∞ Grade 11
For curious 16 and 17 year olds. The grade 11 course with the reasons left in, ideas from infinite descent to max flow 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 11 course, with the why
Radians, sine waves, the addition formulas, the sine and cosine rules, inverse functions, trigonometric equations, log scales, polynomial graphs, piecewise functions, resolving vectors, three unknowns, complex arithmetic, recursive sequences, random variables, the computed line of best fit and the margin of error.
Why radians
Measure an angle by the arc it cuts off and arc length is just radius times angle. Degrees are a habit; radians are what the circle itself uses.
Reading a sine wave
In y = a sin(b(x − c)) + d, each letter does one job: height, speed, slide, lift. Read all four off a graph, and write the equation of a wave you can see.
sin(A + B), from a picture
It is not sin A + sin B. One diagram of two stacked right triangles gives the true formula, and the double-angle formulas drop out.
Triangles without a right angle
The sine rule and the cosine rule solve any triangle. The cosine rule is Pythagoras with a correction term, and you can see where the term comes from.
Undoing a function that is not one-to-one
sin 30° and sin 150° are both a half, so "the angle whose sine is a half" needs a choice. Restrict the domain, and see what the calculator chose for you.
All the solutions
sin x = 0.5 has infinitely many answers. Find the two in one turn from the unit circle, then add whole turns, and never lose a solution again.
Straight lines hiding in curved data
Plot the logarithm instead and exponential growth becomes a line; plot both logarithms and a power law does. The slope tells you the rate or the power.
Sketching a polynomial at sight
The leading term decides the ends, the roots decide the crossings, and a double root touches without crossing.
Functions in pieces
A tax schedule, a phone plan and |x| are all functions with different rules on different intervals. Graph them, compose them, and find where they join up.
Splitting an arrow in two
Any force is the sum of a horizontal part and a vertical part. Resolve, add the parts, and recombine: that is how every bridge and aeroplane is analysed.
Three equations, three unknowns
Eliminate one unknown at a time until the system is a staircase, then climb back up. The method scales to a million unknowns.
Dividing by a complex number
Multiply top and bottom by the conjugate. Add, multiply and divide a + bi, and solve the quadratics that used to have no answer.
Sequences that feed on themselves
Each term is made from the one before. Some settle down, some run away, some cycle, and you can often predict the limit before computing a single term.
The average outcome and how far it strays
Expected value and variance of a random number, computed from a table. Why a fair bet can still be a bad idea.
The line of best fit, computed
Not by eye. The least-squares line passes through the point of means, and its slope is one formula. What r² tells you, and what it hides.
Plus or minus three points
A poll of 1,000 people reports 52% ± 3%. Where the 3 comes from, why it barely depends on the size of the country, and what it does not cover.
Around the corner new ideas
Infinite descent, the rearrangement inequality, modular division, decimal periods, the cubic formula, rational roots, linear recurrences, sums of powers, counting twice, necklaces, linearity of expectation, Buffon's needle, error-correcting bounds, counting trees, max flow, mixed strategies, the power of a point, complex geometry, the isoperimetric problem, Newton's method, Simpson's rule, the overhang puzzle, finite fields and fractal dimension.
Proof by infinite descent
If a solution exists, a smaller one does too, and whole numbers cannot shrink for ever. Fermat's favourite weapon, used on √2 and on a² + b² = 3c².
Big with big, small with small
To make a sum of products as large as possible, pair the largest with the largest. The rearrangement inequality, and three famous inequalities that fall out of it.
Dividing in clock arithmetic
You can divide by 3 on a 7-hour clock but not by 2 on a 6-hour one. When an inverse exists, how to find it, and why primes make everything work.
Why 1/7 repeats after six digits and 1/13 after six too
The length of a repeating decimal is the first power of 10 that leaves remainder 1. It always divides p − 1, and nobody can predict when it equals p − 1.
The cubic formula, and the numbers it forced into existence
Solve x³ = 15x + 4 by Cardano's method and the answer, 4, arrives wrapped in square roots of negative numbers. This is why complex numbers were invented.
Hunting for fraction roots
If a polynomial with whole-number coefficients has a root p/q in lowest terms, then p divides the last coefficient and q the first. A finite list to test, and a quick proof that √2 + √3 is irrational.
Guess rⁿ
To solve aₙ = aₙ₋₁ + 6aₙ₋₂, try a power. A quadratic appears, its roots give the answer, and Fibonacci gets a formula.
Summing squares and cubes without guessing
Find a function whose differences are n², and the sum telescopes. A calculus of differences that works exactly like the one with derivatives.
Count it twice
Count the same thing in two ways and you have proved an identity. Committees with a chair, the hockey stick in Pascal's triangle, and Vandermonde's sum.
How many different necklaces?
Six beads, two colours, and rotations do not count as different. Average the number of colourings each rotation leaves alone, and the answer is exact.
Expectation adds, even when nothing is independent
The expected number of people who get their own hat back is exactly 1, for any number of people. One line of reasoning replaces pages of counting.
Dropping needles to find π
Drop a needle on a lined floor. If it is as long as the gap between lines, the chance it crosses one is 2/π, and a noodle of any shape shows why.
How many errors can a code fix?
Codewords must be far apart so that a damaged one is still closest to the truth. Counting how many fit gives a hard limit, and a few codes meet it exactly.
Counting trees
There are 16 ways to join four labelled towns into a tree, and nⁿ⁻² ways for n. A code that turns every tree into a list of numbers proves it.
The bottleneck theorem
The most traffic a network can carry equals the capacity of its narrowest cut. Push flow along paths until no path is left, and the cut reveals itself.
Where should the penalty taker aim?
If you always shoot left the keeper always dives left. The best plan is a fixed set of odds, found where two lines cross.
The power of a point
Draw any line through a point to cut a circle twice, and multiply the two distances. The product never changes. Similar triangles are why.
Geometry by multiplication
Multiplying by a complex number rotates and scales. Use it to rotate about any point, and to prove that equilateral triangles built on a triangle's sides have centres forming another.
Why bubbles are round
Of all shapes with the same perimeter, the circle encloses the most. Compare regular polygons, then follow an argument that any dent or lopsidedness can be improved.
Newton's method
Slide down the tangent to the axis, and repeat. Each step roughly doubles the number of correct digits, until a bad start sends it somewhere wild.
Areas from a table of values
Join the points with straight lines and you get the trapezium rule. Join them with parabolas and halving the step cuts the error by sixteen, not four.
How far can a stack of books lean?
One book can overhang half its length. Two can reach three quarters. With enough books the stack reaches as far as you like, because the harmonic series diverges.
A number system with five numbers
On a 5-hour clock you can add, subtract, multiply and divide. Lines and parabolas still make sense there, and a line still meets a parabola at most twice.
A shape of dimension 1.26
Double a line and you get 2 copies, a square 4, a cube 8. Triple the Koch curve and you get 4. Solve 3ᵈ = 4 and dimension stops being a whole number.
Zoom out on a random picture
Colour a 9 by 9 grid at random. Replace each 3 by 3 block with its majority colour, then do it again. Zooming out turns a slight lean towards black into a strong one.
Press cos until it stops
Set a calculator to radians, type any number and keep pressing cos. The display always settles at 0.739085... Why must there be such a number, and why does it pull everything in?
Where does the frog end up?
A frog hops between two lily pads with fixed chances. Wherever it starts, its chances settle to the same two numbers. One list explains why, and one number says how fast.
One jump or smooth growth?
Money grows at 10 percent a year. Paid in one jump, in 12 monthly jumps, or smoothly all the time, you get slightly different amounts. One pair of rules handles jumps, smooth stretches and mixtures of the two.
Solving x times 2 to the x equals 10
x times 2 to the power x equals 8 when x is 2. Change the 8 to 10 and logs alone will not isolate x. So mathematicians gave the answer a name.
Why a folded pizza slice stays stiff
Hold a floppy pizza slice by the crust and the tip droops. Fold the crust into a U and the slice stays out straight. The reason is a theorem that Gauss himself called remarkable.
Numbers where i times j is not j times i
In 1843 Hamilton carved i² = j² = k² = ijk = −1 into a bridge in Dublin. He had found numbers where the order of multiplying matters, and you can still divide.
Connections tides · surveying · chemistry · chips · insurance · graphics · rainbows · power laws
The same math, found in the wild.
Predicting the tide
High water to low water and back in about twelve and a half hours. Fit a sine wave to a tide table and say when a boat can cross the bar.
Mapping a country with one measured line
Measure one baseline and two angles and the sine rule gives every other distance. Repeat, and triangles cover a continent.
What pH measures
Each step on the pH scale is ten times the acidity. A logarithm turns a range of a hundred trillion into the numbers 0 to 14.
Fifty years of doubling
Plot transistor counts on ordinary axes and you see nothing; on a log axis you see a straight line. Read the doubling time off the slope.
Why insurance works
You pay a little more than your expected loss and are glad to. Pooling many independent risks shrinks the uncertainty, and the square root explains by how much.
How a screen turns a cube
A 3D model is a list of points. Multiply by a rotation matrix, drop one coordinate, and you have the picture. Every game does this millions of times a second.
Why the rainbow is at 42 degrees
Light bends entering a raindrop, reflects, and bends again. Follow many rays and they pile up at one angle.
The few that are huge
City sizes, word frequencies and earthquake energies follow power laws: a straight line on log-log axes. Why the average of such data can mislead badly.
Savings with luck
Your savings grow 5 percent a year, then a fair coin flip adds or takes away 10 pounds. Every player ends up somewhere different, but the average over all the possible players follows the smooth curve.
Hitting the target
Throw a ball at 20 m/s to land exactly 30 m away. There are two ways to do it, a low throw and a high lob. Move the target and there may be one way, or none.
Can you feel the difference?
Many people can tell 100 g from 110 g in the hand. Hardly anyone can tell 1000 g from 1010 g. To a good approximation, our senses notice ratios, not differences.
Demos play with it
Eight new interactive demos go with this grade, from a sine wave explorer to max flow on a network, alongside the other grades' demos. All of them live in the demo gallery.