∞ Grade 10
For curious 15 and 16 year olds. The grade 10 syllabus derived rather than memorised, the ideas waiting just past it, and where all of it shows up in the world. Three tracks: Sharpen (the school course, with reasons), Around the corner (new mathematics), Connections (where it shows up). One sheet a night, ten to fifteen minutes, and a starred problem if you want more.
Sharpen the grade 10 course, derived not memorised
Quadratics, functions, logs, polynomials, trigonometry, circle theorems, coordinate geometry, sequences, probability and statistics, each with the why and problems a notch harder than school sets.
Why the quadratic formula works
Complete the square once, slowly, and the formula falls out. Then never memorise it again.
What the roots are telling you
The discriminant counts solutions. Vieta says the roots add to −b/a and multiply to c/a, and that answers questions without solving the equation.
Functions as machines
Domain, range, composition and inverses, and why f⁻¹ is a reflection in y = x.
Shift, stretch, flip
Why f(x − 3) moves the graph right, not left, and how to read any transformed graph at sight.
A logarithm is an exponent
The log laws are the exponent laws in disguise. Solve 2ˣ = 10, and find out how many digits 2¹⁰⁰⁰ has.
Where curves meet
Lines, circles and parabolas intersecting: how many solutions, and how to find them without guessing.
Factor, remainder, and the cubes
The factor theorem, synthetic division, the sum and difference of cubes, and a polynomial detective problem.
Rational functions and asymptotes
Holes versus vertical asymptotes, what happens far away, and why 1/x is the shape of so many things.
Roots, rational exponents and rationalising
x to the power one half is a square root because the exponent laws insist. Nested radicals and a famous one that collapses.
Similar triangles and the proof habit
AA, SAS and the altitude to the hypotenuse, which proves Pythagoras in three lines.
SOHCAHTOA grows up: the unit circle
Exact values, angles past 90°, and why sin² + cos² = 1 is just Pythagoras.
Circle theorems, with proofs
The inscribed angle is half the central angle. Prove it, then use it to find a dozen angles fast.
Geometry with coordinates
Distance, midpoint, perpendicular slopes, the equation of a circle, and proving that the diagonals of a parallelogram bisect each other, by algebra alone.
Sequences, series, and Gauss's trick
Arithmetic and geometric sums derived, not quoted, and the infinite geometric series that equals 1.
Conditional probability and trees
With and without replacement, independence, P(A given B), and the two-children problem, which fools most adults.
Standard deviation and the bell curve
What σ measures, 68–95–99.7, z-scores, and why correlation is not causation (with a real example).
Inequalities, absolute value, and sign charts
Absolute value as distance, quadratic inequalities by sign chart, and the AM–GM inequality.
Volumes: why the cone is a third
Cavalieri's principle, the cone, the sphere via Archimedes' argument, and scaling: double the size, eight times the volume.
Around the corner new ideas
Proof, induction, number theory, RSA, complex rotations, e, the first derivative and integral, infinite series, Cantor, matrices, conics, Bayes, Markov chains, chaos, entropy, algorithms, curved geometry, groups.
Proof by contradiction
Assume the opposite and watch it collapse: √2 is irrational, and the primes never run out.
Induction: infinitely many dominoes
Prove a statement for every whole number in two steps. Sum formulas, 2ⁿ > n², and the Tower of Hanoi.
The pigeonhole principle
Obvious, and yet it proves two Londoners have the same number of hairs and that some multiple of 7 is all 1s and 0s.
Counting without counting
Permutations, combinations and the binomial theorem: how many poker hands are a flush?
Fermat's little theorem
Raise a to the power p − 1 and the remainder mod the prime p is 1, unless p divides a. Why, what it does to huge powers, and how it tests primes.
RSA: a lock anyone can close
Encrypt a message with public numbers, decrypt with a private one, and do a whole round by hand with small primes.
Multiplying is rotating
Complex numbers as arrows: multiply to rotate and scale, De Moivre, and the n roots of 1 sitting on a circle.
e to the i pi
Why e to the iθ equals cos θ + i sin θ, what it does to trig identities, and the most famous equation in mathematics.
The number e
Compound interest compounded forever, (1 + 1/n)ⁿ, and the function that is its own slope.
The slope of a curve
Secants becoming tangents: the derivative of x², x³ and 1/x from first principles.
Area under a curve, Archimedes' way
Slice, sum, and squeeze: the area under x² from 0 to 1 is exactly 1/3, and Archimedes knew.
Adding up forever
Geometric series converge, the harmonic series does not, and 1 + 1/4 + 1/9 + … equals π²/6.
Some infinities are bigger
Count the fractions, then prove the decimals cannot be counted. Cantor's diagonal in one page.
Matrices that move the plane
Rotations, shears, determinants as area, inverses, and solving two equations with one matrix.
Vectors and the dot product
The dot product measures alignment: angles, projections, work, and why perpendicular means zero.
Slicing a cone
Ellipse, parabola, hyperbola from one cone, focus and directrix, and why whispering galleries work.
Pythagorean triples and Fermat's margin
Euclid's recipe for every triple, rational points on a circle, and the note that took 358 years.
Continued fractions and the golden ratio
The best fractions for π and √2, why 355/113 is so good, and why φ is the hardest number to approximate.
Graphs: trees, planarity and Ramsey
Trees have n − 1 edges, K₅ cannot be drawn flat, and among any six people three are mutual friends or strangers.
Game theory: the prisoner's dilemma
Dominant strategies, Nash equilibrium, chicken, and why rational players can both lose.
Bayes' theorem
Update a belief with evidence. Base rates, the prosecutor's fallacy, and a spam filter in four lines.
Markov chains and how Google ranked the web
Transition matrices, steady states, and the random surfer that made a search engine.
Random walks and the gambler's ruin
Expected position zero, typical distance √n, and the exact probability of going broke.
The logistic map and chaos
x → r x (1 − x). Steady, then periodic, then chaos, and a constant that shows up everywhere.
Information and entropy
How many yes/no questions does a message need? Bits, Huffman codes, and why compression has a limit.
Algorithms and big-O
Binary search, sorting, and why some problems blow up exponentially while others stay tame.
Geometry on a sphere and beyond
Triangles with three right angles, the parallel postulate, and the geometry Einstein needed.
Groups: the mathematics of symmetry
Symmetries compose, every element has an inverse, and Lagrange's theorem says subgroups divide.
Area and slope undo each other
One thin strip explains the fundamental theorem of calculus. Find areas by checking slopes, then send a car home.
Why π knows whether a fraction cancels
Shade a 6 × 6 grid, count by inclusion–exclusion, and watch the chance of a coprime pair settle towards 6/π².
Lights Out: a puzzle you can prove impossible
Press a light and its neighbours flip. Clearing the board is Ax = b over bits, and a parity mask proves some boards hopeless.
Functions that turn adding into multiplying
A function obeys f(x + y) = f(x) f(y) and f(1) = 2. Knowing only that rule, you can work out f(3), f(0), f(−1) and even f(1/2).
Remove everything, keep a lot
Remove the middle third of a line of length 1, then the middle third of each piece left, for ever. The lengths you remove add up to 1, yet the points left over are too many to list.
Watch a hyperbola break
Graph xy = t for t = 4, 1, 0.1. As t shrinks the curve hugs the axes more tightly. At t = 0 it is no longer a hyperbola: it is two crossing lines.
Rows first or columns first?
An infinite grid of 1s, −1s and 0s adds up to 0 if you add the rows first, and to 1 if you add the columns first. Which is right?
A machine you can undo only one way
Push every item in an endless list one place along and put 0 at the front. Nothing is lost, and a second machine undoes it. Run the two in the other order and something is lost for good. This is Hilbert's hotel as a machine.
How many shuffles mix a deck?
Take the top card and push it back into the deck at a random place. Keep going. By watching one card you can spot a moment when the deck is perfectly random, and you can see it coming.
Time averages equal space averages
Keep adding 0.6180339... and keep only the part after the decimal point. In the long run the values spread out evenly between 0 and 1. One long journey visits every stretch in fair proportion.
Do turns commute?
Spin a book a quarter turn on the table, then tip it up to face you. Now do those two turns in the other order. The book ends up somewhere else. But tiny turns almost commute.
Connections music · astronomy · physics · money · medicine · epidemics · art · sport · GPS · elections
The same mathematics, found in the wild.
Why a violin doesn't sound like a flute
Harmonics, Fourier's idea that every sound is a sum of sines, and beats between two close notes.
Kepler's laws and measuring the universe
T² ∝ a³, parallax, standard candles, and a cosmic distance ladder built out of logarithms.
Parabolas in flight
Why a thrown ball traces a parabola, the 45° rule and when it is wrong, and the same kick on the Moon.
Compound interest, mortgages and why the casino wins
e in your savings account, the mortgage formula, expected value, and the house edge made visible.
What a positive test really means
Sensitivity, specificity, base rates, and why a 99% accurate test can be wrong most of the time.
R₀ and herd immunity
Exponential starts, the SIR model, and the threshold 1 − 1/R₀.
Perspective, projective geometry and the golden ratio myth
Vanishing points, why parallel lines meet, and a sceptical look at φ in art.
Elo ratings and regression to the mean
How chess and football ratings work, expected scores from a logistic curve, and the sophomore slump explained.
How GPS knows where you are
Spheres intersecting, four satellites, clocks, and the relativity correction that keeps you on the road.
Fair shares: apportionment and Arrow's theorem
Why the US House had a paradox, Hamilton versus Jefferson, and why no voting system is perfect.
Why the air does not gather in one corner
All the air in a room could rush into one corner, but it never does. With 10 coins, all heads comes up 1 time in 1024. With 100 coins, a billion tries a second since the universe began would almost certainly not be enough.
The average of 350 degrees and 10 degrees
Two wind readings: 350 degrees and 10 degrees, both nearly north. Their ordinary average is 180, due south. Averaging arrows instead gives the right answer.
Why a straight line on a spinning plate comes out curved
Move a pen straight across a spinning paper plate. The line it leaves on the plate is curved. Earth is a spinning surface too, and the same effect bends winds into swirls.
The colours of hydrogen
Make hydrogen gas glow and split the light with a prism. You get a few sharp colours: red, blue-green, violet. A one-line formula built on 1 over n squared predicts where each one sits.
Outrunning your own sound
A jet flying faster than sound leaves its sound behind in a cone. The faster it goes, the narrower the cone. You hear the boom only when the cone sweeps past you.
Find the image with two rays
Where does a lens put the picture of a candle? Draw just two rays with a ruler. In the thin-lens model, the point where they cross is where the image forms.
A bump that slides
Draw a bump, then replace x by x − t. As t grows, the whole bump slides right without changing shape. The same trick follows sand on a conveyor belt and a wave on a string.
Demos play with it
Fifteen new interactive demos go with this grade, from a quadratic explorer to an epidemic in a box, alongside the grade 6 ones. All of them live in the demo gallery.