The map of mathematics
Mathematicians sort their research into the Mathematics Subject Classification: 63 areas, split into 534 sections. This page lists every section, says its key idea in one sentence, suggests an elementary way in for a curious kid, and links the sheets and demos where the idea is already planted.
The bet behind it: almost every part of mathematics has an intuitive example a 12-year-old can hold. Where we think that is hard, we say so.
How to read this page
- ● Main idea: at least one sheet or demo is mostly about this section. ◐ Touched: a sheet uses the idea in an exercise or stretch. ○ Not yet: no sheet plants it yet.
- Way in is a suggested elementary entry point, with the lowest grade we think it works for. Natural fit means a normal sheet could do it. Stretch means a glimpse, or real care. Hard means an honest elementary taste is difficult.
- Shaded links are sheets where the section is the main idea; outlined links only touch it. G8 means grade 8.
- Plain school arithmetic has no section of its own, so sheets on fractions, percentages and estimation mostly sit under 00A or 11A.
- The sorting was done with AI help and checked by eye, so some calls are judgement. Corrections are welcome as GitHub issues.
00 General and overarching topics; collections 1 of 2
General topics that cut across all of math(s), like recreational math(s), popular articles and math(s) education material.
Natural fit Way in (grade 5): Pick any classic recreational puzzle, such as the Monty Hall problem, and let kids play it 20 times with cups and a coin to see who should switch.
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Bookkeeping: conference proceedings and collected volumes, not a topic in its own right.
01 History and biography 1 of 1
Math(s) has a history: ideas were discovered by real people, in many cultures, often slowly and with mistakes along the way.
Natural fit Way in (grade 5): Multiply 23 by 47 the ancient Egyptian way, by doubling and adding, then compare it with the method you learned in school.
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03 Mathematical logic and foundations 4 of 8
What is a proof, what makes math(s) true, and are mathematical objects discovered or invented?
Stretch Way in (grade 7): Debate: if nobody had ever counted, would 7 still be prime? Then read the liar sentence "This sentence is false" and decide if it is true.
Logic studies which arguments are valid by their form alone, regardless of what the words mean.
Natural fit Way in (grade 5): Knights always tell the truth, knaves always lie. A says "B is a knave." B says "We are both knights." Who is who? Build a truth table to check.
Model theory asks which structures make a set of rules true, and what the rules alone can and cannot force.
Stretch Way in (grade 9): Write three rules for a "friendship" relation, then find two very different groups of people (or dot pictures) that both obey every rule.
Some well-posed yes/no questions can never be answered by any computer program, no matter how clever.
Natural fit Way in (grade 8): Play the halting game: a program that does the opposite of what a predictor says about it. Show by a short story why no perfect predictor can exist.
Set theory builds all of math(s) from collections, and shows some infinities are strictly bigger than others.
Natural fit Way in (grade 7): Try to list every infinite string of 0s and 1s. Flip the diagonal digits of any list to build a string that is missing: Cantor's diagonal trick.
Proof theory studies proofs themselves as objects, and constructive math(s) asks for proofs that actually build what they claim exists.
Stretch Way in (grade 10): Prove there exist irrational a, b with a^b rational using root 2 to the root 2, then notice the proof never tells you which case holds.
Algebraic logic turns logical reasoning into algebra, so AND, OR and NOT obey equations you can calculate with.
Stretch Way in (grade 7): Let 1 mean true and 0 false. Check that AND is multiplication and that NOT x is 1 - x, then simplify a knight-and-knave puzzle by algebra.
Nonstandard models add infinitely small and infinitely large numbers to ordinary numbers while keeping all the usual rules working.
Stretch Way in (grade 10): Is 0.999... less than 1 by an infinitely tiny amount? Argue both sides, then see how a world with infinitesimals makes the slope of x squared exactly 2x.
05 Combinatorics 5 of 5
Counting cleverly: find how many arrangements there are without listing them all.
Natural fit Way in (grade 5): How many ways can you climb 10 stairs taking 1 or 2 steps at a time? Count small cases, spot the Fibonacci pattern, and explain why it works.
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Designs are arrangements where every pair or group appears together exactly the same number of times.
Natural fit Way in (grade 6): Arrange 9 kids into groups of 3 over 4 days so every pair of kids shares a group exactly once (a small Kirkman schoolgirl puzzle). The card game Dobble works the same way.
Graph theory studies dots joined by lines, capturing networks, maps, friendships and routes.
Natural fit Way in (grade 5): Try to draw the envelope shape without lifting your pencil or retracing a line. Count lines at each corner to predict when it is possible, as Euler did for Königsberg's bridges.
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Extremal combinatorics asks how big something can be before a pattern is forced to appear.
Natural fit Way in (grade 6): Among any 6 people, three are mutual friends or three are mutual strangers. Colour the 15 lines between 6 dots red or blue and try to avoid a single-coloured triangle.
Algebraic combinatorics uses algebra, symmetry and polynomials to count and to reveal hidden structure in arrangements.
Stretch Way in (grade 8): How many different necklaces of 4 beads in 2 colours are there if rotations count as the same? Count by averaging over the 4 rotations.
06 Order, lattices, ordered algebraic structures 4 of 6
An order says when one thing comes before another, even if some pairs cannot be compared at all.
Natural fit Way in (grade 5): Draw the getting-dressed diagram: socks before shoes, underwear before trousers. Find every valid dressing order and notice some items are incomparable.
A lattice is an order where every two elements have a best common lower bound and a best common upper bound.
Natural fit Way in (grade 6): Draw the divisors of 12 in layers, with a line up when one divides the next. For any two, find their gcd below and their lcm above.
Modular and complemented lattices are orders with extra balance rules, like those followed by subspaces in geometry.
Hard Way in (grade 11): List the subspaces of the plane: just the origin, every line through it, and the whole plane. Check that each line has a "complement" line, and many choices work.
Distributive lattices are orders where meet and join distribute over each other, just like AND and OR do.
Stretch Way in (grade 8): Using divisors of 30 with gcd and lcm, check that gcd(a, lcm(b,c)) equals lcm(gcd(a,b), gcd(a,c)) for several triples. Then find a lattice drawing where it fails.
Boolean algebra is the algebra of true and false, or of subsets, and it runs every digital circuit.
Natural fit Way in (grade 6): Shade Venn diagrams to check that NOT (A or B) is the same as (NOT A) and (NOT B). Then build that rule from switches in a light circuit.
Ordered algebraic structures combine arithmetic with an order, asking which ordering rules fit together with adding and multiplying.
Stretch Way in (grade 8): Show that the clock numbers mod 5 cannot be ordered so that adding 1 always makes things bigger: going round five times returns you to the start.
08 General algebraic systems 1 of 3
Universal algebra studies any set with operations, looking for laws shared by numbers, symmetries, words and more.
Natural fit Way in (grade 6): Invent an operation on {rock, paper, scissors}: the winner of the pair. Test whether it is commutative and associative, and compare with ordinary addition.
A variety is the family of all structures obeying the same list of equations, such as all groups or all lattices.
Stretch Way in (grade 8): Pick the single law a*b = b*a. Find three very different things that obey it (adding numbers, mixing paints, taking the max) and one that fails (subtraction).
Other kinds of algebraic systems: partial operations, structures with relations as well as operations, and more general ones.
Hard Way in (grade 9): Colour mixing as a partial operation: some pairs combine and some are not allowed. Make a table and ask which algebra laws still make sense.
11 Number theory 17 of 20
The basic structure of whole numbers: primes, divisibility, remainders and the fact that every number factors uniquely into primes.
Natural fit Way in (grade 5): Run the sieve of Eratosthenes on 1 to 100 and circle the primes. Then explain why the divisibility test for 9 works by adding digits.
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Number sequences and special sets of integers, and the patterns, growth and recurrences they follow.
Natural fit Way in (grade 5): Write Fibonacci numbers and divide each by the one before. Watch the ratios settle near 1.618, and check every third Fibonacci number is even.
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Polynomials and matrices with whole-number or rational entries, and what number-theoretic rules they obey.
Natural fit Way in (grade 9): Can 2x squared + 3x + 7 be factored using whole numbers? Use the rational root test: only a few candidates like 1, 7, 1/2, 7/2 need trying.
Diophantine equations ask for whole-number solutions to equations, which can be surprisingly hard or impossible.
Natural fit Way in (grade 5): With only 5p and 7p stamps, what postage amounts can you make? Find the largest you cannot (23p) and show why everything bigger works.
Quadratic forms ask which numbers can be written as expressions like x squared + y squared, and how such forms behave.
Natural fit Way in (grade 7): Which numbers up to 50 are a sum of two squares? Mark them and spot that primes leaving remainder 3 when divided by 4 never appear.
Automorphic forms are functions with an enormous amount of symmetry, and they secretly encode deep number patterns.
Hard Way in (grade 11): Look at an Escher-style tiling of the disc by triangles. A modular form is a function that repeats across every tile; colour a few tiles to see the symmetry.
Arithmetic geometry studies rational points on curves, using the shape of a curve to understand its whole-number or fraction solutions.
Stretch Way in (grade 9): Find all rational points on the circle x squared + y squared = 1 by drawing lines of rational slope through (-1, 0). Out pop the Pythagorean triples.
The geometry of numbers counts and uses lattice points, the dots with whole-number coordinates, inside shapes.
Natural fit Way in (grade 6): On dot paper draw a polygon with corners on dots. Count interior and boundary dots and check Pick's formula: area = interior + boundary/2 - 1.
How well can irrational numbers be approximated by fractions, and which numbers are not roots of any polynomial?
Natural fit Way in (grade 8): Find fractions close to pi with small denominators. Why is 22/7 so good, and 355/113 astonishingly good? Use continued fractions on a calculator.
Studies how sequences spread out modulo 1, and which digit patterns are typical for almost every number.
Natural fit Way in (grade 7): Write the first digits of 2, 4, 8, 16, ... up to 2^30 and tally them. Leading 1s appear about 30% of the time: Benford's law.
Exponential and character sums add up spinning arrows whose angles depend on numbers, and measure how random those numbers look.
Hard Way in (grade 10): Draw arrows of length 1 at angles n squared times 10 degrees for n = 1 to 36, joined head to tail. Watch a curly pattern rather than a random walk.
Zeta and L-functions package information about primes into a single function whose zeros control how primes are spread out.
Stretch Way in (grade 9): Add 1 + 1/4 + 1/9 + 1/16 + ... on a calculator and watch it approach pi squared over 6, a surprise linking squares to circles.
How primes and divisors are distributed among the integers, such as roughly how many primes lie below a given number.
Natural fit Way in (grade 9): Count primes below 100, 1000 and 10000 (from a table) and compare with n divided by the natural log of n. The ratio creeps towards 1.
Additive number theory asks how numbers can be written as sums, such as sums of primes, squares or smaller whole numbers.
Natural fit Way in (grade 5): List all ways to write 1 to 7 as sums of positive whole numbers, ignoring order. Check that partitions into odd parts equal partitions into distinct parts.
Algebraic number theory enlarges the integers, for example with square roots, and studies how primes factor in these bigger number systems.
Stretch Way in (grade 10): In numbers a + b root(-5), show 6 = 2 x 3 = (1 + root(-5))(1 - root(-5)). Unique factorisation breaks, and math(s) needed ideals to fix it.
Local fields like the p-adic numbers measure size by divisibility by a prime, so huge powers of p count as tiny.
Stretch Way in (grade 9): In 10-adic arithmetic, check that ...9999 + 1 = 0, so ...9999 behaves like -1. Then find the last digits of a number whose square ends in itself, like 76.
Finite fields are number systems with finitely many elements where you can add, subtract, multiply and divide, like clock arithmetic mod a prime.
Natural fit Way in (grade 7): Make a multiplication table mod 7 and find every number's inverse. Then try mod 6 and see some numbers have no inverse.
Logic meets number theory: which questions about whole numbers can be decided by an algorithm, and which cannot.
Hard Way in (grade 9): Hilbert asked for a recipe to decide if any polynomial equation has whole-number solutions. Try a few, then learn the 1970 answer: no recipe can exist.
Computational number theory finds fast algorithms for primes, factoring and big number calculations.
Natural fit Way in (grade 8): Find 3 to the power 100 mod 7 by hand using repeated squaring, in under ten steps, and see why computers can check huge primes quickly.
Applications of number theory to the real world, such as codes, check digits, cryptography and music.
Natural fit Way in (grade 6): Check the last digit of a book's ISBN using the weighted sum mod 11, then show that swapping two neighbouring digits always gets caught.
12 Field theory and polynomials 5 of 8
Real and complex number fields, especially where polynomial roots lie and why every polynomial has complex roots.
Natural fit Way in (grade 9): Sketch y = x cubed - 3x + 1 and count real roots by sign changes. Then plot complex cube roots of 1 as points round a circle.
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A field is any number system where the four operations work; this section studies fields in general and polynomials over them.
Stretch Way in (grade 10): Build a 4-element field from {0, 1, a, a+1} with a times a = a + 1 and all coefficients mod 2. Fill in the full multiplication table.
Field extensions add new numbers like root 2 to a field and track the symmetries of roots; this is Galois theory.
Stretch Way in (grade 10): Why can't you trisect 60 degrees with ruler and compass? Constructible lengths only use square roots, but trisecting needs a cube root. Try it and fail.
Homological methods use layered algebraic bookkeeping to measure how field extensions fit together, and what obstructs certain constructions.
Hard Way in (grade 11): Honestly very abstract. Best glimpse: the carry in adding two-digit numbers is a "cocycle", the bookkeeping that glues the digit place values together.
Differential algebra treats differentiation as an algebraic operation, and asks which functions have antiderivatives in elementary terms.
Stretch Way in (grade 12): Differentiate x e^x and x squared e^x, then guess an antiderivative of x cubed e^x. Learn that e^(-x squared) has no elementary antiderivative at all.
Topological fields are number systems where you also have a notion of closeness, like reals or p-adics.
Hard Way in (grade 10): Compare two distance rules on fractions: the usual one, and "close if the difference is divisible by a high power of 2". Under the second, 1, 3, 7, 15, 31 approach -1.
Generalisations of fields relax some rules, for example allowing multiplication that is not commutative or associative.
Stretch Way in (grade 9): Rotate a book 90 degrees about two axes in each order. Different results show multiplication of rotations need not commute, which quaternions capture.
Connections between field theory and logic, such as which statements about fields can be decided mechanically.
Hard Way in (grade 11): Tarski showed any statement about real numbers using +, x and < can be settled by an algorithm. Test one: does every cubic have a real root?
13 Commutative algebra 6 of 13
Commutative rings are number systems where you can add, subtract and multiply in any order, but maybe not divide.
Natural fit Way in (grade 7): Compare clock arithmetic mod 12 with ordinary integers: in mod 12, 3 x 4 = 0 even though neither is 0. Find all such zero divisors.
Ring extensions study what happens when you enlarge a ring by adding new elements, like adding i to the integers.
Stretch Way in (grade 9): Explore Gaussian integers a + bi on a grid. Show 5 = (2 + i)(2 - i), so 5 is no longer prime once i is allowed.
Ideals and modules: an ideal is a set closed under adding and multiplying by anything, like all multiples of 6.
Stretch Way in (grade 8): Show that numbers of the form 12a + 18b are exactly the multiples of 6. The ideal generated by 12 and 18 is just the multiples of the gcd.
Homological methods measure how far a ring or module is from simple, using chains of maps and how badly they fail to be exact.
Hard Way in (grade 11): Honestly abstract. Glimpse: a counting puzzle where each stage's output is the next stage's input, and Euler's V - E + F = 2 appears as an alternating sum.
Chain conditions say you cannot climb forever: every increasing chain of ideals must stop, which keeps rings manageable.
Stretch Way in (grade 9): Start with multiples of 64, then 32, 16, ...: each set contains the last. Why must this chain of multiples stop after a few steps in the integers?
Special rings with good arithmetic, such as those with unique factorisation, Euclidean division or polynomial structure.
Stretch Way in (grade 10): Do long division with remainder for Gaussian integers on a grid: find the nearest lattice point to a/b. It always lands within distance less than 1.
Integral domains are rings with no zero divisors, where ab = 0 forces a = 0 or b = 0, like ordinary integers.
Natural fit Way in (grade 8): Solve x squared = x in ordinary numbers (0 and 1) and in mod 6 (also 3 and 4). Zero divisors create the extra answers.
Local rings focus on one point, ignoring everything far away, like looking at a curve through a microscope.
Hard Way in (grade 11): Zoom into the curve y squared = x cubed + x squared at the origin on graphing software. Near 0 it looks like two crossing lines y = x and y = -x.
Topological rings add a notion of closeness to rings, like power series where higher powers count as smaller.
Stretch Way in (grade 11): Treat 1/(1 - x) as 1 + x + x squared + ... and multiply by (1 - x) term by term. Everything cancels, so the infinite sum really is an inverse.
Applications of logic to commutative algebra, using ideas like ultraproducts to transfer results between number systems.
Hard Way in (grade 12): Honestly hard. Glimpse: a fact true mod every large prime, like "x squared + 1 = 0 has a solution for half the primes", hints at a statement about all fields.
Finite commutative rings: number systems with finitely many elements, like clock arithmetic mod n.
Natural fit Way in (grade 7): Write the multiplication tables mod 4 and mod 5 side by side. Find which numbers have inverses in each, and link it to sharing factors with n.
Differential algebra in commutative rings treats taking derivatives as an algebraic rule, with the product rule as the key axiom.
Stretch Way in (grade 9): Define D on numbers by D(prime) = 1 and D(ab) = aD(b) + bD(a). Compute the "arithmetic derivative" of 6, 12 and 30.
Computational commutative algebra solves polynomial systems by systematic elimination, such as Gröbner bases.
Stretch Way in (grade 9): Solve x + y = 5 and xy = 6 by eliminating y to get a single polynomial in x. That elimination step is the seed of Gröbner bases.
14 Algebraic geometry 8 of 17
Foundations of algebraic geometry: shapes defined by polynomial equations, studied through the algebra of those equations.
Natural fit Way in (grade 8): Graph x squared + y squared = 25 and list its whole-number points. The shape and the equation are two views of the same object.
Local theory zooms in on one point of a polynomial curve or surface, especially points where it crosses or forms a cusp.
Stretch Way in (grade 10): Plot y squared = x cubed and y squared = x squared (x + 1). One has a sharp cusp at the origin, the other a crossing. Describe each by zooming in.
Cycles are formal sums of curves and points on a shape, used to count intersections in a consistent way.
Stretch Way in (grade 9): Count how many points two straight lines meet in. Then a line and a circle, then two circles. Guess the rule: degrees multiply (Bézout).
Families of shapes that vary continuously, like a curve that changes as a parameter moves, and what happens when they degenerate.
Natural fit Way in (grade 9): Graph xy = t for t = 4, 1, 0.1 and 0. The hyperbola squashes into two crossing axes at t = 0. Sketch the whole family.
Birational geometry studies shapes up to transformations that may squash or blow up a few points, to find the simplest model.
Stretch Way in (grade 10): Draw lines through one point of a circle to all others. Each slope gives exactly one circle point, so the circle is "the same" as a line, except one point.
Cohomology in algebraic geometry counts holes and other hidden features of polynomial shapes, often via clever counting.
Hard Way in (grade 11): Count solutions of y squared = x cubed + 1 mod p for p = 5, 7, 11. Weil's insight: such counts are governed by the shape's holes.
Arithmetic problems on polynomial shapes: finding rational or whole-number points, and how many there can be.
Stretch Way in (grade 11): On y squared = x cubed - 2 start from (3, 5). Draw the tangent there, find where it hits again, and get a new rational point (129/100, 383/1000), as Fermat did.
Algebraic curves, the solution sets of one polynomial in two variables, classified by their degree and number of holes.
Natural fit Way in (grade 9): On a graphing app draw y squared = x cubed - x + c for several c. See the curve split into one or two pieces, and count where lines hit it.
Surfaces and higher-dimensional shapes cut out by polynomials, and how to classify them.
Stretch Way in (grade 7): Build a hyperboloid of one sheet from straws between two paper circles twisted apart. A curved surface made of straight lines.
Abelian varieties are polynomial shapes that are also groups: you can add their points, the simplest being elliptic curves.
Stretch Way in (grade 11): On y squared = x cubed - x + 1, add two points by drawing the line through them, finding the third crossing and reflecting. Check P + Q = Q + P.
Algebraic groups are symmetry groups defined by polynomial equations, like rotations or invertible matrices.
Stretch Way in (grade 11): Rotation matrices with entries cos and sin satisfy a squared + b squared = 1. Multiply two and check you get another rotation, a group living on a circle.
Special varieties, like Grassmannians and toric varieties, are very symmetric polynomial shapes that can be described by pictures and counting.
Hard Way in (grade 11): Toric glimpse: list the monomials x^a y^b z^c with a + b + c = 3 as dots in a triangle. The triangle of dots is the picture of a toric surface.
Enumerative geometry counts geometric objects, like how many lines meet four given lines, and projective geometry adds points at infinity.
Stretch Way in (grade 6): Two lines always meet in projective geometry: parallel railway tracks meet at the horizon. Draw a perspective picture with a vanishing point.
Real algebraic geometry studies polynomial shapes over the real numbers, including how many ovals a curve can have and where polynomials are positive.
Stretch Way in (grade 10): Graph (x squared - 1)(y squared - 1) = c for c small positive and negative. Count the separate ovals and pieces as c changes.
Computational algebraic geometry: algorithms to compute with polynomial shapes, such as finding intersections or simplifying equations.
Natural fit Way in (grade 9): Find where the circle x squared + y squared = 25 meets the line y = x + 1 by substitution. Then let a graphing app check the two points.
Affine geometry studies polynomial maps of ordinary space, including the famous unsolved question of when they can be reversed.
Stretch Way in (grade 11): Check that (x, y) goes to (x, y + x squared) can be undone by another polynomial map. The Jacobian conjecture asks when this always happens.
Tropical geometry replaces + with max and x with +, turning curved polynomial shapes into piecewise straight graphs.
Natural fit Way in (grade 8): Do tropical arithmetic: 3 "plus" 5 = 5 and 3 "times" 5 = 8. Graph the tropical line max(x, y, 0) and see three rays meeting at a point.
15 Linear and multilinear algebra; matrix theory 2 of 2
Many problems are just systems of straight-line equations, and matrices let us solve them all at once and see when answers exist.
Natural fit Way in (grade 7): Lights Out on a 3x3 grid: pressing a button flips it and its neighbours. Work out which presses turn every light off; pressing twice undoes, so it is algebra mod 2.
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Matrices with special patterns, like magic squares, permutation grids or stochastic tables, behave in special, predictable ways.
Natural fit Way in (grade 8): Write a 3x3 table of chances for tomorrow's weather given today's (rows add to 1). Multiply it by itself to forecast two days ahead and watch rows settle to the same numbers.
16 Associative rings and algebras 10 of 16
A ring is any number system where you can add and multiply with the usual rules, except multiplication might not commute.
Natural fit Way in (grade 6): Clock arithmetic mod 12: make the full multiplication table, spot that 3 times 4 is 0, and ask which numbers have a partner that multiplies to 1.
Ideals are the multiples of something inside a number system, and they let you build new number systems by ignoring those multiples.
Natural fit Way in (grade 6): List all even numbers, then all multiples of 3. Show that a multiple of 3 times anything is still a multiple of 3, which is why clock arithmetic mod 3 works.
Homological methods measure how far a structure is from being simple by building chains of approximations and counting what is left over.
Hard Way in (grade 11): Best glimpse: the chain where each step's output is used up by the next (like 0 to 2Z to Z to Z/2 to 0); check every arrow's image is exactly the next arrow's zero set.
Representation theory studies an abstract algebra by making it act on arrows and dots, so it becomes a picture of linear maps.
Stretch Way in (grade 10): Draw two dots with one arrow and put a number of beads on each dot; classify all ways to connect beads by strings, finding only three basic building blocks.
Some number systems are built like vectors with a multiplication, and the best known new one is the quaternions.
Natural fit Way in (grade 9): Hamilton's quaternions: use the rules i times j is k but j times i is minus k. Multiply small combinations and see that order matters, just like turning a book.
A division ring lets you divide by anything nonzero even when multiplication does not commute, and there are surprisingly few of them.
Natural fit Way in (grade 8): Check that every nonzero number mod 7 has an inverse, but mod 6 some do not. Then find the inverse of a quaternion by flipping the signs of i, j, k.
A local ring has exactly one way to fail to be invertible, so it zooms in on behaviour near a single point.
Natural fit Way in (grade 8): Work mod 9: list which numbers have inverses. Notice the non-invertible ones are exactly multiples of 3, one single family.
A radical collects the elements that are nearly zero, such as things that become zero when multiplied by themselves enough times.
Natural fit Way in (grade 7): Mod 8, find numbers whose powers eventually hit 0 (2, 4, 6). Show that adding two such numbers gives another one.
Finiteness conditions ask whether chains of ever bigger pieces must stop, which is what makes many proofs about algebra terminate.
Stretch Way in (grade 7): Start with multiples of 64, then 32, 16, 8, 4, 2, 1: each set contains the last. Why must every such growing chain of 'multiples of' sets stop?
Some non-commuting systems still obey a hidden polynomial rule, like 2x2 matrices satisfying a fixed identity, which tames their behaviour.
Stretch Way in (grade 11): For 2x2 matrices A and B, compute AB minus BA several times and check that its square is always a plain number times the identity matrix.
New rings are made from old ones: polynomials, matrices with ring entries, or rings where variables do not commute.
Stretch Way in (grade 9): Build 'twisted polynomials' where x times a number n equals 2n times x. Multiply (x+1)(x+1) and compare with the ordinary answer.
Hopf algebras are structures that can both combine and split things, which is the algebra behind symmetry and quantum groups.
Hard Way in (grade 10): Count ways to split a word into two pieces in order (ABC: '', ABC; A, BC; ...). Then shuffle two words together, and see splitting and shuffling fit like opposite operations.
Special elements of a ring, like units that can be undone, idempotents that square to themselves, and zero divisors, reveal its structure.
Natural fit Way in (grade 6): Mod 10, find all x with x times x equal x. You get 0, 1, 5, 6. Why do 5 and 6 add to 1 and multiply to 0?
Rings with extra structure, like a grading by degree or a symmetry such as complex conjugation, carry more information than bare rings.
Stretch Way in (grade 9): Sort polynomial terms by degree and check the degree of a product is the sum of degrees. Then check conjugating a product of complex numbers reverses nicely.
Loosening the rules of rings gives new systems, like semirings where you cannot subtract, which still have useful algebra.
Natural fit Way in (grade 8): Tropical arithmetic: let 'plus' mean min and 'times' mean ordinary plus. Find the shortest route through a small road map by doing a 'matrix multiplication' with these rules.
Computers can decide questions about non-commuting algebra by rewriting expressions into a standard form.
Natural fit Way in (grade 7): Rewriting game: in words of a and b, the rule is 'ba becomes ab'. Rewrite any word to its standard form and count how many standard forms of length 4 exist.
17 Nonassociative rings and algebras 2 of 4
Nonassociative systems drop the rule that (xy)z equals x(yz), so the order of grouping changes the answer.
Natural fit Way in (grade 5): Rock paper scissors 'winner' operation: compute (rock, paper) then scissors, versus rock with (paper, scissors). Different results, so grouping matters.
Lie algebras capture tiny symmetries, measuring how much doing two small moves in different orders fails to agree.
Stretch Way in (grade 7): Hold a book: rotate a quarter turn forward then a quarter turn sideways, versus the reverse. The difference between the two outcomes is a third rotation.
Jordan algebras use the symmetric product (xy+yx)/2, the multiplication that matches how quantum measurements combine.
Hard Way in (grade 11): Take two 2x2 matrices, compute (AB+BA)/2 and AB. Check the first keeps symmetric matrices symmetric while the second usually does not.
Other nonassociative systems include the octonions, an eight-dimensional number system where even grouping order matters.
Stretch Way in (grade 10): Use the Fano plane picture (seven points, seven lines) as a multiplication table for octonion units and find three units where (ab)c differs from a(bc).
18 Category theory; homological algebra 3 of 9
Category theory studies math(s) through arrows between objects, focusing on how things connect rather than what they are made of.
Natural fit Way in (grade 7): Draw towns as dots and one-way roads as arrows; any two roads end-to-start join into a route. Check route joining is associative and every town has a 'stay put' route.
Many familiar worlds, like sets with functions or shapes with continuous maps, are particular categories with their own patterns.
Natural fit Way in (grade 7): Count all functions from a 2-element set to a 3-element set (9) and from 3 to 2 (8). Which ones can be undone from the other side?
Algebraic theories, like 'groups' or 'monoids', can themselves be treated as mathematical objects that generate their examples.
Stretch Way in (grade 9): List all words using letters a, b that you can make by sticking words together. This 'free' set is the most general monoid; any example is a squashed copy of it.
Categories can carry extra structure, such as ways to combine objects side by side or arrows between arrows.
Stretch Way in (grade 9): Draw string diagrams: boxes with wires in and out. Show that putting two boxes side by side then stacking gives the same as stacking then side by side.
Categorical algebra studies limits, exactness and adjunctions: universal best approximations that appear everywhere in math(s).
Stretch Way in (grade 8): Rounding as a 'best approximation': for real x, the floor is the largest whole number at most x. Check that n at most floor(x) exactly when n at most x.
Gluing local information into global information, as with sheaves, is captured by categories attached to spaces.
Stretch Way in (grade 8): Patch a map: each friend knows the temperature in overlapping parts of a town. When do their readings agree on overlaps so they glue into one town-wide map?
Homological algebra measures hidden holes and obstructions using chains of maps where doing two steps in a row gives zero.
Stretch Way in (grade 7): Count vertices minus edges plus faces on a cube, a donut-shaped block and a tetrahedron. The answer detects holes, the first shadow of homology.
Monoidal categories and operads describe all the ways to combine several inputs into one output, like plugging in trees of operations.
Stretch Way in (grade 7): Count the ways to bracket a times b times c times d (5 ways). Draw each as a tree and see how trees plug into trees, the idea of an operad.
Higher categories have arrows between arrows between arrows, used to treat 'the same up to deformation' precisely.
Hard Way in (grade 9): Two paths across a field from A to B: if you can slide one into the other without crossing a pond, the slide is an arrow between arrows. Find two paths with no slide.
19 K-theory 1 of 11
K-theory starts by turning things you can add but not subtract, like piles of objects, into a group with negatives.
Natural fit Way in (grade 6): Build integers from pairs of natural numbers: (5,3) means 5 minus 3. Decide when two pairs are the same, the trick Grothendieck used for K0.
K1 asks which invertible matrices can be reduced to simple ones by row operations, measuring what is left over.
Stretch Way in (grade 10): Using only 'add a multiple of one row to another', try to turn a 2x2 matrix into the identity. Show you can exactly when its determinant is 1.
K2 records the hidden relations among elementary row operations, the ways they cancel unexpectedly.
Hard Way in (grade 11): Best glimpse: on a 2x2 grid, 'add row 1 to row 2' then 'subtract row 2 from row 1' and so on; find a sequence of such moves that returns to the start.
Higher algebraic K-theory builds an infinite ladder of invariants of rings using topology.
Hard Way in (grade 11): Honest glimpse only: K-groups of the integers are linked to Bernoulli numbers. Compute sums of powers 1+2+...+n and 1^2+...+n^2 to meet those numbers.
K-theory in geometry counts ways to lay bundles of directions over a shape, giving invariants of spaces.
Stretch Way in (grade 8): Make a paper strip into a cylinder and a Mobius band; draw arrows across the strip. One band lets arrows flip direction going around. These are two different line bundles.
K-theory in number theory links algebraic invariants to deep facts about primes and values of zeta functions.
Hard Way in (grade 11): Honest glimpse: check that 1 + 1/4 + 1/9 + ... approaches pi squared over 6 by adding terms on a calculator; K-theory explains patterns in such values.
K-theory of forms classifies quadratic forms, like sums of squares, up to change of variables, using group addition.
Stretch Way in (grade 7): Which numbers are sums of two squares? Test 1 to 50. Then check that the product of two such sums is again one, so sums of squares can be combined.
Obstructions from topology use K-theory to decide whether a shape can be simplified or built from finitely many pieces.
Hard Way in (grade 9): Best glimpse: can you untangle this loop of string into a circle? Some knots refuse; an invariant (like counting colourings of the diagram) proves the obstruction exists.
K-theory of operator algebras counts infinite-dimensional pieces like projections, linking analysis to topology.
Stretch Way in (grade 8): Hilbert's hotel: shift every guest up one room. Shifting is undoable one way but not the other, and the one missing room is the index, a K-theory number.
Topological K-theory classifies vector bundles on spaces, and gives famous results like which spheres admit independent direction fields.
Stretch Way in (grade 6): Hairy ball: try to comb a fuzzy tennis ball flat with no crown. You cannot, yet on a donut you can. K-theory tells exactly when such combing works.
Miscellaneous applications of K-theory, from physics of materials to index theorems, use it to count robust features.
Stretch Way in (grade 7): Count how many times a wiggly loop drawn around a point winds around it; small wiggles do not change the count, the kind of robust number K-theory produces.
20 Group theory and generalizations 11 of 14
A group is a set of moves you can combine and undo, and the axioms capture exactly what symmetry needs.
Natural fit Way in (grade 6): List the symmetries of a square (4 turns, 4 flips). Fill in the 8x8 table of 'do this then that' using a cut-out paper square.
Permutation groups study shuffles of a set, and which arrangements can be reached by given moves.
Natural fit Way in (grade 7): The 15 puzzle: show, by counting swaps, that switching just the 14 and 15 tiles is impossible. Sam Loyd's prize was safe.
Representation theory turns abstract symmetries into matrices or number patterns, making groups computable.
Stretch Way in (grade 9): Rotations of a square as 2x2 grids of 0, 1, -1: write each and multiply them to check they follow the same table as the turns themselves.
Finite groups can be broken into simple building blocks, like primes for numbers, and the simple ones are fully classified.
Natural fit Way in (grade 8): Find all ways to arrange 6 symmetries: compare turns of a hexagon with symmetries of a triangle. One commutes, one does not, so two different groups of size 6.
Infinite and finite groups are classified by how they are assembled from smaller pieces and what they act on.
Natural fit Way in (grade 6): Wallpaper patterns: hunt for rotations and mirrors in floor tiles. There are only 17 types of wallpaper symmetry; sort a few patterns into their types.
Groups given by generators and rules can be studied by their geometry, like the street maps of the moves.
Natural fit Way in (grade 8): Draw the map of moves for 'a' and 'b' where ab equals ba: you get a grid. Where nothing commutes you get an ever-branching tree. Compare how fast they grow.
Linear algebraic groups are symmetry groups defined by polynomial equations, like rotations or invertible matrices.
Stretch Way in (grade 9): Count invertible 2x2 matrices with entries 0 or 1 under arithmetic mod 2 (there are 6). Check they shuffle the three nonzero vectors in all 6 ways.
Groups of matrices, such as those with integer entries, act on the plane or space and generate beautiful patterns.
Natural fit Way in (grade 8): Apply the matrices [[1,1],[0,1]] and [[1,0],[1,1]] repeatedly to (1,1). Every pair of coprime whole numbers appears, linking to fractions in the Stern-Brocot tree.
Homological tools measure how groups fit together, for example whether a big group splits into simpler parts.
Stretch Way in (grade 9): Clock mod 4 contains mod 2 as the even hours, with mod 2 left over. Yet it is not two mod-2 clocks side by side. Find the element that shows why.
Abelian groups, where order never matters, are completely described by lists of clock sizes.
Natural fit Way in (grade 7): Show that mod-6 arithmetic is the same as a mod-2 clock and a mod-3 clock running together, but mod 4 is not two mod-2 clocks.
Groupoids are like groups but with moves only allowed from certain states, keeping track of where you are.
Stretch Way in (grade 8): Fifteen puzzle states as dots, legal slides as two-way arrows. Each move is undoable, but only joinable when it starts where the last ended: a groupoid, not a group.
Semigroups are moves you can combine but not necessarily undo, modelling processes that forget information.
Natural fit Way in (grade 7): Functions on {1,2,3}: 'send everything to 1' cannot be undone. Combine a few such functions and find which combinations are still reversible.
Loops and quasigroups relax group rules, keeping the ability to solve equations but not associativity.
Natural fit Way in (grade 5): A completed Sudoku row-column table (a Latin square) is a quasigroup's multiplication table. Fill a 4x4 Latin square and test whether (ab)c equals a(bc).
Random methods answer questions about groups, like the chance that two random symmetries commute or how fast shuffles mix.
Natural fit Way in (grade 8): Pick two random symmetries of a triangle; what is the chance they commute? (1/2.) Then ask how many riffle shuffles mix a deck: about seven.
22 Topological groups, Lie groups 2 of 6
Some number systems let you both combine things and move smoothly between them, so algebra and nearness live together.
Stretch Way in (grade 7): Turn a clock hand: rotations add like numbers, and a tiny turn is close to no turn. Compare with a 12-hour clock where you can only jump.
Repeating patterns, like sound waves, can be built from pure tones, and the same idea works on circles, lines and clocks.
Stretch Way in (grade 10): Add the columns of a 4-hour clock table using 'secret codes' 1, i, -1, -i, and see a lopsided list of numbers split into four pure rhythms.
Groups that are finite in size but smooth, like all rotations of a sphere, can be averaged over fairly.
Natural fit Way in (grade 6): Average a doodle over all 4 quarter-turns of a square to make it symmetric; then imagine averaging over every angle of a circle.
Big symmetry groups can be studied by how they act as matrices or on functions, turning shape into algebra.
Stretch Way in (grade 9): Write the 6 symmetries of a triangle as ways to shuffle three corners, then as 2-by-2 rotation and flip pictures, and check both multiply the same way.
Smooth symmetries, like all rotations, are understood from their tiny infinitesimal nudges.
Natural fit Way in (grade 6): Turn a book 90 degrees forward then 90 sideways, then undo in the other order: it ends up tilted. Small turns almost commute, big ones do not.
Infinite symmetry groups, like all translations and stretches, push points around a space forever without coming back.
Stretch Way in (grade 8): Mark a dot on graph paper and apply 'double x, halve y' over and over: the dot slides along a hyperbola and never returns. Which curves are left unchanged?
26 Real functions 5 of 5
How a single real function behaves: is it continuous, does it have a slope, can we add up its area?
Natural fit Way in (grade 7): Draw a staircase and a ramp between the same points. Which can you draw without lifting your pencil? Which has a slope everywhere? Zoom in on corners.
50 more
Functions with several inputs, like height on a hill map, have slopes in every direction and volumes underneath.
Natural fit Way in (grade 8): On a contour map, find the steepest way up from a dot: it always crosses contour lines at right angles. Check with a ruler.
Polynomials have hidden rules about where they cross zero and how many times they can wiggle.
Natural fit Way in (grade 9): Descartes' rule of signs: count sign changes in x^3 - 2x^2 - x + 2 and guess how many positive roots; then find them by trying 1, 2, -1.
Inequalities compare quantities without computing them, like the fact that an average of numbers is at least their geometric mean.
Natural fit Way in (grade 7): Of all rectangles with perimeter 20, which has the biggest area? Try 1x9, 2x8, 5x5, then see why (a+b)/2 is at least sqrt(ab).
A grab bag of unusual ways to do calculus: infinitesimals, fuzzy values, set-valued functions, and different kinds of averages.
Natural fit Way in (grade 7): Compare four averages of 2 and 8: arithmetic 5, geometric 4, harmonic 3.2, and root-mean-square. Which fits average speed on a round trip?
28 Measure and integration 2 of 5
Measure theory gives a careful meaning to length, area and probability, even for very ragged sets.
Natural fit Way in (grade 8): Remove the middle third of a line segment, then the middle thirds of what is left, forever. Total removed is 1, yet uncountably many points remain.
Measures and integrals can produce vectors or other objects instead of plain numbers.
Stretch Way in (grade 9): Spread mass along a bent wire and add up mass times position as arrows: the result is the balance point, a vector-valued 'integral'.
On spaces with symmetry, there is a natural fair way to measure size that every symmetry respects.
Stretch Way in (grade 9): Why is 'pick a random angle' fair but 'pick a random slope' not? Throw darts at a circle and compare angle and slope histograms.
If a system keeps mixing for long enough, averaging over time gives the same answer as averaging over all states.
Natural fit Way in (grade 9): Keep multiplying by 2 and keeping the fractional part, starting at 0.1234. Tally which tenth each value lands in: the time average matches the space average.
Odd corners of measure theory: infinitesimal measures, fuzzy measures, and sets so strange they cannot be measured at all.
Hard Way in (grade 11): Vitali's puzzle in words: sort numbers in [0,1) into piles that differ by a rational. Pick one from each pile. Why can that set have no sensible length?
30 Functions of a complex variable 3 of 11
Complex functions, which turn points of a plane into points of a plane, are amazingly rigid once they are smooth.
Natural fit Way in (grade 10): Square each point on a grid of complex numbers and plot the results: right angles between grid lines stay right angles everywhere except at 0.
A smooth complex function equals an infinite power series, and the series works out to the nearest trouble spot.
Natural fit Way in (grade 10): Add 1 - x^2 + x^4 - ... for x = 0.5 and x = 1.5. It fits 1/(1+x^2) only for |x|<1, because of hidden trouble at x = i.
Complex functions bend regions of the plane while keeping angles, and any simple blob can be bent into a disc.
Natural fit Way in (grade 10): Draw a grid on a balloon, then squash it: small squares turn into small squares of new sizes. Check with z -> z^2 on graph paper.
Functions smooth on the whole complex plane, like e^z, are tied by how fast they grow to how many zeros they have.
Stretch Way in (grade 10): A polynomial of degree n has n roots and grows like |z|^n. Plot |z^3 - 1| along big circles and count roughly how the growth gives three roots.
Puzzles of fitting and approximating functions in the complex plane, like how well polynomials can mimic given shapes.
Stretch Way in (grade 9): Find the straight line through 3 given points that misses by the least worst-case amount; this is the simplest version of best approximation.
Some functions, like square root, are many-valued, and become single-valued on a cleverly glued multi-sheeted surface.
Stretch Way in (grade 10): Walk once around 0 tracking sqrt(z) with a pencil: you come back with the opposite sign. Make a two-sheet paper model with a slit glued crosswise.
The rules of complex functions can be copied to new settings, like grids, quaternions or p-adic numbers.
Stretch Way in (grade 10): Discrete analytic functions: fill a grid so each square's diagonal differences match (a-c = i(b-d)). Check that z works and z^2 nearly does.
Families of nice functions on a disc can be treated like vector spaces with sizes and distances.
Stretch Way in (grade 10): Treat polynomials as vectors: add them, scale them, and measure their size as the largest value on a circle. Is |p+q| at most |p|+|q|?
Functions on the unit disc are controlled by their boundary values and by how they shrink the disc.
Stretch Way in (grade 11): Schwarz lemma glimpse: if a map keeps the disc inside itself and 0 at 0, it can't stretch anything. Try z -> z^2 and z/2 on points.
One single function can, by shifting or zooming, approximate every possible function: a universal pattern hiding everything.
Stretch Way in (grade 6): The digits 0.123456789101112... (Champernowne) contain every finite digit string, including your birthday. Find it. Universal functions do the same for shapes.
Calculus ideas like slope and stretch can be done on spaces with only distances, no coordinates.
Stretch Way in (grade 7): On a taxicab grid, measure distances and find all points equidistant from two corners. Shapes change but ideas like 'midpoint' still make sense.
31 Potential theory 2 of 5
Harmonic functions, like steady temperature on a plate, always equal the average of their neighbours.
Natural fit Way in (grade 7): Fill a grid with fixed edge numbers, then repeatedly replace each inside cell by the average of its 4 neighbours. It settles to a smooth steady state.
The same averaging rule in 3D describes gravity, electricity and steady heat, with no hills or pits inside.
Stretch Way in (grade 10): Why is there no gravity inside a hollow ball? Compare pulls from near and far patches of shell using area ~ distance squared.
Averaging and potentials extend to graphs, random walks and other settings beyond flat space.
Natural fit Way in (grade 7): On a path of 5 squares, a random walker stops at either end. The win chance from each square is the average of its neighbours: 0, 1/4, 1/2, 3/4, 1.
Potential theory boiled down to a few axioms about averaging, so it applies to many different equations at once.
Hard Way in (grade 11): List three rules a 'smooth out' operation must follow (no new peaks, averaging, stability), then test them on grid averaging and on random walks.
Heat flow and averaging can be done on fractals like the Sierpinski triangle, giving strange new dimensions.
Stretch Way in (grade 9): Random walk on the Sierpinski gasket graph: count steps to cross level 1, 2, 3. Each level multiplies time by 5, not 4, so walks are slow.
32 Several complex variables and analytic spaces 1 of 20
Functions of several complex variables behave very differently from one variable: for example, holes can force themselves to fill in.
Hard Way in (grade 12): Hard to show directly. Glimpse: in one variable 1/z blows up at a point; in two variables, try to make a function blow up only at a single point and fail.
Near a point, solution sets of complex equations look like a few simple pieces glued together.
Stretch Way in (grade 9): Plot real solutions of y^2 = x^2 and y^2 = x^3 near the origin: a cross and a cusp. Zoom in: these are the local shapes.
Shapes cut out by complex equations, including singular ones, can be studied like smooth surfaces with care.
Hard Way in (grade 10): Sketch y^2 = x^2(x+1): a loop with a self-crossing. Discuss how a spaceship moving along it could pass through the crossing twice.
How far a smooth complex function can be extended; in several variables, functions often extend automatically into holes.
Hard Way in (grade 10): Analytic continuation in one variable: 1+x+x^2+... = 1/(1-x) lets you 'extend' the sum's meaning to x = 2, giving -1. Debate whether that is fair.
Which regions of complex space are natural homes for functions, judged by a special kind of convexity.
Hard Way in (grade 12): Warm-up with real convexity: test shapes by checking whether every segment between two points stays inside. Then note complex space uses a twisted version.
Convexity in several complex variables uses complex lines and curves instead of straight segments.
Hard Way in (grade 12): Real convexity on paper: which of these shapes is convex? A star, a disc, a crescent. The complex version is the same game with curved test lines.
Complex shapes can be continuously wiggled into each other, and we study all possible wiggles at once.
Stretch Way in (grade 11): A flat torus from a parallelogram: slide one corner and the donut changes shape. Which slides give the same donut? Tile the plane to check.
Maps between regions of complex space are very rigid: there are far fewer than you would expect.
Hard Way in (grade 12): Glimpse: in one variable a disc can be mapped onto any simple blob, but in two variables a ball and a 'polydisc' cannot be matched. Only words reach this.
Closed complex shapes with no edges, like the torus, carry very few global functions.
Hard Way in (grade 10): A function on a donut made by gluing a square's edges must repeat both ways. Try to draw a smooth repeating height map with no highest point: impossible.
Extensions of complex analytic spaces to infinite dimensions and other more general settings.
Hard Way in (grade 12): Honestly very hard. Glimpse: a polynomial in infinitely many variables x1 + x2^2/2 + x3^3/3 + ...; when does it add up to a finite number?
Complex spaces can carry a family of attached spaces, like a fibre at every point, twisting globally.
Stretch Way in (grade 7): Make a Möbius strip: a line attached at every point of a circle with a twist. Cut it down the middle and see why the twist can't be undone.
Complex shapes with lots of symmetry, where a group moves every point to every other.
Hard Way in (grade 11): The disc's symmetries: rotations and maps that slide any point to the centre. Draw circles through a point and see how they move.
Functions of several complex variables that repeat under a large symmetry group, generalizing wallpaper patterns.
Hard Way in (grade 11): Start with a wallpaper pattern that repeats in two directions; then imagine repeats in four real directions at once. Only a glimpse is possible.
Doing analysis with p-adic numbers, where closeness means divisible by a high power of p.
Natural fit Way in (grade 7): In 10-adic land 1, 11, 111, 1111... home in on a number. Show ...9999 + 1 = 0, so ...9999 behaves like -1.
Smooth shapes that look locally like complex space, with extra geometry like curvature.
Stretch Way in (grade 9): The Riemann sphere: put a ball on the plane and project from the north pole. Every point of the plane, plus one 'infinity', gets a spot.
Points where complex curves pinch or cross, and how small loops around them knot together.
Stretch Way in (grade 9): Draw y^2 = x^3 and zoom on the cusp. Then a teaser: around the cusp in complex space, the curve meets a small sphere in a trefoil knot.
Regions in complex space shaped well enough that every reasonable equation can be solved on them.
Hard Way in (grade 12): Honestly very hard. Glimpse via real convexity: on a convex region, a straight path from any point to any other stays inside, so problems spread nicely.
Potential theory in several complex variables, using functions that are below average on every complex line.
Hard Way in (grade 12): Take log of distance to the origin on a grid and test the averaging rule around circles. Pluripotential theory asks this on every complex slice.
Boundaries of complex regions carry their own leftover complex structure, a mix of real and complex directions.
Hard Way in (grade 12): Very hard. Glimpse: the 3-sphere in two complex variables has, at each point, one 'real only' direction and one complex plane. Use a picture of Hopf circles.
Special differential equations in several complex variables, especially the one that tests whether a function is complex-smooth.
Hard Way in (grade 11): Check the Cauchy-Riemann test: for f = x^2 - y^2 + i(2xy), compare slopes in x and y; for x^2 + y^2 it fails.
33 Special functions 4 of 5
Exponentials, logarithms, trig functions and the gamma function are the basic building blocks used everywhere.
Natural fit Way in (grade 9): Fill in 1, 1, 2, 6, 24, ... (factorials) and ask what 'half factorial' should be. Plot points and guess; the answer is sqrt(pi)/2.
13 more
Hypergeometric series are a single family that includes most famous functions as special cases.
Stretch Way in (grade 10): Ratios of consecutive terms: for 1+x+x^2+... the ratio is x; for e^x it is x/(n+1). Any series with a nice ratio rule is hypergeometric.
Basic hypergeometric series are q-versions of classical functions, linked to counting partitions.
Natural fit Way in (grade 8): Count ways to write 5 as sums of positive whole numbers (7 ways). Then expand (1+q)(1+q^2)(1+q^3)... and read off partitions into distinct parts.
Other named functions, like Bessel-type, elliptic and Lambert W, that arise from specific problems.
Natural fit Way in (grade 9): Solve x * 2^x = 8 by trial: x = 2. Now try x * 2^x = 10. No algebra works, so the Lambert W function is named to answer it.
How to compute special functions fast and accurately on a computer.
Natural fit Way in (grade 8): Compute e by adding 1 + 1 + 1/2 + 1/6 + ... and count how many terms give 6 correct digits. Compare with (1+1/n)^n.
34 Ordinary differential equations 9 of 12
A differential equation describes how something changes, and its solution traces out the whole future from a starting point.
Natural fit Way in (grade 9): Population doubles each year when growth is proportional to size. Draw a slope field for y' = y and sketch curves from different starting heights.
Sometimes we know conditions at both ends, not just the start, which can give no solution, one, or many.
Natural fit Way in (grade 8): Throw a ball to land exactly at a target: adjust launch speed by trial and error (the shooting method). Some targets have two arcs, high and low.
We can describe the long-run shape of solutions, like cycles and spirals, without solving the equation.
Natural fit Way in (grade 7): Rabbits and foxes: rabbits up means foxes rise, then rabbits fall, then foxes fall. Sketch a loop in a rabbits-versus-foxes picture.
Stability asks whether a small push dies away or grows, like a marble in a bowl versus on a hilltop.
Natural fit Way in (grade 6): Balance a pencil on its tip versus hang it from a string. Then iterate 'y -> y/2' and 'y -> 2y' from 0.01 and watch.
Approximate solutions when some quantity is very large or very small, often revealing thin layers of fast change.
Stretch Way in (grade 10): Cooling coffee with a tiny parameter: a fast drop at first, then slow drift. Sketch two zoom levels of the same curve and match them.
Differential equations with random kicks, such as a population shaken by luck each year.
Natural fit Way in (grade 8): Simulate savings growing 5 percent a year plus a coin flip of +10 or -10. Run several players and compare spread with the smooth curve.
Differential equations whose unknowns are themselves whole functions or infinite lists.
Stretch Way in (grade 9): A row of 10 cups passing water to neighbours each minute: write one change rule for the whole list. Imagine infinitely many cups.
Control theory: choosing inputs to steer a changing system where you want it.
Natural fit Way in (grade 6): Park a toy car: you can only go forward/back and steer. Show you can still move sideways by wiggling, like parallel parking.
Equations where the rate of change depends on the past, like a delay, which can cause oscillations.
Natural fit Way in (grade 6): Shower temperature with a 5-second delay: adjust based on what you feel now and overshoot back and forth. Simulate with a table.
Differential equations viewed as operators with special 'eigen' solutions, like the notes of a vibrating string.
Natural fit Way in (grade 8): Pluck a string held at both ends: only waves that fit a whole number of half-wavelengths survive. Draw the first three and find their pitches.
Differential equations with complex numbers, where going around a singular point can change the solution.
Stretch Way in (grade 11): Follow sqrt(z), a solution of 2zy' = y, once around 0: you return with a minus sign. The equation remembers the loop.
A single theory covering both smooth change and step-by-step change, and mixtures of the two.
Stretch Way in (grade 8): Compare money growing 10 percent a year in one jump with continuous growth, then a calendar with jumps on weekdays and smooth weekends.
35 Partial differential equations 10 of 17
A PDE links how a quantity changes in several directions at once; the first question is whether a solution exists and is unique.
Natural fit Way in (grade 7): On a grid, each cell must equal the average of its four neighbours, with the border numbers fixed. Fill in a 3x3 interior by trial and see there is exactly one answer.
Without solving a PDE you can still prove things about solutions, like where the biggest value must sit or how fast things settle.
Natural fit Way in (grade 7): On a grid where each inside cell is the average of its neighbours, argue the largest number must lie on the border. Try to break this and fail.
Some PDE solutions can be written as explicit formulas, such as travelling waves, sums of simple waves, or similarity shapes.
Natural fit Way in (grade 9): A wave on a rope keeps its shape and slides along: plot y=f(x-t) for a bump at t=0,1,2 and see it move right at speed 1.
Real solutions can have corners and jumps, so mathematicians widen what counts as a solution to include these rough shapes.
Stretch Way in (grade 8): Cars in a traffic jam: draw the speed along a road where fast cars meet a queue. The sharp jump is a shock that no smooth formula describes.
When the rules of a PDE do not change from place to place, waves of the form sine and exponential turn it into plain algebra.
Stretch Way in (grade 11): Guess y=e^(kx) in y''=4y and get k^2=4. The derivative rule becomes a number equation; repeat with sines for y''=-9y.
First-order PDEs carry information along lines, called characteristics, so you can solve them by following each line.
Natural fit Way in (grade 8): A conveyor belt carries sand piles at constant speed. Mark where a pile at x=2 is after 3 seconds; the pile shape just rides along each line x-vt.
Higher-order PDEs, like the bending of a plate or beam, involve fourth derivatives and need more boundary conditions.
Stretch Way in (grade 9): Clamp a ruler at one end and push the tip: it curves smoothly. Compare with a string, which can bend sharply. Stiffness means higher derivatives.
Some equations smooth out rough data even though they only push in certain directions, as long as those directions combine to reach everywhere.
Stretch Way in (grade 8): Parallel parking: you can only drive forward/back or turn, yet wiggling lets you slide sideways. Moves that combine to reach every direction are the key idea.
Elliptic equations describe balanced, steady states, like a soap film or heated plate at rest, where each point equals the average around it.
Natural fit Way in (grade 6): Bend a wire loop and dip it in soap: the film is the smoothest surface. On paper, relax a grid where each cell becomes the average of its neighbours.
Parabolic equations describe spreading and smoothing over time, like heat flowing or ink diffusing, with sharp features fading fast.
Natural fit Way in (grade 5): Row of numbers 0,0,100,0,0: each step replace each by the average of itself and neighbours. Watch the spike spread and flatten, and the total stay fixed.
Hyperbolic equations describe waves that travel at finite speed and keep their sharpness, like sound, light, or a plucked string.
Natural fit Way in (grade 6): Two people shake a long rope from opposite ends. The pulses pass through each other unchanged. Sketch them before, during and after meeting.
Some equations behave like waves in one region and like steady balance in another, as in air flowing past a plane near the sound barrier.
Stretch Way in (grade 8): A boat moving slower than ripples sends waves ahead; faster than ripples it leaves a V wake. Measure the V angle in a photo and relate to speed.
Some problems impose more conditions than needed, so solutions exist only for special shapes; a classic answer is that the shape must be a ball.
Stretch Way in (grade 9): Which shapes of a frying pan cook evenly so the edge always gets the same heat flow? Argue a circle works by symmetry and a square cannot.
A drum or string can only vibrate at special frequencies set by its shape, and studying these notes reveals the shape's geometry.
Natural fit Way in (grade 5): Tap glasses filled with different water levels or press a guitar string at half and a third its length: notes at 2x and 3x the base frequency.
PDEs are the language of physics, from fluids and heat to quantum particles and finance, each with its own famous equation.
Natural fit Way in (grade 6): Drop food colouring into still water versus stirred water. Describe in words what the heat equation and the fluid equation must each say.
Odd PDE settings: shapes with moving boundaries, unknown insides found from outside measurements, random noise, or equations on networks.
Stretch Way in (grade 6): Melting ice cube: the boundary between ice and water moves. Measure an ice cube each 5 minutes and plot how the edge recedes.
Pseudodifferential operators treat a derivative as multiplication by frequency, so you can define half a derivative or any frequency filter.
Hard Way in (grade 12): Hard to make elementary. Glimpse: if differentiating sin(kx) multiplies its wiggle size by k, what operation would multiply it by the square root of k?
37 Dynamical systems and ergodic theory 11 of 14
Ergodic theory asks whether one long journey visits every region in fair proportion, so time averages equal space averages.
Natural fit Way in (grade 7): Keep adding 0.618 and keep only the fractional part. Tally how many land in each tenth from 0 to 1 after 50 steps. Every bin fills evenly.
Topological dynamics studies repeated maps through what returns, what mixes, and what stays fixed, ignoring exact distances.
Natural fit Way in (grade 6): Shuffle a deck with perfect riffle shuffles: after 8 shuffles it returns to the start. Try with 10 cards and find the return time.
Smooth dynamical systems follow points moving under a smooth rule, looking for fixed points, cycles and how nearby paths behave.
Natural fit Way in (grade 9): Press cos repeatedly on a calculator in radians. Any start converges to 0.739. Draw the cobweb diagram showing why it is attracting.
In hyperbolic systems nearby points separate exponentially in some directions while being squeezed in others, causing chaos that is still structured.
Natural fit Way in (grade 7): Baker's map: stretch a square of dough to double width, cut, stack. Track two raisins that start close; after a few folds they are far apart.
In one or two dimensions, simple maps already show period doubling and chaos, and order follows strict rules like period three implies chaos.
Natural fit Way in (grade 8): Iterate x to 3.2x(1-x), then 3.5x(1-x), then 3.9x(1-x) on a calculator. See a 2-cycle, a 4-cycle, then no pattern.
Iterating maps like z to z^2+c on complex numbers creates the Julia and Mandelbrot sets, whose boundaries are fractal.
Natural fit Way in (grade 9): With c=-1, iterate z to z^2+c from 0 and from 1. With c=1 it blows up. Test real c values and shade those that stay bounded.
Bifurcation theory studies how the long-run behaviour of a system suddenly changes as a dial is slowly turned.
Natural fit Way in (grade 6): Slowly tilt a book with a coin on it: nothing, nothing, then it slides. Or bend a ruler by squeezing its ends until it snaps sideways.
Random dynamical systems apply a randomly chosen rule each step, and can still settle into predictable patterns.
Natural fit Way in (grade 5): Chaos game: pick a triangle's corner by die roll and move halfway there, repeat 50 times. The dots draw a Sierpinski triangle.
Hamiltonian systems conserve energy and phase-space area, which forbids attractors and gives regular orbits mixed with chaos.
Stretch Way in (grade 9): A pendulum swings forever without friction. Plot angle against speed for several energies: nested loops, never spiralling in.
Some wave equations have infinitely many conserved quantities, giving solitary waves that pass through each other unchanged.
Stretch Way in (grade 9): Solitons: a single wave travelled miles along a Scottish canal in 1834. Simulate with a row of balls on springs or recall Newton's cradle.
Systems with friction in infinitely many dimensions, like turbulent fluids, often settle onto a finite-dimensional attractor.
Hard Way in (grade 8): Stir a cup of tea and watch: the swirl fades to a few slow eddies. Count how many big eddies remain after 10 seconds.
Computers simulate dynamics in tiny steps, and the choice of method decides whether errors stay small or blow up.
Natural fit Way in (grade 9): Simulate a ball orbiting with Euler steps on graph paper. The orbit spirals outward. Then update speed first and position second, and it stays closed.
Dynamical systems model populations, epidemics, economies and planets, revealing cycles, tipping points and chaos in the real world.
Natural fit Way in (grade 6): Rabbits and foxes: start 100 rabbits, 10 foxes, with simple yearly rules. Tabulate 10 years and see the boom and bust cycle.
Iterate maps on whole numbers, fractions or numbers mod p, and ask which starts cycle and how long cycles are.
Natural fit Way in (grade 8): Iterate x to x^2+1 mod 11 from 0. List values until one repeats, giving a rho shape. This is how Pollard factors numbers.
39 Difference and functional equations 2 of 2
Difference equations give each term from earlier terms, the discrete cousin of differential equations.
Natural fit Way in (grade 5): Fibonacci rabbits: each term is the sum of the previous two. Compute ratios of terms and watch them settle near 1.618.
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A functional equation describes a function only by a rule it obeys, like f(x+y)=f(x)+f(y), and asks what functions fit.
Natural fit Way in (grade 8): Find all f with f(x+y)=f(x)f(y) and f(1)=2. Work out f(2), f(3), f(0), f(-1), f(1/2) and spot powers of 2.
40 Sequences, series, summability 5 of 9
An infinite process may settle to a limit or not, and tests tell which, sometimes with surprising answers.
Natural fit Way in (grade 7): Harmonic series: group 1/3+1/4, then 1/5 to 1/8 and so on. Each group exceeds 1/2, so the sum grows forever, though slowly.
With double sequences and series, the order in which you add rows and columns can change the answer.
Natural fit Way in (grade 8): Grid with 1s on the diagonal and -1s just right of it. Sum rows first: 0. Sum columns first: 1. Find why the order matters.
Summability methods assign sensible values to divergent series, for example by averaging partial sums.
Natural fit Way in (grade 7): Grandi's series 1-1+1-1... has partial sums 1,0,1,0. Average them: 1, 1/2, 2/3, 1/2, 3/5... The averages approach 1/2.
Abelian theorems say a good averaging method gives the true sum whenever the series already converges.
Stretch Way in (grade 9): Partial sums approach 3. Show that running averages of the partial sums also approach 3. Test with sums 2.5, 2.9, 2.99, 2.999.
Tauberian theorems run the other way: if the average converges and the terms are not too wild, the series itself converges.
Stretch Way in (grade 10): Partial sums 1,0,1,0 average to 1/2 but do not converge, because the terms stay size 1. Find a series whose shrinking terms fix that.
Absolute summability asks that the averaged sums settle firmly enough that their total wobble is finite, not just eventually small.
Stretch Way in (grade 10): Compare 1-1/2+1/3-1/4... with the same signs all positive. One converges, one does not. Rearrange the first to approach 2 instead.
Special methods like Abel, Borel and Euler summation each tame divergent series in their own way, sometimes giving the same value.
Natural fit Way in (grade 10): Abel's trick: 1-x+x^2-x^3... equals 1/(1+x). Put x=0.9, 0.99, 0.999 and see the answer approach 1/2 for 1-1+1-1...
Summation methods can be seen as infinite matrices acting on sequences, and functional analysis tells which ones are reliable.
Stretch Way in (grade 10): Write the averaging rule as a triangle of numbers: row n has n entries 1/n. Apply rows to 1,0,1,0,... to get the averaged sequence.
Summability ideas extend to sequences of vectors, functions, or elements of groups and other abstract settings.
Hard Way in (grade 9): Hard to make concrete. Glimpse: average the positions of a point hopping around a square's corners; the average heads to the centre.
41 Approximations and expansions 1 of 1
Approximation theory asks how well simple functions like polynomials can mimic complicated ones and how fast the error shrinks.
Natural fit Way in (grade 9): Approximate sin(0.5) with x, then x-x^3/6, then add x^5/120. Compare to the calculator and count correct digits each time.
42 Harmonic analysis on Euclidean spaces 2 of 3
Any periodic signal can be built by adding sine waves of different frequencies, and the recipe of amounts tells you its sound.
Natural fit Way in (grade 9): Add sin x + sin(3x)/3 + sin(5x)/5 at a few points and plot. A square wave appears, with little overshoots at the jumps.
Fourier ideas in several dimensions break pictures into stripe patterns, which explains image compression and blurring.
Stretch Way in (grade 6): Look through a fine mesh curtain at another mesh: moire stripes appear. Draw two sets of slightly tilted lines and see the big wave.
Signals can be built from building blocks other than sines, such as wavelets or Walsh square waves, each suited to different jobs.
Natural fit Way in (grade 7): Haar wavelet: take 8 numbers, replace pairs by average and half-difference, repeat. Zero the small differences and rebuild a close copy.
43 Abstract harmonic analysis 0 of 1
Fourier analysis works on any symmetry group, splitting functions into pieces that behave simply under the group's moves.
Stretch Way in (grade 11): Clock arithmetic mod 4: split the pattern 3,1,4,1 into constant, alternating and rotating parts by adding with signs +1,-1,+i,-i.
44 Integral transforms, operational calculus 0 of 1
Integral transforms turn hard problems such as differential equations into easier algebra, then transform back.
Stretch Way in (grade 8): Logarithms as a transform: multiply 23x47 by adding logs and taking back. Laplace transforms do the same for calculus.
45 Integral equations 5 of 16
In a linear integral equation the unknown function appears inside an integral, adding up its values weighted by a kernel.
Stretch Way in (grade 9): Savings: today's balance is starting money plus the sum of past balances times interest. Write it as a running sum and solve by steps.
Fredholm equations integrate over a fixed range, and like matrix systems they either have one solution or a special resonance.
Stretch Way in (grade 11): Chop the interval into 3 points: the integral becomes a 3x3 matrix equation. Solve it and note when the matrix is singular.
Eigenvalue problems for integral equations find special functions that the integral just rescales, like natural modes.
Stretch Way in (grade 11): Matrix version: find a vector that a 2x2 averaging matrix only stretches. Try (1,1) and (1,-1) for [[0.7,0.3],[0.3,0.7]].
Volterra equations integrate only over the past, so the solution is built step by step forward in time.
Natural fit Way in (grade 8): Population where births depend on everyone born in the last 3 years. Tabulate year by year from 10 founders and see growth.
Singular integral equations have kernels that blow up, like 1/(x-y), yet still give meaning and model cracks and airfoils.
Hard Way in (grade 11): Hard to make faithful. Glimpse: sum 1/(x-k) over k from -10 to 10, skipping k=x; nearby terms cancel in pairs though each is huge.
Systems of integral equations couple several unknown functions whose histories feed into each other.
Stretch Way in (grade 9): Two bank accounts each earning interest on the other's past balance. Tabulate both year by year and watch them grow together.
Nonlinear integral equations have the unknown inside the integral in a nonlinear way, giving multiple or no solutions.
Stretch Way in (grade 9): Rumour spread: new hearers today depend on past hearers times those not yet told. Tabulate a town of 100 and watch an S curve.
Integral equations with special kernels, such as differences of arguments, can be solved by tricks like Fourier or convolution.
Stretch Way in (grade 8): Convolution: blurring a row of pixels by averaging with neighbours. Given the blurred row 1,2,3,2,1, try to recover the original.
Integro-differential equations combine rates of change with memory of the whole past, as in materials that remember stress.
Stretch Way in (grade 7): Memory foam: press a hand in and let go; the dent fills slowly. Time it and plot depth versus seconds.
Integro-PDEs mix spatial change with long-range or past-dependent effects, like animals that jump far as well as wander.
Stretch Way in (grade 8): Seeds spread by wind: most land nearby, some far. On a line of cells, move each seed by dice and compare with neighbour-only steps.
When integral equations cannot be solved exactly, you approximate by sums over grid points and check the error shrinks.
Natural fit Way in (grade 9): Estimate the area under y=x^2 from 0 to 1 with 4 and then 8 rectangles. Watch the error halve each time.
We study how solutions of integral equations behave in the long run: growth, decay, stability or oscillation.
Stretch Way in (grade 9): In the birth-memory population model, change the birth rate and see when the population dies out versus grows.
Integral equations can be posed for functions taking values in abstract spaces, using operator theory instead of formulas.
Hard Way in (grade 10): Hard to make elementary. Glimpse: solve x = 1 + 0.5x by repeating x to 1+0.5x from 0; the same repetition works for functions.
Integral operators turn one function into another by weighted averaging, and their size and smoothing power are the main questions.
Natural fit Way in (grade 6): Running average of 7 days smooths a jumpy temperature list. Apply it twice and see the result is smoother still.
Inverse problems recover a hidden cause from smeared measurements, often unstable so small errors are greatly amplified.
Natural fit Way in (grade 6): CT scan puzzle: a 3x3 grid of hidden numbers with known row, column and diagonal sums. Reconstruct the grid.
Random integral equations have noisy kernels or inputs, and ask how randomness in the memory affects the answer.
Stretch Way in (grade 8): Savings with dice interest: each year roll to set the rate. Simulate a few accounts and compare their spread over 10 years.
46 Functional analysis 4 of 14
Spaces where you can add things and scale them, and also say when things are getting close, even with infinitely many directions.
Stretch Way in (grade 10): Treat infinite lists of numbers as points. Is (1,1/2,1/3,...) 'close' to all zeros? Try three different ways to measure the gap and see they disagree.
A norm is a way to measure the size of a vector; different norms give different shapes for the 'unit ball'.
Natural fit Way in (grade 7): On grid paper draw all points at distance 1 from the origin using taxi distance, ordinary distance and 'biggest coordinate' distance: a diamond, a circle, a square.
Hilbert spaces carry angles and lengths into infinitely many dimensions, so Pythagoras and projection still work.
Stretch Way in (grade 10): Add 1 + 1/4 + 1/9 + ... as the squared length of the infinite arrow (1, 1/2, 1/3, ...). Pythagoras in infinitely many dimensions gives a finite length.
Whole collections of functions can be treated as spaces, and we can measure how far apart two functions are.
Natural fit Way in (grade 9): Draw y=x and y=x² on [0,1]. Measure their distance as the biggest vertical gap, then as the area between them. Which pairs of functions are closest?
Some useful objects, like a perfect instant kick or point mass, are not functions but can be handled by how they act when averaged.
Stretch Way in (grade 9): Make narrower and taller rectangles with area 1: width 1 height 1, width 1/2 height 2, width 1/10 height 10. The limit is a 'spike' with zero width but area 1.
Measuring, adding up and differentiating still make sense when the values are themselves points in infinite-dimensional spaces.
Hard Way in (grade 11): Hard to make faithful. Glimpse: average a whole family of curves (say each day's temperature graph over a month) to get one 'average curve'; the answer is itself a function.
Some spaces of functions can be multiplied as well as added, and the size of a product is at most the product of sizes.
Stretch Way in (grade 9): Check with polynomials on [0,1] that the biggest value of f·g is at most (biggest of f)·(biggest of g). Find an example where it is strictly less.
In commutative algebras of functions, the space can be rebuilt from the algebra: points correspond to ways of 'evaluating'.
Hard Way in (grade 11): Hard to reach. Glimpse: the rule 'plug in x=2' turns polynomials into numbers and respects + and ×. Find all such rules: each one secretly is a point.
Algebras with a 'mirror' operation, like complex conjugation, where the mirror of a product is the product of mirrors reversed.
Stretch Way in (grade 10): Conjugate complex numbers: check that the conjugate of zw is (conj z)(conj w) and z·conj z is never negative. Then try transposing 2x2 matrices: the order flips.
Algebras of operators, modelling quantum observables, where order of multiplication matters and size is tightly linked to algebra.
Hard Way in (grade 10): Hard to reach faithfully. Glimpse: multiply the 2x2 matrices for 'flip' and 'rotate 90°' in both orders and see AB ≠ BA, the seed of quantum uncertainty.
Building new function spaces by combining old ones (tensor products, limits) and tracking how constructions fit together systematically.
Hard Way in (grade 10): Hard to reach. Glimpse: a 2-by-3 times table grid built from (a,b) and (x,y,z) is a 'tensor product'; check which 2x3 grids of numbers arise this way.
Functional analysis gives tools for physics, economics, signal processing and more by treating functions as points.
Natural fit Way in (grade 8): Compress a short list of 8 numbers by keeping only the averages and differences of pairs (a Haar step), then rebuild it. Dropping small differences barely changes the signal.
Doing functional analysis with unusual number systems: p-adic numbers, infinitesimals, fuzzy sets, or only constructive proofs.
Stretch Way in (grade 9): Glimpse via p-adic size: call a number small if divisible by a high power of 2. Then 1, 2, 4, 8, ... shrinks to 0, and 1+2+4+8+... 'equals' -1.
Solving nonlinear equations where the unknown is a whole function, using fixed points and infinite-dimensional calculus.
Stretch Way in (grade 11): Find a function equal to its own running area plus 1: start with f=1, repeatedly replace f by 1 + area under f from 0 to x, and watch e^x appear.
47 Operator theory 7 of 12
A linear operator transforms vectors while respecting addition and scaling; its eigenvalues tell its essential behaviour.
Natural fit Way in (grade 9): Apply the matrix [[2,1],[1,2]] to many arrows on grid paper. Most arrows turn, but (1,1) and (1,-1) just stretch. Find the stretch factors.
Special operators, like symmetric ones, shifts or compact ones, have extra structure that makes them easier to understand.
Natural fit Way in (grade 8): Play with the shift on infinite lists: (a,b,c,...) to (0,a,b,c,...). It never loses information, yet nothing undoes it from both sides. Hilbert's hotel as an operator.
Single operators studied as members of algebraic systems, asking what algebra they generate and how they sit inside it.
Hard Way in (grade 10): Hard to reach. Glimpse: start with the matrix for 'rotate 90°' and list every matrix you can build from it with + and ×. You get a copy of complex numbers.
Families of operators indexed by time, like T(s+t)=T(s)T(t), describe how systems evolve step by step.
Natural fit Way in (grade 8): Money doubling every year: the 'grow for t years' rule satisfies grow(s) then grow(t) equals grow(s+t). Find grow(1/2) and check it with a calculator.
Differential equations in one variable seen as operators, whose eigenvalues give the natural frequencies of strings and springs.
Stretch Way in (grade 7): Pluck a guitar string or a ruler hanging off a desk at different lengths. Halve the length and the note jumps an octave: eigenvalues you can hear.
Operators built from partial derivatives in several variables, describing heat, waves and quantum particles in space.
Stretch Way in (grade 9): Hard to make elementary. Glimpse: on a grid, replace each number by the average of its four neighbours, repeat, and watch heat spread out: a discrete Laplacian.
Operators that act by weighted averaging (integrals) rather than differencing, plus hybrids used for signals and waves.
Natural fit Way in (grade 7): Blur a row of pixel values by replacing each with the average of itself and its neighbours. Repeat. A blur is an integral operator; un-blurring is the hard direction.
Nonlinear maps between spaces, and when they must have fixed points or behave monotonically.
Natural fit Way in (grade 7): Crumple a copy of a map and drop it on the original: some point lies exactly above itself. Or press cos repeatedly on a calculator and land on 0.739...
Solving equations and inequalities with nonlinear operators, often by iteration or by minimising something.
Natural fit Way in (grade 7): Find the square root of 10 by repeating x → (x + 10/x)/2 from x=3. Count correct digits each step; they roughly double.
Collections of operators that form linear spaces or algebras, studied as whole structures.
Stretch Way in (grade 9): Glimpse: list all 2x2 matrices that are upper triangular. Check sums and products stay upper triangular. Which other shapes of zero pattern are closed like this?
Operator theory applied to quantum physics, probability, signals and other fields.
Natural fit Way in (grade 8): A Markov chain as an operator: a frog jumps between 3 lily pads with given chances. Multiply the probability list by the transition table repeatedly; it settles down.
Operator theory over unusual number systems or logics: quaternions, p-adic numbers, infinitesimals, fuzzy or constructive settings.
Hard Way in (grade 10): Hard to reach. Glimpse: multiply quaternions i·j and j·i using a table, and see that order matters even for these 'numbers'.
49 Calculus of variations and optimal control; optimization 5 of 8
When does a best shape, path or strategy actually exist, rather than just getting better forever?
Stretch Way in (grade 8): Ask for the shortest path from A to B that touches a wall but whose length can never be reached: e.g. the smallest positive number. Contrast with a closed wall, where a best point exists.
At an optimum, small changes cannot help: this gives equations like 'derivative zero' and balance of marginal gains.
Natural fit Way in (grade 7): Fold a square of card into an open box by cutting corners of size x. Tabulate volumes and find the best x; notice the volume barely changes near the best.
The best cost from any starting point satisfies a local equation; working backward from the goal finds it everywhere.
Natural fit Way in (grade 5): On a grid maze, write 0 at the exit, then fill each square with 1 + smallest neighbour. You get the shortest distance from every square, and the best route.
Computer methods for finding optimal controls, such as stepping downhill or splitting time into small steps.
Natural fit Way in (grade 7): Steer a toy car on a number line: each second choose push +1, 0 or -1. Find by trial the fewest pushes to stop exactly at 10. Compare greedy and planned strategies.
Assorted problems on choosing the best curve, shape or control: games, inverse problems, and more.
Natural fit Way in (grade 6): Fence the largest field along a straight river with 100 m of fencing. Try rectangles, then semicircles. Which shape wins?
Finding shapes of least area or length, like soap films, and measuring rough sets in a geometric way.
Natural fit Way in (grade 6): Dip wire frames (a cube, a triangular prism) into soapy water and see the minimal surfaces. Or connect 4 corners of a square with the shortest road network: 120° junctions.
Natural frequencies and other eigenvalues can be found as minimum energy values over all possible shapes of vibration.
Stretch Way in (grade 7): Shake a skipping rope slowly to get one hump, then faster for two humps. The lowest shaking speed is the 'least energy' mode.
Physics laws come from nature choosing the path that makes some quantity least, like light taking the fastest path.
Natural fit Way in (grade 8): Lifeguard puzzle: run on sand at 5 m/s, swim at 1 m/s. Where should you enter the water? Measure with a ruler to find light's bending rule.
51 Geometry 7 of 14
Geometry built only from points, lines and which points lie on which lines, without distance or angle.
Natural fit Way in (grade 7): Draw two triangles in perspective from a point and extend matching sides. The three meeting points always line up (Desargues). Try it with a ruler only.
Incidence geometry where 'lines' are curves like circles, as in Möbius, Laguerre or Minkowski planes.
Stretch Way in (grade 7): Treat circles and straight lines as 'lines': check that through any 3 points there is exactly one. Try with compass on paper, including 3 points in a row.
Geometry over number systems with zero-divisors, where two points can be 'neighbours' joined by many lines.
Stretch Way in (grade 9): Draw a 'geometry' on a 4x4 grid using arithmetic mod 4. Find two points that more than one line passes through, unlike ordinary geometry.
Abstract geometry where you only know which sets are 'closed' (like spans), as in matroids and linear algebra.
Stretch Way in (grade 7): Given some points, keep adding any point on a line through two you have. Which starting sets of 3 points give the whole plane, and which just a line?
Tiny geometries with finitely many points where every two points share exactly one line, linked to designs and codes.
Natural fit Way in (grade 6): Draw the Fano plane: 7 points, 7 lines, 3 points per line. Use it to schedule 7 kids in groups of 3 so every pair meets exactly once. Dobble cards work the same way.
Geometry from distance and reflections: what geometries come from choosing a way to measure lengths?
Natural fit Way in (grade 6): Taxicab geometry: on city-block grid paper draw all points 3 blocks from a corner. Find all points equidistant from two given points; the 'perpendicular bisector' looks strange.
Geometry built from 'betweenness': which point lies between which, and what that alone forces.
Stretch Way in (grade 7): Pasch's axiom: draw a triangle and a line entering one side not through a corner. Show it must leave through another side. Try to draw a counterexample.
Geometries where points and lines form continuous spaces, so lines can wiggle and meet continuously.
Hard Way in (grade 9): Hard to reach faithfully. Glimpse: on a sphere let 'lines' be great circles (rubber bands on a ball). Every two lines meet, and sliding one line moves the meeting points smoothly.
Geometries whose points form a group, with lines tied to the group operation, as in kinematic spaces of motions.
Hard Way in (grade 10): Hard to reach. Glimpse: label the 9 points of a 3x3 grid by pairs mod 3; 'lines' are cosets. Check that adding a fixed pair moves lines to lines.
Recovering shapes from distances alone, and which tables of distances can actually be drawn.
Natural fit Way in (grade 6): Given a table of distances between 4 towns, try to place them on paper with a ruler. Find a table with no possible drawing (break the triangle inequality).
Counting how often curves can cross lines or circles, giving theorems like every closed oval has at least four 'vertices'.
Stretch Way in (grade 8): Draw any closed egg-shape. Find points where the curve is most and least sharply bending. Kids always find at least four: the four-vertex theorem.
Classical geometry of Euclidean, spherical and hyperbolic spaces: lengths, angles, areas, constructions and inequalities.
Natural fit Way in (grade 6): Draw a triangle on an orange with a marker, measure its angles with a protractor, and find the angle sum is more than 180°.
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Geometry with coordinates and projections: describing shapes by equations and drawing 3D objects on flat paper.
Natural fit Way in (grade 6): Draw a road going to a vanishing point and fence posts along it. Where should each post go so they look equally spaced? Use diagonals to find out.
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Using geometry to build physical theories, such as spacetime geometry in relativity.
Stretch Way in (grade 9): Light clock puzzle: a ball bounces up and down on a moving train. Use Pythagoras to see the outside path is longer, so moving clocks must tick slower.
52 Convex and discrete geometry 3 of 3
Convex shapes, with no dents, have rich properties like supporting lines and simple intersection rules.
Natural fit Way in (grade 7): Helly's theorem: draw several convex blobs where every three overlap. Show all of them share a point. Find a non-convex example where it fails.
Polytopes, solid shapes with flat faces, satisfy counting laws like V - E + F = 2.
Natural fit Way in (grade 5): Count vertices, edges and faces of cube, tetrahedron, prisms and a soccer ball. Spot V - E + F = 2 every time, then try a picture frame shape.
Arranging discrete objects in space: packing, covering, tilings, lattice points and point configurations.
Natural fit Way in (grade 6): Pick's theorem: draw polygons on dot paper, count boundary dots B and inside dots I, and check area = I + B/2 - 1.
53 Differential geometry 3 of 6
Curves and surfaces in ordinary space and how they bend, measured by curvature.
Natural fit Way in (grade 7): Fit the biggest circle you can inside a curve at different points (bike track, car on a bend). Tight bends need small circles: curvature is 1/radius.
Measuring distance and bending in small regions of curved spaces using local coordinates.
Stretch Way in (grade 9): Hard to give a full taste. Glimpse: on a map of the globe, a 1 cm step means different real distances at the equator and near the poles. Make a rescaling table.
How the overall shape of a curved space is tied to its curvature, as in the angle sum of triangles on a sphere.
Natural fit Way in (grade 8): Walk a triangle on a globe: north from the equator to the pole, back down along another meridian. Add the angles and compare the excess to the area covered.
Geometry of position-and-momentum spaces, where area is preserved by motion, underlying classical mechanics.
Stretch Way in (grade 10): Draw a swinging pendulum's position versus speed: circles. Start a small blob of starting states and track it: it shears but its area stays the same.
Shapes changing over time by rules tied to curvature, smoothing them out, like curve-shortening or Ricci flow.
Stretch Way in (grade 8): Take a wiggly closed curve on grid paper and repeatedly move each point to the average of its neighbours. It rounds out and shrinks: curve shortening.
Differential geometry applied to physics, engineering, biology, vision and other sciences.
Natural fit Way in (grade 6): Why pizza slices stay stiff when folded: bending a slice one way forbids bending the other way (Gauss's curvature rule). Try it with paper too.
54 General topology 5 of 9
Topology studies closeness without distance: which sets count as 'open' neighbourhoods.
Natural fit Way in (grade 5): Decide which capital letters are 'the same' if you can stretch but not tear: sort A to Z into groups by holes and ends.
Making new spaces from old: gluing edges, taking products, subspaces and quotients.
Natural fit Way in (grade 5): Take a paper strip and glue the ends straight to get a band, or with a half-twist to get a Möbius strip. Cut each down the middle and predict the result.
Continuous maps, those without tearing, and the kinds of spaces they define and preserve.
Natural fit Way in (grade 7): Walk up a mountain on day one, down on day two, same path and times. Show there is a spot you pass at the same clock time both days.
General properties of spaces like compactness and connectedness that guarantee things like 'a best point exists'.
Stretch Way in (grade 8): Compare a closed interval [0,1] with an open one (0,1). In which can you always find the biggest value of x? Then try 'the highest point of a hill'.
Spaces with extra structure, like a way to measure distance or uniform closeness.
Natural fit Way in (grade 6): Find the distance between two cities as the crow flies, by road, and by train lines. Each is a valid distance; check the triangle rule for each.
Special properties of spaces, like dimension and connectedness of strange shapes such as continua.
Natural fit Way in (grade 7): Build the Cantor set: remove the middle third of a line, then of each piece, several times. Total length removed is 1, yet uncountably many points remain.
Weird spaces and counterexamples that test which properties imply which.
Stretch Way in (grade 10): Topologist's sine curve: plot sin(1/x) for small x on a calculator. Is the wiggly curve plus the line segment at x=0 one piece? You cannot walk from one to the other.
Connections between topology and other parts of math, like fixed point theorems and dynamics.
Natural fit Way in (grade 6): Stir a cup of coffee and let it settle. Some point is back where it started (Brouwer). Or: a hairy ball cannot be combed flat, so somewhere on Earth the wind is still.
Topology using infinitesimals, where 'close' means 'differs by an infinitely small amount'.
Hard Way in (grade 10): Hard to reach faithfully. Glimpse: imagine a number ε smaller than every 1/n but bigger than 0. Two points are 'infinitely close' if they differ by a multiple of ε. Is continuity easier to state?
55 Algebraic topology 3 of 8
Some facts are forced by shape alone, like a stirred liquid always having a point that ends up where it started (fixed points).
Natural fit Way in (grade 7): Crumple a copy of a map and drop it on the same map: some point lies exactly above itself. Test it with a city map and a pin.
Count holes of every dimension by an alternating tally, so shapes can be told apart by numbers that stretching cannot change.
Natural fit Way in (grade 6): Count vertices minus edges plus faces for a cube, a pyramid and a donut-shaped frame of sticks. Why do spheres always give 2 and the donut 0?
Two shapes are the same if one can be squashed onto the other without tearing, so a mug and a donut agree.
Natural fit Way in (grade 5): Sort the letters of the alphabet into groups that can be shrunk to the same skeleton: O and D together, B with 8, X with K, I with L.
Loops drawn on a shape, and how they combine, record its holes; some loops can be pulled tight to a point, others cannot.
Stretch Way in (grade 7): Tie a string around a pole and through a ring; which loops can slide off? Then the picture-hanging puzzle: hang a picture on two nails so pulling either nail drops it.
A space can look like a simple product in every small patch yet be twisted overall, like a Mobius strip versus a band.
Natural fit Way in (grade 5): Make a paper band and a Mobius strip, then cut each down the middle. Predict, cut, and explain why the twisted one stays in one piece.
Obstructions are invariants that prove a construction is impossible, such as combing all the hair on a ball flat.
Stretch Way in (grade 7): Try to draw a wind arrow at every point of a ball (orange with toothpicks) with no calm spot. Hairy ball: there is always a cowlick. A donut works.
Spectral sequences compute a hard count by successive approximations, fixing small errors page by page until the answer settles.
Hard Way in (grade 11): Hard to reach faithfully. Glimpse: count a grid's colored cells first by rows, then correct by columns, then correct the corrections, watching the totals stabilise in a table.
Algebraic machines (chains of maps where each step's output is killed by the next) turn shapes into calculable algebra.
Hard Way in (grade 10): Glimpse: a chain of maps where doing two steps in a row always gives zero, like 'boundary of a boundary is empty': the edge of a disk is a circle, and a circle has no ends.
57 Manifolds and cell complexes 6 of 9
Knots, surfaces and 3D spaces have special behaviour; for knots the question is when two tangles are really the same.
Natural fit Way in (grade 6): Tie a knot in a rope, join the ends, and try to untangle. Then color strands of a knot drawing with 3 colors by the tricolorability rule to prove it is knotted.
Low-dimensional shapes like surfaces can be classified by simple features such as holes and twists, built by gluing polygons.
Natural fit Way in (grade 6): Glue a paper square's edges in different patterns: opposite sides straight gives a torus (Asteroids-style wraparound screen), one flipped gives a Klein bottle. Play tic-tac-toe on a torus.
A manifold looks flat up close everywhere, like the Earth looking flat to someone standing on it, but can be curved or looped globally.
Natural fit Way in (grade 5): Why did people think the Earth was flat? Ant on a ball, ant on a donut: each sees a flat patch. Make an atlas of a cube's surface from paper maps that overlap.
Generalized manifolds are spaces that behave like manifolds in their counting of holes even though they may be wild at small scales.
Hard Way in (grade 9): Hard to reach faithfully. Glimpse: a figure 8 looks like a line everywhere except the crossing; find the 'bad' points in letters A, X, T by looking at tiny neighbourhoods.
Shapes can be built from flat triangles and studied by cutting and regluing pieces, piecewise-linear like a polyhedron.
Natural fit Way in (grade 6): Triangulate a square and a sphere-like octahedron with as few triangles as possible. Then cut a paper square into pieces and reassemble it into a triangle (Dudeney dissection).
Smooth shapes carry surprising rigid facts, like spheres with exotic smoothness or curves on surfaces that can never be untwisted.
Stretch Way in (grade 8): Turning a sphere inside out is possible with self-passing, but a circle cannot: draw a loop's winding number (count full turns of your pencil) and see it never changes when you smooth it.
Symmetry groups acting on shapes; the shape left after folding up by a symmetry (the orbit space) reveals structure.
Natural fit Way in (grade 5): Fold a paper snowflake: one cut on the folded wedge appears six times. Predict the pattern before opening. The wedge is the quotient by the symmetry.
Groups that are also shapes, like the rotations of 3D space, have their own holes and loops worth counting.
Stretch Way in (grade 8): The plate trick: hold a cup on your palm and rotate it a full turn, your arm is twisted; turn again the same way and it untwists. Two turns undo, one does not.
Topology of knots and surfaces is used in science, from tangled DNA to the shape of data and robot arm configurations.
Natural fit Way in (grade 7): Untangle earphone cables: how often does random shaking make a knot? Then discuss DNA tangles. Or list all positions of a two-joint robot arm; it is a donut.
58 Global analysis, analysis on manifolds 4 of 9
Calculus can be done on curved spaces using flat local maps glued together, so speeds and directions make sense on a sphere.
Natural fit Way in (grade 6): On a globe, push a pencil 'straight ahead' along a great circle. Fly London to Tokyo on a flat map vs globe with string: why do routes look curved?
Some spaces have infinitely many directions, like the space of all possible curves, and calculus still partly works there.
Stretch Way in (grade 9): Every wiggly path from home to school is one point in a giant space. Describe a path by numbers: how many numbers are needed? Try 3, 10, 100 sample points.
Calculus extended to curved spaces and to maps between spaces of functions, including fixed points of nonlinear maps.
Natural fit Way in (grade 9): Press cosine repeatedly on a calculator starting anywhere: it settles at 0.739. Why must some fixed point exist, and why does this one attract?
The collection of all maps from one shape to another is itself a space with its own geometry.
Stretch Way in (grade 8): List all ways to map a 3-dot set to a 2-dot set (8 of them). Then think of all ways to lay a loop of string on a donut: which can slide into each other?
Find the best shape among infinitely many by making a quantity like length or energy as small as possible, as soap films do.
Natural fit Way in (grade 5): Dip a wire frame of a cube or two linked rings in soapy water and see the minimal surface. Or find the shortest road network joining four towns at square corners.
Pseudogroups and groupoids track symmetries that only hold locally, like matching patches rather than whole-shape symmetries.
Hard Way in (grade 11): Glimpse: in a tiling with a defect, local moves match small patches even where no global symmetry exists. Find all the ways one Penrose-style patch reappears.
The shape of a space controls the waves and heat it can carry, so you can 'hear' properties of a drum from its notes.
Stretch Way in (grade 7): Pluck a rubber band at different lengths and hear the pitch. Can you hear the shape of a drum? Sprinkle salt on a vibrating plate to see Chladni patterns.
Smooth changes can cause sudden jumps; catastrophes are the few standard ways a gradual change produces an abrupt flip.
Natural fit Way in (grade 6): Push down slowly on the middle of a bent ruler held at both ends: it suddenly snaps to the other side. Or the bright cusp of light inside a coffee cup.
Global analysis applied to physics, biology and engineering, such as how the shape of possible states governs motion.
Stretch Way in (grade 10): A pendulum's states (angle and speed) form a cylinder. Draw the phase picture: swinging loops and over-the-top whirls. Where is the dividing line?
60 Probability theory and stochastic processes 10 of 11
Probability needs careful rules about which events can be measured and how chances add up, especially with infinitely many outcomes.
Natural fit Way in (grade 8): Pick a 'random whole number': is it even with chance 1/2? Can every number be equally likely? Compare with spinning a pointer for a random angle.
Randomness on groups and shapes, like shuffling cards as random walks on arrangements, and when the mix becomes uniform.
Natural fit Way in (grade 6): How many riffle shuffles make a deck random? Shuffle 4 cards by top-to-random moves and track the order. Seven shuffles for 52 cards.
Chances computed by careful counting of equally likely arrangements, often with surprising answers.
Natural fit Way in (grade 5): The birthday problem: how many people before two likely share a birthday? Guess, then check in class. Or the hat-check problem: chance nobody gets their own hat.
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Random points, lines and shapes in space, and the chance they hit or cover things.
Natural fit Way in (grade 6): Buffon's needle: drop toothpicks on lined paper and count crossings to estimate pi. Or: break a stick in two random places, chance the pieces form a triangle.
Each random quantity has a distribution, a shape describing how likely each value is, and these shapes combine in rules.
Natural fit Way in (grade 5): Roll two dice 50 times and tally the sums: why is 7 the peak? Then three dice: watch the triangle become a bell.
Averages of many random things become predictable: they settle down and their spread looks like a bell curve.
Natural fit Way in (grade 5): Flip a coin 10, 100, 1000 times (combine the class): the fraction of heads closes in on 1/2. Plot it. Then Galton board with pegs.
Random processes unfold over time, like a gambler's fortune; fair-game rules limit what any strategy can achieve.
Natural fit Way in (grade 7): Gambler's ruin: start with 3 coins, win or lose 1 per flip, stop at 0 or 10. Chance of reaching 10? Show no stopping rule beats a fair game.
Calculus for jittery random paths, where tiny squared wiggles add up to something you cannot ignore.
Stretch Way in (grade 9): Random walk: after n coin-flip steps, how far from start? Average squared distance is n, so distance grows like square root of n. Test with 16 steps.
Processes where the future depends only on the present state, not the past, like a board game driven by dice.
Natural fit Way in (grade 6): Snakes and Ladders: average number of turns to finish? Or weather that is sunny or rainy with given switch chances: after many days, what share is sunny?
Specific random models like queues, renewals and percolation, where simple rules give sharp collective behaviour.
Natural fit Way in (grade 6): Percolation: color grid squares at random with chance p and see when a path crosses the grid. Try p=0.4, 0.6, 0.8. Or the waiting-time paradox at a bus stop.
Rough paths: describe very jagged paths by their increments plus their swept areas, so equations still make sense.
Hard Way in (grade 10): Glimpse: two zigzag routes with the same start and end but enclosing different areas. Draw them and compute signed area; the area is the extra information.
62 Statistics 15 of 16
Statistics asks how to learn about the world from data, and what it even means to be confident about a conclusion.
Natural fit Way in (grade 6): Does a coin seem fair after 7 heads in 10 flips? Debate what 'evidence' means, then simulate fair coins to see how often 7 or more happens.
A summary can capture all the useful information in data; for coin flips, the number of heads is all that matters.
Natural fit Way in (grade 8): Two friends flip a coin 10 times; one reports the full sequence, one only the head count. Is anything about the coin's bias lost? Argue why not.
Choosing actions under uncertainty by weighing the cost of each kind of mistake, not just being right most often.
Natural fit Way in (grade 6): Umbrella decision: chance of rain 30 percent, cost of carrying it vs getting wet. Make a table of costs and pick the rule that loses least on average.
A small, well-chosen random sample can tell you about a huge population, and how you sample matters.
Natural fit Way in (grade 5): Capture-recapture: estimate the beans in a jar by marking 20, mixing, and drawing 30. Then discuss why polling only your friends misleads.
Statistics of data follow known distributions, which tell you how surprising a result really is.
Natural fit Way in (grade 7): Simulate class averages of 5 dice rolls and draw the histogram: averages bunch tighter than single rolls. How much tighter?
Estimate an unknown number in a model from data and say how sure you are, for example with a confidence range.
Natural fit Way in (grade 7): German tank problem: serial numbers 14, 47, 61, 83 captured; estimate how many tanks exist. Compare guesses like max, max plus gap, twice the mean.
Draw conclusions without assuming a specific curve shape, often using ranks or resampling the data itself.
Natural fit Way in (grade 7): Are boys and girls' jump lengths different? Shuffle the labels on cards many times and see how often the gap is as big as the real one (permutation test).
Data with many measured features at once, finding the few directions or groups that explain most of the variation.
Natural fit Way in (grade 7): Plot heights and arm spans of the class: the cloud lies along a line. Find the line by eye; that single direction captures most of the information.
Fit lines to data to predict one quantity from others, and notice traps like regression to the mean.
Natural fit Way in (grade 7): Tallest parents tend to have tall children but less extreme. Simulate: score = skill + luck dice; top scorers' second scores drop. Why?
Plan experiments so that a few cleverly chosen tests answer many questions fairly, using randomisation and balance.
Natural fit Way in (grade 6): Find the one poisoned bottle among 8 with 3 tasters (binary). Or test 3 paper plane tweaks in 4 flights using a balanced design.
Look at data as it arrives and stop as soon as you have enough evidence, saving time and cost.
Natural fit Way in (grade 7): Is this coin biased? Flip and keep a running score, stopping when it crosses +5 or -5. Compare the average number of flips with a fixed 50.
Learn about a process from one long sequence of dependent observations, like reading patterns in weather or stock prices.
Natural fit Way in (grade 6): Can you tell a human-made 'random' coin sequence from a real one? Count runs and streaks: humans avoid long runs. Try it with the class.
Estimate how long things last when some stories are cut short, counting what you know without throwing away incomplete data.
Natural fit Way in (grade 8): Light bulbs tested for 6 months, some still alive at the end. How do you estimate the average lifetime fairly? Build a Kaplan-Meier staircase with 8 bulbs.
Statistics applied to medicine, sport, finance, law and more, turning real data into decisions.
Natural fit Way in (grade 7): Simpson's paradox: a player hits better each season but worse overall. Make a small batting table where this happens.
Statistical tables give precomputed values of distributions; bookkeeping for lookup rather than a topic in itself.
Statistics for data that live on curved or structured spaces, like directions, rankings or shapes.
Natural fit Way in (grade 7): Average wind directions 350 and 10 degrees: the naive mean says 180, south! Fix it by averaging arrows instead. Plot class birthdays on a circle.
65 Numerical analysis 16 of 20
Tables in numerical analysis: bookkeeping for published numerical tables rather than a topic in itself.
Speed up slowly converging sums or sequences by spotting their error pattern and cancelling it.
Natural fit Way in (grade 8): Add 1 - 1/3 + 1/5 - ... times 4 to approach pi: painfully slow. Average two consecutive partial sums and watch it jump closer.
Use random sampling to compute things too hard to calculate exactly, like areas, integrals or risks.
Natural fit Way in (grade 5): Throw rice on a square with a quarter circle drawn in it; the fraction inside times 4 estimates pi. How many grains for two correct digits?
Approximate curves and shapes with simple pieces like polynomials, and do geometry tasks fast with clever algorithms.
Natural fit Way in (grade 6): Draw a smooth curve through 4 points using a bendy ruler (a spline). Or find the convex hull of 15 dots by wrapping an imaginary rubber band, gift-wrapping step by step.
Numerical methods for functions of complex numbers, such as computing flows and fields in the plane.
Hard Way in (grade 11): Glimpse: count how many times the path of z squared winds around 0 as z goes around a circle; draw it. Winding counts locate roots numerically.
Solve big systems of linear equations quickly and accurately, by elimination or by repeated guess-and-improve steps.
Natural fit Way in (grade 8): Solve three equations by elimination. Then the guess-and-improve way: each unknown takes the average of its neighbours, repeat, watch a temperature grid settle.
Every computer calculation carries rounding error; track it, or keep an interval guaranteed to hold the true answer.
Natural fit Way in (grade 6): Calculator trap: compute (1/3)*3 - 1 or sqrt(2)^2 - 2. Then interval arithmetic: if length is 3.1 to 3.2 and width 1.9 to 2.0, what range for area?
Find solutions of equations that cannot be solved by formula, by repeatedly zooming in on the answer.
Natural fit Way in (grade 7): Find the square root of 10 by bisection (guess, square, adjust), then by Babylonian averaging. Count the correct digits per step: one doubles them.
Numerical methods framed in infinite-dimensional spaces, proving algorithms converge for whole families of problems at once.
Hard Way in (grade 11): Glimpse: approximate a whole curve by repeatedly applying an averaging rule to 10 sample points, then 20, 40, and see the answers agree more and more.
Computer methods to find the best choice, like walking downhill in a landscape to reach the lowest point.
Natural fit Way in (grade 6): Blindfold hill game: find the lowest point of a bowl-shaped table of numbers by only checking neighbours. When does stepping downhill get stuck?
Step a changing quantity forward in small time steps; small steps are accurate, too-big steps can blow up.
Natural fit Way in (grade 8): Euler's method for money growing 10 percent per year compounded in steps: 1, 2, 12 steps a year. Watch the answer approach e times the start.
Simulate how things like heat or waves spread over time on a grid, keeping steps small enough to stay stable.
Natural fit Way in (grade 7): A row of 10 cells, hot in the middle. Each step every cell moves toward its neighbours' average. Track by hand; then try too large a step and watch nonsense.
Compute steady states, like temperature in a plate with fixed edges, by dividing the region into small cells.
Natural fit Way in (grade 7): Grid with edges fixed at 0 and 100; replace each inner cell by its neighbours' average until nothing changes. The final picture is the steady heat.
Computers simulate long-term motion where tiny errors can grow wildly (chaos), so methods must respect the underlying structure.
Natural fit Way in (grade 8): Iterate x to 4x(1-x) from 0.3 and 0.3001 on a calculator: after 20 steps they disagree completely. What can a computer ever predict?
Compute sequences defined by recurrence rules, where innocent-looking formulas can amplify errors catastrophically.
Natural fit Way in (grade 8): Compute Fibonacci forward, fine. Then go backward from two rounded values and watch errors explode, or the recurrence x(n+1)=10 x(n) - 1 starting from 0.1 on a calculator.
Numerically solve equations where the unknown function appears inside an integral or transform.
Stretch Way in (grade 10): Glimpse: area under a curve by trapezoids, then reverse: given running totals of rainfall, recover the daily amounts and see how small errors in totals ruin it.
Graphical methods: solving numerical problems by drawing, like reading answers off nomograms and graph intersections.
Natural fit Way in (grade 6): Build a nomogram: three parallel scales where a ruler line gives a+b. Or solve x squared = x + 1 by drawing both graphs and reading the crossing.
Fast computation with waves, breaking a signal into frequencies, famously with the fast Fourier transform.
Stretch Way in (grade 9): Hidden beat: add two sine waves on graph paper and see the beating. Then multiply large numbers by splitting into parts, the trick behind fast algorithms.
How computer hardware and software shape algorithms: parallel work, memory, and floating point numbers.
Natural fit Way in (grade 6): Add 100 numbers with one person vs ten people in parallel: how many rounds? Pairwise adding tree in 7 rounds. Also why 0.1 + 0.2 is not 0.3 in a computer.
Numerical methods applied to science and engineering, from weather forecasts to bridge design.
Natural fit Way in (grade 6): Model a cooling cup of tea: each minute lose 10 percent of the gap to room temperature. Tabulate and compare with a real thermometer.
68 Computer science 8 of 9
How a computer's parts (memory, processors, networks) are arranged so many tasks share them quickly and reliably.
Natural fit Way in (grade 6): Act out a cache: one student is slow 'memory' across the room, another keeps a sticky note of the 3 most recent answers. Count how many trips the sticky note saves on a list of requests.
How to write programs and prove they do exactly what we intended, using precise rules for what each instruction means.
Natural fit Way in (grade 5): Write step-by-step instructions for making a sandwich, then a partner follows them literally. Fix the bugs, then prove a tiny loop 'add 2, five times' always ends at 10.
How to store, organise and search information so questions can be answered fast, and how to keep it small or private.
Natural fit Way in (grade 5): Find a word in a 1000-page dictionary in at most 10 looks by halving each time. Compare with checking pages one by one.
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What computers can and cannot do in principle, and how much time or memory a problem truly needs.
Natural fit Way in (grade 8): Sorting 8 cards by comparing pairs: why 7 comparisons cannot be enough (8! orders, each comparison splits two ways, 2^15 < 8!). A first lower-bound proof.
Graphs, words and counting used as the raw material of computing: networks, strings and combinatorial structures behind algorithms.
Natural fit Way in (grade 5): Colour a map of 8 countries with as few colours as possible, then turn it into dots and lines and see it becomes a scheduling problem for exams.
Making machines learn patterns and make choices, usually by adjusting numbers to reduce mistakes on examples.
Natural fit Way in (grade 6): Build a matchbox machine that learns noughts and crosses: add a bead to moves that win, remove one from moves that lose. Play 20 games and watch it improve.
Using computation for pictures, documents, simulations and other applications, turning real-world tasks into clean steps on numbers.
Natural fit Way in (grade 6): Draw a straight line on graph paper using only whole-number squares (pixels): decide square by square which to shade, the idea behind Bresenham's line algorithm.
Computers helping mathematicians: checking proofs, searching huge cases, and doing exact algebra too big for paper.
Stretch Way in (grade 7): The four colour theorem was proved by computer checking. Kids check by hand a mini case: all 2-colourings of a 5-cycle fail, so it needs 3 colours.
Designing step-by-step methods that solve problems correctly and as efficiently as possible, and comparing their speed.
Natural fit Way in (grade 5): Race two ways to sort 16 shuffled cards: pick-the-smallest repeatedly versus merging sorted piles. Count comparisons and see merging wins as piles grow.
70 Mechanics of particles and systems 11 of 14
Choosing the few basic laws (like Newton's) from which all of mechanics follows, and asking what they really assume.
Stretch Way in (grade 8): Is a frame 'non-accelerating'? Drop a ball in a smoothly moving train versus a braking one and predict where it lands. Newton's laws only hold in the first.
Describing motion itself (position, velocity, acceleration, rotation) without asking what causes it.
Natural fit Way in (grade 6): A bike wheel's valve traces a cycloid. Roll a coin along a ruler with a dot marked and sketch its path; notice the dot stops at the bottom.
When forces balance, nothing moves; statics finds the forces in bridges, levers and stacks at rest.
Natural fit Way in (grade 7): Stack identical books off a table edge, each overhanging the one below by 1/2, 1/4, 1/6... The overhang grows without limit (harmonic series).
How solid objects spin, tumble and wobble, and why some spins are steady while others flip over.
Natural fit Way in (grade 7): Tennis racket theorem: toss a phone or book spinning about each of its three axes. Two spins are steady, the middle one always flips.
How several bodies move under mutual forces, from colliding balls to planets; two bodies are solvable, three generally are not.
Natural fit Way in (grade 7): Newton's cradle: count balls in and out. Momentum and energy both conserved force exactly as many balls to fly out as came in.
General tools for mechanics: symmetry, conserved quantities and geometry that work for any system of particles.
Natural fit Way in (grade 8): A skater pulling in her arms spins faster. Compute: radius halves, so speed of spin quadruples to keep angular momentum (mass times radius squared times spin) fixed.
Nature picks the path that makes a certain quantity (action) stationary; mechanics can be rewritten as an optimisation.
Stretch Way in (grade 8): Lifeguard problem: run on sand then swim to a swimmer. Find the fastest landing point by trying distances; it obeys the same rule as light bending.
Small wiggles around rest add up from simple 'modes', each with its own natural frequency.
Natural fit Way in (grade 7): Hang two pendulums from one string; start one swinging and watch energy pass back and forth. Two hidden modes, together and opposite, explain it.
When forces are not proportional to displacement, motion can jump, lock in rhythm, or become chaotic.
Natural fit Way in (grade 7): A double pendulum made of two rulers and a pin: start it twice from nearly the same spot and record how soon the motions disagree.
Mechanical systems shaken by random forces, where we predict averages and spreads rather than exact paths.
Natural fit Way in (grade 6): Brownian motion by coin flips: a 'pollen grain' steps left or right each flip. After 100 flips it is typically about 10 steps away, not 0 or 100.
How satellites and spacecraft move under gravity, and how to change orbits cheaply.
Stretch Way in (grade 8): To catch a space station ahead of you, you slow down: a lower orbit is faster. Draw two circles and compare lap times using Kepler's rule.
How things move when they gain or lose mass, like rockets throwing fuel backwards.
Natural fit Way in (grade 8): Rocket on a skateboard: throw 10 balls backwards one at a time and compute your speed after each throw. Later balls help more, giving the rocket equation's logarithm.
Steering mechanical systems toward a goal with feedback, like balancing a broom or parking a car.
Natural fit Way in (grade 5): Balance a broom upright on your palm: you must move your hand toward where it leans. Then try with a short pencil and see why short is harder.
Treating fields that fill space (like a vibrating string's shape) with the same action principles used for particles.
Stretch Way in (grade 7): A guitar string: halve its length and pitch goes up an octave. List which wave shapes fit a fixed string: 1, 2, 3... half-waves.
74 Mechanics of deformable solids 10 of 17
Treating solids as continuous stuff where every tiny piece feels stretching and stress from its neighbours.
Stretch Way in (grade 7): Draw a grid on a balloon or rubber glove, stretch it, and see each little square become a parallelogram: that local change of shape is 'strain'.
Elastic materials spring back, and for small stretches the force is proportional to stretch (Hooke's law).
Natural fit Way in (grade 5): Hang coins from a rubber band and measure the stretch for 1 to 6 coins. Plot it: a straight line, until it suddenly is not.
Some materials bend permanently once pushed past a limit; plasticity describes that yield and the memory it leaves.
Stretch Way in (grade 5): Bend a paperclip slightly and it springs back; bend more and it stays bent. Bend it back and forth and count bends until it snaps.
Materials whose response depends on how fast and how long you push them, mixing springiness with slow flow.
Stretch Way in (grade 5): Silly putty or cornflour slime: pull slowly and it stretches, yank it and it snaps. Time how long a ball of it takes to flatten.
Special material behaviours: different stiffness in different directions, layers, crystals, foams and other structures.
Stretch Way in (grade 5): Tear a sheet of newspaper top to bottom, then side to side. One tear is straight and easy: paper has a grain, it is anisotropic.
Solids interacting with heat, electricity, fluids or chemistry, such as metals expanding when warm.
Natural fit Way in (grade 7): Why do bridges have gaps? A 100 m steel bridge grows about 1.2 mm per degree; compute the growth from winter (-10) to summer (35).
Finding the shape and stresses of a loaded solid that has settled down, including when it suddenly buckles.
Natural fit Way in (grade 5): Press down on a drinking straw or a long strand of spaghetti from both ends. It holds, then suddenly bows sideways: buckling. Shorter pieces hold more.
How solids vibrate and respond to sudden or changing loads over time.
Natural fit Way in (grade 6): Chladni plate: sprinkle salt on a metal tray and tap or bow it. The salt gathers on lines that do not move, drawing the vibration shapes.
Waves travelling through solids, like sound in steel or earthquakes, with speeds set by stiffness and density.
Natural fit Way in (grade 7): Locate an earthquake: fast P-waves and slower S-waves arrive 20 seconds apart. With speeds 6 and 3.5 km/s, compute the distance.
Thin things (beams, plates, shells, membranes) whose shape gives them strength far beyond their material.
Natural fit Way in (grade 5): A flat sheet of paper cannot hold a pencil over a gap; fold it into a zigzag or curl it into a tube and it holds a book.
Solid mechanics applied in special settings like soils, ice, bones and geological layers.
Natural fit Way in (grade 5): Build a sandcastle with dry, damp and soaked sand. Damp sand stands tallest: tiny water bridges glue grains. Find the steepest dry sand pile angle.
Special problems such as contact, friction and impact between solid bodies.
Natural fit Way in (grade 5): Drop a ball and measure each bounce height. Each bounce keeps the same fraction, so the heights shrink geometrically. Add them up to find the total distance travelled.
Solids that switch internal structure, like shape-memory metals or ice changing crystal form.
Stretch Way in (grade 6): Nitinol wire, bent out of shape, springs back when dipped in hot water. With no wire: why does a frozen water bottle bulge? Ice is 9% bigger.
Designing the best shape or material layout so a structure is strongest or lightest.
Natural fit Way in (grade 5): Bridge from one sheet of paper across two books: compare flat, folded, arched and tube designs by counting coins held. Then draw the winning layout.
Replacing a finely mixed material by one uniform material with the same overall behaviour.
Natural fit Way in (grade 7): Two springs in a row versus side by side: hang weights and find each pair's single 'effective' stiffness. Composites average in these two different ways.
How cracks start and grow, and why a small notch can make strong things fail.
Stretch Way in (grade 5): A tiny cut makes a crisp packet or ketchup sachet easy to tear. Try tearing paper with and without a nick and count the force in coins.
Computing how solids deform by cutting them into many small simple pieces, the finite element idea.
Stretch Way in (grade 8): Approximate a sagging rope by 4 straight sticks joined at hinges; balance forces at each joint by hand. More sticks, closer to the true curve.
76 Fluid mechanics 12 of 23
The basic laws for fluids: mass, momentum and energy balanced in every tiny parcel, plus how 'runny' each fluid is.
Natural fit Way in (grade 5): Race honey, oil and water down a tilted plate and time them. Then tilt twice as steeply: does the time halve? That ratio defines viscosity.
Idealised fluids with no friction and no squeezing, which explain waves, vortices and lift surprisingly well.
Natural fit Way in (grade 5): Hold a spoon's back loosely under a running tap: it gets pulled into the stream. Faster flow, lower pressure (Bernoulli).
Sticky fluids that cannot be squeezed: how friction shapes flow in pipes, around small objects and in syrup.
Natural fit Way in (grade 6): Drop a marble into a tall glass of syrup and time it: it soon falls at a steady speed. Double the marble's size and compare.
When does a smooth flow stay smooth, and when does a tiny wobble grow into something new?
Natural fit Way in (grade 6): A thin stream from a tap breaks into drops. Measure how long the smooth part is at different tap speeds; small ripples grow and pinch it off.
Turbulence: chaotic swirling flow with eddies of every size, where energy cascades from big swirls to small ones.
Stretch Way in (grade 6): Pour milk into tea or ink into water and sketch the swirls: big ones break into smaller ones. Compare with a slow trickle that stays smooth.
How air flows around wings, cars and balls at ordinary speeds, giving lift and drag.
Natural fit Way in (grade 5): Make paper planes with different wing areas and folds; measure glide distance. Then: curved paper strip lifts when you blow over its top.
Flow near the speed of sound, where some parts move faster than sound and others slower, causing shocks.
Stretch Way in (grade 8): Mach number arithmetic: sound is about 340 m/s. A plane at 900 km/h is at what Mach? Wing tops go faster: find where air reaches Mach 1.
Flow faster than sound, where disturbances cannot travel ahead, so a cone-shaped shock trails the object.
Natural fit Way in (grade 6): A boat's V-shaped wake is a 'Mach cone' on water. Draw ripple circles left at each second by a fast boat; their outer edge forms the V.
Extremely fast flow (above about 5 times sound speed) where air heats so much it changes chemically.
Hard Way in (grade 8): Meteors glow because air in front is squashed and heated, not mainly by rubbing. Compute: a 20 km/s meteor is Mach 60. Honest glimpse only.
Shock waves are near-instant jumps in pressure and density, from sonic booms to explosions.
Stretch Way in (grade 6): Traffic jam shock: cars at a red light. Line up toy cars, let the front move, and watch the 'jam front' travel backwards, just like a shock.
General methods for solving fluid problems: numerical grids, particles, and scaling with dimensionless numbers.
Stretch Way in (grade 8): Shrink a ship model 25 times: what speed matches the real wave pattern? Froude number says scale speed by the square root, so 5 times slower.
Gases that squeeze: how pressure, density and temperature change together in moving air.
Natural fit Way in (grade 6): Pump up a bike tyre and feel the pump warm. Squeeze a syringe of air with your finger blocking it: halve the volume, double the push.
Gases so thin that molecules rarely collide, so we track collisions statistically instead of treating air as smooth.
Stretch Way in (grade 7): Simulate a gas with coins on a grid: each 'molecule' moves randomly; count how many hit a wall per turn. Fewer molecules, a jumpy count.
How moving fluids make sound, from whistles and jets to wind howling round buildings.
Natural fit Way in (grade 5): Blow across a bottle with different water levels and note the pitches. Less air inside, higher note. Tune 5 bottles to a scale.
How heat and substances spread, by random mixing (diffusion) and by the flow carrying them (convection).
Natural fit Way in (grade 6): Drop food colouring in cold and hot water and time spreading. Then square-root rule: four times the distance takes sixteen times as long.
Fluid seeping through holes in sand, rock or coffee, where flow rate is proportional to pressure difference (Darcy's law).
Natural fit Way in (grade 5): Fill identical cups with sand, gravel and soil, pour equal water and time how long it drips through. Double the water height and compare.
Flows mixing several phases or substances, like bubbles in water, oil and vinegar, or sand in wind.
Natural fit Way in (grade 5): Shake oil and vinegar and time the separation. Smaller droplets take longer: measure with a fork-shaken versus a hard-shaken jar.
Flow on spinning things, like Earth, where rotation bends motion sideways and organises weather and ocean currents.
Natural fit Way in (grade 6): Stir tea, stop, and watch tea leaves gather in the middle at the bottom. Or draw a straight line on a spinning paper plate: it comes out curved.
Flows where chemical reactions happen as fluid moves, like flames, combustion and reacting mixtures.
Stretch Way in (grade 7): A candle flame is a flow: hot gas rises, fresh air comes in. In space flames are round. Predict and sketch a candle flame with no 'up'.
Electrically conducting fluids pushed and steered by magnetic or electric fields, as in the Sun or Earth's core.
Hard Way in (grade 8): Saltwater 'boat': a battery's current through salty water over a magnet makes the water move. Kid version: diagram why Earth's molten core makes a compass work.
Plasma, gas so hot that atoms lose electrons, moving under electric and magnetic fields, as in auroras and fusion.
Hard Way in (grade 7): Auroras: charged particles spiral along magnetic field lines into the poles. Draw field lines of a bar magnet with iron filings and mark where they bunch.
Fluids where quantum effects or near-light speeds matter, like superfluid helium that flows with zero friction.
Hard Way in (grade 9): Superfluid helium climbs walls and never stops swirling. Honest glimpse: compare a spinning bucket of water slowing down with one that would spin forever.
Fluids in living things: blood in vessels, air in lungs, and how swimmers and fliers push through fluid.
Natural fit Way in (grade 7): Why is blood pressure fragile in narrow vessels? Flow scales with radius to the 4th power: compute flow if an artery narrows to half width (1/16).
78 Optics, electromagnetic theory 2 of 2
Light and other electromagnetic waves reflect, bend and interfere in ways predicted exactly by geometry and a few equations.
Natural fit Way in (grade 8): Put a pencil in a glass of water, see it bend, then use Snell's law with a protractor to predict where a coin at the bottom appears to sit.
Optics problems are solved by tracing rays, adding waves, or letting a computer split space into tiny cells and step the fields forward.
Natural fit Way in (grade 7): Trace rays through a drawn lens with a ruler: draw the ray through the centre and the ray parallel to the axis, and find where they cross to locate the image.
80 Classical thermodynamics, heat transfer 1 of 2
Heat flows from hot to cold, energy is conserved, and some energy always becomes unusable, which limits every engine.
Natural fit Way in (grade 6): Mix 100 g of water at 80 degrees with 300 g at 20 degrees and predict the final temperature as a weighted average, then test it with a kitchen thermometer.
Heat problems are solved by chopping objects into small pieces and repeatedly averaging each piece's temperature with its neighbours.
Natural fit Way in (grade 6): A row of 6 cells holds temperatures 100,0,0,0,0,0. Each step replace each inner cell by the average of its neighbours and watch the heat spread.
81 Quantum theory 2 of 7
Quantum states carry information that cannot be copied or read without disturbing it, which enables unbreakable key sharing and new kinds of computing.
Stretch Way in (grade 8): Play BB84 with coins and two card 'filters': sender and receiver pick random bases, keep only matching ones, and see how an eavesdropper leaves detectable errors.
Quantum systems are described by operators whose eigenvalues are the only allowed measurement results, like the allowed notes of a guitar string.
Stretch Way in (grade 7): Pluck a string held at both ends: only wavelengths fitting a whole number of half-waves survive. List the allowed wavelengths for a 60 cm string, like energy levels.
Symmetries of quantum systems form groups and algebras, and these symmetries alone predict patterns such as how many states share an energy.
Hard Way in (grade 10): Spin a square and count its 8 symmetries; then see why a square box has energy levels that come in pairs, since swapping x and y gives a second state.
Quantization turns smooth classical motion into discrete levels and probability amplitudes, where outcomes come from squaring complex numbers that can cancel.
Stretch Way in (grade 9): In a two-path 'interferometer' add arrows: two paths with arrows pointing opposite cancel, same direction double. Square lengths to get probabilities and see light vanish at a detector.
Quantum field theory treats particles as ripples in fields filling space, combining relativity and quantum mechanics with bookkeeping by Feynman diagrams.
Hard Way in (grade 9): Count Feynman-style diagrams: how many ways can you connect 4 dots in pairs with lines? (3.) Then 6 dots (15). This pairing count is the heart of the calculations.
Scattering theory works out what comes out when a particle bounces off a target, inferring the hidden shape from the pattern of deflections.
Stretch Way in (grade 6): Roll marbles at a hidden shape under a cardboard sheet and record where they exit; guess the shape from the bounce angles, like Rutherford discovering the nucleus.
Quantum theory explains real systems like atoms, molecules and solids, for example why atoms emit light only at particular colours.
Natural fit Way in (grade 8): Compute hydrogen colours: energy levels go like 1/n^2. Find 1/4 minus 1/9 and 1/4 minus 1/16, compare their ratio with red and blue-green Balmer lines from a chart.
82 Statistical mechanics, structure of matter 4 of 4
Huge numbers of random particles settle into predictable equilibrium, where the most likely arrangements dominate and sudden phase transitions can appear.
Natural fit Way in (grade 6): Toss 10 coins many times: 5 heads happens 252 ways out of 1024, all heads only once. With a million coins, near-half is almost certain. That is equilibrium.
Systems out of equilibrium evolve by random moves, and simple local rules explain spreading, mixing and the arrow of time.
Natural fit Way in (grade 5): Ehrenfest urn: put 10 numbered tokens in box A. Each turn roll a number and move that token to the other box. Watch it drift to 5-5 and rarely return.
Statistical mechanics explains specific materials such as magnets, polymers, liquids and gases from the random behaviour of their parts.
Natural fit Way in (grade 6): Model a polymer as a random walk: flip a coin 16 times, step left or right, and measure how far the chain's end got. Typical distance is about 4, not 16.
Many-particle systems are studied by sampling random configurations on a computer, using tricks like Monte Carlo and renormalisation to zoom out.
Natural fit Way in (grade 7): Block-spin zoom: colour a 9x9 grid of random black and white cells, replace each 3x3 block with its majority colour, and repeat to see large-scale structure.
83 Relativity and gravitational theory 4 of 6
Space and time mix together so that the speed of light is the same for everyone, making moving clocks run slow.
Natural fit Way in (grade 8): Light clock: a light pulse bounces between mirrors 3 m apart. On a moving train it travels a slanted path; use Pythagoras to find how much longer each tick takes.
Relativity is tested by careful measurements, like clocks on aircraft, bending starlight and the timing corrections GPS needs every day.
Natural fit Way in (grade 7): GPS clocks run fast by about 38 microseconds a day. Multiply by the speed of light to find the position error per day (about 11 km) if nobody corrected it.
Gravity is the curving of spacetime by mass, and objects in free fall simply follow the straightest possible paths through that curve.
Stretch Way in (grade 6): Draw a triangle on a ball or orange: its angles add to more than 180 degrees. Straight lines on curved surfaces behave like orbits in curved spacetime.
Physicists build rival theories of gravity to test Einstein's, changing the equations and checking which predictions survive experiment.
Stretch Way in (grade 8): Compare predictions: Newton gives light bending by 0.87 arc seconds at the sun, Einstein 1.75. The 1919 eclipse measured about 1.6 to 1.98. Which theory wins and why?
Unified theories try to combine all forces, often by adding extra hidden dimensions curled up too small to see.
Hard Way in (grade 7): An ant on a garden hose sees two dimensions but from far away the hose looks like a line. Draw a 'curled up' dimension and compute its circumference for radius 1 mm.
The universe expands, so distant galaxies recede faster the farther away they are, and running the film backwards points to a hot beginning.
Natural fit Way in (grade 6): Draw dots on a balloon, measure gaps, inflate it, measure again. Every dot sees others recede, with speed proportional to distance. Estimate the universe's age as distance over speed.
85 Astronomy and astrophysics 1 of 1
Astronomy uses geometry, light and gravity to measure distances, masses and lives of stars and galaxies we can never visit.
Natural fit Way in (grade 5): Measure parallax: hold your thumb at arm's length, blink each eye, and see it jump. Measure the shift against a wall and compute your arm length, as astronomers do for stars.
86 Geophysics 1 of 1
Geophysics uses physics and data from waves, gravity and magnetism to infer what lies deep inside the Earth and how it moves.
Natural fit Way in (grade 7): Locate an earthquake: given P and S wave arrival gaps at three towns on a map, draw distance circles with a compass and find where they meet.
90 Operations research, mathematical programming 2 of 2
Operations research finds the best way to schedule, route, queue and stock things when time, money and capacity are limited.
Natural fit Way in (grade 5): Plan a morning: shower, toast, boil kettle, dress. Some tasks can overlap. Draw a task chart and find the shortest possible finish time, the critical path.
Optimisation finds the best value of something subject to constraints, and for linear problems the best answer always sits at a corner.
Natural fit Way in (grade 8): A bakery makes cakes (profit 3) and pies (profit 2) with limited flour and oven time. Shade the allowed region on graph paper and test its corners.
91 Game theory, economics, finance, and other social and behavioral sciences 7 of 7
Game theory studies strategic choices where each player's best move depends on the others, leading to equilibria where nobody wants to switch.
Natural fit Way in (grade 5): Play prisoner's dilemma in pairs for 10 rounds with points, then try tit-for-tat. Find why both defecting is stable even though both cooperating scores more.
Economics models how people trade, price and share scarce things, and how individual choices combine into markets and votes.
Natural fit Way in (grade 6): Run a classroom market: buyers hold cards with values, sellers with costs. Trade by haggling, record prices, and see them settle near where supply meets demand.
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Social science uses math to study networks, measurement and how individual choices add up to group patterns like segregation.
Natural fit Way in (grade 6): Schelling's model: two colours of counters on a grid; any counter with fewer than a third like neighbours moves. Even mild preferences give strong clustering.
Math models of society track friendships, kinship and influence as networks, revealing small worlds and how ideas spread.
Natural fit Way in (grade 6): Ask: how many friends do your friends have on average? Draw a small friendship network and discover the friendship paradox: your friends usually have more friends than you.
Mathematical psychology models perception, memory and decisions, for instance showing we sense ratios rather than differences.
Natural fit Way in (grade 6): Weber's law: can you tell 100 g from 110 g? Now 1000 g from 1010 g? Test with bags of rice and find the just noticeable difference is a fixed percentage.
Math is used in linguistics, history, law and other social sciences, for example finding striking regularities in how words are used.
Natural fit Way in (grade 5): Count words in a page of a book: the most common word appears about twice as often as the second, three times the third. Check Zipf's law with a tally.
Actuarial science and finance price risk and time, from insurance premiums to options, using probability and the time value of money.
Natural fit Way in (grade 6): Insurance pool: 100 friends each own a 200 dollar bike, 3 get stolen a year. What fair yearly premium covers everyone? Then add compound interest to savings.
92 Biology and other natural sciences 5 of 5
Biology can be described by math models that predict growth, patterns and shapes of living things from simple rules.
Natural fit Way in (grade 5): Count spirals on a pinecone or sunflower head in each direction: you get neighbouring Fibonacci numbers like 8 and 13. Explore why golden-angle packing does this.
Math models of the body describe heartbeats, drug doses, nerve signals and disease spread inside and between cells.
Natural fit Way in (grade 7): A pill halves in the blood every 6 hours and a 100 mg dose is taken every 12 hours. Tabulate the amount and see it level off near 133 mg.
Populations and genes change by births, deaths, chance and selection, and simple equations predict booms, crashes and inheritance.
Natural fit Way in (grade 6): Punnett squares with coins: two Bb parents each flip for B or b. Over 40 flips see about a quarter bb, and compare with the 1:2:1 prediction.
Chemistry uses math to count molecules, balance reactions and enumerate the shapes molecules can take.
Natural fit Way in (grade 7): Count isomers: how many different ways can 4 carbon atoms join in a chain or branch (2)? For 5 carbons (3)? Draw them as trees and avoid duplicates.
Math helps other natural sciences, such as ecology, agriculture and environment, through models of resources and spreading.
Natural fit Way in (grade 7): Fish pond: the population grows 50 percent a year minus overcrowding; fishing takes 20 fish yearly. Tabulate years and find a catch level that never empties the pond.
93 Systems theory; control 5 of 5
Systems theory treats anything with inputs and outputs as a box, and studies how boxes connected in loops behave.
Natural fit Way in (grade 6): Shower knob game: the water temperature responds 5 seconds late. Turn the knob each step by half the error from your target and see overshoot and settling in a table.
Controllability asks whether you can steer a system anywhere using the controls you have; observability asks whether you can see its hidden state.
Natural fit Way in (grade 6): Parallel parking: you can only drive forward/back and steer, yet you can move a car sideways. Act it out with a toy car and list the moves.
Control theory builds simple models of systems like cars, thermostats and robots, describing how the state changes with each control input.
Natural fit Way in (grade 6): Thermostat: the room gains 1 degree a minute when the heater is on and loses 0.5 when off. Switch at 19 and 21 degrees and sketch the zigzag graph.
A control system is stable if small disturbances die out rather than growing, which depends on how strongly it corrects itself.
Natural fit Way in (grade 7): Balance a broom on your hand versus hanging a pendulum. Then iterate x goes to -k times x for k=0.5, 1 and 1.5: corrections that are too strong blow up.
Stochastic control steers systems hit by random noise, estimating the hidden state from noisy readings and acting on the best guess.
Natural fit Way in (grade 7): Guess a hidden number: noisy readings are 52, 47, 50, 55, 46. Keep a running average, updating each time, and see estimates settle, the core of a Kalman filter.
94 Information and communication theory, circuits 3 of 4
Information can be measured in bits, and there is a hard limit on how much any message can be compressed or sent through a noisy channel.
Natural fit Way in (grade 5): Twenty questions: how many yes/no questions to find a number from 1 to 1000? Ten, since 2^10 is 1024. Then compress a message with a Huffman tree.
Adding cleverly chosen extra bits to a message lets the receiver detect and even fix errors without asking again.
Natural fit Way in (grade 5): Card trick: lay out a 5x5 grid of cards, add a parity row and column, let a friend flip one card, and find it by checking which row and column are odd.
Circuits and networks follow rules like conservation of flow, letting us compute currents, flows and the effect of switches in series and parallel.
Natural fit Way in (grade 6): Switches in series and parallel: draw circuits of switches and find when the light is on. Series is AND, parallel is OR. Build a two-way staircase light switch.
This covers other information topics such as fuzzy logic, where statements can be partly true instead of just true or false.
Stretch Way in (grade 7): Rate 'tall' from 0 to 1 for heights 150 to 190 cm. Combine 'tall AND fast' by taking the smaller score, and compare with ordinary yes/no logic.
97 Mathematics education 0 of 14
Bookkeeping: history and social context of math education, not a math topic itself.
Bookkeeping: educational policy and school systems, not a math topic itself.
Bookkeeping: psychology of learning math and research on teaching, not a math topic itself.
Bookkeeping: general teaching methods, assessment and problem solving in math education.
How this part of school math(s) is taught and learned. Every sheet in the matching area plants it, so it is not tracked separately.
How this part of school math(s) is taught and learned. Every sheet in the matching area plants it, so it is not tracked separately.
How this part of school math(s) is taught and learned. Every sheet in the matching area plants it, so it is not tracked separately.
How this part of school math(s) is taught and learned. Every sheet in the matching area plants it, so it is not tracked separately.
How this part of school math(s) is taught and learned. Every sheet in the matching area plants it, so it is not tracked separately.
How this part of school math(s) is taught and learned. Every sheet in the matching area plants it, so it is not tracked separately.
How this part of school math(s) is taught and learned. Every sheet in the matching area plants it, so it is not tracked separately.
How this part of school math(s) is taught and learned. Every sheet in the matching area plants it, so it is not tracked separately.
How this part of school math(s) is taught and learned. Every sheet in the matching area plants it, so it is not tracked separately.
Bookkeeping: educational materials, media and technology for teaching math.
Section names are from MSC2020 (msc2020.org, CC BY-NC-SA). Key ideas, ways in and the matching of sheets are ours.