∞ Grade 9
For curious 14 and 15 year olds. The grade 9 course with the reasons left in, ideas from proof to algorithms that the course never reaches, and where the math shows up in the world. Three tracks: Sharpen (school math, and why it works), Around the corner (new ideas), Connections (where it shows up). One sheet a night, ten to fifteen minutes, and a starred problem if you want more.
Sharpen the grade 9 course, with the why
Factoring, quadratic equations, parabolas, surds, special triangles, trigonometric ratios, congruence, arcs and sectors, inverse variation, exponential functions, sequences, rearranging formulas, algebraic fractions, box plots, two-way tables and precision.
Un-multiplying: the sum and product hunt
To factor x² + 7x + 12, find two numbers that add to 7 and multiply to 12. A rectangle of tiles shows why those are the right two questions.
If a product is zero
Then one of the factors is. That single fact turns a quadratic equation into two easy linear ones, and it works for no other number.
The parabola's mirror line
Every parabola is symmetric. Find where it crosses the axis and the turning point sits exactly half way between.
Why √50 is 5√2
Pull the square factors out from under the root. Exact answers stay short, and √2 turns up everywhere.
Two triangles worth knowing by heart
Half a square and half an equilateral triangle. Their sides are 1, 1, √2 and 1, √3, 2, and they unlock exact answers for 30°, 45° and 60°.
Three ratios that measure an angle
In every right triangle with a 35° angle, opposite over hypotenuse is the same number. Give that number a name and you can measure heights you cannot reach.
Three facts fix a triangle (usually)
Three sides do. Two sides and the angle between them do. Two sides and a different angle may not, and a drawing shows the two triangles that fit.
A slice of a circle
An arc is a fraction of the circumference and a sector the same fraction of the area. The fraction is the angle out of 360.
When one doubles, the other halves
Speed and time for a fixed journey, pressure and volume, workers and days. The product stays constant and the graph is a curve that never touches the axes.
Adding versus multiplying
A linear function adds the same amount each step; an exponential one multiplies by the same amount. Tell them apart from a table, and write y = a·bˣ.
The nth term
Jump straight to the hundredth number of a sequence without writing the first ninety-nine. Constant difference gives a linear rule, constant ratio an exponential one.
Change the subject
A formula is an equation with several letters. Solve it for the one you want, once, and you never have to repeat the algebra.
Fractions with letters
They obey the same rules as fractions with numbers. Cancel common factors, find a common denominator, and never cancel across a plus sign.
Five numbers that draw a data set
Smallest, lower quartile, median, upper quartile, largest. A box plot shows centre, spread and skew at a glance, and flags outliers.
Two questions at once
Sort people by two things at the same time and a table shows whether the two are linked. Row percentages and column percentages answer different questions.
How precise is your answer?
A length given as 12 cm could be anything from 11.5 to 12.5. Carry the bounds through a calculation and see how many digits you can honestly keep.
Around the corner new ideas
Proof with letters, counterexamples, telescoping sums, the missing square, Catalan numbers, stars and bars, probability as area, Monte Carlo, timetabling, shortest paths, cheapest networks, the travelling salesman, stable matching, recursion, the Game of Life, logic gates, circles through three points, Heron's formula, great circles, the doughnut, the fifteen puzzle, game trees and the one-time pad.
Proof with letters
Call an odd number 2n + 1 and you can prove things about every odd number at once. The sum of two odds, the square of an odd, and a calendar trick.
One counterexample is enough
n² + n + 41 is prime for n = 0, 1, 2, … 39. Forty cases are not a proof, and n = 40 is why.
Why n³ − n always divides by 6
Factor it and you find three numbers in a row. One of any three in a row is a multiple of 3, and one of any two is even.
Sums that collapse
Write each term as a difference and the middle cancels, like a telescope closing. A sum of ninety-nine fractions becomes one subtraction.
The puzzle where 64 = 65
Cut an 8 by 8 square into four pieces and rearrange them into a 5 by 13 rectangle. The trick is a Fibonacci identity and a sliver you cannot see.
Counting balanced brackets
(()) and ()() are the two ways to balance two pairs. For three pairs there are 5, then 14, then 42. The same numbers count mountain ranges and handshakes across a table.
Stars and bars
How many ways can three children share ten identical sweets? Line up ten stars and two dividers and it is a counting problem you already know.
Probability you can measure with a ruler
Two friends each arrive at a random moment in the same hour and wait ten minutes. The chance they meet is an area.
Finding an area by throwing darts
Scatter random points over a square and count how many land under a curve. Luck, averaged, becomes a measurement.
Timetabling by colouring
Exams that share a student cannot share a slot. Draw the clashes as a graph, colour it, and the number of colours is the number of slots.
How a map app finds the shortest route
Dijkstra's method: always settle the nearest unsettled place next. Run it by hand on a small map and see why it never has to go back.
The cheapest way to connect everything
Join towns with cable using as little as possible. Being greedy, always adding the cheapest safe link, gives the best answer every time.
The problem nobody can solve quickly
Visit every town once and come home by the shortest route. The greedy idea that built the cheapest network fails here, and no fast method is known.
Matching without regrets
Pair applicants with schools so that no applicant and school would both rather have each other. A simple round of proposals always finds such a matching.
How many ways up the stairs?
Take one step or two at a time. The count for ten stairs comes from the counts for nine and eight, and a familiar sequence appears.
A universe with four rules
Cells on a grid live or die by counting neighbours. From that come still lifes, blinkers, and a glider that walks.
Adding with switches
AND, OR and NOT are enough to build a circuit that adds. Wire a half adder on paper and see how a chip does arithmetic.
The circle through three points
Any three points not in a line lie on exactly one circle. Its centre is where two perpendicular bisectors cross, and the third always passes through the same spot.
Area from three sides
No height needed. Heron's formula gives the area of a triangle from its sides alone, and you can check it against one you already know.
Why planes fly over Greenland
On a globe the shortest path is part of a great circle, and on a flat map it looks bent. A string on a ball shows it.
A doughnut is not a ball
Corners minus edges plus faces is 2 for anything shaped like a ball. For a doughnut it is 0, and that number cannot be changed by stretching.
The puzzle that cannot be solved
Swap two tiles of the fifteen puzzle and no amount of sliding will fix it. A quantity that every move leaves unchanged proves it.
Playing noughts and crosses perfectly
Draw every possible game as a tree, score the endings, and work backwards. With best play the game is always a draw.
The unbreakable code
Add a random key, bit by bit, and the message could be anything. Reuse the key once and it falls apart.
When the calculator lies
Start at one third and keep applying x → 10x − 3. The exact answer is one third for ever. A calculator soon gives nonsense. Tiny errors can grow.
How many terms to compute e?
Add 1 + 1 + 1/2 + 1/6 + 1/24 + ... and after ten terms you have the number e to six decimal places. The famous recipe (1 + 1/n)^n needs n in the millions to do as well.
Grid or tree?
Two moves, a and b. If ab = ba, every route lives on a grid. If no rule links them at all, the routes branch like a tree, and the tree grows far faster.
The biggest number is on the edge
Fill a grid so every inside cell is the average of its four neighbours. Try to make an inside number bigger than everything on the border. You cannot.
Mod 9, every failure is a multiple of 3
Mod 9, the numbers you cannot divide by are 0, 3 and 6, and each of them becomes 0 when you square it. That is not true mod 10.
Whole-number points on a circle
The circle x² + y² = 25 passes through exactly 12 points with whole-number coordinates. A shape and its equation are two views of the same object.
The number ...9999 is minus one
Add 1 to ...99999, a number with nines going on forever to the left. Every column carries, and you get 0. So ...99999 behaves exactly like −1.
Three ways to measure distance
Draw every point at distance 1 from the origin. With ordinary distance you get a circle. With taxi distance you get a diamond, and with king's-move distance a square.
Tropical arithmetic
Make "plus" mean "take the smaller" and "times" mean ordinary adding. Then 3 plus 5 is 3 and 3 times 5 is 8. This strange arithmetic finds shortest routes, and its quadratics have graphs made of straight pieces.
Connections money · medicine · bridges · navigation · building · fuel · population · cameras · memory
The same math, found in the wild.
How long until your money doubles?
Divide 72 by the interest rate. At 6% it takes about 12 years. Where the rule comes from, and how well it works.
How long does a medicine last?
Half the dose is gone after one half-life, three quarters after two. Repeated doses climb towards a steady level you can calculate.
The shape of a bridge
A suspension bridge's cable hangs in a parabola. Fit y = ax² to a span and work out the length of every hanger.
Finding your way with angles
A bearing is an angle clockwise from north. Two legs of a journey, a right triangle, and trigonometry tells you how far you are from home.
How steep is too steep?
A wheelchair ramp may rise 1 in 12. A roof pitch, a road sign's 10% and a staircase are all the same idea, and tangent converts between them.
The miles-per-gallon illusion
Swapping a 10 mpg car for a 20 mpg one saves far more fuel than swapping 30 for 60. Inverse variation fools almost everyone.
Will food keep up with people?
In 1798 Malthus argued that population multiplies while food only adds. Compare an exponential with a line, find where they cross, and see what he missed.
Why camera settings go 2, 2.8, 4, 5.6
Each step halves the light. The numbers are powers of √2, because the light depends on the area of the opening.
Births that remember
If animals have babies at ages 1, 2 and 3, this year's births depend on everyone born in the last three years. The present remembers the past, and you can build the future one year at a time.
Einstein in your phone
Clocks on GPS satellites run fast by about 38 millionths of a second a day. Ignore that, and after one day the distances your phone measures would be wrong by more than 11 km.
A robot arm lives on a doughnut
A robot arm with two spinning joints is described by two angles. Plot them in a square, glue the opposite edges, and every position of the arm becomes a point on a doughnut.
Why the skater spins faster
A skater pulls in her arms and spins faster, though nothing pushes her. Toss a book spinning about each of its three axes and one of the spins will not stay steady. Spinning things follow surprising rules.
Why a tap stream breaks into drops
A thin stream from a tap starts smooth and then breaks into drops. Surface tension is trying to shrink the water's surface, and a long cylinder has more surface than a row of balls.
The heat simulation that exploded
Simulate heat spreading along a bar on a computer. With small time steps it looks right. Make the steps too big and, three steps later, one cell is at −700 °C, far below absolute zero.
Demos play with it
Eight new interactive demos go with this grade, from factoring with tiles to an unbeatable game of noughts and crosses, alongside the other grades' demos. All of them live in the demo gallery.