∞ Grade 7
For curious 12 and 13 year olds. The grade 7 course with the reasons left in, puzzles and ideas 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 7 course, with the why
Negative numbers, proportion, rates, percent error, like terms, two-step equations, inequalities, scale drawings, circumference, angles, triangles, surface area, cross-sections, chance in the long run, sampling and comparing groups.
Why minus times minus is plus
Not a rule to swallow. Follow the pattern down a times table past zero, or cancel a debt, and the sign has no choice.
Is it proportional?
Double one, the other doubles. Test a table, a graph and a story, and find the one number that ties them together.
Half a mile in a quarter of an hour
A rate is "per one". When the amounts are fractions, find the amount for one whole and the question answers itself.
How wrong is wrong?
Off by 2 cm on a pencil is bad; off by 2 cm on a house is superb. Percent error measures a mistake against the size of the thing.
Apples and oranges: collecting like terms
3x + 2x is 5x for the same reason three apples and two apples are five apples. And 3x + 2 is not 5x.
Undo it backwards
A two-step equation is a number that was multiplied, then added to. To find it, undo the last step first, like taking off shoes then socks.
At least, at most
"You must be at least 130 cm tall" is not an equation. Write it as an inequality, solve it, and draw every answer at once on a number line.
Blueprints
A scale drawing shrinks every length by the same factor. Lengths scale once, areas scale twice, and that surprises people.
Roll a wheel once
The distance a wheel rolls in one turn is a little more than three diameters, whatever its size. That "little more than three" is π.
Angle chasing
Angles on a line make 180°, angles at a point make 360°, opposite angles at a crossing are equal. Three facts unlock a whole diagram.
Can three sticks make a triangle?
3, 4 and 8 cannot. The two short sides must be able to reach each other, and that one test decides every case.
Wrapping a box
Surface area is the paper needed to cover a solid. Unfold the box into its net and it is six rectangles in three matching pairs.
Slicing solids
Cut a cube straight and you get a square. Tilt the knife and you can get a triangle, a rectangle, even a hexagon.
What "one in six" really promises
Roll a die six times and you may see no six at all. Roll six thousand and about a thousand will be. Chance is a promise about the long run.
Ask 50 people, learn about 5,000
A fair sample tells you about everyone. A lopsided one tells you only about the people you asked.
Are they really different?
Class A averages 2 marks more than class B. Whether that matters depends on how spread out the marks are.
Around the corner new ideas
The 1001 trick, square endings, letter sums, peasant multiplication, happy numbers, pentominoes, border patterns, wheels that are not round, shearing, inside and outside, party puzzles, tournaments, fair division, collecting sets, the barber, look and say, sums in a row, calendar magic, sorting, the abacus, dividing by zero, four fours, a triangle with no area and clock hands.
The 1001 trick
Type any three digits twice, like 482482. It always divides by 7, by 11 and by 13. One multiplication explains all three.
No square number ends in 7
Squares end only in 0, 1, 4, 5, 6 or 9. Ten cases prove it, and that is enough to spot a fake square at a glance.
SEND + MORE = MONEY
Each letter is a digit. Carrying, column by column, forces the answer, and there is only one.
Multiplying by halving and doubling
Halve one number, double the other, cross out the evens, add what is left. It always works, and binary is the reason.
Happy numbers
Square the digits and add. Keep going. Some numbers arrive at 1 and stay; the rest fall into the same loop of eight.
The twelve pentominoes
Five squares joined edge to edge make exactly twelve shapes. Together they cover 60 squares, and fitting them into a rectangle is harder than it looks.
Seven kinds of border
Slide, flip, turn, glide. Every repeating border pattern ever drawn is one of exactly seven kinds.
A wheel that is not round
A curved triangle rolls as smoothly as a circle and fits a square hole at every angle. Some coins are this shape, on purpose.
Slide the top, keep the area
Push a stack of cards sideways and its area does not change. That is why every triangle is half a rectangle.
Inside or outside?
A tangled loop on paper, and a dot. Count how many times a straight path to the edge crosses the loop: odd means inside.
The party puzzle
At any party, the number of people who shook an odd number of hands is even. Always. Count the hands two ways to see why.
Everyone plays everyone
Eight teams, seven rounds, nobody idle. Sit them round a table, pin one, and rotate the rest.
I cut, you choose
Two people can share a cake so neither envies the other. Three is harder, and the method shows what "fair" means.
How many boxes to collect all six?
Six toys, one per box, at random. Roll a die to simulate it, and find the answer nobody guesses: about fifteen.
The barber who shaves everyone
He shaves all those who do not shave themselves. Who shaves the barber? A puzzle that shook the foundations of mathematics.
Look and say
1, 11, 21, 1211, 111221. Read each line aloud to get the next. No digit bigger than 3 ever appears, and you can prove it.
Sums of numbers in a row
9 is 4 + 5 and also 2 + 3 + 4. Some numbers can never be written this way, and they are exactly the powers of 2.
Calendar magic
Box any 3 by 3 square of dates. The sum is nine times the middle date. Algebra with one letter shows why, and gives you more tricks.
How to sort a hand of cards
Swap neighbours until nothing is out of order. Count the swaps, find the worst case, and meet a faster way.
Calculating with beads
In practised hands a Japanese soroban adds faster than most people can type. Each rod is a digit, and carrying is something you can feel.
Why you cannot divide by zero
12 ÷ 0 is not infinity and not zero. Ask what multiplication would have to do, and no answer survives.
Four fours
Use exactly four 4s and any signs to make every number from 0 to 20. Order of operations becomes a toolbox.
The triangle with no area
Remove the middle of a triangle, then the middle of each piece, for ever. What is left has no area and an endless edge.
When do the clock hands meet?
Not at 1:05. The minute hand chases the hour hand, and the hour hand does not wait. The answer is a fraction of a minute.
Which numbers can you undo on a clock?
On a 5-hour clock every number except 0 has a partner that multiplies to 1. On a 12-hour clock, 3 times 4 is 0. What decides which numbers can be undone?
The family tree of 12's divisors
Draw the divisors of 12 in layers, with a line going up when one divides another and nothing sits in between. Any two of them have a best meeting point below and a best one above.
Five ways to put the brackets
In rock-paper-scissors, (rock vs paper) vs scissors gives scissors, but rock vs (paper vs scissors) gives rock. When grouping matters, how many answers can a chain of four have?
The monk on the mountain
A monk walks up a mountain one day and down the next, on the same path. Is there a spot he passes at exactly the same time of day on both days?
Digits that double
Guess the square root of 10 as 3. Average 3 with 10 divided by 3. Do it again. In two steps you have 3.1623, and the number of correct digits roughly doubles each time.
Multiples that swallow everything
Multiply a multiple of 3 by any whole number and you get a multiple of 3. That fact, with its twin for adding, is why clock arithmetic works.
Connections bikes · journeys · money · models · polls · screens · the globe · rhythm
The same math, found in the wild.
Why a bike has gears
48 teeth at the front and 16 at the back turn the wheel three times for each turn of the pedals. Ratios you can feel in your legs.
Reading a journey
A distance-time graph tells a story: steep is fast, flat is stopped, downhill is coming home.
Changing money
Say 1 dollar buys 0.9 euros. Convert there and back, and see who keeps the difference.
How big is a toy car, really?
A 1:64 model car is 64 times shorter. A perfectly scaled copy using the same materials would be 262,144 times lighter. Scale in one, two and three dimensions.
How can 1,000 people speak for a country?
A poll of a thousand is usually within about three points, however big the country is. Why size of sample matters and size of country does not.
Why films have black bars
4:3, 16:9, 21:9. Fit one rectangle inside another of a different shape and something has to give.
An address for every place on Earth
Two angles pin down any spot on the globe. How far north, how far round, and why a degree of longitude shrinks towards the poles.
Fractions you can clap
A whole note, halves, quarters, eighths, and a dot that adds half again. Rhythm is fractions that must add up to a bar.
How far away was the earthquake?
After an earthquake, fast P-waves arrive first and slower S-waves follow. The gap between them tells you how far away it happened.
Dido's field by the river
With 100 m of fence and a straight river as one side, what shape of field encloses the most land? A rectangle does well. A semicircle does better.
Why do tiny drops take so long to settle?
Shake oil and vinegar gently and they soon separate. Shake hard and it takes far longer. Tiny drops creep along for the same reason a bead sinks steadily through honey.
Why narrow pipes matter so much
Halve the width of a pipe and, for smooth flow at the same pressure, only one sixteenth as much gets through. Water seeps slowly through fine sand for a related reason.
Smoothing a jumpy week
Replace each day's temperature by the average of it and its two neighbours. The jumpy list becomes smoother. Undoing a blur, though, is much harder than doing it.
Soap films find the shortest road
Four towns sit at the corners of a square. What is the shortest road network joining them? A soap film between two plates can find it for you.
Two springs in a row
One spring stretches 2 cm for each 100 g. Two such springs hanging in a row stretch twice as much. Side by side, half as much. A simple model of a layered material averages in these same two ways.
Squeeze the air
Block the end of a syringe and push the plunger. Halve the volume of the trapped air and its pressure doubles. Gases squeeze in a precise way.
How hot is the middle of the plate?
Heat a metal plate along one edge and keep the others cold. Once things settle, every spot is the average of the spots around it.
Which notes fit on a string?
Pluck a guitar string and only waves that fit a whole number of half-waves between the ends can survive. Sprinkle sand on a ringing plate and you see the same idea in two dimensions.
Why spaghetti snaps sideways
Press a strand of dry spaghetti from both ends. It holds, then suddenly bows sideways. A smooth push can cause a sudden jump.
Demos play with it
Eight new interactive demos go with this grade, from a balance that solves equations to the twelve pentominoes, alongside the other grades' demos. All of them live in the demo gallery.