many of this piece fit in that one, and why invert and multiply works.
- [Dividing fractions: how many fit](https://math.wiki/fraction-division-measurement-meaning/): Asking how many copies of one amount fit inside another amount.
- [Dividing fractions: sharing into equal parts](https://math.wiki/fraction-division-partitive-meaning/): Asking how much one whole group gets when an amount is shared out.
- [Dividing mixed numbers](https://math.wiki/divide-mixed-numbers/): Rewriting each mixed number as an improper fraction before dividing.
- [Equal shares and naming the whole](https://math.wiki/equal-shares-in-a-fraction/): The parts a fraction counts have to be the same size, or the name of the fraction means nothing.
- [Equal shares that look different](https://math.wiki/equal-shares-can-look-different/): Equal shares of the same whole can have different shapes and still be the same size.
- [Equivalent fractions](https://math.wiki/equivalent-fractions/): Different names for the same number: cut every piece the same way.
- [Equivalent ratios and ratio tables](https://math.wiki/equivalent-ratios/): Scale both parts by the same factor and the ratio stays the same.
- [Estimating with fractions](https://math.wiki/estimate-fraction-sums/): Replacing each fraction with zero, one half or one whole to judge the answer quickly.
- [Finding a fraction between two fractions](https://math.wiki/fractions-between-fractions/): Between any two fractions there is always another one, so the line is never full.
- [Finding a missing value in a pair of equal ratios](https://math.wiki/missing-value-in-equivalent-ratios/): Working out the scaling number and applying it to the amount you already know.
- [Finding a percent of a quantity](https://math.wiki/percent-of-a-number-with-a-decimal/): Changing the percent into a decimal and multiplying by it.
- [Finding the whole from a part and a percent](https://math.wiki/find-the-whole-from-part-and-percent/): Working backwards when the amount you are given is only a slice of the total.
- [Finding the whole when only a fraction of it is known](https://math.wiki/find-the-whole-from-a-fraction/): Working backwards from a part and the fraction of the whole it represents.
- [Finding what percent one number is of another](https://math.wiki/find-the-percent-one-number-is-of-another/): Dividing the part by the whole and reading the answer as hundredths.
- [Fraction of a quantity](https://math.wiki/fraction-of-a-quantity/): Three fifths of 40: divide into parts, take some.
- [Fractions](https://math.wiki/fractions/): A number that names part of a whole, and everything you can do with it.
- [Fractions greater than one (improper fractions)](https://math.wiki/fractions-greater-than-one/): A fraction whose top number is bigger than its bottom number, naming an amount past one whole.
- [Fractions on the number line](https://math.wiki/fractions-on-number-line/): A fraction is a point on the line, a number like any other.
- [Fractions, ratio and percent](https://math.wiki/fractions-ratio-percent/): Parts of a whole, comparisons by multiplying, and per-hundred.
- [Inversely proportional relationships](https://math.wiki/inversely-proportional-relationships/): When one amount doubles the other halves, so the two always multiply to the same number.
- [Locating a fraction on the number line](https://math.wiki/locate-a-fraction-on-the-number-line/): Counting unit fraction jumps from zero to find where a fraction lands.
- [Making an equivalent fraction by multiplying](https://math.wiki/multiply-to-make-equivalent/): Multiplying the top and the bottom by the same number renames a fraction in smaller pieces.
- [Markup and margin](https://math.wiki/markup-percent/): The amount added to what something cost, given as a percent of that cost.
- [Mixed numbers and improper fractions](https://math.wiki/mixed-numbers/): A whole plus a part, and the same number written as one fraction.
- [Mixed numbers and improper fractions: rewriting and comparing](https://math.wiki/mixed-to-improper/): Turning the whole part into pieces and adding them to the pieces already there.
- [Multiplying as scaling](https://math.wiki/predict-size-of-a-fraction-product/): Judging a product against its starting number without doing the arithmetic.
- [Multiplying fractions](https://math.wiki/multiply-fractions/): A fraction of a fraction, and why the numerators and denominators multiply.
- [Multiplying mixed numbers](https://math.wiki/multiply-mixed-numbers/): Rewriting each mixed number as an improper fraction before multiplying.
- [Part-to-part and part-to-whole ratios](https://math.wiki/part-to-part-ratio/): A comparison of one group with another group, not with the total.
- [Percent](https://math.wiki/percent/): Per hundred: a ratio with a fixed second part.
- [Percent change](https://math.wiki/percent-change/): Increase or decrease, measured against where you started.
- [Percent error](https://math.wiki/percent-error/): How far a measured or predicted value sits from the true one, as a percent of the true one.
- [Percent in money situations](https://math.wiki/percent-applications/): The everyday uses of percent in prices, wages, loans and taxes.
- [Percent increase and decrease](https://math.wiki/percent-increase/): How much something grew, measured against what it was before.
- [Percents above 100 and below 1](https://math.wiki/percents-over-one-hundred/): A percent above one hundred names more than the whole, which happens whenever something grows.
- [Proportion](https://math.wiki/proportion/): Two ratios set equal, and solving for the unknown part.
- [Proportional relationships](https://math.wiki/proportional-relationships/): When one quantity is always a fixed multiple of another, and how to spot it in a table, graph or equation.
- [Rate and unit rate](https://math.wiki/rate-and-unit-rate/): A ratio of different kinds of things, scaled to per one.
- [Rates and unit rates](https://math.wiki/unit-rate/): How much of one amount goes with exactly one of the other.
- [Ratio](https://math.wiki/ratio/): Comparing two amounts by how many times one fits in the other.
- [Ratio tables](https://math.wiki/build-a-ratio-table/): Listing equivalent ratios in rows so the pattern between them is easy to see.
- [Recognizing a proportional relationship in a graph](https://math.wiki/proportional-in-a-graph/): A straight line through the origin, and nothing else, is proportional.
- [Recognizing a proportional relationship in an equation](https://math.wiki/proportional-in-an-equation/): The form y equals k times x, with nothing added on the end.
- [Scale drawings](https://math.wiki/scale-drawings/): Maps and blueprints: the same shape at a different size, by a fixed factor.
- [Scale factor](https://math.wiki/scale-drawing-scale-factor/): The single number every length is multiplied by when one figure is enlarged or shrunk into a similar one.
- [Scaling a ratio up or down](https://math.wiki/scale-a-ratio-up/): Multiplying both amounts by the same number so the comparison stays the same.
- [Seeing equivalent fractions in pictures and on the number line](https://math.wiki/equivalent-fractions-area-model/): Cutting every piece of a shaded shape in two doubles both numbers and changes nothing.
- [Setting up proportions from word problems](https://math.wiki/set-up-a-proportion-from-words/): Choosing what goes on top and what goes underneath so both sides line up.
- [Simple interest](https://math.wiki/simple-interest-formula/): Multiplying principal, rate and time to get the interest earned or owed.
- [Simplifying a fraction to simplest form](https://math.wiki/divide-to-make-equivalent/): Dividing the top and the bottom by a common factor renames a fraction in bigger pieces.
- [Solving a proportion](https://math.wiki/solve-a-proportion/): Finding the unknown amount that makes the two ratios equal.
- [Solving percent problems with an equation or a proportion](https://math.wiki/percent-as-an-equation/): Turning part equals percent times whole into an equation that can be solved.
- [Speed, distance and time](https://math.wiki/speed-as-a-rate/): Distance covered for each unit of time, the most familiar rate there is.
- [Subtracting mixed numbers](https://math.wiki/subtract-mixed-numbers/): Taking whole from whole and fraction from fraction.
- [Tape diagrams and sharing in a ratio](https://math.wiki/tape-diagram-for-ratios/): Equal blocks drawn for each part of a ratio, which makes a total easy to split.
- [The constant of proportionality and proportional tables](https://math.wiki/constant-of-proportionality/): The single number that every input is multiplied by to give its output.
- [The double number line](https://math.wiki/double-number-line/): Two number lines drawn side by side whose matching points show equal ratios.
- [The multiplier for a percent change](https://math.wiki/percent-change-multiplier/): A rise of twenty percent is one single multiplication by one point two.
- [The three percent questions](https://math.wiki/percent-find-the-whole/): If 12 is 40% of something, what is the something?
- [Two percent changes one after the other](https://math.wiki/successive-percent-changes/): Multiplying the two multipliers together, which is not the same as adding the percents.
- [Unit fractions](https://math.wiki/unit-fractions/): One piece when a whole is split into equal parts, the building block every other fraction is made of.
- [Unit price and comparing rates](https://math.wiki/unit-price/): The cost of one item or one unit of measure, used to compare package sizes.
- [Unit rates when the amounts are fractions](https://math.wiki/unit-rate-with-fractions/): Finding the amount per one when both sides of the rate are fractions.
- [Using a scale on maps, plans and models](https://math.wiki/read-a-map-or-plan-scale/): Turning a stated scale such as one inch to fifty miles into a rate you can use.
- [What a proportion says](https://math.wiki/proportion-meaning/): A statement that two ratios name the very same comparison.
- [What a ratio is](https://math.wiki/ratio-concept/): Two quantities in a fixed relationship, written 3 to 2 or 3:2.
- [What a ratio is: language and notation](https://math.wiki/ratio-language-and-notation/): The ways a comparison of two amounts is written down and read out loud.
- [What percent means: a rate per hundred](https://math.wiki/percent-as-rate-per-hundred/): A percent is a ratio whose second amount is always one hundred.
- [What the numerator and the denominator each tell you](https://math.wiki/numerator-and-denominator/): The bottom number names the size of the piece and the top number counts how many there are.
- [Whole numbers written as fractions](https://math.wiki/whole-numbers-as-fractions/): Any whole number is a fraction with denominator one, and three thirds is exactly one.
- [Why flipping the second fraction works](https://math.wiki/why-inverting-works/): Dividing by a number and multiplying by its reciprocal undo exactly the same thing.
- [Word problems that add or subtract fractions](https://math.wiki/add-subtract-fraction-word-problems/): Reading a situation to decide whether pieces are being joined or taken away.
- [Word problems that divide fractions](https://math.wiki/divide-fraction-word-problems/): Deciding whether a situation asks how many fit or how much each one gets.

## Functions

- [A reasonable domain for a real situation](https://math.wiki/reasonable-domain-in-context/): In a real setting the domain shrinks to the inputs that make sense, such as whole numbers of people.
- [Adding, subtracting and multiplying functions](https://math.wiki/sum-of-two-functions/): The sum function gives, at each input, the sum of the two outputs.
- [Asymptotes](https://math.wiki/asymptotes/): Lines the graph approaches but never reaches: vertical, horizontal, slant.
- [Average rate of change](https://math.wiki/average-rate-of-change/): How much the output changes per unit of input across an interval, the slope of the line joining two points.
- [Average rate of change from a formula](https://math.wiki/average-rate-of-change-from-a-formula/): Evaluating the rule at both ends of an interval and dividing the change in output by the width.
- [Average rate of change from a table or graph](https://math.wiki/average-rate-of-change-from-a-table/): Using two rows of a table to find the change in output divided by the change in input.
- [Building a model from a description](https://math.wiki/model-from-a-verbal-description/): Turning a written account of a situation into a rule, with each quantity named and its units stated.
- [Building a polynomial from its zeros](https://math.wiki/write-a-polynomial-from-its-zeros/): Turning each zero into a factor and choosing a leading coefficient to match one known point.
- [Building an exponential model from two points](https://math.wiki/exponential-model-from-two-points/): Using two known values to work out both the starting amount and the factor.
- [Change of base](https://math.wiki/change-of-base/): Compute any logarithm from the ones your calculator has.
- [Choosing a function family for data](https://math.wiki/choose-a-function-family/): Using differences, ratios and the shape of a scatter plot to decide which family fits the data.
- [Combining and composing functions](https://math.wiki/composition-of-functions/): Feed one function's output into another.
- [Combining transformations and the general form](https://math.wiki/combining-transformations/): Applying a shift, a flip and a stretch to one parent graph together.
- [Comparing linear functions given different ways](https://math.wiki/compare-linear-functions-across-representations/): Deciding which of two linear functions grows faster when one is a graph, one a table and one a rule.
- [Composing functions algebraically](https://math.wiki/compose-two-functions-algebraically/): Substituting the whole inner rule for the input of the outer rule and simplifying.
- [Compound interest and compounding periods](https://math.wiki/compounding-periods/): Splitting a yearly rate over more periods changes how much a balance grows in a year.
- [Concavity: when the rate of change itself changes](https://math.wiki/concavity-and-changing-rates/): A graph is concave up where its rate of change is increasing and concave down where it is decreasing, with points of inflection between.
- [Constant rate of change and linear tables](https://math.wiki/constant-rate-of-change/): A function is linear exactly when equal steps in the input always give equal steps in the output.
- [Continuous graphs, jumps and holes](https://math.wiki/continuous-and-broken-graphs/): A graph drawn without lifting the pen, compared with one that jumps or has a gap in it.
- [Cube root functions and roots as powers](https://math.wiki/cube-root-functions/): An odd root accepts every input, so its graph runs through the origin in both directions.
- [Cubic, quartic and higher degree functions](https://math.wiki/cubic-functions/): Degree three functions, whose graphs run from one corner of the plane to the opposite corner.
- [Definition of a function](https://math.wiki/function-definition/): Each input gets one output. Tables, mappings, and why the rule matters.
- [Definition of a logarithm](https://math.wiki/exponential-and-log-forms/): Rewriting a logarithm statement as the power statement it stands for, and back again.
- [Degree, end behavior and the shape of a polynomial graph](https://math.wiki/degree-and-graph-shape/): The degree limits how often a polynomial graph can cross the axis and how many bends it can have.
- [Dividing functions and the domain of a combination](https://math.wiki/quotient-of-two-functions/): The quotient function divides one output by the other and is undefined where the bottom output is zero.
- [Domain and range](https://math.wiki/domain-and-range/): Which inputs are allowed, and which outputs come out.
- [Domain and range from a graph](https://math.wiki/domain-from-a-graph/): Reading how far a graph stretches left and right to describe every input it accepts.
- [Domain and range from pairs and tables](https://math.wiki/domain-from-a-set-of-pairs/): Collecting the inputs into the domain and the outputs into the range for a finite list of pairs.
- [Domain and zeros of a rational function](https://math.wiki/domain-of-a-rational-function/): Removing every input that would make the denominator zero.
- [Domain of a composition](https://math.wiki/domain-of-a-composition/): The allowed inputs are those the inner function accepts whose outputs the outer function also accepts.
- [Doubling time and half life](https://math.wiki/doubling-time-and-half-life/): The fixed length of time a growing quantity takes to double or a decaying one takes to halve.
- [End behavior](https://math.wiki/end-behavior-from-a-graph/): Describing in words where the two ends of a drawn graph are going.
- [Evaluating a composition from numbers, tables and graphs](https://math.wiki/evaluate-a-composition-at-a-number/): Running a number through the inner function and feeding the answer into the outer one.
- [Evaluating a function at a number](https://math.wiki/evaluate-a-function-at-a-number/): Putting a number in for the input and working out the output the rule gives.
- [Evaluating a function at an expression](https://math.wiki/evaluate-a-function-at-an-expression/): Putting a whole expression, such as x plus 2, in for the input and simplifying.
- [Evaluating a piecewise function](https://math.wiki/evaluate-a-piecewise-function/): Choosing the branch whose condition the input satisfies and using only that rule.
- [Evaluating logarithms, common and natural](https://math.wiki/evaluate-a-logarithm/): Asking what power of the base gives the number, for bases and numbers that work out whole.
- [Even and odd functions](https://math.wiki/test-symmetry-algebraically/): Replacing the input by its opposite, or both variables by their opposites, and seeing whether the equation survives.
- [Expanding and condensing logarithms](https://math.wiki/expand-a-log-expression/): Breaking one logarithm of a complicated expression into a sum and difference of simpler ones.
- [Exponential functions](https://math.wiki/exponential-functions/): The variable is in the exponent: growth by a constant factor.
- [Exponential growth and decay](https://math.wiki/exponential-growth-and-decay/): Doubling, halving, and everything in between.
- [Features of a graph](https://math.wiki/features-of-a-graph/): The named landmarks any function graph can have, such as intercepts, turning points and asymptotes.
- [Finding an inverse from a rule](https://math.wiki/find-an-inverse-algebraically/): Swapping the roles of input and output in the equation and solving for the new output.
- [Finding the rule of a transformed graph or table](https://math.wiki/write-an-equation-from-a-transformed-graph/): Identifying the parent graph and every change applied to it, then assembling the rule.
- [Function notation](https://math.wiki/function-notation/): f(x) means the output of f at x; reading and writing it.
- [Function notation in a real situation](https://math.wiki/function-notation-in-context/): Saying in words what a statement such as C of 12 equals 40 means for the quantities involved.
- [Functions](https://math.wiki/functions/): Rules that turn one number into exactly one other, and the graphs they draw.
- [Functions as input-output rules](https://math.wiki/input-output-machine/): A function pictured as a machine that takes one number in and gives exactly one number back.
- [Graphing a function from a table](https://math.wiki/plot-a-function-from-a-table/): Making a table of inputs and outputs and plotting each pair as a point.
- [Graphing a piecewise function](https://math.wiki/graph-a-piecewise-function/): Drawing each branch only over the inputs its condition allows.
- [Graphing by transforming a parent graph](https://math.wiki/graph-using-transformations/): Drawing a new graph by moving the parent graph instead of plotting many points.
- [Graphing polynomial functions](https://math.wiki/graph-polynomial-functions/): Put zeros, multiplicity and end behavior together.
- [Graphing rational functions](https://math.wiki/graph-rational-functions/): Find the holes, the asymptotes and the intercepts, then sketch.
- [Graphing square root functions](https://math.wiki/graph-a-square-root-function/): Plotting the half curve that begins at its endpoint and rises more and more slowly.
- [Graphs of exponential functions](https://math.wiki/graph-exponential-functions/): The curve that starts flat and takes off, and its horizontal asymptote.
- [Graphs of functions](https://math.wiki/graphs-of-functions/): Plot input against output; the picture of the rule.
- [Graphs of logarithmic functions](https://math.wiki/graph-log-functions/): The exponential's mirror image across y = x.
- [Graphs that tell a story](https://math.wiki/qualitative-graph-reading/): Describing in words what a graph says happened, without reading exact numbers off it.
- [Growth and decay factors](https://math.wiki/growth-factor-and-decay-factor/): The number the quantity is multiplied by each period, above one for growth and below one for decay.
- [Horizontal asymptotes](https://math.wiki/horizontal-asymptotes/): A horizontal line the graph settles toward far out to the left and to the right.
- [Increasing, decreasing, constant, positive and negative intervals](https://math.wiki/increasing-and-decreasing-intervals/): The stretches of input where a graph rises as you read left to right, and the stretches where it falls.
- [Independent and dependent variables](https://math.wiki/independent-and-dependent-variables/): The input variable is chosen freely and the output variable depends on what was chosen.
- [Intercepts and zeros of a function](https://math.wiki/x-intercepts-of-a-function/): The inputs where a graph meets the horizontal axis, the places where the output is zero.
- [Interpreting the numbers in a model](https://math.wiki/interpret-model-parameters/): Saying what the starting value, the rate or the factor means in the units of the situation.
- [Inverse from a table or list of pairs](https://math.wiki/inverse-from-a-table/): Swapping the input and output columns to get the inverse pairing.
- [Inverse functions](https://math.wiki/inverse-functions/): Swap input and output; the function that undoes another.
- [Judging and comparing models](https://math.wiki/judge-a-models-fit/): Comparing predicted outputs with the real ones to see where the model starts to drift.
- [Laws of logarithms](https://math.wiki/product-law-of-logs/): The logarithm of a product is the sum of the logarithms, because powers with the same base add.
- [Linear and nonlinear functions](https://math.wiki/linear-and-nonlinear-functions/): A function is linear when equal steps in the input give equal steps in the output; tell linear from nonlinear in tables, graphs and equations.
- [Linear functions](https://math.wiki/linear-functions/): Constant rate of change, straight-line graph, and modeling with them.
- [Linear functions in context](https://math.wiki/linear-function-in-context/): Naming what the starting value and the rate of change mean for the quantities in a situation.
- [Linear versus exponential growth](https://math.wiki/compare-rates-of-change-of-families/): Linear functions grow by equal differences over equal intervals, exponentials by equal factors, and an increasing exponential eventually passes any linear function.
- [Linear, polynomial and exponential growth compared](https://math.wiki/exponential-versus-linear-growth/): Linear growth adds the same amount each step while exponential growth multiplies by the same factor.
- [Logarithmic scales](https://math.wiki/logarithmic-scales/): Scales such as pH, decibels and earthquake magnitude, where each step up multiplies the quantity.
- [Logarithms](https://math.wiki/logarithms/): The exponent you need: the inverse of the exponential.
- [Logistic growth: when growth levels off](https://math.wiki/logistic-growth-models/): Growth that starts nearly exponential and slows toward a carrying capacity, its S-shaped graph and its horizontal asymptotes.
- [Maximum, minimum and range of a quadratic function](https://math.wiki/maximum-or-minimum-of-a-quadratic/): The vertex gives the largest output for a downward parabola and the smallest for an upward one.
- [Modeling with functions](https://math.wiki/function-modeling/): Choosing a function family and fitting its numbers so that one rule describes real data.
- [Modeling with polynomial functions](https://math.wiki/polynomial-modeling/): Fitting a polynomial to data that rises and falls more than once, such as the volume of a folded box.
- [One to one functions and the horizontal line test](https://math.wiki/one-to-one-functions/): A function is one to one when no two different inputs share the same output.
- [Parent functions](https://math.wiki/parent-functions/): The simplest member of each family, the plain graph every other member is a moved or stretched copy of.
- [Piecewise and absolute value functions](https://math.wiki/piecewise-and-absolute-value-functions/): Different rules on different stretches of the input.
- [Piecewise functions](https://math.wiki/piecewise-functions/): Read which rule applies, then evaluate or graph it.
- [Piecewise functions in context](https://math.wiki/piecewise-functions-in-context/): Rules that change at a threshold, such as tax bands or shipping charges by weight.
- [Polynomial functions](https://math.wiki/polynomial-functions/): Graphs of polynomials: their shape, their ends, and where they cross zero.
- [Quadratic functions](https://math.wiki/quadratic-functions/): The family whose graphs are parabolas, where the output depends on the square of the input.
- [Quadratic functions in context](https://math.wiki/quadratic-function-in-context/): Areas, profits and thrown objects, whose graphs rise to a peak and then fall.
- [Radical functions](https://math.wiki/radical-functions/): The family whose rule takes a root of the input, with graphs that start at an edge and curve away.
- [Radical functions in context](https://math.wiki/radical-functions-in-context/): Pendulum periods, braking distances and other quantities that grow like a root.
- [Rates at an instant from shrinking intervals](https://math.wiki/estimate-a-rate-at-an-instant/): Estimating the rate of change at a single point by computing average rates over smaller and smaller intervals around it.
- [Rational functions](https://math.wiki/rational-functions/): One polynomial over another, and the gaps and asymptotes that creates.
- [Rational functions in context](https://math.wiki/rational-function-in-context/): Average cost, concentration and shared work, where one quantity is divided by another that changes.
- [Reading a linear function from its graph](https://math.wiki/linear-function-from-a-graph/): Taking the starting value and the steady change off the picture and writing the rule.
- [Reading a quadratic in its three forms](https://math.wiki/quadratic-function-representations/): Standard, vertex and factored form each show a different feature of the very same parabola.
- [Reading values off a graph](https://math.wiki/read-a-value-from-a-graph/): Finding the output for a given input, and the inputs for a given output, from the picture alone.
- [Reflecting a graph across the axes](https://math.wiki/reflection-across-the-x-axis/): Taking the opposite of the whole output turns the graph upside down.
- [Relations and functions](https://math.wiki/relation-concept/): A relation is any pairing of inputs with outputs, with no rule about how many outputs one input may have.
- [Restricting a domain to get an inverse](https://math.wiki/restrict-a-domain-to-get-an-inverse/): Cutting a function down to a one to one piece so that an inverse can exist.
- [Semi-log plots: straightening exponential data with logarithms](https://math.wiki/semi-log-plots/): Taking the logarithm of the output turns exponential data into a line, and logs of both variables turn a power law into one.
- [Shifting a graph up, down, left and right](https://math.wiki/vertical-shift/): Adding a number to the whole output slides the graph up or down by that amount.
- [Slant asymptotes](https://math.wiki/slant-asymptotes/): When the numerator is one degree higher than the denominator, the graph follows a slanted line far out.
- [Solving by taking a logarithm of both sides](https://math.wiki/take-a-log-of-both-sides/): Applying a logarithm to both sides brings an unknown exponent down where it can be solved for.
- [Solving exponential equations by matching bases](https://math.wiki/solve-when-bases-match/): If both sides are powers of the same base, then the two exponents must be equal.
- [Solving f(x) = g(x) with graphs and tables](https://math.wiki/solve-f-equals-g-graphically/): The x-coordinates where two graphs meet are the solutions of f(x) = g(x), found from a graph, a table or successive approximation.
- [Solving logarithmic equations](https://math.wiki/extraneous-log-solutions/): An answer must be dropped if it makes any logarithm in the original equation take a value that is not positive.
- [Step functions and the greatest integer function](https://math.wiki/greatest-integer-function/): The function that rounds every input down to the nearest whole number.
- [Stretches and compressions](https://math.wiki/vertical-stretch-and-compression/): Multiplying the whole output by a number pulls the graph away from the horizontal axis or squashes it toward it.
- [Symmetry about the vertical axis and the origin](https://math.wiki/symmetry-about-the-y-axis/): A graph with this symmetry matches itself when the page is folded along the vertical axis.
- [Testing graphs and equations for functions](https://math.wiki/vertical-line-test/): If a vertical line hits the graph twice, one input has two outputs, so it is not a function.
- [The absolute value graph](https://math.wiki/graph-the-absolute-value-parent/): Plotting the V shaped graph whose point sits at the origin.
- [The difference quotient](https://math.wiki/difference-quotient/): Writing the average rate of change over an interval of width h as a single expression.
- [The graph of an inverse](https://math.wiki/graph-of-an-inverse/): The inverse graph is the original graph reflected across the line where output equals input.
- [The identity and constant functions](https://math.wiki/identity-function/): The function that gives back its input unchanged, drawn as the line through the origin at slope one.
- [The implied domain of a formula](https://math.wiki/implied-domain/): When no domain is stated, the domain is every input the formula can accept without breaking.
- [The number e and the natural exponential function](https://math.wiki/natural-exponential-function/): The exponential function with base e, the one whose growth rate always equals its own height.
- [The square root and reciprocal functions](https://math.wiki/reciprocal-parent-function/): The function one over x, whose graph is two curves hugging the two axes.
- [The squaring and cubing functions](https://math.wiki/squaring-parent-function/): The function that squares its input, whose graph is the plain upward parabola.
- [Transformations of functions](https://math.wiki/transformations-of-functions/): Shift, flip and stretch a graph by changing the formula.
- [Transforming an exponential graph](https://math.wiki/transform-exponential-graphs/): Shifting and stretching the curve, which also moves its horizontal asymptote.
- [Transforming the absolute value graph](https://math.wiki/transform-absolute-value-graphs/): Shifting, flipping and stretching the V to match a given rule.
- [Transforming the reciprocal graph](https://math.wiki/transform-the-reciprocal-function/): Shifting and stretching one over x, which moves both of its asymptotes.
- [Turning points, maximums and minimums](https://math.wiki/relative-maximum-and-minimum/): An output that is the largest or smallest in its own neighborhood, even if not for the whole function.
- [Vertical asymptotes](https://math.wiki/vertical-asymptotes/): A vertical line the graph shoots up or down along, sitting at an input the function cannot accept.
- [What a function is](https://math.wiki/function-concept/): A rule that gives exactly one output for each input.
- [What an asymptote is](https://math.wiki/asymptote-concept/): A line the graph gets ever closer to as it heads off without end, showing where the graph is heading.
- [Where a model holds: domain, interpolation and extrapolation](https://math.wiki/interpolation-and-extrapolation/): Reading a value inside the range of the data is far safer than predicting one outside it.
- [Writing a piecewise rule from a graph](https://math.wiki/write-a-piecewise-rule-from-a-graph/): Finding the equation of each segment and the interval of inputs it covers.
- [Writing an exponential function from a table or graph](https://math.wiki/exponential-function-from-a-table-or-graph/): Reading the starting value and the constant ratio from a table or graph and writing the exponential rule.
- [Zeros and multiplicity](https://math.wiki/zero-crosses-or-touches/): A zero of odd multiplicity crosses the axis and one of even multiplicity touches it and turns back.

## Geometry

- [AA similarity](https://math.wiki/aa-similarity/): Two pairs of congruent angles are enough to make two triangles similar.
- [Acute, right, obtuse and straight angles](https://math.wiki/acute-right-obtuse-straight/): Angles sorted by size, from smaller than a square corner up to a full half turn.
- [Adjacent angles and linear pairs](https://math.wiki/linear-pairs/): Two adjacent angles whose outer sides form a straight line, so their measures add to 180 degrees.
- [Alternate and same side angles](https://math.wiki/alternate-interior-angles/): Angles on opposite sides of the transversal and between the two lines.
- [Angle bisectors of a triangle](https://math.wiki/angle-bisector-distance-theorem/): A point is the same distance from both sides of an angle exactly when it lies on the bisector.
- [Angle sums of polygons](https://math.wiki/polygon-interior-angle-sum/): Splitting a polygon into triangles shows its interior angles add to 180 degrees times two fewer than its sides.
- [Angle types and pairs](https://math.wiki/angle-types-and-pairs/): Acute, right, obtuse; complementary, supplementary, vertical.
- [Angles and how to name them](https://math.wiki/naming-angles/): Reading and writing an angle by its vertex or by three points, with the vertex in the middle.
- [Angles made by chords, secants and tangents](https://math.wiki/angles-outside-a-circle/): Two secants, a secant and a tangent, or two tangents give an angle equal to half the difference of the arcs.
- [Arc length](https://math.wiki/arc-length-formula/): Taking the fraction of the full turn an arc covers and applying it to the circumference.
- [Arcs and central angles](https://math.wiki/minor-and-major-arcs/): A chord splits a circle into a shorter arc and a longer one, named with two or three letters.
- [Area of a circle](https://math.wiki/area-of-a-circle/): Cut into wedges, lay them into a near-rectangle, read off pi r squared.
- [Area of a polygon from coordinates](https://math.wiki/area-of-a-polygon-on-the-coordinate-plane/): Boxing a plotted polygon inside a rectangle and subtracting the corner pieces.
- [Auxiliary lines](https://math.wiki/auxiliary-lines/): A segment added to a figure to create the triangles or parallel lines an argument needs.
- [Bisecting a segment and an angle](https://math.wiki/construct-a-perpendicular-bisector/): Two equal arcs from the endpoints cross on the perpendicular bisector, which also locates the midpoint.
- [Building shapes from sticks and clay](https://math.wiki/build-shapes-from-sticks-and-clay/): Shapes are built from sticks for sides and clay balls for corners, so the parts that make a shape can be counted.
- [Cavalieri's principle](https://math.wiki/cavalieris-principle/): Two solids whose cross sections match in area at every height have the same volume, even if one leans.
- [Centroid](https://math.wiki/centroid/): Where the three medians meet, two thirds of the way along each from its vertex, and the balance point of the triangle.
- [Chord theorems](https://math.wiki/perpendicular-from-center-bisects-chord/): A radius drawn perpendicular to a chord cuts that chord and its arc in half.
- [Chords, secants and tangents](https://math.wiki/chord/): A segment whose two endpoints both sit on the circle.
- [Circles](https://math.wiki/circles/): Circumference, area, and the angle and chord theorems inside a circle.
- [Circumcenter](https://math.wiki/circumcenter/): Where the perpendicular bisectors meet, the one point the same distance from all three vertices.
- [Circumference](https://math.wiki/circumference/): Pi as the ratio of around to across, and the formula that follows.
- [Classifying figures from coordinates](https://math.wiki/classify-a-quadrilateral-on-the-plane/): Using slopes and distances to decide which quadrilateral four plotted points form.
- [Classifying shapes in a hierarchy](https://math.wiki/two-dimensional-shape-hierarchy/): Flat shapes, triangles and four-sided shapes included, are sorted into a family tree where each group sits inside the one above it.
- [Classifying triangles](https://math.wiki/classify-triangles-by-angles/): Sorting triangles into acute, right and obtuse.
- [Compass and straightedge constructions](https://math.wiki/constructions/): Copy a segment, bisect an angle, drop a perpendicular, with no ruler markings.
- [Complementary and supplementary angles](https://math.wiki/complementary-angles/): Two angles whose measures add to a right angle.
- [Composing and decomposing shapes](https://math.wiki/compose-and-decompose-shapes/): Building a new flat shape out of smaller ones and cutting a shape into pieces.
- [Composing transformations](https://math.wiki/composition-of-transformations/): Applying one transformation and then another, where the order can change the result.
- [Composite solids and density](https://math.wiki/volume-of-composite-solids/): Splitting a shape such as a silo into familiar solids and then adding or subtracting the pieces.
- [Congruence statements and correspondence](https://math.wiki/congruence-statements-and-correspondence/): Writing a congruence statement so that the order of the letters names which parts match.
- [Congruent angles, angle addition and bisectors](https://math.wiki/congruent-angles/): Two angles are congruent when they have equal measure.
- [Congruent triangles](https://math.wiki/congruent-triangles/): Same shape, same size: every side and angle matches.
- [Conic sections](https://math.wiki/conic-sections/): Slice a cone and you get a circle, an ellipse, a parabola, or a hyperbola.
- [Conic sections from a double cone](https://math.wiki/the-double-cone/): Every conic comes from cutting a double cone at a different angle.
- [Constructing a triangle from given parts](https://math.wiki/construct-a-triangle-from-three-sides/): Arcs of the two other lengths swung from the ends of the first meet at the missing vertex.
- [Constructions with circles](https://math.wiki/construct-the-circumscribed-circle/): Two perpendicular bisectors locate the circumcenter, which is the center of the circle.
- [Converse of the Pythagorean theorem](https://math.wiki/pythagorean-converse/): If the squares add up, the angle is right.
- [Converses and biconditional definitions](https://math.wiki/theorem-and-its-converse/): A true theorem can have a false converse, so each direction needs its own proof.
- [Coordinate geometry](https://math.wiki/coordinate-geometry/): Geometry done with numbers on a grid.
- [Coordinate proofs with variables](https://math.wiki/placing-a-figure-on-the-coordinate-plane/): Choosing axes and simple coordinates so the figure stays general and the algebra stays short.
- [Coordinates and distance in three dimensions](https://math.wiki/three-dimensional-coordinates/): Locating a point in space with three numbers instead of two.
- [Copying a segment and an angle](https://math.wiki/construct-a-congruent-segment/): Setting the compass to a given length and marking that same length on a new ray.
- [Corresponding angles](https://math.wiki/corresponding-angles/): Angles in matching positions at the two crossings, congruent when the lines are parallel.
- [Cross sections of solids](https://math.wiki/cross-sections-of-prisms-and-pyramids/): Naming the flat shape left behind when a plane slices through a polyhedron.
- [Cylinders and cones](https://math.wiki/cylinders/): Two congruent parallel circular bases joined by a curved surface.
- [Deductive reasoning](https://math.wiki/deductive-reasoning/): Drawing a conclusion that must follow from statements already accepted as true.
- [Defining and non-defining attributes of shapes](https://math.wiki/defining-attributes-of-shapes/): Three straight sides make a triangle, but its color, size and which way it turns do not.
- [Dilations](https://math.wiki/center-of-dilation/): The fixed point that every other point moves directly toward or directly away from.
- [Distance from a point to a line](https://math.wiki/distance-from-a-point-to-a-line/): The shortest distance is measured along the perpendicular segment from the point to the line.
- [Distance on a number line](https://math.wiki/distance-on-a-number-line/): Finding how far apart two marked points are by taking the absolute value of their difference.
- [Dividing a segment into equal parts](https://math.wiki/divide-a-segment-into-equal-parts/): Marking equal steps along a slanted ray and joining back with parallel lines.
- [Drawing triangles from given sides and angles: one, many or none](https://math.wiki/draw-triangles-from-conditions/): Draw triangles from given side lengths and angle measures with ruler and protractor, and decide whether the conditions give one triangle, many or none.
- [Equation of a circle](https://math.wiki/circle-equation-from-center-and-radius/): Turning a stated center and radius into the standard equation.
- [Equilateral triangle, square and hexagon in a circle](https://math.wiki/construct-an-equilateral-triangle/): Two circles of the same radius drawn through each other's centers meet at the third vertex.
- [Faces, edges and vertices](https://math.wiki/faces-edges-and-vertices/): The flat surfaces of a solid, the lines where two faces meet, and the corners where edges meet.
- [Finding the center and radius](https://math.wiki/find-center-and-radius-by-completing-the-square/): Rewriting an expanded circle equation into standard form so the center and radius can be read off.
- [Finding unknown angle measures](https://math.wiki/find-unknown-angle-measures/): Using named angle pairs in a figure to work out a measure that was not given.
- [Flowchart proofs and choosing a format](https://math.wiki/choosing-a-proof-format/): Two column, paragraph, flowchart and coordinate forms all carry the same argument in different clothes.
- [Focus and directrix](https://math.wiki/focus-directrix-definition/): Each conic is the set of points whose distances to a fixed point and a fixed line keep a fixed ratio.
- [Geometric means in a right triangle](https://math.wiki/geometric-mean-altitude-relation/): The altitude is the geometric mean of the two pieces it cuts the hypotenuse into.
- [Geometric proof](https://math.wiki/geometric-proof/): Arguing from definitions and accepted facts to a certain conclusion.
- [Geometry](https://math.wiki/geometry/): Shapes, space, and the reasoning that proves facts about them.
- [Given, prove and the diagram](https://math.wiki/given-and-prove/): Every proof starts by sorting what is assumed from what must be shown, beside a labeled figure.
- [Hinge theorem](https://math.wiki/hinge-theorem/): Two triangles with two pairs of congruent sides have the longer third side where the included angle is larger.
- [Horizontal and vertical distance](https://math.wiki/horizontal-and-vertical-distance/): Finding the length of a segment that runs straight across or straight up by subtracting coordinates.
- [Hypotenuse leg congruence](https://math.wiki/hl-congruence/): Right triangles are congruent when their hypotenuses match and one pair of legs matches.
- [Identifying and translating conics](https://math.wiki/identify-a-conic-from-its-equation/): Reading the squared terms and their signs to tell circle, ellipse, parabola and hyperbola apart.
- [Incenter](https://math.wiki/incenter/): Where the angle bisectors meet, the one point the same distance from all three sides.
- [Indirect proof](https://math.wiki/indirect-proof-in-geometry/): Assuming the opposite of what is to be shown and following it until something impossible appears.
- [Inductive reasoning](https://math.wiki/inductive-reasoning/): Noticing a pattern in examples and proposing that it always holds.
- [Inscribed polygons and cyclic quadrilaterals](https://math.wiki/cyclic-quadrilateral/): A quadrilateral with all four vertices on one circle has opposite angles that add to 180 degrees.
- [Isosceles and equilateral triangles](https://math.wiki/base-angles-theorem/): The two angles opposite the congruent sides are congruent.
- [Line symmetry](https://math.wiki/line-symmetry/): A figure has line symmetry when a reflection maps it exactly onto itself.
- [Lines and planes in space](https://math.wiki/intersections-of-lines-and-planes/): Two lines meet in at most one point and two planes meet in a line.
- [Locus of points](https://math.wiki/locus-of-points/): A locus is the set of every point that meets a stated condition, such as staying a fixed distance from a point.
- [Medians and altitudes](https://math.wiki/medians-of-a-triangle/): A median joins a vertex to the midpoint of the opposite side.
- [Midpoints and segment bisectors](https://math.wiki/midpoint-of-a-segment/): The point that cuts a segment into two congruent pieces.
- [Midsegments of triangles and trapezoids](https://math.wiki/triangle-midsegment/): The segment joining two side midpoints is parallel to the third side and half as long.
- [Modeling with solids: density and design](https://math.wiki/modelling-with-solids/): Describing real objects as solids and using volume, surface area and density to answer design questions.
- [Naming and classifying polygons](https://math.wiki/naming-polygons-by-sides/): Pentagon, hexagon, octagon and the rest, each named by how many sides it has.
- [Naming and describing solid shapes](https://math.wiki/name-solid-shapes/): Telling solid shapes apart by name and spotting them in everyday objects.
- [Naming circles, triangles, squares and rectangles](https://math.wiki/name-basic-shapes/): Telling everyday flat shapes apart and calling each one by name.
- [Nets of solids](https://math.wiki/net-of-a-prism/): Unfolding a prism into its two bases and the rectangle that wraps around them.
- [Ordered pairs and the first quadrant](https://math.wiki/ordered-pairs/): Two numbers in a fixed order that name one point, the first across and the second up.
- [Orthocenter and the Euler line](https://math.wiki/orthocenter/): Where the three altitudes meet, a point that lands outside the triangle when the triangle is obtuse.
- [Parallel lines](https://math.wiki/parallel-lines/): Two lines in a flat plane that never meet, however far they go, are parallel.
- [Parallel lines and transversals](https://math.wiki/parallel-lines-and-transversals/): A line crossing two parallels makes equal and supplementary angle pairs.
- [Parallelograms, rectangles, rhombuses and squares: what each one guarantees](https://math.wiki/properties-of-special-quadrilaterals/): Each special four-sided shape promises certain things, and anything true of every parallelogram is also true of every rectangle, rhombus and square.
- [Partitioning a segment in a given ratio](https://math.wiki/partition-a-segment-in-a-ratio/): Finding the point that divides a segment so its two pieces have a chosen ratio.
- [Parts of a triangle](https://math.wiki/parts-of-a-triangle/): Vertices, sides and interior angles, and which side is included between two named angles.
- [Perimeter and area from coordinates](https://math.wiki/perimeter-and-area-on-the-coordinate-plane/): Computing the perimeter and area of polygons drawn on the coordinate plane with the distance formula and decomposition.
- [Perpendicular and parallel lines through a point](https://math.wiki/construct-a-perpendicular-at-a-point-on-a-line/): Marking equal distances either side of the point and then bisecting the segment between them.
- [Perpendicular bisector of a segment](https://math.wiki/perpendicular-bisector-of-a-segment/): The line that crosses a segment at its midpoint at a right angle.
- [Perpendicular bisectors of a triangle](https://math.wiki/perpendicular-bisector-theorem/): A point is the same distance from two endpoints exactly when it lies on the perpendicular bisector between them.
- [Perpendicular lines](https://math.wiki/perpendicular-lines-geometry/): Two lines that meet at a right angle.
- [Planning a congruence proof](https://math.wiki/plan-a-congruence-proof/): Marking the figure, listing what is given, and choosing which criterion those given parts can reach.
- [Plotting in all four quadrants](https://math.wiki/plot-points-with-negative-coordinates/): Extending plotting to all four quadrants.
- [Points that differ only in sign: reflections across the axes](https://math.wiki/points-reflected-across-the-axes/): Changing the sign of one coordinate reflects a point across an axis, and changing both reflects it through the origin.
- [Points, lines and planes](https://math.wiki/points-lines-planes/): The undefined starting objects, and what is built from them.
- [Points, lines and planes](https://math.wiki/undefined-terms-in-geometry/): Point, line and plane are accepted without definition so that everything else can be defined from them.
- [Points, lines, rays and segments](https://math.wiki/points-lines-rays-and-segments/): A line goes on forever both ways, a ray goes on one way from an endpoint, and a segment stops at two endpoints.
- [Polygons](https://math.wiki/polygons/): Closed shapes with straight sides, named by how many.
- [Polyhedra](https://math.wiki/polyhedra/): Solids whose faces are all flat polygons.
- [Position words: above, below, beside, in front of, behind](https://math.wiki/position-words/): Words like above, below, beside, in front of and behind say where one object is compared with another.
- [Prisms and pyramids](https://math.wiki/prisms/): Two congruent parallel bases joined by parallelogram faces.
- [Proofs about circles](https://math.wiki/proofs-about-circles/): Arguments that lean on radii being congruent and on the isosceles triangles they create.
- [Proofs about parallelograms](https://math.wiki/proofs-about-parallelograms/): Congruent triangles made by a diagonal supply almost every parallelogram property.
- [Proofs about similar figures](https://math.wiki/proofs-about-similarity/): Arguments that establish proportional sides once two pairs of angles have been matched.
- [Properties of a parallelogram](https://math.wiki/parallelogram-properties/): Opposite sides are parallel and congruent, opposite angles are congruent, and neighboring angles are supplementary.
- [Properties of equality and congruence](https://math.wiki/properties-of-congruence/): Congruence of segments and angles behaves like equality, which is why a shared side can be cited.
- [Proportions in similar triangles](https://math.wiki/similar-triangle-proportions/): Set up the ratio of matching sides and solve for the missing one.
- [Proving a parallelogram is a rectangle or a rhombus](https://math.wiki/prove-a-parallelogram-is-a-rectangle/): Congruent diagonals or one right angle forces all four angles to be right.
- [Proving a quadrilateral is a parallelogram](https://math.wiki/prove-a-quadrilateral-is-a-parallelogram/): Any one of several conditions on the sides, the angles or the diagonals is enough.
- [Proving a tangent is perpendicular to the radius](https://math.wiki/prove-the-tangent-radius-theorem/): Every other point of the tangent line lies outside the circle, so the radius has to be the shortest path.
- [Proving alternate interior angles congruent](https://math.wiki/prove-alternate-interior-angles/): Combining the corresponding angles postulate with a pair of vertical angles.
- [Proving angle pair theorems](https://math.wiki/prove-vertical-angles-congruent/): Using two linear pairs that share an angle to show the opposite angles must match.
- [Proving lines parallel](https://math.wiki/proving-lines-parallel/): Use the transversal angle pairs in reverse.
- [Proving parts congruent after the triangles](https://math.wiki/proof-with-cpctc/): Proving two triangles congruent first so that a matching side or angle can then be claimed.
- [Proving properties of a parallelogram](https://math.wiki/prove-opposite-sides-congruent/): One diagonal splits the parallelogram into two congruent triangles.
- [Proving properties of an isosceles trapezoid](https://math.wiki/prove-isosceles-trapezoid-properties/): Base angles and diagonals shown congruent by dropping altitudes or by using congruent triangles.
- [Proving the base angles theorem](https://math.wiki/prove-base-angles-congruent/): Adding a bisector or a median to split an isosceles triangle into two congruent halves.
- [Proving the inscribed angle theorem](https://math.wiki/prove-the-inscribed-angle-theorem/): Handling the case through the center first, then reaching the others by adding or subtracting.
- [Proving the power of a point relationships](https://math.wiki/prove-power-of-a-point/): Similar triangles formed by the chords or secants give the equal products.
- [Proving the Pythagorean theorem by rearrangement](https://math.wiki/prove-the-pythagorean-theorem-by-area/): Fitting four copies of one right triangle into a square in two different ways.
- [Proving the Pythagorean theorem with similar triangles](https://math.wiki/prove-the-pythagorean-theorem-with-similarity/): The altitude to the hypotenuse makes two similar copies whose ratios combine into the theorem.
- [Proving the triangle angle sum](https://math.wiki/prove-the-triangle-angle-sum/): Drawing a line through one vertex parallel to the opposite side and reading the angles off it.
- [Proving the triangle proportionality theorem](https://math.wiki/prove-the-triangle-proportionality-theorem/): A line parallel to one side makes a smaller similar triangle, and that similarity gives the proportion.
- [Proving triangles congruent](https://math.wiki/proving-triangles-congruent/): Pick the criterion, line up the parts, conclude the rest match.
- [Proving two triangles similar](https://math.wiki/prove-triangles-similar/): Finding two congruent pairs of angles, often from parallel lines or from a shared angle.
- [Pythagorean triples](https://math.wiki/pythagorean-triples/): Whole number side lengths that fit the Pythagorean relationship, such as 3, 4 and 5.
- [Quadrilaterals](https://math.wiki/quadrilaterals/): Squares, rectangles, parallelograms, rhombuses, trapezoids, and how they nest.
- [Radius and diameter](https://math.wiki/radius-and-diameter/): The distance from the center out to the edge, and the distance all the way across through the center.
- [Rectangles, rhombuses and squares](https://math.wiki/rectangle-properties/): A parallelogram with four right angles, whose two diagonals are also equal in length.
- [Reflections](https://math.wiki/reflect-over-the-axes/): Flipping a figure over the x axis or the y axis by changing the sign of one coordinate.
- [Regular polygons and their angles](https://math.wiki/interior-angle-of-a-regular-polygon/): Sharing the interior angle sum equally among angles that are all congruent.
- [Regular polygons and their circles](https://math.wiki/central-angle-of-a-regular-polygon/): The angle at the center between two neighboring vertices, equal to 360 degrees divided by the side count.
- [Rhombuses, rectangles and squares](https://math.wiki/rhombuses-rectangles-and-squares/): A rhombus has four equal sides, a rectangle four right angles, and a square has both.
- [Rigid motions and congruence](https://math.wiki/congruence-by-rigid-motion/): Two figures are congruent exactly when some sequence of slides, flips and turns carries one onto the other.
- [Rigid motions defined precisely, and what they preserve](https://math.wiki/rigid-motion-definitions/): Translations, reflections and rotations defined from segments, perpendicular bisectors and circles, and why each keeps distances and angles.
- [Rotating about a point other than the origin](https://math.wiki/rotate-about-any-point/): Turning a figure around a chosen center while every distance to that center stays the same.
- [Rotational and point symmetry](https://math.wiki/rotational-symmetry/): A figure has rotational symmetry when a turn of less than a full circle maps it onto itself.
- [Rotations](https://math.wiki/center-and-angle-of-rotation/): A rotation is fixed by the point it turns about, the number of degrees, and the direction.
- [Scale factor and perimeter, area and volume](https://math.wiki/ratio-of-perimeters-and-areas/): In similar figures perimeters scale like the scale factor and areas scale like its square.
- [Sectors, segments and their areas](https://math.wiki/sector-area-formula/): Taking that same fraction of the full turn and applying it to the area of the whole circle.
- [Segment length, addition and congruence](https://math.wiki/congruent-segments/): Two segments are congruent when they have the same length.
- [Segment lengths in circles](https://math.wiki/chord-chord-product/): The products of the two pieces of each chord are equal.
- [Segments, rays and length](https://math.wiki/segments-and-rays/): Pieces of a line, and adding lengths along one.
- [Shapes in more than one category, and quadrilaterals in none](https://math.wiki/shapes-in-more-than-one-category/): A square is also a rectangle and a rhombus, while some four-sided shapes belong to none of those groups.
- [Sides and angles in order](https://math.wiki/side-angle-inequality-in-a-triangle/): The longest side of a triangle lies opposite the largest angle and the shortest lies opposite the smallest.
- [Sides, corners and sorting shapes](https://math.wiki/sort-shapes-by-attributes/): Grouping flat shapes by number of sides, number of corners, or whether the sides are equal.
- [Similar figures](https://math.wiki/similar-figures/): Matching angles equal, matching sides in one ratio.
- [Similarity](https://math.wiki/similarity/): Same shape, different size.
- [Similarity transformations](https://math.wiki/similarity-transformations/): A dilation followed by rigid motions, which is what it means for two figures to be similar.
- [Solids](https://math.wiki/solids/): Three-dimensional shapes, their surfaces and their insides.
- [Solids of revolution](https://math.wiki/solids-of-revolution/): The solid swept out when a flat region is spun about a line.
- [Special right triangles](https://math.wiki/45-45-90-triangles/): The isosceles right triangle, whose hypotenuse is a leg times the square root of two.
- [Special segments in a triangle](https://math.wiki/triangle-segments/): Medians, altitudes, angle bisectors and perpendicular bisectors, the four families of segments drawn in a triangle.
- [Spheres](https://math.wiki/spheres/): A surface every point of which is the same distance from the center.
- [SSS and SAS similarity](https://math.wiki/sss-similarity/): Triangles are similar when all three pairs of sides are in the same ratio, or when two pairs are and the angles between them are equal.
- [SSS, SAS, ASA, AAS](https://math.wiki/congruence-criteria/): The short lists of matching parts that force two triangles to be congruent.
- [Surface area](https://math.wiki/surface-area/): Add up the area of every face.
- [Surface area of a sphere](https://math.wiki/surface-area-of-a-sphere/): A sphere has four times the surface area of a circle with the same radius.
- [Surface area of prisms and cylinders](https://math.wiki/surface-area-of-a-prism/): Adding the two bases to the rectangles that wrap around them.
- [Surface area of pyramids and cones](https://math.wiki/surface-area-of-a-pyramid/): Adding the base to the triangular faces, which use the slant height rather than the vertical height.
- [Tangent properties](https://math.wiki/tangent-radius-perpendicular/): A tangent line is perpendicular to the radius drawn to the point where it touches.
- [Tessellations](https://math.wiki/tessellations/): A tiling that covers the plane with copies of a shape leaving no gaps and no overlaps.
- [The apothem of a regular polygon](https://math.wiki/apothem/): The perpendicular distance from the center of a regular polygon out to one of its sides.
- [The coordinate plane](https://math.wiki/coordinate-plane/): Two number lines at right angles; every point gets a pair of numbers.
- [The distance formula](https://math.wiki/distance-formula/): The Pythagorean theorem between two points.
- [The ellipse](https://math.wiki/foci-of-an-ellipse/): An ellipse is every point whose distances to two fixed points add to the same total.
- [The five regular solids](https://math.wiki/platonic-solids/): Only five solids have identical regular faces meeting the same way at every vertex.
- [The hyperbola](https://math.wiki/foci-of-a-hyperbola/): A hyperbola is every point whose distances to two fixed points differ by the same amount.
- [The inscribed angle theorem](https://math.wiki/inscribed-angles-on-the-same-arc/): Any two inscribed angles opening onto the same arc are congruent.
- [The midpoint formula](https://math.wiki/endpoint-from-a-midpoint/): Working backwards from a midpoint and one endpoint to the other endpoint.
- [The parabola as a locus](https://math.wiki/parabola-focus-and-directrix/): A parabola is every point that is the same distance from a fixed point as from a fixed line.
- [The parallel postulate](https://math.wiki/the-parallel-postulate/): Through a point not on a line there is exactly one parallel line, the assumption Euclidean geometry rests on.
- [The Pythagorean theorem](https://math.wiki/pythagorean-theorem/): In a right triangle, the squares on the legs add to the square on the hypotenuse.
- [The Pythagorean theorem in three dimensions](https://math.wiki/pythagorean-theorem-in-three-dimensions/): Use the Pythagorean theorem twice to find the space diagonal of a box and other lengths in solid figures.
- [The quadrilateral family tree](https://math.wiki/quadrilateral-hierarchy/): How square, rectangle, rhombus, parallelogram, trapezoid and kite sit inside one another.
- [Three-dimensional shapes](https://math.wiki/three-dimensional-shapes/): Prisms, pyramids, cylinders, cones, spheres: faces, edges, vertices.
- [Transformational proof](https://math.wiki/transformational-proof/): Proving a claim by exhibiting the rigid motion or dilation that carries one figure onto another.
- [Transformations](https://math.wiki/transformations/): Moving figures: slide, flip, turn, and what stays the same.
- [Translations](https://math.wiki/translation-with-a-vector/): Describing a slide by how far the figure moves across and how far it moves up or down.
- [Trapezoids and kites](https://math.wiki/trapezoid-definition/): A quadrilateral with at least one pair of parallel sides, which are called its bases.
- [Triangle angle bisector theorem](https://math.wiki/triangle-angle-bisector-proportion/): An angle bisector splits the opposite side in the same ratio as the two sides that form the angle.
- [Triangle angle sum](https://math.wiki/triangle-angle-sum/): The three angles add to a straight line, and why.
- [Triangle centers](https://math.wiki/triangle-centers/): Each family of special segments meets at one point, giving the triangle four classic centers.
- [Triangle inequality](https://math.wiki/triangle-inequality/): Any two sides together are longer than the third.
- [Triangle proportionality theorem](https://math.wiki/triangle-proportionality/): A line parallel to one side splits the other two in the same ratio.
- [Triangles](https://math.wiki/triangles/): The simplest polygon, and most of geometry's theorems.
- [Triangles, quadrilaterals, pentagons and hexagons from their sides and angles](https://math.wiki/draw-shapes-with-given-attributes/): Shapes are named and drawn from what they must have, such as five sides or three corners, whatever their size.
- [Two number patterns side by side and graphing their pairs](https://math.wiki/two-patterns-and-their-graph/): Two patterns are built from two rules, their matching terms are paired, and the pairs are graphed to see how the patterns relate.
- [Two reflections in a row](https://math.wiki/two-reflections-make-a-rotation/): Reflecting twice gives a translation when the mirrors are parallel and a rotation when they cross.
- [Two-column and paragraph proofs](https://math.wiki/justifying-each-step/): Matching each statement to a definition, postulate, theorem or property that is already accepted.
- [Two-dimensional shapes](https://math.wiki/shapes-2d/): Naming and sorting flat shapes by their sides and corners.
- [Using the Pythagorean theorem](https://math.wiki/pythagorean-theorem-applications/): Finding a missing distance in a real figure by spotting a right triangle inside it.
- [Vertical angles](https://math.wiki/vertical-angles/): The opposite pair of angles formed when two lines cross, always equal in measure.
- [Volume of a cylinder](https://math.wiki/volume-of-a-cylinder/): The same stacking idea carried out with a circular base.
- [Volume of a sphere](https://math.wiki/volume-of-a-sphere/): A sphere holds two thirds of the cylinder that just contains it.
- [Volume of prisms and cylinders](https://math.wiki/volume-of-a-prism/): Stacking copies of the base area all the way up to the height.
- [Volume of pyramids and cones](https://math.wiki/volume-of-a-pyramid/): A pyramid holds one third of the prism with the same base and height.
- [Volume of solids](https://math.wiki/volume-of-pyramids-cones-spheres/): One third of the matching prism or cylinder, and the sphere formula.
- [What a circle is](https://math.wiki/circles-concept/): Every point the same distance from the center; radius, diameter, chord.
- [What a dilation does to a line](https://math.wiki/dilations-of-lines/): A line through the center stays exactly where it is and every other line maps to a line parallel to it.
- [What rigid motions preserve](https://math.wiki/invariance-under-rigid-motion/): Lengths, angle measures, betweenness and parallelism all survive a rigid motion unchanged.
- [Why a construction works](https://math.wiki/justify-a-construction/): Every compass and straightedge step can be defended by the congruent triangles or circles it creates.
- [Why AAA and SSA fail](https://math.wiki/why-ssa-is-not-a-criterion/): Two sides and an angle that is not between them can build two different triangles, so they do not force congruence.
- [Why the volume formulas hold: dissection, slicing and Cavalieri](https://math.wiki/derive-volume-formulas/): Informal arguments for the cylinder, pyramid, cone and sphere volume formulas, by dissection, by slicing and by Cavalieri's principle.

## Linear algebra

- [Adding and scaling vectors in n dimensions](https://math.wiki/vector-addition-in-rn/): Adding two lists of numbers one position at a time.
- [Adding, subtracting and scaling matrices](https://math.wiki/add-subtract-matrices/): Entry by entry, and only when the sizes match.
- [Augmented matrices and row operations](https://math.wiki/elementary-row-operations/): Swapping rows, scaling a row, or adding a multiple of one row to another, none of which change the solutions.
- [Basis and dimension](https://math.wiki/dimension-of-a-space/): The number of vectors in a basis, which comes out the same for every basis of the same space.
- [Complex eigenvalues and rotation-scaling matrices](https://math.wiki/complex-eigenvalues/): A real matrix with no real eigenvectors rotates and scales, and its complex eigenvalues give the angle and the scale factor.
- [Composing transformations as a matrix product](https://math.wiki/composition-as-matrix-product/): Doing one transformation and then another is the same as multiplying their two matrices.
- [Computing a determinant](https://math.wiki/determinant-2x2/): Multiply the diagonals and subtract.
- [Coordinates in a basis and change of basis](https://math.wiki/coordinates-and-change-of-basis/): The same vector has different coordinates in different bases, and an invertible matrix converts one set of coordinates into the other.
- [Cramer's rule](https://math.wiki/cramers-rule/): Solve a system with a ratio of determinants.
- [Determinants by cofactor expansion](https://math.wiki/cofactor-expansion/): A determinant of any size breaks into smaller determinants along any row or column, with alternating signs.
- [Diagonalization](https://math.wiki/diagonalization/): Using a basis of eigenvectors, a transformation turns into plain scaling along each direction.
- [Dot product, length and orthogonality in n dimensions](https://math.wiki/dot-product-and-length-in-rn/): Multiplying matching entries and adding gives the dot product, which measures length, angle and perpendicularity in any number of dimensions.
- [Echelon forms and Gaussian elimination](https://math.wiki/row-echelon-form/): A staircase shape where each row starts further right than the one above and zero rows sit at the bottom.
- [Eigenvalues and eigenvectors](https://math.wiki/eigenvalues-and-eigenvectors/): A vector that a transformation only stretches, together with the number by which it stretches.
- [Finding an eigenvector](https://math.wiki/find-an-eigenvector/): Putting an eigenvalue back in and solving the system for the vectors it leaves in place.
- [Finding an inverse by row reducing A next to I](https://math.wiki/inverse-by-row-reduction/): Row reduce A with the identity beside it; when A has become I, the identity has become the inverse of A.
- [Finding the matrix of a transformation](https://math.wiki/matrix-of-a-rotation/): Where a transformation sends each unit vector gives the columns of the matrix that performs it.
- [Homogeneous systems and solutions in parametric vector form](https://math.wiki/solution-sets-of-linear-systems/): The solutions of Ax = 0 form a span, and every solution of Ax = b is one particular solution plus a solution of Ax = 0.
- [Image](https://math.wiki/image-of-a-matrix/): All the vectors a transformation can produce, which is the span of its columns.
- [Inverse of a 2 by 2 matrix, and solving with it](https://math.wiki/inverse-2x2/): Swap, negate, divide by the determinant.
- [Kernel](https://math.wiki/kernel-of-a-matrix/): All the vectors that a transformation sends to zero.
- [Least squares: the best approximate solution of Ax = b](https://math.wiki/least-squares-solutions/): When Ax = b has no solution, the normal equations give the x that brings Ax as close to b as possible.
- [Linear algebra](https://math.wiki/linear-algebra/): Vectors, matrices and the linear transformations they describe: solving many equations at once, and what a matrix does to space.
- [Linear combination](https://math.wiki/linear-combination-of-vectors/): A sum of scaled vectors, which is the single move the whole of linear algebra is built on.
- [Linear independence](https://math.wiki/linear-independence/): A set is independent when no vector in it can be built from the others.
- [Linear systems in use: networks, chemical equations, fitting curves](https://math.wiki/applications-of-linear-systems/): Balancing a chemical equation, the flows through a network and a polynomial through given points each become one linear system.
- [Linear transformation](https://math.wiki/linear-transformation/): A function on vectors that respects addition and scaling, which is exactly what a matrix does.
- [Matrices](https://math.wiki/matrices/): Rectangular arrays of numbers, with their own arithmetic.
- [Matrices as linear transformations](https://math.wiki/matrices-as-transformations/): A matrix moves every vector in the plane; columns show where the basis goes.
- [Matrix multiplication](https://math.wiki/matrix-multiplication-rule/): Each entry is a row of the first dotted with a column of the second.
- [Matrix powers, dynamical systems and steady states](https://math.wiki/dynamical-systems-and-steady-states/): Applying a matrix again and again is simple in a basis of eigenvectors, which shows where a system settles in the long run.
- [Multiplying a matrix by a vector](https://math.wiki/matrix-times-a-vector/): The answer is a linear combination of the columns, with the vector entries as the weights.
- [Orthogonal bases and projection onto a subspace](https://math.wiki/orthogonal-projection/): With an orthogonal basis, the point of a subspace closest to a vector is a sum of simple projections onto the basis vectors.
- [Orthogonality and least squares](https://math.wiki/orthogonality-and-least-squares/): Perpendicular vectors, projection onto a subspace, and the best approximate solution of a system that has no exact one.
- [Properties of determinants: row operations, products, transposes](https://math.wiki/properties-of-determinants/): A row swap flips the sign, scaling a row scales the determinant, and the determinant of a product is the product of the determinants.
- [Rank](https://math.wiki/rank-of-a-matrix/): The number of leading entries left after reduction, which says how many rows carry new information.
- [Rank and nullity](https://math.wiki/rank-and-nullity/): The dimensions of the image and of the kernel always add up to the number of columns.
- [Reduced row echelon form](https://math.wiki/reduced-row-echelon-form/): The staircase taken further, so each leading entry is 1 and is the only non-zero entry in its column.
- [Row reduction](https://math.wiki/row-reduction/): Systematic elimination on a matrix to solve any linear system.
- [Rows, columns and dimensions](https://math.wiki/matrix-vocabulary/): Naming a matrix by its size, and naming an entry by its address.
- [Solving a system by row reduction: one, none or infinitely many solutions](https://math.wiki/solve-a-system-by-row-reduction/): Reducing the augmented matrix and reading the solution off the final rows.
- [Span](https://math.wiki/span-of-a-set-of-vectors/): Every vector that can be reached from a given set using linear combinations.
- [Subspaces](https://math.wiki/subspaces-of-rn/): A set of vectors closed under adding and scaling, such as a line or plane through the origin, is a subspace.
- [The characteristic equation and finding eigenvalues](https://math.wiki/characteristic-equation/): Eigenvalues are the numbers that make the shifted matrix fail to be invertible.
- [The Gram-Schmidt process](https://math.wiki/gram-schmidt-process/): Any basis becomes an orthogonal one by subtracting from each vector its projections onto the vectors before it.
- [The invertible matrix theorem](https://math.wiki/invertible-matrix-theorem/): For a square matrix, a dozen conditions, from a nonzero determinant to a kernel holding only zero, all say the same thing.
- [The transpose and symmetric matrices](https://math.wiki/transpose-and-symmetric-matrices/): Flipping a matrix across its diagonal swaps its rows and columns, and a matrix equal to its flip is symmetric.
- [Vectors in n dimensions](https://math.wiki/vectors-in-rn/): Lists of numbers with addition and scaling; linear combinations and span.
- [What the determinant means](https://math.wiki/determinant-meaning/): It is the area or volume scaling factor, and zero means the matrix flattens everything.
- [When a matrix or transformation can be undone](https://math.wiki/invertible-transformations/): A matrix can be undone exactly when nothing but zero lands on zero, which is when its determinant is not zero.
- [Writing a system as a matrix equation](https://math.wiki/system-as-a-matrix-equation/): Coefficients, variables and constants, split into three matrices.

## Logic, proof and discrete math

- [A conditional and its contrapositive](https://math.wiki/contrapositive-equivalence/): Swapping the two parts of an if-then statement and negating both gives a statement with the same meaning.
- [Adjacency matrices and their powers](https://math.wiki/adjacency-matrix-powers/): Each entry of the nth power counts the walks of length n between one vertex and another.
- [And and or: conjunction and disjunction](https://math.wiki/conjunction-and/): A statement joined by and, true only when both of its parts are true.
- [Axioms, definitions and theorems](https://math.wiki/what-an-axiom-is/): A statement accepted without proof, chosen as a starting point for everything built after it.
- [Bezout's identity and the extended Euclidean algorithm](https://math.wiki/extended-euclidean-algorithm/): Running the Euclidean algorithm backwards to find the combination that actually gives the divisor.
- [Biconditional](https://math.wiki/biconditional/): A statement saying two things hold together and fail together, written as if and only if.
- [Bipartite graphs](https://math.wiki/bipartite-graphs/): A graph whose vertices split into two groups with every edge running between the groups.
- [Building a truth table for a compound statement](https://math.wiki/build-a-compound-truth-table/): Working outward from the small parts of a statement, one column at a time, to reach its whole value.
- [Cantor's theorem: no set matches its power set](https://math.wiki/cantors-theorem/): No set can be matched one to one with its power set, so there is no largest infinity.
- [Cardinality of a finite set](https://math.wiki/cardinality-of-a-finite-set/): How many members a set has, counting each distinct member exactly once.
- [Cartesian product](https://math.wiki/cartesian-product/): The set of all ordered pairs made by taking one member from each of two sets.
- [Closed form versus recursive form](https://math.wiki/closed-form-versus-recursive/): A recursive rule needs every earlier term, while a closed formula gives any term straight away.
- [Combinatorial proof: counting one set two ways](https://math.wiki/combinatorial-proof/): An identity is proved by showing that both sides count the same collection in two different ways.
- [Composing functions between sets, and when an inverse exists](https://math.wiki/composition-and-inverses-on-sets/): Following one function with another gives a composite, and a function has an inverse exactly when it is a bijection.
- [Conditionals, converse, inverse, contrapositive](https://math.wiki/conditionals-and-converse/): If p then q, and the three ways to turn it around; only one keeps the truth.
- [Congruence modulo n and its arithmetic](https://math.wiki/congruence-notation/): Two numbers are congruent when they leave the same remainder on division by a fixed number.
- [Conjecture, and why a pattern is not a proof](https://math.wiki/conjecture/): A statement someone believes is true but that nobody has proved yet.
- [Countable sets](https://math.wiki/countable-sets/): An infinite set is countable when its members can be lined up in a list that goes on forever.
- [Counterexample](https://math.wiki/counterexample/): One case that fails kills a universal claim.
- [De Morgan's laws for statements](https://math.wiki/de-morgans-laws-logic/): Negating an and turns it into an or, and negating an or turns it into an and.
- [Degree, the handshake lemma and complete graphs](https://math.wiki/degree-of-a-vertex/): How many edge ends meet at a vertex.
- [Direct proof](https://math.wiki/direct-proof/): Start from what is given and reason step by step to the claim.
- [Divisibility as a relation](https://math.wiki/divisibility-as-a-relation/): One integer divides another when the second is an integer multiple of the first.
- [Elements and membership](https://math.wiki/element-and-membership/): An object either belongs to a set or does not, and a set is decided entirely by what belongs to it.
- [Euler paths and circuits](https://math.wiki/condition-for-an-euler-circuit/): A connected graph has a closed route using every edge once exactly when every vertex has even degree.
- [Exclusive or and logic circuits](https://math.wiki/exclusive-or-and-logic-circuits/): Exclusive or is true when exactly one part is true, and gates for and, or and not wire statements into circuits that compute.
- [Existence proof](https://math.wiki/existence-proof/): A proof that at least one object with a stated property exists.
- [Fermat's little theorem](https://math.wiki/fermats-little-theorem/): Raising a number to the power one less than a prime gives 1 modulo that prime, unless the prime divides it.
- [Finding a shortest path](https://math.wiki/shortest-path-in-a-weighted-graph/): Growing a set of settled vertices outward, each time taking the cheapest edge that reaches something new.
- [For all](https://math.wiki/universal-quantifier/): A claim that a statement holds for every object in a stated collection.
- [Generating a sequence from a rule](https://math.wiki/sequence-from-a-rule/): Starting from the given first terms and applying the rule again and again to get the next ones.
- [Graph coloring](https://math.wiki/chromatic-number/): The smallest number of colors a given graph can manage with.
- [Graph isomorphism: when two drawings are the same graph](https://math.wiki/graph-isomorphism/): Two graphs are the same when some matching of their vertices keeps every edge, and a difference in degrees or cycles proves they are not.
- [Graphs, vertices and edges](https://math.wiki/graphs-vertices-and-edges/): A network as a set of dots joined by lines.
- [Hamiltonian paths and cycles](https://math.wiki/hamiltonian-paths-and-cycles/): Routes that visit every vertex once, which unlike Euler routes have no simple test.
- [How induction works: base case, hypothesis and step](https://math.wiki/inductive-step/): The part of an induction that carries the statement from one value to the next one.
- [Infinite sets](https://math.wiki/infinite-sets/): A set whose members never run out, so its size cannot be given by a counting number.
- [Introductory graph theory](https://math.wiki/graph-theory-intro/): Dots and lines, and the puzzles they solve.
- [Introductory number theory](https://math.wiki/number-theory-intro/): Properties of the integers, proved.
- [Logic](https://math.wiki/logic/): Statements, how they combine, and what follows from what.
- [Logic, proof and discrete math](https://math.wiki/logic-and-proof/): How mathematical truth is argued, plus sets, counting structures and a first look at linear algebra.
- [Logical equivalence](https://math.wiki/logical-equivalence/): Two statements are equivalent when their truth tables agree on every single row.
- [Matching two sets one to one](https://math.wiki/matching-sets-one-to-one/): Two sets are the same size when their members pair off with none left over on either side.
- [Modular arithmetic](https://math.wiki/modular-arithmetic/): Clock arithmetic: only the remainder matters.
- [Modular inverses and linear congruences](https://math.wiki/modular-inverse/): A number that multiplies with a given one to give 1 modulo n, which exists only when the two are coprime.
- [Modus ponens, chaining, and a common fallacy](https://math.wiki/modus-ponens/): Given if p then q, and given p, the conclusion q follows.
- [Modus tollens, ruling out, and a common fallacy](https://math.wiki/modus-tollens/): Given if p then q, and given that q is false, it follows that p is false.
- [Negating a quantified statement](https://math.wiki/negate-a-quantified-statement/): The opposite of a for-all claim is a there-exists claim about the opposite, and the other way round.
- [Nested quantifiers](https://math.wiki/nested-quantifiers/): Statements with one quantifier inside another, where changing their order changes what is being claimed.
- [Numbers in other bases](https://math.wiki/convert-to-base-ten/): Multiplying each digit by the value of its place and adding the results.
- [One-to-one, onto and bijective functions](https://math.wiki/injective-surjective-bijective/): A function is one to one when no two inputs share an output, onto when every target is reached, and bijective when it is both.
- [Planar graphs and Euler's formula](https://math.wiki/eulers-formula-for-planar-graphs/): For any connected drawing with no crossings, vertices minus edges plus faces always equals 2.
- [Power set](https://math.wiki/power-set/): The set whose members are all the subsets of a given set, the empty set and the whole set included.
- [Powers modulo n](https://math.wiki/modular-exponentiation/): Reducing at every step, and squaring repeatedly, keeps a huge power manageable.
- [Proof by cases](https://math.wiki/cases-from-parity/): Using the fact that every integer is either even or odd to cover everything with two cases.
- [Proof by contradiction](https://math.wiki/proof-by-contradiction/): Assume the opposite and reach something impossible.
- [Proof by contraposition](https://math.wiki/proof-by-contraposition/): Proving if p then q by instead proving that q being false forces p to be false.
- [Proof by induction](https://math.wiki/proof-by-induction/): Prove the first case, then that each case forces the next.
- [Proof techniques](https://math.wiki/proof-techniques/): The standard ways of showing something must be true.
- [Proving a set identity](https://math.wiki/prove-a-set-identity/): Showing each side is a subset of the other by following one element through the definitions.
- [Proving an if and only if](https://math.wiki/if-and-only-if-proof/): Splitting the claim into two conditionals and proving each direction on its own.
- [Proving statements about divisibility](https://math.wiki/proof-with-divisibility/): Replacing the words a divides b by the equation b equals a times an integer, then working with that.
- [Proving statements about even and odd numbers](https://math.wiki/proof-with-even-and-odd/): Writing an even number as two times an integer and an odd number as two times an integer plus one.
- [Quantifiers](https://math.wiki/quantifiers/): For all, there exists, and how to negate each.
- [Recurrence relations](https://math.wiki/recurrence-relations/): A rule that builds each term of a sequence out of the terms that came before it.
- [Relations and their properties: reflexive, symmetric, antisymmetric, transitive](https://math.wiki/properties-of-relations/): A relation is a set of ordered pairs, and four properties sort relations into equivalences and orders.
- [Relations, equivalence classes and partitions](https://math.wiki/equivalence-classes/): The groups an equivalence relation sorts a set into, where everything inside a group counts as the same.
- [RSA public-key encryption](https://math.wiki/rsa-encryption/): Anyone can lock a message with a public power mod n, but unlocking it needs a private exponent that only the prime factors of n reveal.
- [Saying a conditional without if-then](https://math.wiki/conditional-as-or/): If p then q says the same thing as not p or q, which is what makes its negation easy to write.
- [Set identities: distributive and De Morgan laws](https://math.wiki/distributive-set-laws/): An intersection spread across a union, and a union spread across an intersection.
- [Sets](https://math.wiki/sets/): Collections of things, and the ways they combine.
- [Setting up a recurrence from a counting problem](https://math.wiki/recurrences-from-counting-problems/): Sort the cases by what happens first or last, and the count for n becomes a sum of counts for smaller n.
- [Solving and checking a two-term recurrence](https://math.wiki/solve-a-linear-recurrence/): Trying a power as a guess turns the rule into a quadratic whose roots build the closed formula.
- [Spanning trees and minimum spanning trees](https://math.wiki/minimum-spanning-tree/): Choosing the cheapest set of edges that keeps everything connected, adding safe edges cheapest first.
- [Statements and negation](https://math.wiki/statements-and-negation/): A sentence that is true or false, and its opposite.
- [Strong induction and well-ordering](https://math.wiki/strong-induction/): Assuming the statement for all earlier values, not only the one just before, to prove the next.
- [Subsets, proper subsets and equal sets](https://math.wiki/set-equality/): Two sets are equal when each one is a subset of the other, which is how set identities get proved.
- [Tautologies and contradictions](https://math.wiki/tautology/): A statement that comes out true on every row of its truth table.
- [Testing an argument with a truth table](https://math.wiki/test-an-argument-with-a-truth-table/): Checking every row where the given statements are all true to see whether the conclusion is ever false.
- [The Chinese remainder theorem](https://math.wiki/chinese-remainder-theorem/): Remainders on division by numbers with no common factor pin down a number uniquely up to their product.
- [The division algorithm](https://math.wiki/the-division-algorithm/): Dividing any integer by a positive one leaves a quotient and a remainder smaller than the divisor.
- [The Euclidean algorithm and coprime numbers](https://math.wiki/euclidean-algorithm/): Find a greatest common factor by repeated division, and why it works.
- [The Fibonacci sequence](https://math.wiki/fibonacci-sequence/): The sequence starting 1, 1 in which each later term is the sum of the two before it.
- [The pigeonhole principle](https://math.wiki/pigeonhole-in-proofs/): Choosing what counts as the items and what counts as the containers so the conclusion falls out.
- [There are infinitely many primes](https://math.wiki/infinitude-of-primes/): Euclid's argument, the first great proof by contradiction.
- [There exists, and there is exactly one](https://math.wiki/existential-quantifier/): A claim that at least one object in a stated collection makes a statement true.
- [Trees](https://math.wiki/tree-edge-count/): Every tree with n vertices has exactly n minus 1 edges, which induction on the vertex count shows.
- [Triangular numbers](https://math.wiki/triangular-numbers/): The running totals 1, 3, 6, 10 and onward, each counting the dots in a triangle.
- [Truth table for the conditional](https://math.wiki/truth-table-for-the-conditional/): Showing that if-then counts as false only when the first part is true and the second is false.
- [Truth tables](https://math.wiki/truth-tables/): Every combination of true and false, and what the compound statement does.
- [Two-term recurrences with a repeated root](https://math.wiki/repeated-characteristic-roots/): When the characteristic equation has a double root r, the solutions are built from r to the n and n times r to the n.
- [Uncountable sets and the diagonal argument](https://math.wiki/cantor-diagonal-argument/): Any list of real numbers misses one, built to differ from the first number in the first digit and so on.
- [Union, intersection and complement](https://math.wiki/union-and-intersection/): Either, both, or not.
- [Valid and sound arguments](https://math.wiki/valid-and-sound-arguments/): Valid means the conclusion must follow, and sound adds that the given statements are actually true.
- [Venn diagrams and counting a union](https://math.wiki/inclusion-exclusion-two-sets/): Adding the sizes of two sets counts the overlap twice, so the overlap is subtracted once.
- [Walks, paths, cycles and connected graphs](https://math.wiki/walks-paths-and-cycles/): Routes through a graph, a path repeating no vertex and a cycle returning to where it began.
- [Weighted and directed graphs](https://math.wiki/weighted-graphs/): Graphs whose edges carry a number, such as a distance, a cost or a time.
- [What an argument is](https://math.wiki/logical-argument/): A list of statements taken as given, followed by a conclusion drawn from them.
- [What may be assumed and what must be shown](https://math.wiki/anatomy-of-a-proof/): A proof starts from stated givens and definitions and ends at the exact claim, with nothing else assumed.
- [Where the divisibility tests come from](https://math.wiki/divisibility-tests-from-congruences/): Because 10 leaves remainder 1 modulo 9, a number and its digit sum leave the same remainder.
- [Writing a proof as prose](https://math.wiki/how-a-proof-ends/): A proof stops when the sentence on the page is the claim itself, often marked by a small square or QED.
- [Writing sets: roster, rule and the empty set](https://math.wiki/roster-notation/): Describing a set by listing its members inside braces.

## Measurement

- [An angle is an amount of turn](https://math.wiki/angle-as-an-amount-of-turn/): The size of an angle is how far one arm has to swing round to land on the other
- [Angle measures add, and finding a missing angle](https://math.wiki/angle-measures-add/): An angle split into parts has a measure equal to the parts added together
- [Area and perimeter](https://math.wiki/area-and-perimeter/): Around the edge, and how much surface inside.
- [Area as covering](https://math.wiki/area-as-covering/): How much flat space a shape takes up, found by covering it with copies of a unit shape
- [Area of a regular polygon](https://math.wiki/area-of-a-regular-polygon/): Splitting a regular polygon into equal triangles from the center and adding their areas
- [Area of any polygon by cutting it into triangles and rectangles](https://math.wiki/area-of-polygons-by-decomposing/): Find the area of an irregular polygon by cutting it into rectangles and triangles, or by subtracting pieces from a surrounding rectangle.
- [Area of composite figures and shaded regions](https://math.wiki/area-of-composite-figures/): Split into pieces you know, add or subtract.
- [Area of parallelograms and trapezoids](https://math.wiki/area-of-parallelograms-and-trapezoids/): Cut and slide to make a rectangle.
- [Area of rectangles and squares](https://math.wiki/area-of-rectangles/): Rows of unit squares: length times width, and why.
- [Area of triangles](https://math.wiki/area-of-triangles/): Half a parallelogram.
- [Attributes you can measure](https://math.wiki/attributes-you-can-measure/): Length, area, volume, mass, angle and time are separate things about an object that can each get a number
- [Balancing to compare mass](https://math.wiki/balance-scales-and-equal-mass/): Deciding which of two objects has more matter by seeing which side of a balance goes down
- [Bearings and navigation](https://math.wiki/navigation-with-bearings/): Tracking a position after one or more legs of travel, each given as a bearing and a distance.
- [Benchmarks, estimating and choosing a unit](https://math.wiki/benchmark-referents-for-units/): Keeping a familiar object in mind, like a doorway for two meters, to feel how big a unit is
- [Calendars and dates](https://math.wiki/calendars-and-dates/): How days are grouped into weeks, months and years and named by a date
- [Comparing two objects](https://math.wiki/direct-comparison-of-two-objects/): Deciding which of two things is taller, longer, or holds more by lining them up or pouring from one into the other
- [Comparing with a middle object](https://math.wiki/indirect-comparison-by-transitivity/): Using a third object as a go between to compare two things that cannot be put side by side
- [Conversion factors and dimensional analysis](https://math.wiki/conversion-factors-equal-one/): Twelve inches over one foot equals one, so multiplying by it changes the unit and not the amount
- [Converting between Celsius and Fahrenheit](https://math.wiki/converting-celsius-and-fahrenheit/): Trading one temperature scale for the other with a rule that both stretches and shifts
- [Converting metric units by powers of ten](https://math.wiki/unit-conversion-metric/): Powers of ten: shift the decimal point.
- [Converting square and cubic units](https://math.wiki/converting-square-units/): Changing area units uses the length factor twice, so a hundred per meter becomes ten thousand
- [Counting mixed coins](https://math.wiki/count-mixed-coins/): Adding up a handful of different coins by starting with the biggest values
- [Currency exchange](https://math.wiki/currency-exchange/): Trading an amount of one country's money for another using a rate between them
- [Customary capacity units](https://math.wiki/customary-capacity-units/): Cups, pints, quarts and gallons, the United States units for liquid amounts
- [Customary conversions](https://math.wiki/unit-conversion-customary/): Inches, feet, yards, miles; ounces, pounds; cups to gallons.
- [Customary length units](https://math.wiki/customary-length-units/): Inches, feet, yards and miles, the length units used in the United States
- [Customary weight units](https://math.wiki/customary-weight-units/): Ounces, pounds and tons, the United States units for weight
- [Degrees](https://math.wiki/degrees/): A full turn is 360 of them; a right angle is 90.
- [Density](https://math.wiki/density-as-mass-per-volume/): How much mass is packed into each unit of volume
- [Elapsed time](https://math.wiki/elapsed-time/): How long from then to now.
- [Finding a missing side from the area](https://math.wiki/missing-side-from-area/): Working backwards from a known area and one side to the other side
- [Finding a missing side from the perimeter](https://math.wiki/missing-side-from-perimeter/): Working backwards from a known perimeter and the other sides
- [Flow rate](https://math.wiki/flow-rate/): How much liquid passes a point in each unit of time
- [Iterating a single unit](https://math.wiki/iterating-one-unit/): Measuring by moving one copy of the unit along end to end and keeping count
- [Length and units](https://math.wiki/length-and-units/): Measuring how long, and moving between units.
- [Length on a number line](https://math.wiki/length-on-a-number-line/): The distance between two points on a number line is the difference of the two numbers
- [Liquid volume: liters and milliliters](https://math.wiki/metric-capacity-units/): Milliliters and liters, the metric units for how much liquid a container holds
- [Mass and weight](https://math.wiki/mass-and-weight/): How much matter is in an object, and how hard gravity pulls on it
- [Mass is not the same as weight](https://math.wiki/mass-versus-weight/): Mass does not change when an object is moved, but its weight depends on the pull of gravity there
- [Measurement](https://math.wiki/measurement/): Putting numbers on the world: length, area, volume, angle, time.
- [Measurement error](https://math.wiki/measurement-error/): The gap between a reading and the true amount, which every measurement carries
- [Measurement is approximate: precision of a tool](https://math.wiki/precision-of-a-measuring-tool/): How finely a tool is marked sets the smallest difference it can possibly show
- [Measuring and drawing angles with a protractor](https://math.wiki/read-a-protractor/): Lining the center and base line up with the angle and reading where the other arm crosses
- [Measuring angles](https://math.wiki/angle-measure/): Degrees, and reading a protractor.
- [Measuring area with unit squares](https://math.wiki/tiling-with-unit-squares/): Covering a shape with same size squares that touch without gaps, then counting them
- [Measuring length](https://math.wiki/measure-length/): Lay units end to end; read a ruler.
- [Measuring to the half and quarter inch](https://math.wiki/measure-to-half-and-quarter-inch/): Reading a length that falls between the whole inch marks
- [Measuring tools and reading their scales](https://math.wiki/reading-a-scale-between-marks/): Working out what one space on a marked scale stands for and reading a value that falls inside it
- [Measuring what you cannot reach](https://math.wiki/indirect-measurement-with-shadows/): Finding the height of something too tall to reach by comparing its shadow with a known object
- [Metric length units](https://math.wiki/metric-length-units/): Millimeters, centimeters, meters and kilometers, the length units used almost everywhere else
- [Metric mass units](https://math.wiki/metric-mass-units/): Milligrams, grams, kilograms and tonnes, the metric units for how much matter there is
- [Metric prefixes](https://math.wiki/metric-prefixes/): Kilo, centi, milli and the rest are names for a size, and they mean the same beside any base unit
- [Money](https://math.wiki/money/): Coins, bills, and making change.
- [Money written as decimals](https://math.wiki/money-as-decimals/): A price is a decimal number where the two digits after the point count hundredths of a dollar
- [Morning and afternoon on a twelve hour clock](https://math.wiki/am-and-pm/): The clock face comes round twice a day, so a label says which half of the day you are in
- [Multi-step problems with unit conversions](https://math.wiki/multi-step-conversion-problems/): Real problems that need a unit changed, such as centimeters to meters, partway through several steps.
- [Perimeter](https://math.wiki/perimeter/): Add up the sides.
- [Perimeter versus area](https://math.wiki/perimeter-versus-area/): Perimeter measures the edge in length units and area measures the inside in square units
- [Rates from measurements](https://math.wiki/measured-rates/): Numbers made by dividing one measurement by another, such as speed or density
- [Reading a thermometer](https://math.wiki/reading-a-thermometer/): Taking a temperature off a marked scale where the liquid or the bar stops
- [Scale and size](https://math.wiki/scale-and-size/): What happens to lengths, areas and volumes when a whole object is made bigger or smaller
- [Scale factor multiplies every length](https://math.wiki/scale-factor-and-length/): Doubling a shape doubles every side, every diagonal and the perimeter
- [Scale factor, area and volume](https://math.wiki/scale-factor-and-area/): Doubling every length makes four times the area, because an area needs two lengths
- [Significant figures](https://math.wiki/significant-figures/): The digits in a recorded measurement that the tool actually supported
- [Smaller units give bigger counts](https://math.wiki/unit-size-and-count-are-inverse/): The same object measures as more small units or fewer big ones, so size and count move opposite ways
- [Splitting a figure into rectangles](https://math.wiki/decompose-a-figure-into-rectangles/): Breaking an L shaped or stepped figure into rectangles and adding their areas
- [Splitting a rectangle into rows and columns of squares](https://math.wiki/partition-a-rectangle-into-squares/): A rectangle is cut into rows and columns of same-size squares, and the squares are counted.
- [Story problems with dollars and cents](https://math.wiki/money-story-problems/): Story problems with dollar bills and coins, using the right symbol for dollars and for cents.
- [Story problems with grams, kilograms and liters](https://math.wiki/mass-and-liquid-volume-problems/): One-step story problems with masses and liquid volumes in the same unit, solved with any of the four operations.
- [Surface area, and how it compares with volume](https://math.wiki/surface-area-from-a-net/): Unfolding a solid flat and adding the areas of every face
- [Systems of units](https://math.wiki/unit-systems/): A whole family of units built to fit together, so amounts can be traded between them
- [Telling time](https://math.wiki/tell-time/): Hours, minutes, and the two hands.
- [Telling time to the nearest five minutes](https://math.wiki/tell-time-to-five-minutes/): Reading and writing the time from a clock with hands to the nearest five minutes, counting the minute hand by fives.
- [Temperature](https://math.wiki/temperature/): How hot or cold something is, placed on a numbered scale
- [Temperature scales: Celsius, Fahrenheit and Kelvin](https://math.wiki/celsius-scale/): A scale with zero at the freezing point of water and one hundred at its boiling point
- [Temperatures below zero and temperature change](https://math.wiki/negative-temperatures/): Reading and comparing temperatures under the zero of a scale
- [The customary system](https://math.wiki/the-customary-system/): The inch, pound and gallon family used in the United States, grown out of old trade measures
- [The metric system and SI units](https://math.wiki/the-metric-system-as-a-system/): One base unit for each attribute, with every other unit ten times bigger or smaller
- [Time and money](https://math.wiki/time-and-money/): Reading clocks and counting coins.
- [Time zones](https://math.wiki/time-zones/): The world is cut into zones whose clocks differ by set amounts, mostly whole hours but in some places half or three quarters of an hour
- [Twenty four hour time and crossing midnight](https://math.wiki/twenty-four-hour-time/): Giving every hour of the day its own number so no morning or afternoon label is needed
- [Units of time](https://math.wiki/units-of-time/): Seconds, minutes, hours, days, weeks, months and years, and how they nest inside one another
- [Units, quantities and a sensible level of accuracy](https://math.wiki/reasoning-with-units/): Using units to set up and check a multi-step problem, choosing quantities to model a situation, and reporting a result to a sensible accuracy.
- [Volume and capacity](https://math.wiki/volume-and-capacity/): How much space something takes up, or holds.
- [Volume as filling](https://math.wiki/volume-as-filling/): How much space a solid takes up, found by filling it with copies of a unit cube
- [Volume does not change when it is poured](https://math.wiki/volume-stays-the-same-when-poured/): The same water looks taller in a thin glass and shorter in a wide one, but there is just as much
- [Volume of a box](https://math.wiki/volume-of-a-rectangular-prism/): Finding how much space a rectangular box holds from its three edge lengths
- [Volume of a box with fraction edge lengths](https://math.wiki/volume-with-fractional-edge-lengths/): Packing a box with unit-fraction cubes shows that length times width times height still gives the volume when the edges are fractions.
- [Volume of a solid made of boxes](https://math.wiki/volume-of-a-composite-solid/): Splitting a stepped solid into boxes and adding, or taking a hollow part away
- [Volume with unit cubes](https://math.wiki/volume-with-unit-cubes/): A box holds whole layers of cubes, and each layer holds as many cubes as the base holds squares
- [What a unit is, and why we agree on standard ones](https://math.wiki/what-a-unit-is/): A unit is one chosen amount that you count copies of to get a measurement
- [What measurement is](https://math.wiki/measurement-ideas/): Putting a number on something by picking a unit and counting how many copies of it fit
- [Why the circle area formula works](https://math.wiki/why-the-circle-area-formula-works/): Thin wedges cut from a circle and interlocked make a near rectangle of radius by half the circumference
- [Word problems with distance, time, volume, mass and money](https://math.wiki/measurement-word-problems/): Story problems with measurements, including fractions and decimals, solved with the four operations and converted to one unit.

## Number

- [Absolute value](https://math.wiki/absolute-value-as-distance/): How far a number sits from zero, with no regard for which side.
- [Acting out add and take-away stories within 10](https://math.wiki/story-problems-within-10/): A short story about joining or taking away is acted out with objects or fingers, drawn, and told as an equation.
- [Adding a one-digit number to a two-digit number by making a ten](https://math.wiki/add-within-100-making-a-ten/): Adding a one-digit number to a two-digit number such as 38 + 5 by filling up to the next ten, then adding what is left.
- [Adding and subtracting decimals](https://math.wiki/add-decimals-to-hundredths/): Totals where the places must line up before they are combined.
- [Adding and subtracting integers](https://math.wiki/add-subtract-integers/): Combining signed numbers; subtracting is adding the opposite.
- [Adding and subtracting within 20](https://math.wiki/add-within-20/): Fluent facts, using ten as a landmark.
- [Adding and subtracting within 20 from memory](https://math.wiki/fluency-within-20/): Every sum and difference within 20 is known quickly, by recall or by a fast mental strategy.
- [Adding integers](https://math.wiki/add-integers-with-different-signs/): Totals where the two numbers partly cancel each other out.
- [Adding on an open number line](https://math.wiki/open-number-line-addition/): Jumping by tens and then by ones from one number up to the total.
- [Adding two-digit numbers without regrouping](https://math.wiki/add-two-digit-numbers-no-regrouping/): Tens add to tens and ones add to ones with no bundling needed.
- [Adding within five and ten](https://math.wiki/addition-within-10/): Totals up to ten.
- [Adding, subtracting and multiplying complex numbers](https://math.wiki/multiply-complex-numbers/): Expanding as with binomials, then replacing i squared with negative one.
- [Addition and subtraction](https://math.wiki/addition-and-subtraction/): Putting together and taking apart.
- [Addition and subtraction problems in words](https://math.wiki/addition-and-subtraction-word-problems/): Stories whose structure decides which operation finds the missing amount.
- [Addition as putting together](https://math.wiki/addition-as-joining/): Two amounts combined make a new total.
- [Approximating irrational numbers](https://math.wiki/estimate-a-square-root-between-integers/): Using the perfect squares on either side to bracket the value.
- [Arithmetic](https://math.wiki/arithmetic/): The four operations on whole numbers and decimals, and the rules they obey.
- [Arrays](https://math.wiki/arrays-and-multiplication/): Rows and columns of objects show a product as a rectangle of items.
- [Arrays: rows and columns of objects](https://math.wiki/arrays-of-objects/): Objects set out in equal rows and columns are counted by adding the same number again for each row.
- [Bar models for story problems](https://math.wiki/bar-models-for-story-problems/): A bar split into parts shows how the numbers in a story fit together, which shows whether to add or subtract.
- [Base ten blocks and trading ten for one](https://math.wiki/trade-ten-for-one/): Swapping ten rods for one flat, or one flat back into ten rods.
- [Base, exponent and the value of a power](https://math.wiki/base-and-exponent/): The lower number is the repeated factor and the raised one counts how many times.
- [Calculating in scientific notation](https://math.wiki/multiply-and-divide-in-scientific-notation/): Leading parts combine and the powers of ten add or subtract.
- [Closure](https://math.wiki/closure-property/): A set is closed under an operation when the answer always lands back inside the set.
- [Commutative, associative and distributive properties](https://math.wiki/commutative-property-of-real-numbers/): Addition and multiplication of any real numbers give the same result in either order.
- [Comparing and ordering decimals](https://math.wiki/compare-decimals/): Deciding which decimal is larger by comparing the places from the left.
- [Comparing and ordering rational numbers](https://math.wiki/order-rational-numbers/): Place fractions, decimals and negatives on one line and compare them.
- [Comparing and ordering whole numbers](https://math.wiki/compare-and-order-whole-numbers/): Use place value to decide which number is bigger, and to put numbers in order.
- [Comparing numbers and the symbols >, < and =](https://math.wiki/comparison-symbols/): Written signs that record which of two amounts is larger, or that they match.
- [Comparing numbers in scientific notation](https://math.wiki/compare-numbers-in-scientific-notation/): The power of ten is checked first and the leading part only breaks a tie.
- [Comparing quantities](https://math.wiki/compare-quantities/): Tell which group has more, fewer, or the same.
- [Comparing two numbers to 10](https://math.wiki/compare-numbers-to-10/): Two written numbers from 1 to 10 are compared by counting or matching to say which is greater and which is less.
- [Compensation when adding and subtracting](https://math.wiki/compensation-in-addition/): Adding a round number and then taking the extra back off.
- [Complex numbers](https://math.wiki/complex-numbers/): Numbers built from a square root of negative one.
- [Complex roots come in pairs](https://math.wiki/conjugate-pairs-as-roots/): A polynomial with real coefficients has complex roots that are conjugates of each other.
- [Conjugates and dividing complex numbers](https://math.wiki/divide-complex-numbers/): Multiplying above and below by the conjugate turns the denominator real.
- [Converting between fractions and decimals](https://math.wiki/fraction-to-decimal-by-division/): Dividing the top by the bottom.
- [Counting a large pile by grouping](https://math.wiki/counting-collections-by-grouping/): Making equal groups, usually tens, so a big collection can be counted quickly.
- [Counting a set of objects](https://math.wiki/counting-objects-to-20/): Touching each object once while saying the next counting word.
- [Counting and cardinality](https://math.wiki/counting-and-cardinality/): Numbers name how many. Counting is matching things to number words, one each.
- [Counting by hundreds](https://math.wiki/skip-count-by-hundreds/): Counting in jumps of one hundred.
- [Counting by tens to one hundred](https://math.wiki/skip-count-by-tens/): Saying ten, twenty, thirty and on, one group of ten at a time.
- [Counting by twos, fives, threes and fours](https://math.wiki/skip-count-by-twos/): Counting in jumps of two.
- [Counting on and counting back](https://math.wiki/counting-on-from-a-number/): Starting from a known amount and continuing the count instead of going back to one.
- [Counting on to add and counting back to subtract](https://math.wiki/counting-on-to-add/): Starting at the larger part and counting forward by the smaller one.
- [Counting out a given number of objects](https://math.wiki/count-out-a-given-number/): Asked for a number up to 20, a child counts out exactly that many objects from a larger pile and stops.
- [Counting past one hundred](https://math.wiki/count-to-1000/): Continuing the counting words through the hundreds up to one thousand.
- [Counting to 100 and beyond](https://math.wiki/count-to-100/): Say the number words in order to one hundred, and start counting from any number.
- [Counting up to find a difference](https://math.wiki/count-up-to-subtract/): Starting at the smaller number and adding on until the larger one is reached.
- [Counting, reading and writing numbers to 120](https://math.wiki/numbers-to-120/): Counting on past 100 to 120 from any number, and reading and writing each numeral in that range.
- [Cube roots and higher roots](https://math.wiki/cube-roots/): The number that gives the original when used three times as a factor, which works for negatives too.
- [Decimal arithmetic](https://math.wiki/decimal-arithmetic/): The four operations when numbers have a decimal point.
- [Decimal place value](https://math.wiki/decimal-place-value/): Digits to the right of the decimal point are tenths, hundredths, thousandths.
- [Decimal problems in words](https://math.wiki/decimal-word-problems/): Stories about money, measures and rates that need decimal arithmetic.
- [Decimals on a number line](https://math.wiki/decimals-on-a-number-line/): Placing a decimal between two whole numbers by splitting the gap into ten parts.
- [Dividing a decimal by ten, one hundred or one thousand](https://math.wiki/divide-decimals-by-powers-of-ten/): Each factor of ten moves every digit one place to the right.
- [Dividing by a decimal](https://math.wiki/divide-by-a-decimal-divisor/): Scaling both numbers by a power of ten so the divisor becomes whole.
- [Dividing by a single digit](https://math.wiki/divide-by-one-digit/): Quotients where the divisor is a basic fact away.
- [Dividing by a two-digit number](https://math.wiki/divide-by-two-digit-divisors/): Quotients where each step needs a trial multiple of a larger divisor.
- [Dividing decimals](https://math.wiki/divide-a-decimal-by-a-whole-number/): Quotients where the dividend carries a decimal point.
- [Dividing multi-digit decimals with the standard algorithm](https://math.wiki/divide-decimals-standard-algorithm/): Scale the divisor and dividend by the same power of ten so the divisor is whole, then divide with the long division algorithm.
- [Divisibility and primes](https://math.wiki/divisibility-and-primes/): How whole numbers break into factors.
- [Divisibility rules](https://math.wiki/divisible-by-2-5-and-10/): The last digit alone settles these three.
- [Division facts](https://math.wiki/division-facts/): The quotients that match the known multiplication facts.
- [Doubles and near doubles](https://math.wiki/doubles-facts/): The sums in which both parts are the same number.
- [Doubling, halving and quick products](https://math.wiki/doubling-and-halving/): Reshaping a product into an easier one with the same value.
- [Equal group problems](https://math.wiki/equal-groups-problems/): Stories about a number of groups holding the same amount each.
- [Estimating a huge quantity from small facts](https://math.wiki/fermi-estimation/): Building a rough answer to an enormous question out of a chain of small guesses.
- [Estimating by rounding](https://math.wiki/rounding-to-estimate/): Round first, then compute, to check a result makes sense.
- [Estimating sums and differences](https://math.wiki/estimate-sums-and-differences/): Rounding first so a total or a gap can be judged quickly.
- [Estimation and mental math](https://math.wiki/estimation-and-mental-math/): Getting close fast, and knowing when close is enough.
- [Even and odd numbers](https://math.wiki/pair-up-to-test-even-odd/): Splitting a set into twos to see whether one object is left over.
- [Every number has exactly one prime factorization](https://math.wiki/unique-factorization/): Whatever order the splitting happens in, the same primes appear the same number of times.
- [Expanded form](https://math.wiki/expanded-form/): Writing a number as the sum of what each of its digits is worth.
- [Exponent notation](https://math.wiki/exponent-notation/): A small raised number counts how many times to multiply the base.
- [Exponents and roots](https://math.wiki/exponents-and-roots/): Repeated multiplication, and undoing it.
- [Fact families](https://math.wiki/fact-families/): One trio of numbers gives two addition sentences and two subtraction sentences.
- [Factors and multiples](https://math.wiki/find-all-factors-of-a-number/): Listing the whole numbers that divide it with nothing left over.
- [Facts for seven, eight and nine](https://math.wiki/facts-for-7-8-and-9/): The last and hardest block of the multiplication table.
- [Facts for three, four and six](https://math.wiki/facts-for-3-4-and-6/): The middle block of the multiplication table.
- [Facts for two, five and ten](https://math.wiki/facts-for-2-5-and-10/): The products that come straight out of the easiest skip counts.
- [Finding the missing number in an equation](https://math.wiki/unknown-number-in-an-equation/): The missing number in an addition or subtraction equation, wherever it sits, is found by asking what makes both sides the same.
- [Finding them from prime factorizations](https://math.wiki/gcf-and-lcm-from-prime-factors/): Comparing the powers of each prime in the two factorizations.
- [Friendly numbers and deliberate over and under estimates](https://math.wiki/compatible-numbers/): Swap in numbers that are easy to work with together.
- [Greatest common factor and least common multiple](https://math.wiki/gcf-and-lcm/): The biggest shared factor and the smallest shared multiple.
- [Greatest common factor by listing](https://math.wiki/gcf-by-listing-factors/): Writing out the factors of each number and taking the largest they share.
- [Grouping symbols](https://math.wiki/grouping-symbols/): Brackets that mark a part of an expression to be treated as one number.
- [Higher roots](https://math.wiki/nth-roots/): Roots of any index, and how the index decides whether negatives are allowed.
- [How counting tells how many](https://math.wiki/cardinality-principle/): The final word said while counting a set names the size of the whole set.
- [How many more and how many fewer](https://math.wiki/comparison-word-problems/): Stories that ask for the gap between two amounts.
- [Identity and inverse elements](https://math.wiki/identity-elements/): Zero leaves a sum unchanged and one leaves a product unchanged.
- [Imaginary and complex numbers](https://math.wiki/standard-form-of-a-complex-number/): Every complex number can be written as a real part plus an imaginary part.
- [Integers](https://math.wiki/integers/): Whole numbers and their negatives: the counting numbers extended both ways.
- [Irrational and real numbers](https://math.wiki/irrational-and-real-numbers/): Numbers that are not fractions, and the complete number line they fill in.
- [Irrational numbers](https://math.wiki/square-roots-as-irrational/): Why the square root of 2 cannot be a fraction.
- [Laws of exponents](https://math.wiki/laws-of-exponents/): Multiply powers by adding exponents, divide by subtracting, power of a power by multiplying, and why.
- [Least common multiple by listing](https://math.wiki/lcm-by-listing-multiples/): Writing out multiples of each number and taking the first they share.
- [Listing multiples](https://math.wiki/multiples-of-a-number/): Writing out the numbers a given number divides into.
- [Long division](https://math.wiki/long-division/): Divide place by place, from the biggest, bringing down what is left.
- [Making ten and bridging through ten](https://math.wiki/making-ten-to-add/): Moving part of one addend across to fill a ten before adding the rest.
- [Mental strategies](https://math.wiki/mental-strategies/): Make a ten, double and halve, break apart: computing without paper.
- [Multi-digit addition](https://math.wiki/multi-digit-addition/): Add by place value, regrouping when a column passes nine.
- [Multi-digit multiplication](https://math.wiki/multi-digit-multiplication/): Break numbers by place value, multiply the parts, add them up.
- [Multi-digit subtraction](https://math.wiki/multi-digit-subtraction/): Subtract by place value, regrouping when a column runs short.
- [Multiplication and division](https://math.wiki/multiplication-and-division/): Repeated groups, and splitting into equal groups.
- [Multiplication and division undo each other](https://math.wiki/inverse-relationship-of-multiplication-and-division/): Dividing by a number reverses multiplying by it.
- [Multiplication facts](https://math.wiki/multiplication-facts/): Products through ten times ten, from understanding to fluency.
- [Multiplying a decimal by ten, one hundred or one thousand](https://math.wiki/multiply-decimals-by-powers-of-ten/): Each factor of ten moves every digit one place to the left.
- [Multiplying and dividing by ten, one hundred and one thousand](https://math.wiki/times-ten-shifts-place-value/): Each place is worth ten times the place to its right.
- [Multiplying and dividing integers](https://math.wiki/sign-rules-for-products/): Whether a product is positive or negative depends on how many factors are negative.
- [Multiplying and dividing powers with the same base](https://math.wiki/product-of-powers/): The exponents add because the repeated factors are simply lined up together.
- [Multiplying and dividing rational numbers](https://math.wiki/multiply-and-divide-rational-numbers/): Products and quotients of signed fractions and decimals.
- [Multiplying by tens and hundreds](https://math.wiki/multiply-by-multiples-of-ten/): Products such as 40 times 600, built from a basic fact and a shift of places.
- [Multiplying decimals](https://math.wiki/multiply-a-decimal-by-a-whole-number/): Products such as a price repeated several times.
- [Multiplying multi-digit decimals with the standard algorithm](https://math.wiki/multiply-decimals-standard-algorithm/): Multiply as if the numbers were whole, then place the decimal point by counting decimal places and check with an estimate.
- [Negative fractions and decimals on the number line](https://math.wiki/negative-fractions-and-decimals/): Parts of a whole can sit on the negative side of zero too.
- [Negative numbers](https://math.wiki/negative-numbers/): Numbers less than zero, read as opposites on the number line.
- [Number](https://math.wiki/number/): What numbers are, and what you do with them: counting, place value, the four operations, negatives, fractions as numbers, and the real line.
- [Number lines with different scales](https://math.wiki/number-line-intervals/): Reading a line whose tick marks step by two, five, ten or one hundred.
- [Number names, numerals and zero](https://math.wiki/number-names-and-numerals/): The spoken word, the written symbol and a pile of objects can all name the same amount.
- [Numbers in other bases](https://math.wiki/other-number-bases/): Place value still works when the bundling size is something other than ten.
- [Numbers to one million](https://math.wiki/numbers-to-one-million/): Reading, writing and building numbers in the thousands and beyond.
- [Numerical expressions](https://math.wiki/numerical-expressions/): A written string of numbers and operations that names a single value.
- [One more and one less](https://math.wiki/one-more-one-less/): Adding or removing a single object moves the count one step along the number words.
- [One-to-one correspondence](https://math.wiki/one-to-one-correspondence/): Each object gets exactly one count word. The last word said is how many.
- [Operations with rational numbers](https://math.wiki/add-and-subtract-rational-numbers/): Combining signed fractions and decimals.
- [Opposites and zero pairs](https://math.wiki/opposites-and-zero-pairs/): Every number has a partner the same distance the other side of zero, and the pair adds to nothing.
- [Order of operations](https://math.wiki/order-of-operations/): Grouping first, then powers, then multiply and divide, then add and subtract, and why there has to be a rule.
- [Order of operations with negative numbers](https://math.wiki/order-of-operations-with-negatives/): Expressions mixing signs, powers and brackets.
- [Order of operations with powers](https://math.wiki/order-of-operations-with-exponents/): Expressions where powers are settled before multiplying and dividing.
- [Ordinal numbers](https://math.wiki/ordinal-numbers/): First, second and third describe position in a line rather than how many there are.
- [Partial products](https://math.wiki/partial-products/): Breaking both factors by place, multiplying every pair, then adding the pieces.
- [Partial quotients](https://math.wiki/partial-quotients/): Taking away easy multiples of the divisor and adding up how many were taken.
- [Parts, wholes and number bonds](https://math.wiki/number-bonds-to-10/): Knowing every pair of whole numbers that adds to ten.
- [Patterns in the multiplication table](https://math.wiki/multiplication-table-patterns/): Rows and diagonals in the table repeat in ways that make facts predictable.
- [Perfect squares and perfect cubes](https://math.wiki/perfect-squares-and-perfect-cubes/): Numbers that are a whole number to the second or third power.
- [Perfect, abundant, deficient and amicable numbers](https://math.wiki/amicable-numbers/): A pair of numbers where each one equals the sum of the other one's smaller factors.
- [Place value](https://math.wiki/place-value/): Where a digit sits decides what it is worth: ones, tens, hundreds.
- [Placing a number on a number line](https://math.wiki/locate-numbers-on-a-number-line/): Finding where a whole number sits on a marked line.
- [Placing and comparing integers on a number line](https://math.wiki/integers-on-a-number-line/): Placing positive and negative whole numbers on a line that runs both ways from zero.
- [Powers of powers, products and quotients](https://math.wiki/power-of-a-power/): The exponents multiply because a group of factors is repeated.
- [Powers of ten](https://math.wiki/powers-of-ten/): Ten multiplied by itself, which is what each place value column is worth.
- [Prime and composite numbers](https://math.wiki/composite-numbers/): Whole numbers with more than two factors.
- [Prime factorization](https://math.wiki/factor-trees/): Splitting a number into factors again and again until only primes remain.
- [Problems using several operations](https://math.wiki/multi-step-problems-with-four-operations/): Stories that need two or more of the four operations in a chosen order.
- [Problems with the start, change or result missing](https://math.wiki/change-unknown-problems/): Stories that give a start and an end and ask what happened in between.
- [Problems with the starting amount missing](https://math.wiki/start-unknown-problems/): Stories that give the change and the end and ask where things began.
- [Problems with the total missing](https://math.wiki/result-unknown-problems/): Stories that give both parts and ask for the whole.
- [Properties of addition](https://math.wiki/commutative-property-of-addition/): Two numbers added in either order give the same total.
- [Properties of multiplication](https://math.wiki/commutative-property-of-multiplication/): Turning an array on its side gives the same number of items.
- [Properties of real numbers](https://math.wiki/properties-of-real-numbers/): Commutative, associative, distributive, identities and inverses, named and used.
- [Rational exponents](https://math.wiki/unit-fraction-exponents/): A power of one half means a square root, and one third means a cube root.
- [Rational numbers](https://math.wiki/rational-numbers/): Any number that can be written as one integer over another.
- [Rational numbers in real situations](https://math.wiki/rational-numbers-in-context/): Stories about balances, temperatures and changes that use signed fractions and decimals.
- [Reading and writing numbers to 1,000](https://math.wiki/numbers-to-1000/): Hundreds, tens, ones, in words, digits and expanded form.
- [Regrouping in addition](https://math.wiki/regrouping-in-addition/): Ten ones become one ten so that no place ever holds more than nine.
- [Regrouping in subtraction](https://math.wiki/regrouping-in-subtraction/): One ten is broken into ten ones when a place does not have enough.
- [Remainders and what they mean](https://math.wiki/interpret-remainders-in-context/): Deciding whether a leftover is dropped, rounded up or reported as it stands.
- [Rounding](https://math.wiki/rounding/): Replace a number with a nearby one that is easier to work with.
- [Rounding decimals](https://math.wiki/rounding-decimals/): Naming the nearest tenth, hundredth or whole number.
- [Rounding to any place, including decimals](https://math.wiki/round-to-any-place/): Choosing the nearest number that is zero from a named place onward.
- [Rounding to the nearest ten or hundred](https://math.wiki/rounding-to-the-nearest-ten/): Naming the closest multiple of ten.
- [Saying the counting words to 20](https://math.wiki/rote-counting-to-20/): Reciting the number words in order, before they are attached to any objects.
- [Scientific notation](https://math.wiki/scientific-notation/): Very big and very small numbers as a number between 1 and 10 times a power of ten.
- [Skip counting](https://math.wiki/skip-counting/): Counting in equal jumps rather than one at a time.
- [Sorting a number into its sets](https://math.wiki/classify-real-numbers/): Deciding whether a number is whole, an integer, rational or irrational.
- [Square and cube roots](https://math.wiki/square-and-cube-roots/): The number that, squared or cubed, gives you back the one you started with.
- [Square roots](https://math.wiki/roots-of-perfect-squares/): Reading a root straight off the squares that are already known.
- [Subitizing and ten frames](https://math.wiki/subitizing-small-groups/): Naming the size of a very small group instantly.
- [Subtracting across zeros](https://math.wiki/subtract-across-zeros/): Differences where the places being borrowed from hold nothing.
- [Subtracting integers](https://math.wiki/subtract-integers-as-adding-the-opposite/): Taking a number away gives the same result as adding its opposite.
- [Subtracting multiples of ten](https://math.wiki/subtract-multiples-of-ten/): Taking away whole tens from a multiple of ten, such as 70 minus 30, by counting tens.
- [Subtracting three-digit numbers by place value, including across zero](https://math.wiki/subtract-within-1000/): Subtracting within 1,000 hundreds from hundreds, tens from tens and ones from ones, trading a hundred or a ten when needed, even across a zero.
- [Subtracting two-digit numbers without regrouping](https://math.wiki/subtract-two-digit-no-regrouping/): Each place has enough to take from, so nothing is unbundled.
- [Subtracting within ten](https://math.wiki/subtraction-within-10/): Taking away from ten or less.
- [Subtraction as an unknown addend](https://math.wiki/subtraction-as-unknown-addend/): Taking away and finding the missing part are the same question.
- [Subtraction as taking away](https://math.wiki/subtraction-as-taking-away/): Removing part of an amount leaves the rest.
- [Subtraction as the difference](https://math.wiki/subtraction-as-comparison/): How much more one amount is than another.
- [Tally marks and Roman numerals](https://math.wiki/roman-numerals/): Letters that stand for fixed values and are added or subtracted according to position.
- [Teen numbers as ten and some more](https://math.wiki/teen-numbers-as-ten-and-ones/): Thirteen is one ten and three ones, which the English number word hides.
- [Ten frames](https://math.wiki/ten-frames/): A two by five grid showing a number as how full it is and how far from ten.
- [Ten more and ten less](https://math.wiki/add-ten-to-a-number/): Adding ten makes one more ten: the ones stay the same and the tens go up by one, so 36 becomes 46.
- [Tens and ones](https://math.wiki/tens-and-ones/): A two-digit number is some tens and some ones.
- [Tenths and hundredths](https://math.wiki/tenths-and-hundredths/): The first two places right of the point, worth one tenth and one hundredth of a whole.
- [Terminating and repeating decimals](https://math.wiki/repeating-decimal-notation/): A line drawn above the digits that repeat forever.
- [Tests for three and nine](https://math.wiki/divisible-by-3-and-9/): The sum of the digits settles these two.
- [The area model](https://math.wiki/area-model-of-multiplication/): A rectangle whose side lengths are the factors and whose area is the product.
- [The complex plane, modulus and argument](https://math.wiki/plot-complex-numbers/): The real part reads across and the imaginary part reads up.
- [The distributive property](https://math.wiki/distributive-property/): Multiply a sum by multiplying each part: the idea behind every multiplication method.
- [The equal sign means the same amount](https://math.wiki/equal-sign-means-same-amount/): Both sides of an equation name one amount, so the sign is not an order to write an answer.
- [The number line](https://math.wiki/number-line/): Numbers as positions on a line, evenly spaced, growing to the right.
- [The partial sums method](https://math.wiki/partial-sums-method/): Adding each place on its own and then combining the parts.
- [The place value chart](https://math.wiki/place-value-chart/): A table of columns in which a digit's column decides what it is worth.
- [The real number line](https://math.wiki/real-number-line/): Every point on the line is a real number; rationals and irrationals fill it together.
- [The sieve of Eratosthenes](https://math.wiki/sieve-of-eratosthenes/): Crossing out multiples in order so only the primes are left standing.
- [The standard addition algorithm](https://math.wiki/standard-algorithm-for-addition/): Adding column by column from the right and carrying into the next place.
- [The standard long division algorithm](https://math.wiki/standard-long-division-algorithm/): Working one place at a time, recording a digit of the quotient at each step.
- [The standard multiplication algorithm](https://math.wiki/standard-algorithm-for-multiplication/): Stacking the factors and recording one shortened partial product per digit.
- [The standard subtraction algorithm](https://math.wiki/standard-algorithm-for-subtraction/): Subtracting column by column from the right, unbundling where needed.
- [Thousandths: reading, writing and expanded form](https://math.wiki/thousandths/): Places further to the right, each worth one tenth of the place before it.
- [Times as many](https://math.wiki/multiplicative-comparison/): Describing one amount as a number of copies of another.
- [Turning a repeating decimal into a fraction](https://math.wiki/repeating-decimal-to-fraction/): Multiply a repeating decimal by a power of ten and subtract to cancel the repeating tail, leaving a fraction.
- [Two-step addition and subtraction problems](https://math.wiki/two-step-word-problems-add-subtract/): Stories that need one answer worked out before the next.
- [What division means](https://math.wiki/division-as-sharing/): How many in each group, or how many groups: two questions, one operation.
- [What makes a number rational](https://math.wiki/what-makes-a-number-rational/): A number that can be written as one integer divided by another.
- [What multiplication means](https://math.wiki/multiplication-as-groups/): Three groups of four, as an array, as repeated addition, as area.
- [Where the point goes in a product](https://math.wiki/decimal-places-in-a-product/): The number of decimal places in a product comes from the places in the factors.
- [Which fractions give terminating decimals](https://math.wiki/which-fractions-terminate/): A fraction in lowest terms stops exactly when its denominator has only twos and fives as prime factors.
- [Writing and reading numerical expressions from words](https://math.wiki/write-expressions-from-words/): Turning a phrase like add 8 and 7, then multiply by 2 into an expression, and reading an expression without working it out.
- [Writing and reading scientific notation](https://math.wiki/write-in-scientific-notation/): Recording a number as one digit, a decimal part, and a power of ten.
- [Writing the numerals 0 to 10](https://math.wiki/write-numerals-0-to-10/): Forming the written digits that stand for the first ten amounts.
- [Zero and negative exponents](https://math.wiki/zero-exponent/): The rule for dividing powers forces the zero exponent to stand for one.

## Probability

- [Arrangements and factorials](https://math.wiki/permutations-of-all-objects/): The number of orders for a whole set of different objects is the factorial of how many there are.
- [At least one problems](https://math.wiki/probability-that-none-occur/): Multiplying the chances of each separate failure to get the chance that every attempt misses.
- [Basic probability](https://math.wiki/basic-probability/): Counting what can happen, and the fraction of it you want.
- [Bayes' rule](https://math.wiki/bayes-rule-with-a-tree-diagram/): Finding the wanted branch probability and dividing it by the total probability of all matching branches.
- [Chance in words and the 0 to 1 scale](https://math.wiki/chance-in-words/): Saying how likely something is using impossible, unlikely, even chance, likely and certain, before any number is used.
- [Combinations](https://math.wiki/combination-notation/): The symbol for combinations records how many items there are and how many are being chosen.
- [Combinations with repetition: stars and bars](https://math.wiki/combinations-with-repetition/): Choosing k items from n kinds with repeats allowed is the same as placing k stars among n minus 1 bars.
- [Combining random variables](https://math.wiki/variances-add-for-independent-variables/): If two random variables do not affect each other, the variance of their sum is the sum of the variances.
- [Complements and probabilities that add to one](https://math.wiki/complement-notation/): Writing the event that something does not happen, and reading the rule that the two probabilities add to one.
- [Compound events](https://math.wiki/compound-events/): Two or more things happening, together or in sequence.
- [Conditional probability](https://math.wiki/conditional-probability/): The probability of A, given that B happened.
- [Counting](https://math.wiki/counting-principles/): How many ways, without listing them all.
- [Counting with restrictions](https://math.wiki/counting-with-a-position-restriction/): Filling the position that carries the rule first, then counting the free positions that remain.
- [Cumulative probability](https://math.wiki/probability-of-an-interval-of-values/): Finding the chance the variable lands between two values by subtracting cumulative probabilities.
- [Dependent events and drawing without replacement](https://math.wiki/dependent-events/): The first outcome changes the odds of the second; multiply with the updated probability.
- [Designing a simulation](https://math.wiki/design-a-simulation/): Setting up a chance process whose outcomes match the real situation being studied.
- [Expected value](https://math.wiki/expected-value-as-a-weighted-mean/): Each value is weighted by how likely it is, so likelier values pull the average toward themselves.
- [Experimental probability and relative frequency](https://math.wiki/experimental-probability-from-data/): Working out a probability by counting how often something actually happened in recorded trials.
- [Finding a conditional probability](https://math.wiki/conditional-probability-formula/): Dividing the chance that both happen by the chance of the event being conditioned on.
- [Geometric probability](https://math.wiki/probability-from-an-area-model/): Comparing the area of a target region with the area of the whole region to get a probability.
- [Inclusion-exclusion for three or more sets, and derangements](https://math.wiki/inclusion-exclusion-principle/): Add the single sets, subtract the overlaps of pairs, add back the overlaps of triples, and so on; the same count gives the arrangements that leave nothing in its place.
- [Independent events and the multiplication rule](https://math.wiki/multiplication-rule-for-independent-events/): When one event does not change the chance of the other, multiply the two probabilities together.
- [Joint and marginal probability](https://math.wiki/joint-and-marginal-probability/): A joint probability sits inside a table cell and a marginal probability sits in a row or column total.
- [Linear transformations of a random variable](https://math.wiki/adding-a-constant-shifts-the-mean/): Adding the same amount to every value moves the mean by that amount and leaves the spread alone.
- [Listing a sample space: lists and tables](https://math.wiki/list-a-sample-space/): Writing out every possible outcome of an experiment once each, in an order that shows none was missed.
- [Mean and standard deviation of a binomial](https://math.wiki/binomial-mean/): The expected number of successes is the number of trials times the chance of success on one trial.
- [Mean and standard deviation of a geometric random variable](https://math.wiki/geometric-mean-number-of-trials/): On average the first success arrives after one over p tries, with a standard deviation of the square root of one minus p, over p.
- [Mutually exclusive events and the addition rule](https://math.wiki/mutually-exclusive-events/): Cannot both happen; their probabilities add.
- [Odds](https://math.wiki/odds-in-favor-and-against/): Odds in favor put the successes first and odds against put the failures first, so the two are reversed.
- [Pascal's triangle](https://math.wiki/pascals-triangle-and-combinations/): Every entry of the triangle is a combination count, and the building rule is a fact about choosing.
- [Permutations](https://math.wiki/permutations-of-some-objects/): Counting the ordered ways to pick and place only part of a set, leaving the rest unused.
- [Probability](https://math.wiki/probability/): Measuring how likely something is.
- [Probability models: equally likely or not](https://math.wiki/probability-models/): A probability model is a sample space together with a probability attached to every outcome in it.
- [Probability of an event](https://math.wiki/probability-of-an-event-with-several-outcomes/): Counting how many outcomes make the event happen and comparing that to the total number of outcomes.
- [Probability with counting formulas](https://math.wiki/hypergeometric-counting-problems/): Counting how many ways to draw a set number of each type when items are taken without replacement.
- [Random variables](https://math.wiki/random-variables/): A number attached to each outcome, and the distribution of those numbers.
- [Random variables and probability distributions](https://math.wiki/probability-distribution-table/): Laying out the values in one row and their probabilities in the next so the whole distribution is visible.
- [Running a simulation and estimating a probability](https://math.wiki/estimate-a-probability-from-trials/): Dividing the number of trials that gave the outcome by the number of trials run.
- [Sample spaces, outcomes and events](https://math.wiki/outcome-versus-event/): An outcome is one single result, while an event is any collection of outcomes you care about.
- [Shape of a binomial and the normal approximation](https://math.wiki/normal-approximation-to-a-binomial/): When the counts are large enough, the binomial distribution is close enough to a normal curve to use it.
- [The binomial distribution](https://math.wiki/binomial-distribution/): Count successes in a fixed number of independent tries.
- [The binomial probability formula](https://math.wiki/binomial-probability-of-exactly-k/): Putting the number of trials, the success chance and the wanted count into the binomial formula.
- [The binomial setting](https://math.wiki/recognize-a-binomial-setting/): Testing a situation against each of the four conditions before any binomial formula is used.
- [The binomial theorem](https://math.wiki/binomial-coefficients/): The numbers multiplying each term of an expanded binomial are combination counts.
- [The fundamental counting principle](https://math.wiki/counting-choices-in-stages/): Multiplying the number of options at each step to count all the ways a whole sequence can turn out.
- [The general addition rule](https://math.wiki/why-subtract-the-overlap/): Outcomes in both events get counted twice when the probabilities are added, so one copy is removed.
- [The general multiplication rule](https://math.wiki/general-multiplication-rule/): The chance of both events is the chance of the first times the chance of the second given the first.
- [The geometric distribution](https://math.wiki/geometric-distribution-formula/): The chance the first success lands on a given trial is a run of failures followed by one success.
- [The law of large numbers and the gambler's fallacy](https://math.wiki/law-of-large-numbers/): As the number of trials grows, the relative frequency of an outcome settles near its true probability.
- [The law of total probability](https://math.wiki/law-of-total-probability/): Splitting a population into non-overlapping groups and adding each group's share of the event.
- [Tree diagrams and organized lists](https://math.wiki/tree-diagram-probabilities-on-branches/): Writing a probability on every branch and multiplying along a path to get the chance of that whole path.
- [Union and intersection of events](https://math.wiki/union-and-intersection-of-events/): The union is everything in either event and the intersection is only what is in both.
- [Variance and standard deviation of a random variable](https://math.wiki/variance-of-a-discrete-random-variable/): Weighting each squared distance from the mean by its probability and adding the results.
- [What randomness means](https://math.wiki/what-randomness-means/): An outcome is random when nobody can say which one will happen, even though the list of possible ones is known.

## Statistics

- [Association in a two way table and segmented bar graphs](https://math.wiki/association-in-a-two-way-table/): Two variables are associated when the conditional shares differ from row to row.
- [Bar graphs and pictographs](https://math.wiki/bar-graphs-and-pictographs/): One bar or picture per category.
- [Bias](https://math.wiki/bias-in-sampling/): Ways a sample systematically misses the population.
- [Bias and variability of an estimator](https://math.wiki/unbiased-estimator/): A statistic whose sampling distribution is centered at the parameter it is meant to estimate.
- [Bivariate data](https://math.wiki/bivariate-data/): Two variables at once: is there a relationship, and how strong?
- [Blocking: randomized block and matched pairs designs](https://math.wiki/blocking-in-an-experiment/): Grouping similar units first so a known source of variation does not hide the treatment effect.
- [Box plots](https://math.wiki/box-plots/): Five numbers, one picture: minimum, quartiles, maximum.
- [Box plots that show outliers](https://math.wiki/modified-box-plot/): A box plot whose whiskers stop at the last value inside the fence, with far values plotted alone.
- [Cautions about significance](https://math.wiki/statistical-vs-practical-significance/): A difference can be too small to matter and still be detected by a large enough sample.
- [Center and spread](https://math.wiki/center-and-spread/): One number for the middle, one number for how spread out.
- [Checking whether data look normal](https://math.wiki/assess-normality-with-a-plot/): Comparing the data against what a normal model would give, by graph rather than by assumption.
- [Chi-square tests](https://math.wiki/chi-square-tests/): Do observed counts match what a claim predicts?
- [Choosing a display for a question](https://math.wiki/choose-a-display/): Picking the graph that answers the question asked, given whether the data are categories or numbers.
- [Choosing the right procedure](https://math.wiki/choose-an-inference-procedure/): Matching the number of samples, the kind of variable and the design to the procedure that fits.
- [Circle graphs](https://math.wiki/read-a-circle-graph/): Turning a wedge back into a percent and then into a count of the group.
- [Cluster samples and choosing a design](https://math.wiki/cluster-sample/): Splitting the population into naturally occurring groups and taking every member of a few of them.
- [Collecting data with a survey](https://math.wiki/collect-data-with-a-survey/): Asking the same question of many people and recording every answer.
- [Comparing box plots](https://math.wiki/compare-box-plots/): Putting two boxes on the same scale to compare center, spread and skew.
- [Comparing two distributions](https://math.wiki/compare-two-distributions/): Saying how two groups differ in center, spread and shape, on the same scale, with units.
- [Comparing two populations: the gap between means in MADs](https://math.wiki/compare-populations-with-mad/): Measure how far apart two distributions are by expressing the difference of their means as a multiple of their mean absolute deviation.
- [Comparing two treatments by re-randomizing](https://math.wiki/compare-treatments-by-rerandomizing/): Deciding whether a difference between two treatment groups is bigger than chance assignment alone would produce, by shuffling the data many times.
- [Completely randomized design and replication](https://math.wiki/completely-randomized-design/): Assigning every unit to a treatment purely at random, with no grouping first.
- [Computing a standard deviation, sample and population](https://math.wiki/compute-standard-deviation-by-hand/): Working through the distances from the mean, squaring, averaging and taking the root.
- [Conditional relative frequencies](https://math.wiki/conditional-relative-frequency/): The share of one row or one column that lands in a given cell.
- [Conditions for one mean](https://math.wiki/conditions-for-one-mean-inference/): A random sample, small against the population, from a population that is normal or a sample large enough.
- [Conditions for one proportion](https://math.wiki/conditions-for-one-proportion-inference/): A random sample, a sample small against the population, and enough expected successes and failures.
- [Conditions for regression inference](https://math.wiki/conditions-for-regression-inference/): A true linear relationship, independent observations, equal spread of residuals, and residuals near normal.
- [Confidence intervals](https://math.wiki/confidence-intervals/): A range of plausible values for the population parameter, and what the percent really means.
- [Confidence level and interpreting an interval](https://math.wiki/confidence-level/): The share of all possible samples for which this method would capture the true parameter.
- [Control groups, placebos and blinding](https://math.wiki/control-group/): A group receiving no treatment or the usual one, to compare the others against.
- [Convenience and voluntary response samples](https://math.wiki/convenience-sample/): Taking whoever is easiest to reach, which usually leaves out whole parts of the population.
- [Correlation](https://math.wiki/correlation/): Direction and strength of a linear relationship, and the number r.
- [Correlation from standardized values](https://math.wiki/r-from-standardized-values/): Finding r as the average product of the two variables' z-scores.
- [Correlation is not causation: lurking and confounding variables](https://math.wiki/confounding-variables/): When two possible causes change together so their effects cannot be told apart.
- [Counting each group and ordering groups by how many](https://math.wiki/count-and-order-categories/): After sorting, each group is counted and the groups are put in order from most to fewest.
- [Critical value](https://math.wiki/critical-value/): The multiplier from the reference distribution that turns a standard error into a margin of error.
- [Cumulative frequency graphs and percentiles](https://math.wiki/cumulative-frequency-graph/): A rising graph showing how many values fall at or below each point.
- [Data and variables](https://math.wiki/data-basics/): Recorded facts about a group of individuals, where a variable is one thing recorded about each individual.
- [Data displays](https://math.wiki/data-displays/): Pictures of data: which one to draw, and how to read it.
- [Data tables, tallies and frequency tables](https://math.wiki/frequency-tables/): A table listing each value or category beside the number of times it occurred.
- [Deciding and concluding from a p-value](https://math.wiki/reject-or-fail-to-reject/): Comparing the p-value with the chosen cutoff and saying which hypothesis the evidence supports.
- [Density curves: area, mean and median](https://math.wiki/density-curves/): A smooth curve standing in for a histogram, with total area one under it.
- [Describing a distribution](https://math.wiki/describe-a-distribution-in-context/): Reporting shape, center, spread and unusual features together, in the words of the setting.
- [Describing a scatter plot](https://math.wiki/describe-a-scatter-plot/): Saying the direction, the form, the strength and any unusual points, in context.
- [Distributions](https://math.wiki/distributions/): Models for how values are spread, and the normal curve above all.
- [Dot plots](https://math.wiki/dot-plots-and-line-plots/): One dot per data point, stacked over its value.
- [Expected counts and the chi-square statistic](https://math.wiki/chi-square-statistic/): Adding up, over all cells, the squared gap between observed and expected divided by expected.
- [Experimental design](https://math.wiki/experimental-design/): Control, randomization, replication, blocking.
- [Finding the median](https://math.wiki/median-of-an-even-list/): When two values sit in the middle, the median is halfway between them.
- [Fitting linear and exponential models with technology](https://math.wiki/fit-a-function-to-data-with-technology/): Using a calculator or spreadsheet to fit a least-squares line or an exponential curve to data and reading what its numbers say.
- [Goodness of fit test](https://math.wiki/chi-square-goodness-of-fit-test/): Testing whether one categorical variable follows a claimed set of proportions.
- [Histograms](https://math.wiki/histograms/): Group values into bins, then bar the counts.
- [Hypothesis testing](https://math.wiki/hypothesis-testing/): Assume nothing is going on, see how surprising the data would be.
- [Individuals, variables and units](https://math.wiki/individuals-and-variables/): The rows of a data table are the individuals studied and the columns are the variables recorded about them.
- [Inference for means](https://math.wiki/inference-for-means/): Estimating and testing claims about the average value in a population.
- [Inference for proportions](https://math.wiki/inference-for-proportions/): Estimating and testing claims about the share of a population with some feature.
- [Inference for slopes](https://math.wiki/inference-for-slopes/): Deciding what a sample regression line says about the true relationship in the population.
- [Interpreting a residual](https://math.wiki/interpret-a-residual/): Reading a residual as how far above or below the prediction an actual value fell.
- [Interpreting a standard deviation](https://math.wiki/interpret-standard-deviation-in-context/): Saying in the setting's own units how far a typical value sits from the mean.
- [Interpreting the slope and intercept](https://math.wiki/interpret-the-slope-of-a-regression-line/): Saying what a one unit change in the explanatory variable predicts about the response, in context.
- [Interval for a difference of means](https://math.wiki/two-sample-t-interval-for-a-difference-of-means/): Estimating how far apart two population averages are.
- [Interval for a difference of proportions](https://math.wiki/two-sample-z-interval-for-a-difference-of-proportions/): Estimating how far apart two population shares are.
- [Interval for a slope](https://math.wiki/t-interval-for-a-slope/): Estimating the true rate of change as the fitted slope give or take a margin of error.
- [Joint and marginal frequencies](https://math.wiki/marginal-frequencies/): The row and column totals of a two way table, each describing one variable on its own.
- [Least squares regression](https://math.wiki/least-squares-regression/): The line that makes the squared vertical misses as small as possible.
- [Line graphs](https://math.wiki/read-a-line-graph/): Getting a value at a given time, and saying where the line rises, falls or stays flat.
- [Line of best fit](https://math.wiki/line-of-best-fit/): Eyeball a line through the cloud and use it to predict.
- [Line plots with fractions](https://math.wiki/line-plots-with-fractions/): A dot plot whose number line is marked in halves, quarters or eighths.
- [Log-log transforms and power models](https://math.wiki/log-transform-of-both-variables/): Plotting log against log to straighten a power relationship.
- [Making a dot plot](https://math.wiki/make-a-dot-plot/): Drawing a number line and stacking one mark above each value every time it appears.
- [Making a histogram and choosing the bin width](https://math.wiki/make-a-histogram/): Grouping values into equal intervals and drawing a bar for the count in each.
- [Making and reading box plots](https://math.wiki/make-a-box-plot/): Drawing a box from the lower to the upper quartile with whiskers out to the extremes.
- [Making and reading residual plots](https://math.wiki/residual-plots/): Plotting each residual against the explanatory value to see what the line missed.
- [Making bar graphs](https://math.wiki/make-a-bar-graph/): Turning a frequency table into bars of the right height with a labeled scale.
- [Mean](https://math.wiki/mean/): Add them up, share them out evenly.
- [Mean absolute deviation](https://math.wiki/mean-absolute-deviation/): Average distance from the mean.
- [Mean as balance point](https://math.wiki/mean-as-balance-point/): The place on the number line where the distances above and the distances below cancel out.
- [Mean as fair share: computing the mean](https://math.wiki/mean-as-fair-share/): The amount each individual would get if the total were shared out evenly.
- [Mean from a frequency table and weighted mean](https://math.wiki/mean-from-a-frequency-table/): Finding the mean when each value carries a count instead of being listed one at a time.
- [Mean, median and skew: choosing the center](https://math.wiki/mean-and-median-in-a-skewed-distribution/): The long tail pulls the mean toward it while the median stays near the bulk of the data.
- [Median and mode](https://math.wiki/median-and-mode/): The middle value, and the most common one.
- [Misleading graphs](https://math.wiki/misleading-graphs/): Displays whose scale, cut axis or picture sizes make a difference look bigger or smaller than it is.
- [Mode and bimodal data](https://math.wiki/mode-and-bimodal-data/): The most common value, and what it means when two separate values are both common.
- [Normal calculations: areas and percentiles](https://math.wiki/area-under-a-normal-curve/): Turning a range of values into z-scores and reading off the share of the distribution that falls there.
- [Null and alternative hypotheses](https://math.wiki/null-and-alternative-hypotheses/): A claim of no effect set against the claim the researcher is investigating, both about a parameter.
- [Observational studies and experiments](https://math.wiki/observational-vs-experimental/): Watching versus assigning, and what each can and cannot conclude.
- [One sample interval for a mean and sample size](https://math.wiki/one-sample-t-interval-for-a-mean/): Estimating a population average as the sample average give or take a margin of error.
- [One sample interval for a proportion and sample size](https://math.wiki/one-sample-z-interval-for-a-proportion/): Estimating a population share as the sample share give or take a margin of error.
- [One sample test for a mean](https://math.wiki/one-sample-t-test-for-a-mean/): Testing a claimed population average against what the sample found.
- [One sample test for a proportion](https://math.wiki/one-sample-z-test-for-a-proportion/): Testing a claimed population share against what the sample found.
- [Outliers and the 1.5 IQR rule](https://math.wiki/iqr-outlier-rule/): Marking as an outlier any value more than one and a half interquartile ranges beyond a quartile.
- [Outliers, leverage and influential points](https://math.wiki/outliers-vs-influential-points/): A point far from the pattern is an outlier, while a point that visibly moves the line is influential.
- [p-values and the significance level](https://math.wiki/p-values/): The probability of data at least this extreme if the null were true, and what that does not mean.
- [Paired data: interval and test for a mean difference](https://math.wiki/paired-data/): Two measurements linked on the same individual or matched pair, reduced to one column of differences.
- [Parameters and statistics](https://math.wiki/parameter-vs-statistic/): A parameter describes the whole population and a statistic describes the sample in hand.
- [Percentiles](https://math.wiki/find-the-value-at-a-percentile/): Going from a given percent back to the data value that sits there.
- [Picture graphs and their key](https://math.wiki/picture-graph-key/): The note saying how many things one picture stands for.
- [Plotting paired data: explanatory and response variables](https://math.wiki/plot-paired-data/): Turning two measurements on the same individual into one point.
- [Point estimates and the margin of error](https://math.wiki/margin-of-error/): How far the estimate might reasonably sit from the true value, given the method's variability.
- [Populations, samples and census](https://math.wiki/populations-and-samples/): The whole group you care about, and the part you actually measure.
- [Power and what raises it](https://math.wiki/power-of-a-test/): The chance a test detects an effect of a given size when that effect is really there.
- [Predicting with a regression line, and extrapolation](https://math.wiki/predict-with-a-regression-line/): Putting a value of the explanatory variable into the equation to get a predicted response.
- [Properties of the correlation coefficient](https://math.wiki/properties-of-r/): It has no units, does not change if the variables are swapped, and never leaves the interval from negative one to one.
- [Quartiles and the five number summary](https://math.wiki/quartiles/): The three values that cut an ordered data set into four parts of about equal size.
- [r squared and the standard deviation of the residuals](https://math.wiki/coefficient-of-determination/): The share of the variation in the response that the regression line accounts for.
- [Random assignment and confounding](https://math.wiki/random-assignment/): Using chance to decide which unit gets which treatment, so the groups start out alike.
- [Random sampling](https://math.wiki/random-sampling/): Every member equally likely, so the sample looks like the population.
- [Range and interquartile range](https://math.wiki/range-and-iqr/): How far the data spans, and how far its middle half spans.
- [Reading a bar graph](https://math.wiki/read-a-bar-graph/): Getting a count out of a bar by following it across to the scale.
- [Reading a graph: titles, labels and units](https://math.wiki/read-a-graph-you-did-not-draw/): Working out what a display shows by reading its title, axes and key before reading any value.
- [Reading a histogram](https://math.wiki/read-a-histogram/): Getting the count in an interval from a bar, and seeing that individual values are no longer shown.
- [Reading dot plots: clusters, gaps and peaks](https://math.wiki/read-a-dot-plot/): Counting the stack of dots above a value to see how often that value occurred.
- [Reading regression output](https://math.wiki/regression-computer-output/): Finding the slope, intercept, residual spread and r squared in a printed regression table.
- [Relative frequency histograms](https://math.wiki/relative-frequency-histogram/): A histogram whose bar heights are shares of the total instead of counts.
- [Relative frequency tables](https://math.wiki/relative-frequency-tables/): A table giving each category as a fraction or percent of the total instead of a raw count.
- [Residuals](https://math.wiki/residuals/): Actual minus predicted, and what their pattern reveals.
- [Resistant and non resistant measures](https://math.wiki/resistant-vs-nonresistant-measures/): A summary is resistant when a few extreme values barely move it.
- [Response bias and question wording](https://math.wiki/response-bias/): When people answer untruthfully or inaccurately, whatever the sampling did.
- [Sampling and study design](https://math.wiki/sampling-and-studies/): How data was collected decides what it can tell you.
- [Sampling distribution of a difference](https://math.wiki/sampling-distribution-of-a-difference-of-means/): The pattern made by the gap between two independent sample means.
- [Sampling distribution of a sample mean](https://math.wiki/sampling-distribution-of-a-sample-mean/): The pattern made by the sample mean over many samples of the same size.
- [Sampling distribution of a sample proportion](https://math.wiki/sampling-distribution-of-a-sample-proportion/): The pattern made by the sample share of successes over many samples of the same size.
- [Sampling distribution of the slope](https://math.wiki/sampling-distribution-of-the-slope/): The pattern the fitted slope makes over many samples from the same population.
- [Sampling distributions](https://math.wiki/sampling-distributions/): Take many samples, compute the statistic each time, look at how it varies.
- [Sampling variability and simulating it](https://math.wiki/sampling-variability/): Different samples from the same population give different statistics, and that spread is predictable.
- [Scaled picture graphs](https://math.wiki/scaled-picture-graphs/): A picture graph where one symbol stands for several items, so half a symbol is half that many.
- [Scatter plots](https://math.wiki/scatter-plots/): Two variables, one point per case.
- [Shape of a distribution](https://math.wiki/shape-of-a-distribution/): Symmetric, skewed, uniform, bimodal, and which center to trust.
- [Shifting and rescaling a data set](https://math.wiki/add-a-constant-to-every-value/): Shifting all the data moves the center by that amount and leaves the spread alone.
- [Simple random samples and the sampling frame](https://math.wiki/simple-random-sample/): A sample drawn so that every group of the chosen size is equally likely.
- [Simulating a sampling distribution and a margin of error](https://math.wiki/simulate-a-sampling-distribution/): Drawing many samples by chance and collecting their statistics to see the pattern they make.
- [Slope from r and the spreads, and regression to the mean](https://math.wiki/slope-from-r-and-standard-deviations/): The slope is r times the ratio of the response spread to the explanatory spread.
- [Sorting objects into groups](https://math.wiki/sort-objects-into-categories/): Putting objects into groups by one shared feature and counting how many are in each group.
- [Standard deviation](https://math.wiki/standard-deviation/): The typical distance from the mean, squared and rooted, and why squared.
- [Standard error](https://math.wiki/standard-error/): The estimated spread of a statistic's sampling distribution, computed from the sample.
- [Standardizing a data set](https://math.wiki/standardizing-a-data-set/): Subtracting the mean and dividing by the standard deviation to get a set with mean zero and spread one.
- [Statistical inference](https://math.wiki/inference/): From a sample to a claim about the population, with honest uncertainty.
- [Statistical questions and variability](https://math.wiki/statistical-question/): A question worth asking of data because the answers vary from one individual to the next.
- [Statistics](https://math.wiki/statistics/): Collecting, describing, and drawing careful conclusions from data.
- [Stem and leaf plots](https://math.wiki/back-to-back-stem-and-leaf-plots/): Two groups sharing one column of stems, with leaves running out to the left and the right.
- [Stratified random sample](https://math.wiki/stratified-random-sample/): Splitting the population into similar groups first, then sampling at random inside each.
- [Symmetric and skewed distributions](https://math.wiki/skewed-distributions/): A shape with one long tail, named for the side the tail runs to.
- [Systematic sample](https://math.wiki/systematic-sample/): Picking a random start and then taking every so many individuals down a list.
- [Test for a difference of means](https://math.wiki/two-sample-t-test-for-a-difference-of-means/): Testing whether two groups really differ in average value.
- [Test for a difference of proportions](https://math.wiki/two-sample-z-test-for-a-difference-of-proportions/): Testing whether two groups really differ in the share with some feature.
- [Test for a slope](https://math.wiki/t-test-for-a-slope/): Testing whether there is any real linear relationship, usually against a slope of zero.
- [Test statistic](https://math.wiki/test-statistic/): How many standard errors the observed result sits from what the null hypothesis expected.
- [Tests for independence and homogeneity](https://math.wiki/chi-square-test-for-independence/): Testing whether two categorical variables recorded on one sample are associated.
- [The central limit theorem](https://math.wiki/central-limit-theorem/): Averages of enough independent draws look normal, whatever the population looked like.
- [The correlation coefficient](https://math.wiki/correlation-coefficient-r/): A number between negative one and one measuring how closely points cluster about a straight line.
- [The least squares criterion and its line](https://math.wiki/least-squares-criterion/): The line is chosen to make the total of the squared vertical misses as small as possible.
- [The normal distribution](https://math.wiki/normal-distribution/): The bell curve: symmetric, described by mean and standard deviation.
- [The standard normal distribution](https://math.wiki/standard-normal-distribution/): The normal curve with mean zero and standard deviation one, which every normal curve becomes after standardizing.
- [The t distribution and degrees of freedom](https://math.wiki/t-distribution/): A bell shape with heavier tails than the normal, used when the population spread is estimated from the sample.
- [The ten percent condition](https://math.wiki/ten-percent-condition/): Sampling without replacement behaves like independent draws as long as the sample is a small part of the population.
- [Transforming to linearity: logs and exponential models](https://math.wiki/transforming-to-achieve-linearity/): Re-expressing one or both variables so a curved relationship becomes a straight one.
- [Two-way tables](https://math.wiki/two-way-tables/): Counts for two categorical variables at once, and the relative frequencies inside.
- [Type I and Type II errors](https://math.wiki/type-ii-error/): Failing to reject a null hypothesis that was actually false.
- [Types of data: categorical, numerical, discrete, continuous](https://math.wiki/categorical-vs-numerical-data/): Data that name a group versus data that count or measure an amount.
- [Undercoverage and nonresponse](https://math.wiki/undercoverage/): When part of the population had no chance of being chosen at all.
- [Uniform and bimodal shapes](https://math.wiki/uniform-and-bimodal-distributions/): A flat shape where every value is about as common, and a two peaked shape that often means two groups.
- [Units, treatments, factors and levels](https://math.wiki/experimental-units-and-treatments/): What or who receives a condition, and the conditions being compared.
- [Variance of a data set](https://math.wiki/variance-of-a-data-set/): The average squared distance from the mean, whose square root is the standard deviation.
- [z-scores and the empirical rule](https://math.wiki/z-scores/): How many standard deviations from the mean.

## Trigonometry

- [Adding and subtracting vectors](https://math.wiki/vector-addition-triangle-rule/): Placing the second arrow's tail at the first arrow's head, with the sum running from first tail to last head.
- [Amplitude and reflection](https://math.wiki/amplitude/): Half the distance between the highest and the lowest value of a wave.
- [Amplitude, period and phase shift](https://math.wiki/amplitude-period-phase/): How tall, how long, and how far shifted, read off the formula.
- [Angles and their measure](https://math.wiki/angles-and-their-measure/): How an angle is named and measured once it is placed on the coordinate plane instead of inside a shape.
- [Angles in standard position and coterminal angles](https://math.wiki/standard-position-angle/): An angle whose vertex sits at the origin and whose initial side lies along the positive x axis.
- [Angles of elevation and depression](https://math.wiki/two-observer-elevation-problems/): Using sightings of the same object from two places to find a height or distance neither one gives alone.
- [Angular and linear speed](https://math.wiki/angular-speed/): How fast something turns, measured as angle swept per unit of time.
- [Applications of oblique triangles](https://math.wiki/triangulation/): Locating a point by measuring the angles to it from the two ends of a known baseline.
- [Arc length and sector area in radians](https://math.wiki/arc-length-in-radians/): Finding the distance along a circle by multiplying the radius by the angle in radians.
- [Area from two sides and the included angle](https://math.wiki/derive-area-from-two-sides-and-an-angle/): The sine of the included angle turns one side into the height used by the usual triangle area formula.
- [Cofunction identities](https://math.wiki/cofunction-identities/): Each function of an angle equals its partner function of the complementary angle.
- [Combining a sine and a cosine into one wave](https://math.wiki/linear-combination-as-a-single-sinusoid/): A sine term plus a cosine term of the same period is itself one wave with a new amplitude and shift.
- [Component form of a vector](https://math.wiki/component-form-of-a-vector/): Writing a vector as the horizontal and vertical change from its tail to its head.
- [Composing trig and inverse trig functions](https://math.wiki/simplify-a-composition-to-an-algebraic-expression/): Drawing a reference triangle from the inner inverse so the outer function becomes an expression in x.
- [Converting an equation between the two systems](https://math.wiki/convert-an-equation-between-polar-and-rectangular/): Substituting the conversion relations to rewrite a whole curve in the other coordinate system.
- [Converting between polar and rectangular](https://math.wiki/polar-to-rectangular-conversion/): Turning a distance and an angle into x and y with cosine and sine.
- [Cosine of a sum or difference](https://math.wiki/sum-formula-for-cosine/): The cosine of two angles added is not the sum of the cosines but a mix of both functions.
- [Directed line segments](https://math.wiki/directed-line-segment/): A segment with an arrow on it, carrying both a length and a direction.
- [Double angle formulas](https://math.wiki/double-angle-formula-for-sine/): Setting both angles equal in the sine sum formula gives twice sine times cosine.
- [Eliminating the parameter](https://math.wiki/eliminate-the-parameter/): Combining the two equations to get a single relation between x and y.
- [Exact values on the unit circle](https://math.wiki/exact-values-at-quadrantal-angles/): Reading the values of the six functions where the terminal side lies on an axis.
- [Finding the period from the equation](https://math.wiki/period-from-the-b-value/): Reading how much a wave is squeezed or stretched sideways from the coefficient on the input.
- [Forces and velocities as vectors](https://math.wiki/force-and-velocity-vectors/): Combining pushes or motions acting in different directions, such as a plane in a crosswind.
- [Graphing sine and cosine](https://math.wiki/graph-sine-from-the-unit-circle/): Plotting the height of the terminal point against the angle to see where the wave shape comes from.
- [Graphing tangent and the reciprocal functions](https://math.wiki/asymptotes-of-the-tangent-graph/): The vertical lines the tangent graph races toward at the angles where cosine is zero.
- [Height on a turning wheel](https://math.wiki/ferris-wheel-model/): Describing a rider's height above the ground as a wheel turns at a steady rate.
- [Heron's formula](https://math.wiki/area-of-a-triangle-from-three-sides/): Taking half the perimeter and combining it with each side to get the area.
- [Inverse trig functions](https://math.wiki/restricting-the-domain-for-an-inverse/): A repeating function fails the one to one test, so only a chosen slice of it can be inverted.
- [Inverse trigonometric ratios](https://math.wiki/inverse-trig-ratios/): From a ratio back to the angle.
- [Law of cosines](https://math.wiki/derive-the-law-of-cosines/): Placing the triangle on the coordinate plane and applying the distance formula to the third vertex gives the law.
- [Law of sines](https://math.wiki/derive-the-law-of-sines/): An altitude splits the triangle into two right triangles sharing a height, and equating the heights gives the law.
- [Laws of sines and cosines](https://math.wiki/laws-of-sines-and-cosines/): Trig for triangles that are not right.
- [Limacons and cardioids](https://math.wiki/cardioid/): A heart shaped curve that appears when the constant term and the varying term are equal in size.
- [Magnitude and direction of a vector](https://math.wiki/magnitude-of-a-vector/): Finding a vector's length from its components with the distance formula.
- [Midline and vertical shift](https://math.wiki/vertical-shift-of-a-sinusoid/): Moving the whole wave up or down, which moves the midline without changing the amplitude.
- [Modeling things that repeat](https://math.wiki/modeling-periodic-phenomena/): Using a sine or cosine model for any quantity that rises and falls on a steady cycle.
- [Modulus, argument and polar form](https://math.wiki/modulus-and-argument/): The distance from the origin is the modulus and the direction angle is the argument.
- [Multiplying, powers and De Moivre's theorem](https://math.wiki/powers-of-a-complex-number/): Using the theorem to work out a high power without repeated multiplication.
- [Opposite, adjacent and hypotenuse](https://math.wiki/opposite-adjacent-hypotenuse/): Naming the three sides of a right triangle relative to whichever acute angle is being looked at.
- [Orthogonal vectors](https://math.wiki/orthogonal-vectors/): Two vectors are perpendicular exactly when their dot product is zero.
- [Parametric equations](https://math.wiki/parameter-and-orientation/): The parameter acts like time, so a parametric curve is traced in a definite direction.
- [Periodic functions](https://math.wiki/period-and-cycle/): One full repetition of the pattern is a cycle and its width along the input axis is the period.
- [Periodicity and the periodicity identities](https://math.wiki/periodicity-on-the-unit-circle/): Adding a full turn returns to the same point, so every trig value repeats after one revolution.
- [Phase shift](https://math.wiki/factor-before-reading-a-phase-shift/): The sideways slide can only be read once the input is written as the coefficient times a shifted variable.
- [Plotting polar points](https://math.wiki/plot-a-polar-point/): Locating a point by turning through an angle and then walking out a distance.
- [Polar circles and symmetry tests](https://math.wiki/polar-graph-of-a-circle/): Equations that trace a circle centered at the origin or passing through it.
- [Polar coordinates](https://math.wiki/polar-coordinates/): Locate a point by distance from the origin and angle from the x-axis.
- [Polar coordinates and vectors](https://math.wiki/polar-and-vectors/): Direction and distance instead of x and y; quantities with a direction.
- [Polar equations and their graphs](https://math.wiki/polar-equations/): Curves described by giving the distance from the origin as a function of the direction.
- [Polar form of a complex number](https://math.wiki/polar-form-of-a-complex-number/): Writing a complex number by how far it sits from the origin and in which direction.
- [Power reducing and half angle formulas](https://math.wiki/power-reducing-formulas/): Rewriting a squared trig function as a first power of a double angle.
- [Product to sum and sum to product formulas](https://math.wiki/product-to-sum-formulas/): Turning a product of two trig functions into a sum, which adding the sum and difference formulas produces.
- [Radian measure](https://math.wiki/why-radians-measure-arc-length/): A radian is the angle whose arc equals the radius, so radian measure is arc length divided by radius.
- [Reciprocal and quotient identities](https://math.wiki/quotient-identities/): Tangent equals sine over cosine, and cotangent equals cosine over sine.
- [Reference angles](https://math.wiki/reference-angles-for-negative-and-large-angles/): Bringing an angle inside one turn first, then taking its acute angle to the x axis.
- [Right triangle trigonometry](https://math.wiki/right-triangle-trig/): Ratios of sides in a right triangle, named by an angle.
- [Roots of a complex number and roots of unity](https://math.wiki/roots-of-a-complex-number/): Every nonzero complex number has n different nth roots, spaced evenly around a circle.
- [Rose curves](https://math.wiki/rose-curves/): Petal shaped curves whose petal count depends on whether the multiplier inside is odd or even.
- [Scaling a vector](https://math.wiki/scalar-multiple-of-a-vector/): Multiplying by a number stretches or shrinks a vector, and a negative number also reverses it.
- [Secant, cosecant and cotangent on the unit circle](https://math.wiki/undefined-values-of-tangent-and-secant/): A zero coordinate makes the matching reciprocal or quotient undefined, so some angles have no value at all.
- [Secant, cosecant and cotangent ratios](https://math.wiki/secant-ratio/): The ratio of the hypotenuse to the side next to an acute angle, the reciprocal of cosine.
- [Signs of the trig functions by quadrant](https://math.wiki/determine-a-quadrant-from-two-signs/): Pinning down where an angle ends when told, for instance, that sine is negative and tangent is positive.
- [Simple harmonic motion](https://math.wiki/simple-harmonic-motion/): The steady back and forth of a spring or a pendulum, written as a sine or cosine of time.
- [Simplifying trig expressions](https://math.wiki/simplify-trig-expressions/): Using the basic identities to rewrite a messy expression as a short one.
- [Sine and tangent of a sum or difference](https://math.wiki/sum-and-difference-formulas-for-sine/): The sine versions, which follow from the cosine versions through the cofunction identities.
- [Sine, cosine and tangent](https://math.wiki/sine-ratio/): The ratio of the side opposite an acute angle to the hypotenuse.
- [Sine, cosine and tangent as functions of a real number](https://math.wiki/circular-functions-of-a-real-number/): Once angles are in radians, the input to a trig function can be read as a distance along the circle.
- [Solving basic trig equations](https://math.wiki/solve-a-basic-trig-equation/): Finding the angles whose sine, cosine or tangent equals a given number.
- [Solving equations with a multiple angle](https://math.wiki/solve-multiple-angle-equations/): When the unknown is multiplied inside the function, the extra cycles create extra solutions.
- [Solving right triangles](https://math.wiki/find-a-missing-side-with-a-trig-ratio/): Choosing the ratio that links the known angle, the known side and the side being found.
- [Solving trig equations by factoring, quadratics and identities](https://math.wiki/solve-trig-equations-by-factoring/): Getting one side to zero and factoring so each piece can be set equal to zero on its own.
- [Solving trigonometric equations](https://math.wiki/trig-equations/): Find every angle that works, on an interval or in general.
- [The ambiguous case](https://math.wiki/height-test-for-ssa/): The altitude from the given angle decides whether zero, one or two triangles can close.
- [The cross product](https://math.wiki/cross-product-magnitude-as-area/): Its length is the area of the parallelogram the two vectors form.
- [The dot product](https://math.wiki/dot-product/): Multiply matching parts and add; it measures how aligned two vectors are.
- [The dot product and the angle between vectors](https://math.wiki/dot-product-as-magnitudes-and-cosine/): It equals the two lengths times the cosine of the angle between them, so it measures how aligned they are.
- [The Pythagorean identities](https://math.wiki/derive-the-pythagorean-identity/): The identity is the circle equation with the coordinates renamed as cosine and sine.
- [The unit circle](https://math.wiki/unit-circle/): Trig for any angle, read as coordinates on a circle of radius one.
- [Tides, daylight and sound](https://math.wiki/tide-models/): Describing the height of water at a harbor as a wave with a period of about half a day.
- [Trig functions on the unit circle](https://math.wiki/cosine-and-sine-as-coordinates/): On the unit circle the x coordinate of the terminal point is the cosine and the y coordinate is the sine.
- [Trig values of the special right triangles](https://math.wiki/trig-values-of-special-right-triangles/): Reading exact sine, cosine and tangent values off the thirty sixty ninety and forty five triangles.
- [Trigonometric functions and their graphs](https://math.wiki/trig-functions/): Sine and cosine as waves, tangent as something wilder.
- [Trigonometric identities](https://math.wiki/trig-identities/): Equations true for every angle, and how to use them.
- [Trigonometric inequalities](https://math.wiki/trig-inequalities/): Finding the stretches of angles where a trig function stays above or below a given value.
- [Trigonometry](https://math.wiki/trigonometry/): Angles and triangles, then the waves they turn into.
- [Unit circle symmetry and even-odd identities](https://math.wiki/unit-circle-symmetry/): Reflecting a terminal point across an axis or the origin changes only the signs of its coordinates.
- [Unit vectors and i, j notation](https://math.wiki/i-and-j-notation/): Expressing any plane vector as a multiple of the horizontal unit vector plus a multiple of the vertical one.
- [Vector addition and scalar multiplication](https://math.wiki/vector-operations/): Tip to tail, and scaling.
- [Vector and scalar projection](https://math.wiki/scalar-projection/): The signed length of that shadow, with no direction attached.
- [Vectors](https://math.wiki/vectors/): An arrow with length and direction; the same arrow anywhere is the same vector.
- [Vectors in space](https://math.wiki/three-dimensional-vectors/): Extending components, magnitude and the dot product to three coordinates.
- [Verifying identities](https://math.wiki/verify-by-working-one-side/): Changing only one side of a proposed identity until it matches the other side.
- [What an identity is](https://math.wiki/what-an-identity-is/): An equation true for every value where both sides are defined, rather than for a few solutions.
- [Writing an equation from a wave](https://math.wiki/write-a-sinusoid-from-a-graph/): Reading the amplitude, midline, period and shift off a picture and assembling the matching formula.