∞ Grade 8
For curious 13 and 14 year olds. The grade 8 course explained rather than drilled, ideas the course never reaches, 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 8 course, explained not drilled
Linear equations and lines, systems, exponents, units, roots, Pythagoras, functions, transformations, angles, scatter plots, percent change, inequalities, probability, expanding and factoring, volume, each with the why and problems a notch harder than school sets.
One, none, or infinitely many
Equations with x on both sides. Why 3x + 5 = 3x + 7 has no solution and 2(x + 1) = 2x + 2 has every solution, and how to tell before you finish.
Why slope is the same everywhere on a line
Rise over run between any two points on a line gives the same number. Similar triangles are why, and that is what makes y = mx + b work.
Reading a line at sight
Intercepts, parallel lines, horizontal and vertical lines, and writing the equation from two points without a formula sheet.
Two equations, two unknowns
Substitution, elimination, and the picture: two lines cross once, never, or everywhere.
Why a⁰ = 1 and a⁻³ = 1/a³
The exponent laws follow from counting factors, and the strange cases follow from the laws. Then (2³)⁴, 2⁻⁵, and 10⁶ ÷ 10⁹.
Multiply by 1
1 km / 1000 m equals 1, so multiplying by it changes nothing but the units. Chain these fractions to convert speeds, prices and areas, and use the units to catch a wrong formula.
Squeezing √2
It is between 1.4 and 1.5. Then 1.41 and 1.42. Trap a square root between decimals, then do it faster with the Babylonian method.
Pythagoras, distances, and the diagonal of a box
A rearrangement proof you can see, the distance between two points, and the longest pencil that fits in a box.
Linear or not?
Read the rate of change from a table, a graph, and a story. Constant change means a line; anything else does not.
Slide, flip, turn, scale
Coordinate rules for translations, reflections, rotations and dilations, which ones keep lengths, and what congruent and similar really mean.
Why a triangle's angles add to 180°
One parallel line proves it. Then the exterior angle, the angle sum of any polygon, and the angles of a regular 20-gon.
Scatter plots and the line of best fit
Plot two variables, draw the trend, read the slope as a rate, and learn what an outlier does to the line.
Up 20%, down 20%
Percent changes multiply. Why a 20% rise followed by a 20% fall is a loss, and the one multiplier that undoes any change.
Turning 0.272727… back into a fraction
Call it x, multiply by 100, subtract. Every repeating decimal is a fraction, and 0.999… is exactly 1.
Why the inequality flips
Multiply both sides by a negative and < becomes >. See why on a number line, then solve linear inequalities without ever being unsure.
Turning a story into an equation
Name the unknown, write what you know, solve. Two cyclists, a mixture, and why the average of two speeds is not the average speed.
"At least one": the complement trick
P(at least one six in four rolls) is 1 minus P(no sixes). Independence, the multiplication rule, and the dice bet that cost the gambler de Méré his money.
Expand and factor
The area model for (a + b)(c + d), the difference of two squares, and why 51 × 49 = 2499 in your head.
A cone, a sphere and a cylinder: 1, 2, 3
Same radius, same height. Compute the three volumes and find the ratio Archimedes put on his tomb.
Around the corner new ideas
Euclid's algorithm, divisibility, proofs without words, Platonic solids, expected value, the birthday paradox, a secret agreed in public, fractals, magic squares, unsolved problems, logic, taxicab geometry, trapping π, constructions, strategy stealing, four dimensions, the perfect shuffle.
Euclid's algorithm
The fastest way to a common factor: divide, keep the remainder, repeat. Why it works, and gcd(1071, 462) in four lines.
The Chicken McNugget problem
Boxes of 6, 9 and 20. Which totals can you buy? The largest you cannot is 43, and there is a reason.
Why the digit-sum test works
A number is divisible by 9 when its digits add to a multiple of 9. The reason is that 10 is one more than 9. Then the test for 11.
Proofs without words
1 + 3 + 5 + … + (2n − 1) = n², two triangles make a rectangle, and 1³ + 2³ + … + n³ is a perfect square. Each one is a picture.
The last digit of 7²⁰²⁴
Last digits of powers go round in cycles. Find the cycle, use the remainder, and answer in seconds.
Counting divisors, and perfect numbers
360 has 24 divisors, and you can count them without listing. Then 6, 28, 496, and a question open for 2,300 years.
Why there are exactly five Platonic solids
Each corner needs at least three faces and less than 360°. That rule leaves five shapes, and no more.
What a game is worth
Expected value: multiply each outcome by its chance and add. Fair prices, a lottery ticket, and a game that is worth infinity.
The birthday paradox
In a room of 23 people it is more likely than not that two share a birthday. Multiply the chances of no match and watch it fall.
Agreeing on a secret in public
Two people shout numbers across a room and end up with a shared secret nobody else can find. Diffie–Hellman with small numbers, by hand.
The snowflake with an infinite edge
The Koch snowflake: add bumps forever. The perimeter grows without bound while the area stays under 8/5 of the first triangle.
Egyptian fractions
The Egyptians wrote every fraction as a sum of different unit fractions. Fibonacci's greedy method always works. Why?
Magic squares
Why the centre of a 3 × 3 magic square must be 5, and a method that builds a magic square of any odd size.
Problems nobody has solved
Collatz, Goldbach, twin primes, odd perfect numbers. Each takes one minute to state. Test them, and see why they are hard.
Knights, knaves and truth tables
Knights always tell the truth, knaves always lie. Solve the island puzzles, then build the truth tables that solve them all.
The shoelace formula
The area of any polygon from its corner coordinates, by criss-cross multiplying. Why it works for a triangle, then for everything.
Taxicab geometry
Measure distance the way a taxi drives: along the grid. Circles become squares, and some lines have many midpoints.
Four weights that weigh anything to 40
Weights of 1, 3, 9 and 27 on a two-pan balance measure every whole number up to 40. The reason is a number system with digits −1, 0, 1.
Trapping π between polygons
A hexagon inside the circle and one outside give 3 < π < 3.46. Double the sides and the trap tightens. Archimedes got to 3.1408.
Ruler and compass
Bisect a segment and an angle, drop a perpendicular, copy an angle. Each construction is a proof in disguise: congruent triangles.
The mediant
Add tops and add bottoms: (a + c)/(b + d) always lands between a/b and c/d. Why, and the tree that lists every fraction once.
Strategy stealing
In Chomp the first player can always win, and we can prove it without knowing how. The strangest kind of proof.
The Josephus problem
Forty-one people in a circle, every second one leaves. Where do you stand to be last? The answer is a binary trick.
The shortest path bounces
To get from A to B via a wall, reflect B. Heron's problem, why light obeys it, and a pool shot you can compute.
A cube in four dimensions
A point, a segment, a square, a cube. Keep going: the 4-cube has 16 corners, 32 edges and 8 cubes for faces, and you can count them.
The perfect shuffle
Shuffle a 52-card deck perfectly eight times and it is back in order. Follow one card and the arithmetic shows why.
Half-planes and the best corner
Graph y > 2x − 1 as a shaded region. Add constraints, find the feasible region, and the best answer is always at a corner.
Roads, routes and stay-put arrows
Towns are dots and one-way roads are arrows. Join routes end to start and you have found the basic rules of category theory.
An atlas for a ball
No single flat map can show a whole ball without tearing it somewhere. An atlas gets round this with overlapping pages, which is how mathematicians do calculus on curved spaces.
Counting holes with V − E + F
For a cube, vertices minus edges plus faces is 2. For a picture frame cut into flat four-sided faces it is 0. The number keeps count of the holes, and stretching cannot change it.
Distinct parts and odd parts
There are as many ways to split a number into different parts as into odd parts. Euler proved it works for every number, using a product of brackets.
What is the chance two moves commute?
Pick two symmetries of a triangle at random. Does it matter which you do first? Exactly half the time it does not.
A geometry where parallel lines meet
In a photo, railway tracks run together at the horizon. Mathematicians took this seriously: add points at infinity and every two lines meet. One geometry built this way has just 7 points.
Glue a square four ways
Take a square and glue its edges. Depending on the arrows, you get a tube, a Möbius strip, a doughnut, or a Klein bottle, which cannot be built in our space without passing through itself.
Only seventeen wallpapers
Every pattern that repeats in two directions, on wallpaper, floor tiles or fabric, has one of exactly 17 kinds of symmetry. Why can a wallpaper turn by a quarter or a sixth, but never by a fifth?
Stop exactly at 10
Each second you may push a cart forwards, backwards, or not at all. What is the fastest way to move it from 0 to 10 and have it stop exactly there?
Why mod 10 is two clocks at once
5 × 5 ends in 5 and 76 × 76 = 5776 ends in 76. Numbers that square to themselves reveal that a 10-hour clock is secretly a 2-hour clock and a 5-hour clock running together.
Functions you cannot undo
"Send everything to 1" is a perfectly good rule, but it forgets where things came from. Which rules can be undone, and what happens when you combine one that cannot?
The random walker's chances
A walker on a path of squares flips a fair coin to step left or right, and stops at either end. Its chance of reaching the right end from any square in between is the average of its neighbours' chances.
Two games that are secretly one
One game is played on a road map, the other on a 3 by 3 grid. Match the roads with the cells and they turn out to be the same game.
Connections barcodes · medicine · ecology · surveying · screens · roads · earthquakes · sport · elections · weather
The same mathematics, found in the wild.
Why a typo in your card number gets caught
ISBNs, barcodes and the Luhn check on credit cards: one extra digit that catches almost every slip.
When both groups improve and the total gets worse
Simpson's paradox at Berkeley, in hospitals and in sport. The numbers are right. The conclusion is not.
Counting fish you cannot see
Tag 50 fish, release them, catch 40, find 4 tagged. Estimate the lake. Proportions, and the same trick for counting words a writer knew.
How tall is that?
A tree, its shadow, and a metre stick. Similar triangles, a mirror on the ground, and a clinometer you can make from a protractor.
Sixteen million colours
Why #FF8800 is orange: hexadecimal, three bytes, and 256³ colours on a standard screen.
Why speed squared matters
Double your speed and braking distance quadruples. A nonlinear function that road signs get wrong, with the numbers from the Highway Code.
Earthquakes and decibels
Scales that go up by multiplying: one Richter unit is ten times the shaking, ten decibels is ten times the sound energy. Powers of ten in the news.
Will anyone run 100 m in 9 seconds?
Fit a line to world records and extrapolate. In 2004 a journal predicted women would outsprint men by 2156. What went wrong.
Winning with fewer votes
Draw the districts and you choose the winner. Packing, cracking, and a number that measures it: the efficiency gap.
What "70% chance of rain" means
How to judge a forecaster: calibration, the Brier score, and why always saying 50% is not safe.
How old is the universe?
Distant galaxies are moving away from us, faster the farther they are. Run the film backwards and you can estimate when everything was together.
Stirred or still?
A drop of dye in still water spreads painfully slowly. Stir once and it is everywhere. Two simple rules, one with a square in it, explain the difference.
Two rulers and a pin
A pendulum hanging from a pendulum, given a big swing, moves in a wild way that nobody can predict for long. Start it twice from almost the same spot and soon the two runs look nothing alike.
How many fish can you catch?
A pond refills itself, but only so fast. In a simple model, catching 100 fish a year can go on for ever. Catching 150 empties the pond.
Squash eight numbers into a few
Replace pairs of numbers by their average and half their difference. Throw away the small differences and you can still rebuild a recognisable copy from far fewer numbers.
Find the hidden shape
Roll marbles at a shape hidden under a sheet of card. From the bounces alone you can work out where its walls are, much as Rutherford's laboratory found the atomic nucleus.
Make paper hold 100 coins
A flat sheet of paper across two books sags under one coin. Fold the same sheet into a zigzag and it can hold dozens. The paper has not changed. Only its shape has.
How can a car move sideways?
A car can only drive forwards, drive backwards and steer. Yet you can park it in a space directly beside you. How do wiggles turn into sideways motion?
Mild preferences, strong patterns
Suppose everyone is happy as long as at least a third of their neighbours are like them. Let the unhappy ones move. In this simple model the town can still end up split into blocks.
Will tomorrow be like today?
Rainy days come in runs. From one month of weather you can estimate how sticky the weather is, and the same count helps you spot an invented list of coin flips.
Demos play with it
Twelve new interactive demos go with this grade, from a slope explorer to a game of Chomp against a perfect opponent, alongside the grade 6 and grade 10 ones. All of them live in the demo gallery.