Courses · Mathematics · All grades
Map of Mathematics
Mathematics is a large country with many regions. You do not have to visit them all, but it helps to know they are there. Here are sixteen main areas, what each one is about in plain words, a tiny example, someone from history who helped build it, and where you meet it in School Launch courses.
Explore the map
- Numbers
- Structure
- Shape and space
- Change
- Chance and data
- Reasoning
- Math at work
Numbers
Arithmetic
Grades K–2, 3–5, 6–8Arithmetic is counting and working with numbers: adding, taking away, multiplying and sharing.
Going deeper: It covers whole numbers, fractions, decimals and percents, and the written methods that make big calculations reliable.
Tiny example: 24 cookies shared by 6 children: 24 ÷ 6 = 4 cookies each.
From history: Leonardo of Pisa, called Fibonacci (about A.D. 1170 – about 1250) wrote Liber Abaci (1202), the book that taught European merchants to calculate with the digits 0 to 9 we use today.
Where you meet it at School Launch
- Kindergarten Math 1.1: Counting Objects to 5 Grades K–2
- Grade 2 Math 3.2: Regrouping Ones Grades K–2
- Grade 4 Math 4.2: Long Division Steps Grades 3–5
- Pre-Algebra, Unit 3: Fractions Grades 6–8
- Bear Bakery (Grade 1 practice game) Grades K–2
Main branches
Counting and place value Grades K–2, 3–5, 6–8
Our numbers use ten digits, and where a digit sits tells how much it is worth.
Going deeper: Place value lets ten symbols write any whole number, and it is the reason the column methods for adding and multiplying work.
Tiny example: In 352 the 3 means 3 hundreds, the 5 means 5 tens and the 2 means 2 ones.
Where you meet it at School Launch
- Grade 1 Math 5.1: Tens and Ones Grades K–2
- Grade 4 Math 1.1: Reading and Writing Large Numbers Grades 3–5
- Pre-Algebra 1.1: Place Value and Our Number System Grades 6–8
- Number Grid Tic-Tac-Toe (game) Grades K–2, 3–5
The four operations Grades K–2, 3–5, 6–8
Adding, subtracting, multiplying and dividing are the four tools for working with numbers.
Going deeper: Multiplication is repeated addition and division undoes multiplication; the standard written methods break big problems into small, sure steps.
Tiny example: 7 × 8 = 56, so 56 ÷ 8 = 7.
Where you meet it at School Launch
- Grade 1 Math 2.3: Making Ten Grades K–2
- Grade 3 Math 3.1: Equal Groups and Arrays Grades 3–5
- Grade 4 Math 2.3: The Standard Algorithm Grades 3–5
- Pre-Algebra 1.4: Order of Operations Grades 6–8
Fractions and decimals Grades 3–5, 6–8
Fractions and decimals name parts of a whole, like half a pie or a tenth of a dollar.
Going deeper: Any fraction can be written as a decimal that either stops or repeats, and equivalent fractions name the same amount in different ways.
Tiny example: 1/2 = 2/4 = 0.5, and 3/4 = 0.75.
Where you meet it at School Launch
- Grade 3 Math 6.1: Unit Fractions and Fractions of a Set Grades 3–5
- Grade 4 Math 7.1: Tenths and Hundredths Grades 3–5
- Grade 5 Math 3.2: Adding Unlike Fractions Grades 3–5
- Pre-Algebra 3.6: Dividing Fractions Grades 6–8
Ratios, rates and percent Grades 3–5, 6–8
A ratio compares two amounts, and a percent means "out of a hundred."
Going deeper: Ratios and rates turn into proportions, which handle recipes, maps, unit prices and percent change.
Tiny example: 2 cups of flour for every 1 cup of sugar; 25% of 80 is 20.
Where you meet it at School Launch
- Grade 5 Math 9.1: Ratios and Ratio Tables Grades 3–5
- Grade 5 Math 9.2: Percent of a Number Grades 3–5
- Pre-Algebra 7.1: Ratios Grades 6–8
- Pre-Algebra 8.2: Finding a Percent of a Number Grades 6–8
Negative numbers Grades 3–5, 6–8
Negative numbers are below zero, like a cold winter temperature or money you owe.
Going deeper: Integers extend the number line both ways, with rules such as "subtracting a negative is adding."
Tiny example: 5 − 8 = −3, and −3 + 10 = 7.
From history: Brahmagupta (A.D. 598 – 670) wrote rules for calculating with zero and with negative numbers, which he called debts, in a book of A.D. 628.
Where you meet it at School Launch
- Grade 5 Math 8.3: Below Zero: Negative Numbers Grades 3–5
- Pre-Algebra 5.1: Negative Numbers and Absolute Value Grades 6–8
- Pre-Algebra 5.2: Adding Integers Grades 6–8
- Kitty Sushi Shop (Pre-Algebra practice game) Grades 6–8
Numbers
Number Theory
Grades K–2, 3–5, 6–8Number theory is the study of whole numbers themselves: which are even, which are prime and what patterns they make.
Going deeper: Simple questions about whole numbers can be very hard to answer, and some of its results now protect online banking.
Tiny example: 12 = 2 × 2 × 3, and no other set of primes multiplies to 12.
From history: Carl Friedrich Gauss (1777–1855) published Disquisitiones Arithmeticae (1801), the book that organized number theory as a subject.
Where you meet it at School Launch
- Grade 2 Math 1.3: Even and Odd Grades K–2
- Grade 4 Math 3.3: Factors, Multiples, Primes Grades 3–5
- Pre-Algebra, Unit 2: Factors, Multiples and Primes Grades 6–8
Main branches
Even, odd and divisibility Grades K–2, 6–8
A number is even if it splits into two equal groups, and quick tests tell what a number divides by.
Going deeper: Divisibility rules come from place value: a number is divisible by 3 exactly when the sum of its digits is.
Tiny example: 471: 4 + 7 + 1 = 12, and 12 divides by 3, so 471 does too (471 = 3 × 157).
Where you meet it at School Launch
- Grade 2 Math 1.3: Even and Odd Grades K–2
- Pre-Algebra 2.1: Divisibility Rules Grades 6–8
Prime numbers Grades 3–5, 6–8
A prime number can be divided evenly only by 1 and itself, like 2, 3, 5, 7 and 11.
Going deeper: Every whole number above 1 is a product of primes in exactly one way, and Euclid proved there is no largest prime.
Tiny example: The primes below 20: 2, 3, 5, 7, 11, 13, 17, 19.
From history: Eratosthenes (276–194 B.C.) gave his name to the "sieve," a way to find primes by crossing out multiples.
Where you meet it at School Launch
- Grade 4 Math 3.3: Factors, Multiples, Primes Grades 3–5
- Pre-Algebra 2.2: Prime and Composite Numbers Grades 6–8
- Pre-Algebra 2.3: Prime Factorization Grades 6–8
Common factors and Euclid’s algorithm Grades 6–8
The greatest common factor is the biggest number that divides two numbers evenly.
Going deeper: Euclid’s algorithm finds it by repeated division, one of the oldest step-by-step methods still in use.
Tiny example: GCF of 48 and 18: 48 = 2 × 18 + 12, 18 = 1 × 12 + 6, 12 = 2 × 6, so the answer is 6.
From history: Euclid of Alexandria (lived about 300 B.C.) described this method in Book VII of the Elements.
Where you meet it at School Launch
- Pre-Algebra 2.4: Greatest Common Factor Grades 6–8
- Pre-Algebra 2.5: The Euclidean Algorithm Grades 6–8
- Pre-Algebra 2.6: Least Common Multiple Grades 6–8
Equations with whole-number answers Later: college and beyond
Some puzzles only allow whole-number answers, like paying an exact amount with two kinds of stamps.
Going deeper: These are called Diophantine equations; finding every whole-number solution, or proving there is none, is a central problem of number theory.
Tiny example: Pay 22 cents with 3-cent and 5-cent stamps: four 3s and two 5s (3x + 5y = 22).
From history: Diophantus of Alexandria (about A.D. 200 – about 284) wrote Arithmetica, a collection of such problems.
Later: college and beyond. School Launch does not teach this yet; it is part of college mathematics.
Clock arithmetic (modular arithmetic) Later: college and beyond
On a clock, 5 hours after 9 o’clock is 2 o’clock, because the numbers wrap around.
Going deeper: Modular arithmetic works with remainders; it underlies check digits, calendars and modern codes.
Tiny example: 9 + 5 = 14, and 14 leaves remainder 2 when divided by 12, so 9 + 5 ≡ 2 (mod 12).
From history: Carl Friedrich Gauss (1777–1855) introduced the ≡ sign for this in 1801.
Later: college and beyond. School Launch does not teach this yet; it is part of college mathematics.
A first taste you can find here:
Structure
Algebra
Grades K–2, 3–5, 6–8, 9–12Algebra uses letters to stand for numbers we do not know yet, so we can find them.
Going deeper: It studies equations, functions and their graphs, and it is the language most of higher mathematics and science is written in.
Tiny example: If 3n + 2 = 14, then 3n = 12 and n = 4.
From history: Muhammad ibn Musa al-Khwarizmi (about A.D. 780 – about 850) wrote a book on solving equations whose title gave us the word algebra; his own name gave us algorithm.
Where you meet it at School Launch
- Grade 5 Math 6.2: Letters for Numbers: Expressions and Formulas Grades 3–5
- Pre-Algebra 10.4: Two-Step Equations Grades 6–8
- Algebra I (the whole course) Grades 6–8, 9–12
- Algebra II (the whole course) Grades 9–12
Main branches
Variables and equations Grades K–2, 6–8, 9–12
A letter like n can stand for a mystery number, and an equation is a balance we keep even.
Going deeper: Doing the same thing to both sides of an equation keeps it true, which turns word problems into steps anyone can check.
Tiny example: ___ + 4 = 9 is the same puzzle as n + 4 = 9, so n = 5.
From history: François Viète (1540–1603) began writing letters for known and unknown numbers in equations, as we still do.
Where you meet it at School Launch
- Grade 1 Math 3.4: Missing Numbers Grades K–2
- Pre-Algebra 9.1: Variables and Expressions Grades 6–8
- Algebra I 2.3: Equations with Variables on Both Sides Grades 6–8, 9–12
Functions and graphs Grades 6–8, 9–12
A function is a rule that turns each input into exactly one output, like "double it and add 1."
Going deeper: Graphing a function shows its whole behavior at once: where it rises, falls, crosses zero or levels off.
Tiny example: f(x) = 2x + 1 turns 3 into 7 and 10 into 21.
From history: Leonhard Euler (1707–1783) made the notation f(x) standard.
Where you meet it at School Launch
- Pre-Algebra 11.2: Tables, Rules and Graphs Grades 6–8
- Algebra I 4.2: What Is a Function? Grades 6–8, 9–12
- Algebra II 1.2: Parent Functions and Transformations Grades 9–12
- Graphing Calculator (tool) Grades 6–8, 9–12
Polynomials and their equations Grades 6–8, 9–12
Polynomials are expressions built from a number multiplied by itself, like x × x, plus other terms.
Going deeper: Solving polynomial equations, from the quadratic formula to the cubic and beyond, drove algebra forward for centuries.
Tiny example: x² − 5x + 6 = 0 factors as (x − 2)(x − 3) = 0, so x = 2 or x = 3.
From history: Gerolamo Cardano (1501–1576) published the solution of the cubic equation in Ars Magna (1545).
Where you meet it at School Launch
- Algebra I 9.6: Solving Equations by Factoring Grades 6–8, 9–12
- Algebra I 11.5: The Quadratic Formula Grades 6–8, 9–12
- Algebra II 4.6: Finding Zeros of Polynomials Grades 9–12
Linear algebra: matrices and vectors Grades 6–8, 9–12
Linear algebra solves many equations at once by arranging the numbers in neat boxes called matrices.
Going deeper: Vectors and matrices describe systems of equations, rotations and transformations; computers use them for graphics and data.
Tiny example: x + y = 10 and x − y = 4 give x = 7 and y = 3.
From history: Gabriel Cramer (1704–1752) published a rule for solving systems of equations with determinants (1750).
Where you meet it at School Launch
- Algebra I 6.3: Solving Systems by Elimination Grades 6–8, 9–12
- Algebra II 12.1: Matrices and Matrix Operations Grades 9–12
- Pre-Calculus 7.4: Matrices for Linear Systems Grades 9–12
Exponents and logarithms Grades 6–8, 9–12
An exponent says how many times to multiply a number by itself: 2³ = 2 × 2 × 2 = 8.
Going deeper: Logarithms undo exponents, turning multiplication into addition; they measure growth, sound levels and earthquakes.
Tiny example: log₂ 8 = 3, because 2³ = 8.
From history: John Napier (1550–1617) published the first tables of logarithms (1614), which made long astronomical calculations far quicker.
Where you meet it at School Launch
- Pre-Algebra 6.1: Exponents and Powers Grades 6–8
- Algebra II 8.1: Logarithms Grades 9–12
- Pre-Calculus 3.4: The Number e and Continuous Growth Grades 9–12
Structure
Abstract Algebra
Later: college and beyondAbstract algebra looks at the rules numbers follow, like "order does not matter when you add," and finds the same rules in other places.
Going deeper: It studies structures called groups, rings and fields; they explain symmetry, which equations can be solved by formulas, and how codes work.
Tiny example: Turning a square a quarter turn four times brings it back to the start, just like adding 1 four times on a 4-hour clock.
From history: Évariste Galois (1811–1832) used groups to show which polynomial equations can be solved by a formula.
Later: college and beyond. School Launch does not teach this yet; it is part of college mathematics.
A first taste you can find here:
Main branches
Groups and symmetry Later: college and beyond
A group is a set of moves that can be combined and undone, like the turns and flips that leave a square looking the same.
Going deeper: Group theory measures symmetry exactly, in crystals, patterns and the laws of physics.
Tiny example: A square has 8 symmetries: 4 turns (including no turn) and 4 flips.
From history: Arthur Cayley (1821–1895) gave the first general definition of a group (1854).
Later: college and beyond. School Launch does not teach this yet; it is part of college mathematics.
A first taste you can find here:
- Geometry 7.3: Rotations and Rotational Symmetry Grades 9–12
- Flower Geometry (explainer) Grades 9–12
Rings and fields Later: college and beyond
Rings and fields are number systems with their own adding and multiplying, such as clock numbers.
Going deeper: Fields are systems where you can always divide by anything except zero; finite fields power error-correcting codes.
Tiny example: Counting on a 7-hour clock, 3 × 5 = 15, which is 1 more than 14, so 3 × 5 = 1 there.
From history: Emmy Noether (1882–1935) built much of the modern theory of rings.
Later: college and beyond. School Launch does not teach this yet; it is part of college mathematics.
A first taste you can find here:
Shape and space
Geometry
Grades K–2, 3–5, 6–8, 9–12Geometry is the study of shapes, sizes and positions: lines, angles, triangles, circles and solids.
Going deeper: It is the first part of mathematics built by proof, starting from a few plain assumptions and reasoning to many sure conclusions.
Tiny example: The angles of any triangle add up to 180°.
From history: Euclid of Alexandria (lived about 300 B.C.) wrote the Elements, thirteen books that taught geometry by proof for more than two thousand years.
Where you meet it at School Launch
- Kindergarten Math 3.1: Circles, Squares, Triangles, Rectangles Grades K–2
- Grade 3 Math 7.2: Area = Length × Width Grades 3–5
- Pre-Algebra 12.6: The Pythagorean Theorem Grades 6–8
- Geometry (the whole course) Grades 9–12
Main branches
Euclidean geometry and proof Grades 9–12
In geometry we do not just measure; we explain why something must always be true.
Going deeper: Euclidean geometry builds theorems about congruence, similarity and circles from definitions, postulates and earlier proofs.
Tiny example: Two triangles with all three sides equal are congruent (SSS).
From history: Thales of Miletus (about 624 – about 547 B.C.) is traditionally credited with the first proofs in Greek geometry.
Where you meet it at School Launch
Length, area and volume Grades 3–5, 9–12
Measuring tells how long a fence is, how much carpet a floor needs and how much a box holds.
Going deeper: Formulas for area and volume come from cutting shapes into pieces we already know how to measure.
Tiny example: A box 4 in by 3 in by 2 in holds 4 × 3 × 2 = 24 cubic inches.
From history: Heron of Alexandria (about A.D. 10 – about 75) gave a formula for the area of a triangle from its three sides.
Where you meet it at School Launch
- Grade 3 Math 7.3: Perimeter Grades 3–5
- Grade 5 Math 6.4: Volume = l × w × h Grades 3–5
- Geometry 12.6: Spheres and Similar Solids Grades 9–12
- Slice the Sphere (explainer) Grades 9–12
Coordinate geometry Grades 3–5, 6–8, 9–12
A grid with two number lines lets us find any point with a pair of numbers, like (3, 2).
Going deeper: Coordinates join algebra and geometry: lines and circles become equations, and geometric facts can be proved by calculation.
Tiny example: The distance from (0, 0) to (3, 4) is 5, by the Pythagorean Theorem.
From history: René Descartes (1596–1650) joined algebra to geometry in La Géométrie (1637).
Where you meet it at School Launch
- Grade 5 Math 7.3: The Coordinate Plane and Patterns Grades 3–5
- Pre-Algebra 11.1: The Coordinate Plane Grades 6–8
- Geometry 13.3: Coordinate Proofs Grades 9–12
- Every Conic Is a Slice of One Cone (explainer) Grades 9–12
Transformations and symmetry Grades 3–5, 9–12
Slides, flips and turns move a shape without changing its size or form.
Going deeper: Transformations give a second way to define congruence and similarity, and they describe the symmetry of patterns.
Tiny example: A butterfly’s wings are mirror images across a line of symmetry.
Where you meet it at School Launch
- Grade 5 Math 7.4: Slides, Flips and Turns on the Grid Grades 3–5
- Geometry 7.2: Reflections and Symmetry Grades 9–12
- Flower Geometry (explainer) Grades 9–12
Non-Euclidean geometry Later: college and beyond
On a ball, "straight" paths curve around, and triangles behave differently than on flat paper.
Going deeper: Changing Euclid’s parallel postulate gives other consistent geometries; one of them describes the surface of the earth.
Tiny example: On a globe, a triangle from the North Pole down to two points on the equator can have three right angles.
From history: Nikolai Lobachevsky (1792–1856) published a geometry in which the parallel postulate fails (1829).
Later: college and beyond. School Launch does not teach this yet; it is part of college mathematics.
A first taste you can find here:
Shape and space
Trigonometry
Grades 9–12Trigonometry uses the angles of triangles to find lengths we cannot measure directly, like the height of a tree.
Going deeper: Its sine and cosine functions also describe anything that repeats in cycles: waves, tides, sound and circular motion.
Tiny example: A ladder 10 ft long leaning at 60° reaches 10 × sin 60° ≈ 8.7 ft up the wall.
From history: Hipparchus of Nicaea (190–120 B.C.) made the first known table for trigonometry, to work out the positions of the sun and moon.
Where you meet it at School Launch
- Geometry 9.5: The Tangent Ratio Grades 9–12
- Trigonometry (the whole course) Grades 9–12
- Pre-Calculus 4.1: The Unit Circle in Radians Grades 9–12
Main branches
Right-triangle trigonometry Grades 9–12
In a right triangle, the angles decide the ratios of the sides.
Going deeper: Sine, cosine and tangent are those ratios; with one side and one angle you can solve the whole triangle.
Tiny example: tan 45° = 1, so a 45° sight line rises 1 ft for every foot across.
Where you meet it at School Launch
- Geometry 9.6: The Sine and Cosine Ratios Grades 9–12
- Trigonometry 1.2: Sine, Cosine and Tangent Grades 9–12
- Trigonometry 8.3: Surveying and Triangulation Grades 9–12
The unit circle and wave graphs Grades 9–12
A point going around a circle goes up and down in a smooth wave.
Going deeper: Defining sine and cosine on the unit circle extends them to every angle and makes them the basic functions of periodic motion.
Tiny example: sin 0° = 0, sin 90° = 1, sin 180° = 0, sin 270° = −1.
Where you meet it at School Launch
- Trigonometry 3.6: The Unit Circle Grades 9–12
- Trigonometry 4.1: The Graph of Sine Grades 9–12
- Circles That Draw (explainer) Grades 9–12
Trigonometric identities Grades 9–12
Identities are facts about angles that are true every time, like a rule you can count on.
Going deeper: The Pythagorean and sum-and-difference identities let you rewrite expressions, prove results and compute exact values.
Tiny example: sin² x + cos² x = 1 for every angle x.
From history: Claudius Ptolemy (about A.D. 85 – about 165) used what we now call the sum and difference rules to build the table of chords in his Almagest.
Where you meet it at School Launch
- Trigonometry 5.2: The Pythagorean Identities Grades 9–12
- Trigonometry 5.4: Sum and Difference Formulas Grades 9–12
Spherical trigonometry Later: college and beyond
Spherical trigonometry works with triangles drawn on a globe, which sailors and astronomers need.
Going deeper: It finds great-circle routes and star positions, where the flat-triangle rules no longer hold.
Tiny example: The shortest flight from Chicago to Rome follows a great circle, which looks curved on a flat map.
From history: Regiomontanus (Johann Müller) (A.D. 1436 – 1476) wrote On Triangles, which treated both flat and spherical triangles.
Later: college and beyond. School Launch does not teach this yet; it is part of college mathematics.
A first taste you can find here:
- Trigonometry 8.2: Bearings and Navigation Grades 9–12
- Trigonometry 8.4: Problems in Three Dimensions Grades 9–12
Shape and space
Topology
Later: college and beyondTopology studies what stays the same when a shape is stretched or bent without tearing, like how many holes it has.
Going deeper: It asks which shapes can be deformed into one another; to a topologist a coffee mug and a doughnut are the same, each with one hole.
Tiny example: A ball and a cube are the same in topology; a ball and a ring are not.
From history: Henri Poincaré (1854–1912) founded algebraic topology in a series of papers that began in 1895.
Later: college and beyond. School Launch does not teach this yet; it is part of college mathematics.
A first taste you can find here:
- Twelve Pentagons (explainer) Grades 9–12
Main branches
Euler’s formula for solids Later: college and beyond
For a solid with flat faces, corners minus edges plus faces always comes out to 2.
Going deeper: V − E + F = 2 holds for every solid without holes, which shows that counting can reveal shape.
Tiny example: A cube: 8 corners − 12 edges + 6 faces = 2.
From history: Leonhard Euler (1707–1783) stated this formula in the 1750s.
Later: college and beyond. School Launch does not teach this yet; it is part of college mathematics.
A first taste you can find here:
- Twelve Pentagons (explainer) Grades 9–12
- Geometry 12.1: Solids, Nets and Cross Sections Grades 9–12
Surfaces and the Möbius strip Later: college and beyond
Give a paper strip a half twist and tape the ends: it now has only one side.
Going deeper: Surfaces are classified by their sides and holes; the Möbius strip is the simplest one-sided surface.
Tiny example: Draw a line down the middle of a Möbius strip and you return to the start without lifting your pencil.
From history: August Möbius (1790–1868) discovered the one-sided strip in 1858.
Later: college and beyond. School Launch does not teach this yet; it is part of college mathematics.
Knot theory Later: college and beyond
Knot theory asks when two tangled loops of string are really the same knot.
Going deeper: Knots are closed loops in space; mathematicians find measurements that do not change however the loop is moved.
Tiny example: A trefoil knot cannot be untangled into a plain circle without cutting it.
From history: Peter Guthrie Tait (1831–1901) drew up the first tables of knots.
Later: college and beyond. School Launch does not teach this yet; it is part of college mathematics.
Manifolds Later: college and beyond
A manifold is a space that looks flat up close, like the earth seems flat to someone standing on it.
Going deeper: Manifolds are the setting for curved space in geometry and physics.
Tiny example: The surface of a ball is a 2-dimensional manifold: every small patch looks like a flat map.
From history: Bernhard Riemann (1826–1866) introduced the idea in a lecture in 1854.
Later: college and beyond. School Launch does not teach this yet; it is part of college mathematics.
Change
Calculus
Grades 9–12Calculus is the mathematics of change: how fast something is moving and how much piles up over time.
Going deeper: Derivatives measure rates of change and integrals add up infinitely many small pieces; the Fundamental Theorem shows they undo each other.
Tiny example: A car’s position is 5t² feet after t seconds; its speed at t = 3 is 30 feet per second.
From history: Gottfried Wilhelm Leibniz (1646–1716) developed calculus at the same time as Isaac Newton and gave us the symbols dy/dx and ∫.
Where you meet it at School Launch
- Pre-Calculus 9.4: From Average to Instantaneous Rate of Change Grades 9–12
- Calculus (the whole course) Grades 9–12
- Calculus 6.3: The Fundamental Theorem of Calculus Grades 9–12
Main branches
Limits Grades 9–12
A limit is the value something gets closer and closer to.
Going deeper: Limits make "infinitely close" precise, and every definition in calculus is built on them.
Tiny example: 1/2, 3/4, 7/8, 15/16 … get closer and closer to 1.
Where you meet it at School Launch
- Pre-Calculus 9.1: Limits from Tables and Graphs Grades 9–12
- Calculus 1.1: Limits Numerically and Graphically Grades 9–12
Derivatives Grades 9–12
A derivative tells how fast something is changing at one exact moment.
Going deeper: The derivative is the slope of a curve at a point; it finds speeds, rates and the highest and lowest points of a graph.
Tiny example: The slope of y = x² at x = 3 is 6.
From history: Isaac Newton (1643–1727) developed his method of "fluxions" for rates of change in the 1660s.
Where you meet it at School Launch
- Calculus 2.1: The Derivative as a Limit Grades 9–12
- Calculus 4.2: Related Rates Grades 9–12
Integrals Grades 9–12
An integral adds up many thin slices to find a total, like an area or a volume.
Going deeper: Definite integrals compute areas, volumes, distances and totals from rates.
Tiny example: The area under y = 2x from 0 to 3 is a triangle: ½ × 3 × 6 = 9.
From history: Bonaventura Cavalieri (1598–1647) found volumes by comparing thin slices of solids.
Where you meet it at School Launch
- Calculus 6.1: Riemann Sums and the Definite Integral Grades 9–12
- Calculus 8.3: Volumes by Disks and Washers Grades 9–12
- Slice the Sphere (explainer) Grades 9–12
Infinite series Grades 9–12
Sometimes adding up endlessly many smaller and smaller numbers gives a plain answer.
Going deeper: Series write functions like sin x and eˣ as endless sums, which is how calculators compute them.
Tiny example: 1/2 + 1/4 + 1/8 + 1/16 + … = 1.
From history: Madhava of Sangamagrama (A.D. 1350 – 1425) found infinite series for sine and cosine in southern India.
Where you meet it at School Launch
- Algebra II 9.4: Infinite Geometric Series Grades 9–12
- Calculus 9.3: Convergence Tests for Series Grades 9–12
- Calculus 9.4: Taylor and Maclaurin Series Grades 9–12
Multivariable and vector calculus Later: college and beyond
This is calculus for things that change in more than one direction at once, like wind across a map.
Going deeper: It extends derivatives and integrals to surfaces and three-dimensional space, the language of fluid flow and electromagnetism.
Tiny example: The temperature on a hill can change differently going north than going east.
From history: Josiah Willard Gibbs (1839–1903) developed the vector notation still used in physics.
Later: college and beyond. School Launch does not teach this yet; it is part of college mathematics.
A first taste you can find here:
Change
Analysis
Grades 9–12Analysis is the careful "why" behind calculus: it proves that the methods of calculus really work.
Going deeper: It studies limits, continuity, infinite series and the real and complex numbers with full rigor.
Tiny example: Between any two fractions there is always another fraction: between 1/2 and 1/3 is 5/12.
From history: Augustin-Louis Cauchy (1789–1857) put calculus on careful foundations in his Cours d’analyse (1821).
Where you meet it at School Launch
- Calculus 1.4: Continuity and the Intermediate Value Theorem Grades 9–12
- Calculus 5.1: The Mean Value Theorem Grades 9–12
Later: college and beyond. Our calculus course gives a first taste; the full subject comes in college.
Main branches
Real analysis Later: college and beyond
Real analysis studies the number line itself, including numbers like √2 that are not fractions.
Going deeper: It defines limits and continuity exactly and proves the theorems that calculus uses.
Tiny example: √2 = 1.41421356… never ends and never repeats.
From history: Karl Weierstrass (1815–1897) set the exact definitions of limit and continuity used today.
Later: college and beyond. School Launch does not teach this yet; it is part of college mathematics.
A first taste you can find here:
- Pre-Algebra 6.5: Estimating Roots and Irrational Numbers Grades 6–8
- Algebra I 1.1: The Real Number System Grades 6–8, 9–12
Complex analysis Later: college and beyond
Complex analysis uses numbers built from i, where i × i = −1.
Going deeper: Calculus with complex numbers is surprisingly powerful and is used in electrical engineering and physics.
Tiny example: (1 + i)² = 1 + 2i + i² = 2i.
From history: Leonhard Euler (1707–1783) found the formula eix = cos x + i sin x (1748), which gives eiπ + 1 = 0.
Later: college and beyond. School Launch does not teach this yet; it is part of college mathematics.
A first taste you can find here:
- Algebra II 3.3: Complex Numbers Grades 9–12
- Pre-Calculus 6.2: Complex Numbers in Polar Form Grades 9–12
Fourier analysis Later: college and beyond
Fourier analysis breaks a complicated wave, like a musical chord, into simple waves.
Going deeper: Any repeating signal can be built from sines and cosines; this underlies sound, image and radio technology.
Tiny example: A note on a violin is a main frequency plus quieter overtones at 2, 3 and 4 times that frequency.
From history: Joseph Fourier (1768–1830) used sums of sine waves to study the flow of heat.
Later: college and beyond. School Launch does not teach this yet; it is part of college mathematics.
A first taste you can find here:
- Circles That Draw (explainer) Grades 9–12
Functional analysis Later: college and beyond
Functional analysis treats whole functions like single points, so it can measure how far apart two curves are.
Going deeper: It studies spaces of functions and is the mathematical setting of quantum mechanics.
Tiny example: The "distance" between the curves y = x and y = x² on 0 to 1 can be measured by the biggest gap, ¼.
From history: Stefan Banach (1892–1945) wrote one of its founding books (1932).
Later: college and beyond. School Launch does not teach this yet; it is part of college mathematics.
Change
Differential Equations
Grades 6–8, 9–12A differential equation is a rule about how something changes, and solving it predicts what happens next.
Going deeper: Physics, engineering, biology and economics write their laws as differential equations, from falling objects to cooling coffee.
Tiny example: "The amount grows at a rate of 5% of itself per year" leads to A = A₀ × e0.05t.
From history: Leonhard Euler (1707–1783) devised the step-by-step method for differential equations taught as "Euler’s method."
Where you meet it at School Launch
- Calculus 7.1: Slope Fields Grades 9–12
- Calculus 7.2: Separation of Variables Grades 9–12
Main branches
Growth and decay Grades 6–8, 9–12
Some things grow or shrink in proportion to how much there is, like money earning interest.
Going deeper: The equation dy/dt = ky gives exponential growth or decay; logistic models add a limit to growth.
Tiny example: A culture that doubles every hour grows from 100 to 800 in 3 hours.
Where you meet it at School Launch
- Algebra I 7.5: Exponential Growth and Decay Grades 6–8, 9–12
- Algebra II 7.4: Logistic Functions Grades 9–12
- Calculus 7.3: Exponential Growth and Decay Grades 9–12
Ordinary differential equations Grades 9–12
These equations describe one thing changing over time, like a swinging pendulum.
Going deeper: Slope fields show solutions at a glance, and numerical methods find them when no formula exists.
Tiny example: A small pendulum 1 m long swings back and forth about once every 2 seconds.
Where you meet it at School Launch
- Calculus 7.1: Slope Fields Grades 9–12
- Calculus 7.4: Euler's Method and Logistic Growth Grades 9–12
- Physics 9.1: Oscillations: Huygens and the Pendulum Clock Grades 9–12
Partial differential equations Later: college and beyond
These equations describe things that change in space and time together, like a ripple spreading on a pond.
Going deeper: The heat, wave and fluid equations are partial differential equations.
Tiny example: Pluck a guitar string and the wave equation predicts its shape at every moment.
From history: Sofia Kovalevskaya (1850–1891) proved a key theorem about when such equations have solutions.
Later: college and beyond. School Launch does not teach this yet; it is part of college mathematics.
A first taste you can find here:
- Sound and Waves Explorer (explainer) Grades 9–12
Chance and data
Probability
Grades 3–5, 6–8, 9–12Probability measures how likely something is, from impossible to certain.
Going deeper: It gives exact rules for chance, used in insurance, science, weather forecasts and wise decisions.
Tiny example: A fair die shows a 6 with probability 1/6.
From history: Blaise Pascal (1623–1662) worked out the first rules of probability in letters with Pierre de Fermat in 1654.
Where you meet it at School Launch
- Grade 4 Math 9.2: Likely and Unlikely Grades 3–5
- Playing the Odds (Grades 4–7) 1.3: Pascal and Fermat: The Unfinished Game Grades 3–5, 6–8
- Statistics, Unit 6: Probability Grades 9–12
Main branches
Chance and likelihood Grades 3–5, 6–8
Some events are likely and some are unlikely, and a fraction can say exactly how likely.
Going deeper: When outcomes are equally likely, probability is favorable outcomes over total outcomes.
Tiny example: With two dice, a total of 7 comes up 6 ways out of 36: probability 1/6.
Where you meet it at School Launch
Expected value Grades 6–8, 9–12
Expected value is what you would get on average if you could play a game over and over.
Going deeper: Multiply each outcome by its probability and add; it is the basis of fair prices and good decisions under risk.
Tiny example: Win $6 on a roll of 6, nothing otherwise: expected value 6 × 1/6 = $1 per roll.
From history: Christiaan Huygens (1629–1695) wrote the first printed book on probability (1657).
Where you meet it at School Launch
The law of large numbers Grades 3–5, 6–8, 9–12
Flip a coin a few times and anything can happen; flip it thousands of times and close to half will be heads.
Going deeper: Averages of many independent trials settle near the expected value, which is why casinos and insurers can plan.
Tiny example: 10 flips might give 7 heads; 10,000 flips will very likely give between 4,900 and 5,100.
From history: Jacob Bernoulli (1655–1705) proved the first version in Ars Conjectandi, published in 1713.
Where you meet it at School Launch
- Grade 5 Math 9.3: Probability Experiments Grades 3–5
- Playing the Odds (Grades 8–12) 1.5: The Law of Large Numbers Grades 6–8, 9–12
Conditional probability Grades 6–8, 9–12
New information can change how likely something is.
Going deeper: Conditional probability and Bayes’ theorem update a probability when evidence comes in.
Tiny example: Draw a card: P(king) = 4/52, but if you know it is a face card, P(king) = 4/12.
From history: Thomas Bayes (1702–1761) a Presbyterian minister, wrote the theorem that bears his name, published in 1763.
Where you meet it at School Launch
- Statistics 6.3: Conditional Probability Grades 9–12
- Playing the Odds (Grades 8–12) 1.4: What Do You Know Now? Grades 6–8, 9–12
Random variables and distributions Grades 9–12
A distribution shows all the possible results of a chance process and how likely each one is.
Going deeper: Binomial and normal distributions model counts and measurements; the bell curve appears again and again.
Tiny example: Flip 4 coins: 0, 1, 2, 3 or 4 heads happen 1, 4, 6, 4 and 1 times out of 16.
From history: Abraham de Moivre (1667–1754) found the bell-shaped normal curve as an approximation for coin-flip counts (1733).
Where you meet it at School Launch
Random processes Later: college and beyond
A random process is chance unfolding step by step, like a board game where each move depends on a die.
Going deeper: Markov chains and other random processes model queues, genetics, weather and web search.
Tiny example: If rain today makes rain tomorrow 60% likely, you can chain the days to forecast the week.
From history: Andrey Markov (1856–1922) introduced the chains named for him in 1906.
Later: college and beyond. School Launch does not teach this yet; it is part of college mathematics.
A first taste you can find here:
- Statistics 6.1: Randomness and Simulation Grades 9–12
Chance and data
Statistics
Grades K–2, 3–5, 6–8, 9–12Statistics is collecting and reading data to learn something true, and to spot when numbers are being used to mislead.
Going deeper: It designs surveys and experiments, summarizes data and measures how sure a conclusion can be.
Tiny example: Test scores 70, 80 and 90 have a mean of 80.
From history: Florence Nightingale (1820–1910) used carefully drawn charts to show that most soldiers in the Crimean War died of preventable disease, and won reforms in army hospitals.
Where you meet it at School Launch
- Grade 1 Math 8.4: Tally Charts and Bar Graphs Grades K–2
- Grade 5 Math 8.2: Line Plots and Averages Grades 3–5
- Statistics (the whole course) Grades 9–12
- Statistics 14.1: Graphs That Deceive Grades 9–12
Main branches
Data and graphs Grades K–2, 3–5, 9–12
Tally charts, bar graphs and line plots turn a pile of numbers into a picture.
Going deeper: Choosing the right display, and reading its scale, is the first step of every statistical question.
Tiny example: A bar graph of favorite fruits shows at a glance that apples beat pears 9 to 4.
Where you meet it at School Launch
- Grade 2 Math 8.4: Pictographs and Line Plots Grades K–2
- Grade 3 Math 8.4: Scaled Bar Graphs Grades 3–5
- Statistics 1.2: Displaying Categorical Data Grades 9–12
Averages and spread Grades 6–8, 9–12
An average tells a typical value, and the spread tells how far the data scatter.
Going deeper: Mean, median, quartiles and standard deviation summarize a whole data set in a few numbers.
Tiny example: 2, 3, 3, 4, 13: the median is 3 but the mean is 5, pulled up by the 13.
Where you meet it at School Launch
- Pre-Algebra 13.1: Mean, Median, Mode and Range Grades 6–8
- Statistics 2.4: Standard Deviation Grades 9–12
Sampling and experiments Grades 9–12
You can learn about a whole crowd by asking a fair, random handful of people.
Going deeper: Random sampling and controlled experiments are how statistics avoids bias and finds causes.
Tiny example: Drawing names from a hat gives every student the same chance to be picked.
Where you meet it at School Launch
- Statistics 5.2: Random Sampling Methods Grades 9–12
- Statistics 5.5: Designing Experiments Grades 9–12
Correlation and regression Grades 6–8, 9–12
Some things go up together, like height and shoe size, and a line can show the trend.
Going deeper: Regression fits the best line through data, and correlation measures how tightly the points follow it.
Tiny example: If each extra hour of practice adds about 3 points, the line’s slope is 3.
From history: Adrien-Marie Legendre (1752–1833) published the method of least squares for fitting lines to data (1805).
Where you meet it at School Launch
- Algebra I 13.2: Lines of Best Fit Grades 6–8, 9–12
- Statistics 4.3: The Least-Squares Regression Line Grades 9–12
Statistical inference Grades 9–12
Inference uses a sample to make a careful guess about everyone, with an honest margin of error.
Going deeper: Confidence intervals and significance tests say how much a sample can tell us about the whole population.
Tiny example: "52% ± 3%" means the true value is probably between 49% and 55%.
Where you meet it at School Launch
Reading statistics wisely Grades 6–8, 9–12
Graphs and averages can trick us, so it pays to ask who was counted and how.
Going deeper: Cherry-picking, stretched scales and confusing correlation with cause are common ways numbers mislead.
Tiny example: A bar graph whose scale starts at 90 can make 92 look twice as big as 91.
Where you meet it at School Launch
- Statistics 14.8: Taking Apart a Statistical Claim Grades 9–12
- Logic 8.8: Refuting a Statistical Claim Grades 6–8, 9–12
Reasoning
Logic and Set Theory
Grades K–2, 6–8, 9–12Logic is the study of good reasoning: how to tell when a conclusion really follows from what we know.
Going deeper: Mathematical logic and set theory give exact rules for proof and lay the foundations on which the rest of mathematics is built.
Tiny example: All squares are rectangles; this shape is a square; so it is a rectangle.
From history: Aristotle (384–322 B.C.) wrote the first systematic study of logic, built around the syllogism.
Where you meet it at School Launch
- Logic (the whole course) Grades 6–8, 9–12
- Geometry 1.4: The Laws of Logic Grades 9–12
- Logic 6.2: Truth Tables for the Connectives Grades 6–8, 9–12
Main branches
Classical logic and the syllogism Grades 6–8, 9–12
A syllogism joins two statements to reach a third, like a chain with three links.
Going deeper: Categorical logic sorts arguments into valid and invalid forms and names the common fallacies.
Tiny example: No reptiles have fur; all snakes are reptiles; so no snakes have fur.
From history: Charles Dodgson (Lewis Carroll) (1832–1898) wrote logic textbooks full of playful syllogism puzzles.
Where you meet it at School Launch
- Logic 5.1: The Syllogism Grades 6–8, 9–12
- Logic 5.7: Chains of Reasoning: Sorites and Carroll's Puzzles Grades 6–8, 9–12
- Logic 3.1: What Is a Fallacy? Grades 6–8, 9–12
Symbolic logic Grades 6–8, 9–12
Symbolic logic writes reasoning with letters and signs, the way algebra writes numbers.
Going deeper: Truth tables test arguments mechanically, and the same rules run every computer chip.
Tiny example: "p and q" is true only when p and q are both true.
From history: George Boole (1815–1864) turned logic into an algebra in The Laws of Thought (1854).
Where you meet it at School Launch
Proof Grades 9–12
A proof is an explanation so careful that it shows something must be true every time.
Going deeper: Direct proof, indirect proof and mathematical induction are the main tools for proving theorems.
Tiny example: Induction: if a ladder’s first rung can be reached, and each rung leads to the next, every rung can be reached.
From history: Giuseppe Peano (1858–1932) wrote down axioms for the counting numbers, with induction among them (1889).
Where you meet it at School Launch
Set theory Grades K–2, 6–8, 9–12
A set is a collection of things, and sorting things into groups is the start of set theory.
Going deeper: Sets, unions, intersections and Venn diagrams are the common language of modern mathematics, and set theory even compares the sizes of infinite collections.
Tiny example: {2, 4, 6} and {4, 5, 6} share {4, 6}.
From history: Georg Cantor (1845–1918) founded set theory and showed that some infinite sets are larger than others.
Where you meet it at School Launch
- Kindergarten Math 3.4: Sorting and Patterns Grades K–2
- Logic 4.7: Venn Diagrams for Propositions Grades 6–8, 9–12
- Algebra I 1.1: The Real Number System Grades 6–8, 9–12
Computability Later: college and beyond
Computability asks which problems a step-by-step machine can solve at all.
Going deeper: Some precisely stated problems can never be solved by any algorithm; this sets limits on what computers can do.
Tiny example: No program can check every other program and say for certain whether it will ever stop.
From history: Alan Turing (1912–1954) described a simple imaginary machine that captures what "computable" means (1936).
Later: college and beyond. School Launch does not teach this yet; it is part of college mathematics.
A first taste you can find here:
- Logic 6.9: Logic and Machines Grades 6–8, 9–12
Reasoning
Combinatorics and Graph Theory
Grades K–2, 3–5, 6–8, 9–12This part of math is about counting arrangements and studying networks of dots and lines.
Going deeper: Often called discrete mathematics, it deals with separate, countable things, and it is the mathematics behind computer science.
Tiny example: 3 shirts and 2 pairs of pants make 3 × 2 = 6 outfits.
From history: Leonhard Euler (1707–1783) showed in 1736 that no walk in Königsberg could cross each of its seven bridges exactly once, the first result in graph theory.
Where you meet it at School Launch
Main branches
Counting and arrangements Grades 6–8, 9–12
Clever counting finds how many ways something can happen without listing them all.
Going deeper: Permutations count arrangements where order matters and combinations count choices where it does not.
Tiny example: 3 children can line up in 3 × 2 × 1 = 6 ways.
From history: Thomas Kirkman (1806–1895) posed the "fifteen schoolgirls" puzzle (1850), an early problem in arranging groups.
Where you meet it at School Launch
- Pre-Algebra 13.4: Counting and Compound Events Grades 6–8
- Algebra II 10.2: Combinations Grades 9–12
- Playing the Odds (Grades 8–12) 1.2: Counting the Ways Grades 6–8, 9–12
Pascal’s triangle Grades 9–12
In Pascal’s triangle each number is the sum of the two above it.
Going deeper: Its rows count combinations and give the coefficients of the binomial theorem.
Tiny example: Row 4 is 1, 4, 6, 4, 1, and (a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴.
Where you meet it at School Launch
Graph theory Later: college and beyond
A graph is a set of dots joined by lines, like towns joined by roads.
Going deeper: Graph theory studies routes, networks and connections, from delivery routes to computer networks.
Tiny example: You can trace a square with one diagonal without lifting your pencil or going over a line twice, but not a square with both diagonals.
From history: William Rowan Hamilton (1805–1865) invented a puzzle (1857) about visiting every corner of a twelve-faced solid exactly once.
Later: college and beyond. School Launch does not teach this yet; it is part of college mathematics.
Algorithms Grades K–2, 3–5, 6–8
An algorithm is a list of exact steps that always gets the job done, like a recipe.
Going deeper: Designing fast, correct algorithms is the heart of computer science.
Tiny example: To find the largest number in a list, keep the biggest so far as you read each number.
From history: Ada Lovelace (1815–1852) published a step-by-step method for Charles Babbage’s planned Analytical Engine (1843).
Where you meet it at School Launch
Math at work
Mathematical Physics
Grades 3–5, 6–8, 9–12Mathematical physics uses math to describe how the world works: motion, light, sound and electricity.
Going deeper: Its equations, from Newton’s laws to Maxwell’s, predict how the created world behaves with remarkable accuracy.
Tiny example: Force = mass × acceleration: 2 kg pushed with 10 newtons speeds up by 5 m/s every second.
From history: Isaac Newton (1643–1727) set out the laws of motion and gravity in the Principia (1687).
Where you meet it at School Launch
- Grade 6 Engineering & Physics 1.2: Newton’s Laws with Numbers: F = ma Grades 6–8
- Physics 3.1: The Principia and Newton’s Three Laws Grades 9–12
- How Fast Are We Moving? (explainer) Grades 6–8, 9–12
Main branches
Motion and forces Grades 3–5, 9–12
Mechanics explains how things move and what makes them speed up, slow down or turn.
Going deeper: Position, velocity and acceleration are linked by calculus, and forces explain the changes.
Tiny example: A dropped ball falls about 16 feet in the first second.
From history: Galileo Galilei (1564–1642) studied falling bodies and wrote Two New Sciences (1638).
Where you meet it at School Launch
- Grade 5 Engineering & Physics 1.2: Newton’s Three Laws and a Balloon Rocket Grades 3–5
- Pre-Calculus 5.4: Projectile Motion Grades 9–12
- Calculus 4.1: Motion Along a Line Grades 9–12
- The Fastest Way Down (explainer) Grades 9–12
Gravity and orbits Grades 3–5, 9–12
Gravity keeps the moon circling the earth and the earth circling the sun.
Going deeper: Planets move in ellipses, and Newton’s law of gravity explains why.
Tiny example: The moon takes about 27 days to go once around the earth.
From history: Johannes Kepler (1571–1630) discovered that planets travel in ellipses (1609).
Where you meet it at School Launch
Waves and sound Grades 9–12
Sound, light and ripples on water are waves that carry energy from place to place.
Going deeper: Sine waves model vibrations, and frequency and wavelength are tied together by the wave’s speed.
Tiny example: Sound travels about 343 meters per second in air, so thunder 3 seconds after lightning is about 1 km away.
Where you meet it at School Launch
- Trigonometry 8.5: Modeling Periodic Motion Grades 9–12
- Physics 9.2: Waves and Sound: The Slinky and Chladni’s Plates Grades 9–12
- Sound and Waves Explorer (explainer) Grades 9–12
Electricity and magnetism Grades 3–5, 9–12
Electricity and magnetism are two sides of one force that runs motors, radios and lights.
Going deeper: Four equations describe every electric and magnetic effect and predict that light is an electromagnetic wave.
Tiny example: Ohm’s law: 9 volts across 3 ohms drives 3 amps.
From history: James Clerk Maxwell (1831–1879) wrote the equations of electromagnetism (1865).
Where you meet it at School Launch
Fluids Later: college and beyond
Fluids are liquids and gases, and math explains how they flow around ships and wings.
Going deeper: Fluid dynamics links speed and pressure and is used to design aircraft, pipes and dams.
Tiny example: Water speeds up where a pipe narrows, and its pressure drops there.
From history: Daniel Bernoulli (1700–1782) published Hydrodynamica (1738) on the motion of fluids.
Later: college and beyond. School Launch does not teach this yet; it is part of college mathematics.
A first taste you can find here:
Quantum mechanics Later: college and beyond
Quantum mechanics describes the tiniest things, like atoms, which behave in surprising ways.
Going deeper: Its wave equations predict the colors of light that atoms give off and how electronics work.
Tiny example: A hydrogen atom gives off only certain colors of light, never the ones in between.
Later: college and beyond. School Launch does not teach this yet; it is part of college mathematics.
A first taste you can find here:
- Quantum Explorer (explainer) Grades 9–12
Math at work
Applied Mathematics
Grades 3–5, 6–8, 9–12Applied mathematics puts math to work on real problems: planning, building, deciding and sending messages.
Going deeper: It builds models, finds the best choices, studies strategy and secures and corrects information.
Tiny example: Which is the better buy: 12 oz for $3 (25¢ an ounce) or 20 oz for $4.40 (22¢ an ounce)?
From history: Archimedes of Syracuse (287–212 B.C.) used mathematics to explain levers and floating bodies and to design machines.
Where you meet it at School Launch
- Grade 5 Math 8.4: Money and Budgets Grades 3–5
- Algebra I 5.7: Linear Models Grades 6–8, 9–12
- Calculus 5.4: Optimization Grades 9–12
Main branches
Mathematical modeling Grades 6–8, 9–12
A model is a math picture of a real situation that helps us predict what will happen.
Going deeper: Modeling chooses variables, writes equations, tests them against data and improves them.
Tiny example: A taxi charges $3 plus $2 per mile: cost = 3 + 2m.
Where you meet it at School Launch
- Algebra I 5.7: Linear Models Grades 6–8, 9–12
- Algebra II 3.6: Quadratic Models Grades 9–12
- Trigonometry 8.5: Modeling Periodic Motion Grades 9–12
Optimization Grades 9–12
Optimization finds the best choice: the shortest route, the lowest cost or the biggest garden.
Going deeper: Calculus finds highest and lowest points, and linear programming handles many limits at once.
Tiny example: With 20 m of fence, a 5 m by 5 m square encloses the most area for a rectangle: 25 m².
From history: George Dantzig (1914–2005) invented the simplex method for linear programming (1947).
Where you meet it at School Launch
- Algebra II 2.3: Linear Programming Grades 9–12
- Calculus 5.4: Optimization Grades 9–12
Game theory Grades 6–8, 9–12
Game theory is the math of choices when the other side is also thinking ahead.
Going deeper: It studies strategy in games, business and war, including when to mix up your choices.
Tiny example: In rock-paper-scissors, the best plan is to pick each one a third of the time at random.
From history: John von Neumann (1903–1957) proved the minimax theorem (1928) and later wrote the first big book on the subject with Oskar Morgenstern (1944).
Where you meet it at School Launch
- Strategy: Games, Conflict and Decision 7.1: Zero-Sum Games and Minimax Grades 9–12
- Strategy: Games, Conflict and Decision 7.3: The Prisoner’s Dilemma and the Evolution of Cooperation Grades 9–12
- Playing the Odds (Grades 8–12) 4.3: Keep Them Guessing Grades 6–8, 9–12
- Economics & the History of Money 3.3: Oligopoly and Game Theory: Railroad Pools and the Prisoner’s Dilemma Grades 9–12
Information and coding theory Grades 6–8, 9–12
Coding theory turns messages into 0s and 1s and adds checks so mistakes can be caught.
Going deeper: Information theory measures information in bits; error-correcting codes protect data on phones, discs and spacecraft.
Tiny example: Send 1011 with an extra 1 so the number of 1s is even (10111); one flipped bit is noticed.
From history: Claude Shannon (1916–2001) founded information theory in "A Mathematical Theory of Communication" (1948).
Where you meet it at School Launch
Cryptography Grades 9–12
Cryptography is the math of secret codes.
Going deeper: Modern codes rest on number theory: problems easy to do one way and very hard to undo.
Tiny example: Shift each letter 3 places: CAT becomes FDW.
Where you meet it at School Launch
Numerical analysis and computing Grades 6–8, 9–12
Numerical analysis finds very good approximate answers when an exact one is hard to get.
Going deeper: It designs step-by-step methods for computers and measures how accurate they are.
Tiny example: To estimate √2: guess 1.5, then average 1.5 and 2 ÷ 1.5 to get about 1.4167; √2 ≈ 1.4142.
From history: Thomas Simpson (1710–1761) gave his name to a rule for estimating areas under curves.
Where you meet it at School Launch
- Pre-Algebra 6.6: Square Roots by Hand Grades 6–8
- Calculus 7.4: Euler's Method and Logistic Growth Grades 9–12
- Calculus 6.1: Riemann Sums and the Definite Integral Grades 9–12
How this map is arranged
No map of mathematics is the only right one; the areas overlap and borrow from each other all the time. This one keeps to sixteen main areas grouped in seven families. Number theory has its own area, and prime numbers and equations with whole-number answers belong to it. Logic and set theory share one area because together they are the foundations of mathematics. Combinatorics and graph theory share one area, often called discrete mathematics. Probability and statistics are kept separate: one reasons forward from chance, the other backward from data.
Where School Launch does not teach an area, the map says so plainly: Later: college and beyond. Sometimes a lesson gives a first taste, and that is listed too.
Sources for the dates
Dates and accomplishments were checked against the biographies in the MacTutor History of Mathematics Archive (University of St Andrews). Many early dates are known only roughly, so they are marked "about."
- Euclid of Alexandria (lived about 300 B.C.)
- Diophantus of Alexandria (about A.D. 200 – about 284)
- Heron of Alexandria (about A.D. 10 – about 75)
- Aristotle (384–322 B.C.)
- Stefan Banach (1892–1945)
- Thomas Bayes (1702–1761)
- Jacob Bernoulli (1655–1705)
- Daniel Bernoulli (1700–1782)
- George Boole (1815–1864)
- Brahmagupta (A.D. 598 – 670)
- Georg Cantor (1845–1918)
- Gerolamo Cardano (1501–1576)
- Augustin-Louis Cauchy (1789–1857)
- Bonaventura Cavalieri (1598–1647)
- Arthur Cayley (1821–1895)
- Gabriel Cramer (1704–1752)
- George Dantzig (1914–2005)
- René Descartes (1596–1650)
- Charles Dodgson (Lewis Carroll) (1832–1898)
- Eratosthenes (276–194 B.C.)
- Leonhard Euler (1707–1783)
- Leonardo of Pisa, called Fibonacci (about A.D. 1170 – about 1250)
- Joseph Fourier (1768–1830)
- Galileo Galilei (1564–1642)
- Évariste Galois (1811–1832)
- Carl Friedrich Gauss (1777–1855)
- Josiah Willard Gibbs (1839–1903)
- William Rowan Hamilton (1805–1865)
- Christiaan Huygens (1629–1695)
- Johannes Kepler (1571–1630)
- Thomas Kirkman (1806–1895)
- Sofia Kovalevskaya (1850–1891)
- Adrien-Marie Legendre (1752–1833)
- Gottfried Wilhelm Leibniz (1646–1716)
- Nikolai Lobachevsky (1792–1856)
- Ada Lovelace (1815–1852)
- Andrey Markov (1856–1922)
- James Clerk Maxwell (1831–1879)
- Thales of Miletus (about 624 – about 547 B.C.)
- Abraham de Moivre (1667–1754)
- August Möbius (1790–1868)
- John Napier (1550–1617)
- John von Neumann (1903–1957)
- Isaac Newton (1643–1727)
- Hipparchus of Nicaea (190–120 B.C.)
- Florence Nightingale (1820–1910)
- Emmy Noether (1882–1935)
- Blaise Pascal (1623–1662)
- Giuseppe Peano (1858–1932)
- Henri Poincaré (1854–1912)
- Claudius Ptolemy (about A.D. 85 – about 165)
- Regiomontanus (Johann Müller) (A.D. 1436 – 1476)
- Bernhard Riemann (1826–1866)
- Madhava of Sangamagrama (A.D. 1350 – 1425)
- Claude Shannon (1916–2001)
- Thomas Simpson (1710–1761)
- Archimedes of Syracuse (287–212 B.C.)
- Peter Guthrie Tait (1831–1901)
- Alan Turing (1912–1954)
- François Viète (1540–1603)
- Karl Weierstrass (1815–1897)
- Muhammad ibn Musa al-Khwarizmi (about A.D. 780 – about 850)