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Grade 12

What Math Does a 12th Grader Learn? All 27 Topics, in Order

By Knomi Learning · Updated

The short answer

A 12th grader learns 27 math topics, in 8 units: Functions, Trigonometry, Complex Numbers, Vectors and Parametrics, Matrices, Conic Sections, Probability and Toward Calculus. That is 110 smaller skills in all, taught in the order below, each building on earlier ones.

  • The earlier topics 12th grade leans on most: Rational Expressions (Grade 11), Functions and Notation (Grade 9), Counting Outcomes (Grade 10), Functions (Grade 8) and Pythagorean Theorem (Grade 8).
  • All 110 skills are matched to a Common Core standard.
  • Every one of them has a full lesson on Knomi.

Unit 1: Functions

1

Composite Functions

The kid feeds one function's output into another and reads f(g(x)) from the inside out, in formulas, tables, graphs and stories. Composition is how the year is built: inverses undo it and limits pass through it. Reading f(g(2)) as f(2) × g(2) sends the tutor back to function notation.

What it covers

  • Evaluating Composite Functions
  • Composites from Graphs and Tables
  • Finding Composite Functions
  • Modeling with Composite Functions

Builds on

  • Grade 9: Evaluating with Function Notation
  • Grade 9: Special Products
  • Grade 9: Domain and Range
  • Grade 9: Units in Formulas
  • Grade 8: Function Rules
  • Grade 8: Functions in Tables
  • and 4 more
Composite Functions on the 12th grade map
2

Inverse Functions

Grade 11 found inverses of simple functions; here the kid decides when an inverse exists, reads one from a table or graph, checks a pair by composition, restricts a domain to make one, and sees the logarithm as the exponential undone. Reading f^−1(x) as 1/f(x) is the slip the tutor names.

What it covers

  • Invertible or Not
  • Inverses from Graphs and Tables
  • Verifying Inverses by Composition
  • Restricting the Domain
  • Logarithms as Inverses

Builds on

  • Grade 11: Writing an Inverse Function
  • Grade 11: Simplifying Rational Expressions
  • Grade 11: Combining Transformations
  • Grade 11: Meaning of a Logarithm
  • Grade 9: What Makes a Function
  • Grade 8: Functions on Graphs
  • and 5 more
Inverse Functions on the 12th grade map
3

Rational Functions

The kid graphs functions like (x − 1)/(x^2 − 4) from their factors: zeros, vertical asymptotes, holes, end behavior and the sign between them, then solves polynomial and rational inequalities with the same sign chart. Calling every zero of a denominator an asymptote is the slip the tutor names.

What it covers

  • Zeros, Asymptotes and Holes
  • End Behavior and Slant Asymptotes
  • Graphing Rational Functions
  • Polynomial and Rational Inequalities

Builds on

  • Grade 11: Simplifying Rational Expressions
  • Grade 11: Zeros from Factored Form
  • Grade 11: End Behavior
  • Grade 11: Quotient and Remainder Form
  • Grade 11: Combining Transformations
  • Grade 7: Quotients of Integers
  • and 2 more
Rational Functions on the 12th grade map

Unit 2: Trigonometry

4

Trig Functions and Graphs

The kid makes the special-angle values fluent all the way round the circle, adds secant, cosecant and cotangent, and graphs every trig function with shifts and asymptotes. A kid who gives sin(π/6) = √3/2 has swapped the legs, and the tutor goes back to the special triangles.

What it covers

  • Exact Values Around the Circle
  • Reciprocal Trig Functions
  • All Six from One Ratio
  • Phase Shift and Frequency
  • Graphs of Tangent and Reciprocals

Builds on

  • Grade 11: Special Angle Values
  • Grade 11: Sine and Cosine as Coordinates
  • Grade 11: Values from One Ratio
  • Grade 11: Building a Sinusoidal Model
  • Grade 11: Combining Transformations
  • Grade 10: Sine, Cosine and Tangent
  • and 3 more
Trig Functions and Graphs on the 12th grade map
5

Inverse Trig Functions

Sine, cosine and tangent repeat their outputs, so the kid keeps one piece of each graph and builds arcsin, arccos and arctan, then uses them to turn a ratio into an angle. The slip the tutor names is arcsin(sin 5π/6) = 5π/6: the answer must land in the restricted range.

What it covers

  • Arcsine and Arccosine
  • Arctangent
  • Compositions with Inverse Trig
  • Inverse Trig in Context

Builds on

  • Grade 11: Unwrapping the Unit Circle
  • Grade 10: Finding an Angle
  • Grade 8: Finding a Leg
Inverse Trig Functions on the 12th grade map
6

Trigonometric Identities

The kid proves the sum and difference formulas, derives the double- and half-angle formulas from them, and uses all the identities to find exact values, simplify and prove. The slip the tutor names is sin(a + b) = sin a + sin b, disproved with one pair of angles.

What it covers

  • Sum and Difference Formulas
  • Tangent Sum and Difference
  • Double-Angle Formulas
  • Half-Angle Formulas
  • Simplifying with Identities
  • Proving Identities

Builds on

  • Grade 11: Symmetry and Periodicity
  • Grade 11: Simplifying Rational Expressions
  • Grade 10: Distance Formula
  • Grade 9: Difference of Squares
  • Grade 9: Reasoning Step by Step
  • Grade 8: Roots of Fractions and Decimals
Trigonometric Identities on the 12th grade map
7

Trigonometric Equations

A trig equation has infinitely many solutions because the functions repeat. The kid finds one with an inverse function, a second from the circle's symmetry, all of them by adding periods, and the ones on an interval. Stopping at the calculator's single answer is the slip the tutor names.

What it covers

  • Basic Equations and General Solutions
  • Solutions on an Interval
  • Equations with Multiple Angles
  • Equations Using Identities
  • Trig Equations in Context

Builds on

  • Grade 11: Unwrapping the Unit Circle
  • Grade 11: Building a Sinusoidal Model
  • Grade 9: Zero Product Property
  • Grade 8: Fraction and Decimal Coefficients
  • Grade 7: Solving Two-Step Equations
  • Grade 6: Comparing Rational Numbers
Trigonometric Equations on the 12th grade map
8

Solving Any Triangle

Grade 10 used the laws of sines and cosines on acute triangles; here the kid solves any triangle, obtuse ones included, meets the side-side-angle case that gives two triangles, and works navigation and surveying problems. Keeping only the calculator's acute angle is the slip the tutor names.

What it covers

  • Obtuse Triangles
  • The Ambiguous Case
  • Triangle Word Problems

Builds on

  • Grade 10: Choosing a Law
  • Grade 10: Law of Sines
  • Grade 10: Elevation and Depression
  • Grade 8: Parallel Lines and Transversals
  • Grade 7: One Triangle, Many or None
Solving Any Triangle on the 12th grade map

Unit 3: Complex Numbers

9

Complex Plane

Every complex number gets a place: a + bi is the point (a, b). On that plane adding is sliding, the conjugate is a mirror image, and each number has a length and an angle. Plotting 3 − 2i at (−2, 3) is the slip the tutor names: the real part goes across.

What it covers

  • Adding on the Complex Plane
  • Conjugates and Division
  • Modulus and Argument
  • Distance and Midpoint

Builds on

  • Grade 11: Multiplying Complex Numbers
  • Grade 10: Midpoint Formula
  • Grade 10: Distance Formula
  • Grade 9: Difference of Squares
  • Grade 8: Translations
  • Grade 8: Finding the Hypotenuse
  • and 1 more
Complex Plane on the 12th grade map
10

Polar Form

Every complex number can be named by its length and angle, r(cos θ + i sin θ), and then multiplying is easy: multiply the lengths, add the angles. The kid converts forms, multiplies, divides, raises to powers and finds roots. Multiplying the angles is the slip the tutor names: rotations add.

What it covers

  • Rectangular and Polar Forms
  • Multiplying Graphically
  • Multiplying and Dividing in Polar Form
  • Powers of Complex Numbers
  • Roots of Complex Numbers

Builds on

  • Grade 11: Complex Zeros
  • Grade 8: Rotations
  • Grade 8: Dilations with Coordinates
  • Grade 8: Powers of Powers and Products
Polar Form on the 12th grade map

Unit 4: Vectors and Parametrics

11

Vectors

A vector is an arrow carrying two facts, how far and which way. The kid writes vectors in components, finds length and direction, adds and subtracts them, and scales them. Subtracting the endpoints the wrong way round is the slip the tutor names: components are terminal minus initial.

What it covers

  • Vectors and Components
  • Magnitude and Direction
  • Adding and Subtracting Vectors
  • Scalar Multiples and Unit Vectors

Builds on

  • Grade 10: Finding a Side
  • Grade 8: Translations
  • Grade 8: Scale Factor and Center
  • Grade 7: Adding the Opposite
  • Grade 6: Four Quadrants
  • Grade 6: Absolute Value
Vectors on the 12th grade map
12

Vector Applications

Vectors earn their keep: a plane in a crosswind, a boat in a current, ropes pulling a load. Then the dot product measures how much two vectors point the same way, giving angles, perpendicular tests and work. Adding speeds that point different ways is the slip the tutor names.

What it covers

  • Velocity and Force Problems
  • Dot Product and Angle
  • Projections and Work

Builds on

  • Grade 10: Finding a Side
Vector Applications on the 12th grade map
13

Parametric Equations

A curve can be drawn by a moving point, with x and y each a function of time t. The kid plots parametric curves, removes the parameter, writes lines and circles parametrically, and models projectiles. Plotting t on an axis instead of pairing x(t) with y(t) is the slip the tutor names.

What it covers

  • Plotting Parametric Curves
  • Eliminating the Parameter
  • Parametric Lines
  • Motion and Projectiles

Builds on

  • Grade 9: Evaluating with Function Notation
  • Grade 9: Using the Formula
  • Grade 8: Rearranging Formulas
  • Grade 8: Equations from Two Points
  • Grade 6: Tables from Equations
Parametric Equations on the 12th grade map

Unit 5: Matrices

14

Matrices

A matrix is a rectangle of numbers where every row and column means something: stock in shops, roads between towns, payoffs in a game. The kid reads dimensions, adds, subtracts and scales matrices. Writing dimensions as columns × rows is the slip the tutor names: rows come first.

What it covers

  • Matrices and Data
  • Adding and Scaling Matrices
  • Properties of Matrix Addition

Builds on

  • Grade 8: Reading and Building Two-Way Tables
  • Grade 7: Percent Change as a Multiplier
  • Grade 7: Adding the Opposite
  • Grade 7: Solving Two-Step Equations
  • Grade 6: Four Quadrants
Matrices on the 12th grade map
15

Matrix Multiplication

A 2 × 2 matrix moves every point of the plane, so multiplying matrices means doing one move after another, which is why AB is usually not BA. The determinant says how much area scales. Multiplying entry by entry is the slip the tutor names: an entry is a row times a column.

What it covers

  • Matrices as Transformations
  • Composing Transformations
  • Multiplying Two Matrices
  • Properties of Matrix Multiplication
  • Determinants and Area

Builds on

  • Grade 8: Rotations
  • Grade 8: Sequences of Transformations
  • Grade 8: Scale Factor and Center
  • Grade 7: Order of Operations with Rationals
  • Grade 7: Expanding Expressions
  • Grade 6: Area of Parallelograms
Matrix Multiplication on the 12th grade map
16

Inverse Matrices

Some transformations can be undone and the determinant says which. The kid finds the inverse of a 2 × 2 matrix, writes a system of equations as AX = B, and solves it with X = A^−1B, the way 3x = 12 is solved by multiplying by 1/3. Multiplying on the wrong side is the slip the tutor names.

What it covers

  • Invertible Matrices and Determinants
  • Inverse of a 2 × 2 Matrix
  • Systems as Matrix Equations
  • Solving Systems with Inverses

Builds on

  • Grade 9: Standard Form
  • Grade 9: Number of Solutions of a System
  • Grade 8: Solutions of a System
  • Grade 7: Multiplying Rational Numbers
Inverse Matrices on the 12th grade map

Unit 6: Conic Sections

17

Ellipses

An ellipse is a circle stretched differently across and up. The kid reads its center and radii from (x − h)^2/a^2 + (y − k)^2/b^2 = 1, graphs it, writes its equation and finds the two foci. Reading a radius as a^2 instead of a is the slip the tutor names.

What it covers

  • Ellipse Equations and Graphs
  • Foci of an Ellipse
  • Ellipses Off the Origin

Builds on

  • Grade 10: Equation from Center and Radius
  • Grade 10: Distance Formula
  • Grade 10: Completing the Square for Circles
  • Grade 9: Building a Perfect Square
  • Grade 8: Square Roots of Perfect Squares
  • Grade 8: Finding a Leg
Ellipses on the 12th grade map
18

Hyperbolas

A hyperbola is an ellipse with a minus sign: two branches hugging a pair of asymptotes. The kid graphs and writes hyperbolas, finds their foci, and tells the four conics apart. Deciding the direction from the bigger denominator is the slip the tutor names: the positive term decides.

What it covers

  • Vertices and Asymptotes
  • Foci of a Hyperbola
  • Hyperbolas Off the Origin
  • Identifying Conic Sections

Builds on

  • Grade 11: Combining Transformations
  • Grade 10: Parabola as a Locus
  • Grade 10: Completing the Square for Circles
  • Grade 8: Meaning of y = mx + b
  • Grade 8: Finding the Hypotenuse
Hyperbolas on the 12th grade map

Unit 7: Probability

19

Counting Methods

Grade 10 counted with permutations and combinations; here the kid counts the harder cases: repeated letters, people who must sit together, committees from several groups, "at least one". The question the tutor keeps asking is whether order matters and what is being counted twice.

What it covers

  • Arrangements with Repeats
  • Arrangements with Restrictions
  • Selections with Conditions
  • Probability with Counting

Builds on

  • Grade 10: Counting Principle and Permutations
  • Grade 10: Combinations
  • Grade 10: Probability by Counting
  • Grade 10: The Conditional Probability Formula
  • Grade 6: Meaning of Exponents
Counting Methods on the 12th grade map
20

Binomial Theorem

Expanding (x + y)^5 bracket by bracket is slow. The kid learns that the coefficients are a row of Pascal's triangle and count combinations, then expands any binomial power and finds one term directly. Writing (x + y)^2 = x^2 + y^2 is the slip the tutor names: the first case of the theorem.

What it covers

  • Pascal's Triangle
  • Expanding Binomials
  • Finding One Term

Builds on

  • Grade 10: Combinations
  • Grade 9: Special Products
  • Grade 8: Powers of Powers and Products
  • Grade 8: Powers with Negative Bases
  • Grade 8: Rules with Integer Exponents
Binomial Theorem on the 12th grade map
21

Random Variables

A random variable puts a number on each outcome: heads in four flips, cars per household. The kid builds its probability distribution from theory or from data and meets the binomial distribution. Accepting a table whose probabilities do not add to 1 is the slip the tutor names.

What it covers

  • Discrete Random Variables
  • Theoretical Distributions
  • Empirical Distributions
  • The Binomial Distribution

Builds on

  • Grade 10: Sample Spaces and Events
  • Grade 10: Compound Events in Stages
  • Grade 7: Probability Models
  • Grade 7: Relative Frequency
  • Grade 7: Sample Variability
  • Grade 6: Histograms
Random Variables on the 12th grade map
22

Expected Value

Expected value is the long-run average of a random variable: what a game pays per play over many plays. The kid computes it, judges games and insurance by it, and compares strategies. Ignoring the cost to play is the slip the tutor names.

What it covers

  • Computing Expected Value
  • Expected Payoffs
  • Comparing Strategies

Builds on

  • Grade 10: Probability in Everyday Life
  • Grade 7: Multiplying Rational Numbers
  • Grade 7: Adding Integers Fluently
  • Grade 7: Solving Two-Step Equations
  • Grade 6: Mean
Expected Value on the 12th grade map

Unit 8: Toward Calculus

23

Series and Sigma Notation

A series adds up the terms of a sequence. The kid writes sums in Σ notation, sums arithmetic and geometric series, and meets infinitely many terms adding to a finite number when the ratio is small. Counting the terms wrong is the slip the tutor names: k = 3 to k = 10 is 8 terms, not 7.

What it covers

  • Sigma Notation
  • Arithmetic Series
  • Finite Geometric Series
  • Infinite Geometric Series

Builds on

  • Grade 11: The Geometric Series Formula
  • Grade 9: Sequences as Functions
  • Grade 9: Arithmetic Sequences
  • Grade 8: Repeating Decimals as Fractions
  • Grade 6: Substituting Values
  • Grade 6: Mean
  • and 1 more
Series and Sigma Notation on the 12th grade map
24

Limits

A limit asks where a function is heading as x gets close to a number, whether or not it ever arrives. The kid estimates limits from graphs and tables, reads one-sided limits, and meets limits that grow without bound. Reading the limit as the value at the point is the slip the tutor names.

What it covers

  • Introducing Limits
  • Limits from Graphs
  • One-Sided Limits
  • Limits from Tables
  • Infinite Limits

Builds on

  • Grade 9: Evaluating with Function Notation
  • Grade 9: Piecewise Functions
  • Grade 6: Comparing Rational Numbers
  • Grade 6: Subtracting Decimals
Limits on the 12th grade map
25

Evaluating Limits

The kid finds limits exactly: by substituting when that works and, when substitution gives 0/0, by factoring or by a conjugate, then finds limits at infinity. The slip the tutor names is reading 0/0 as "the limit is 0": it only says to do more algebra.

What it covers

  • Limit Properties and Substitution
  • Limits by Factoring
  • Limits by Rationalizing
  • Limits at Infinity

Builds on

  • Grade 11: Combining Functions
  • Grade 11: Simplifying Rational Expressions
  • Grade 9: Special Products
  • Grade 9: Difference of Squares
  • Grade 9: Simplifying Square Roots
  • Grade 6: Absolute Value
Evaluating Limits on the 12th grade map
26

Continuity

A function is continuous where its graph can be drawn without lifting the pen. The kid learns the exact test, classifies the breaks, fills a hole, and uses the intermediate value theorem to show a solution exists. Calling a function discontinuous outside its domain is the slip the tutor names.

What it covers

  • Types of Discontinuities
  • Continuity at a Point
  • Continuity on an Interval
  • Intermediate Value Theorem

Builds on

  • Grade 9: Domain and Range
  • Grade 9: Solving with Graphs and Tables
  • Grade 7: Solving Two-Step Equations
  • Grade 6: Graphing Inequalities
Continuity on the 12th grade map
27

Intro to the Derivative

The year ends where limits were heading: the rate of change at one instant. The kid moves from secant to tangent, defines the derivative as a limit, finds it for polynomials with the power rule, and writes tangent lines. Polynomials only: no product or chain rule, no integrals.

What it covers

  • Average and Instantaneous Rates
  • The Derivative at a Point
  • The Power Rule
  • Tangent Lines and Turning Points

Builds on

  • Grade 9: Average Rate of Change
  • Grade 9: Point-Slope Form
  • Grade 9: Using the Formula
  • Grade 8: Slope from Two Points
  • Grade 8: Multiplying and Dividing Powers
  • Grade 8: Meaning of y = mx + b
Intro to the Derivative on the 12th grade map

Questions parents ask

How many math topics are in 12th grade?

On Knomi's map, 12th grade math has 27 topics in 8 units, made of 110 smaller skills.

What should a student know before starting 12th grade math?

The earlier topics 12th grade leans on most are Rational Expressions (Grade 11), Functions and Notation (Grade 9), Counting Outcomes (Grade 10), Functions (Grade 8) and Pythagorean Theorem (Grade 8). A student who is quick and sure with those is ready. Each topic below lists exactly what it builds on.

Is this 12th grade math list aligned to Common Core?

Common Core is the spine. All 110 skills carry a Common Core reference, and the order follows the usual US course.

Does Knomi teach 12th grade math?

Every one of them has a full lesson on Knomi. The 12th grade map is open to anyone, with no account needed to look around.

See 12th grade math as a map.

Every topic above, with lines to what it needs first. Open it and look around, no account needed.

Open the Grade 12 map

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