Count it by hand until you can’t. Then find the rule that can.
Draw a tower of squares one square high and count the lines. Then two squares high. Then three. Anyone can do it, and in the first session everyone does.
Then someone asks about a tower a hundred squares high, and nobody is going to draw that one. The case you want is out of reach of counting. So you go back to the small cases you could count, find the rule hiding in them, and let the rule do the counting for you.
And then the better questions: why must the rule be true, and is there a case where it stops working? That is investigation. The rule was there all along. Your job is to find it, and when you find it, you are exactly right.
Every session is a new problem, begun and finished in ninety minutes. Over sixteen of them the rules grow up: from counting what you can see, to a rule in one letter, to a rule in two, to rules you can prove — and to a last problem with no rule at all.
What you need before starting: Arithmetic, and the patience to draw the first few cases carefully. The letters arrive when they are needed, not before.
Nobody is shown the rule first. The problem goes up, everybody draws and counts, and the rest of the ninety minutes is spent finding the rule and testing it.
The questionSquares in a row, then squares stacked into a tower. How many lines does each one take to draw?
CoversDraw, count and fill in a table, counting the upright lines and the flat ones separately. The row and the tower grow at very different speeds, and the table shows it before anyone names it.
LandsEvery student leaves the first session having found a rule themselves.
The questionHow many ways are there to win at noughts and crosses? And on a bigger board?
CoversTry to count them all at once and get lost. Then split them up — across, down, diagonal — and each pile is easy.
LandsOrganising is part of counting. It is the habit that saves the later sessions.
The questionA path of tiles runs around a rectangular lawn. How many tiles, for a lawn of any size?
CoversCount the tiles one by one, then find the shortcut: the big rectangle take away the lawn. Change the length, then the width, and watch the rule follow both.
LandsCounting by taking away — the first idea in the course about structure rather than adding up.
The questionAdd the digits of a number, then the digits of the answer, until one digit is left. What can that digit tell you?
CoversNo picture this time, only numbers. The pattern becomes a tool that answers questions about numbers far too big to work out.
LandsA pattern is not always a picture, and a pattern you understand can be put to work.
The questionDots grow in rings around a centre dot. How many dots in the twentieth picture?
CoversThe pictures soon get too big to draw, so the rule has to be found. It is a squared rule, and you can see where the square comes from in the picture itself.
LandsA rule you can see in the picture is a rule you believe.
The questionSquares are piled up, step by step. Is there one rule for how many there are?
CoversThere is not. The odd steps and the even steps behave differently and need a rule each. Most of the group will resist that, and working through it is the session.
LandsSome patterns need two rules, and that is not a failure.
The questionWhy does 1/8 stop at 0.125, while 1/7 goes on forever?
CoversBreak the bottom number into primes and the answer appears. Then build fractions to order: one that stops after exactly three places, one that never stops.
LandsMoving from describing what you found to making something to a specification.
The questionEvery row, column and diagonal adds up to the same total. How do you build one?
CoversFind the facts every magic square must obey, follow a recipe that makes one, then write the recipe for any size. Finally run it backwards: given the total, find the entries.
LandsA letter can stand for a step in a recipe, and a rule can be used in reverse.
The questionShapes made of dots, with any number of sides and any number of dots on each side. How many dots altogether?
CoversNow two things change at once. Hold one still and explore the other, then swap.
LandsA rule in two letters, met gently — and a way of working that is named only after it has been used.
The questionA frog crosses a row of stones by jumping. How many different ways can it get across?
CoversEach answer can be built from the answers before it. Then a shortcut formula turns up and works perfectly, until the jumps change and it quietly stops working.
LandsA formula is true for some cases, not all. Finding where it stops is part of the answer.
The questionPlace a square window on a number grid. Multiply the opposite corners and subtract. Why is the answer always the same?
CoversTry it on window after window, then let letters explain it. The number trick turns out to be ordinary algebra.
LandsAlgebra explains arithmetic. It does not replace it.
The questionTwo sets of lines cross each other. How many crossing points are there?
CoversFind the rule in two letters. Then find the drawing that breaks it: three lines meeting at a single point.
LandsA rule, and the situation that breaks it, in the same session.
The questionReverse the digits of a number and subtract. What do the answers always have in common?
CoversThe pattern is easy to see. Then a place-value argument shows why it must happen for every number, not only the ones tried.
LandsFrom here on, seeing a pattern is not enough. You show why it is true.
The questionDraw a square ring of numbers on a grid, and another ring inside it. Why is the difference the same wherever you put them?
CoversTest it all over the grid, then prove it, using the algebra built in the last three sessions.
LandsA proof that covers every position, including the ones nobody will ever try.
The questionThree regular shapes meet at a point, with no gaps. Which three can do it?
CoversWork out the angles, then search. The search comes to an end, and you can show that nothing was missed.
LandsProof by checking every possibility — and knowing there are no others.
The questionBoxes of different sizes have to go into bins, and each bin has limits. What is the best way to pack them?
CoversThere is no formula. Try different strategies, compare them, criticise them, and work out what each one costs.
LandsWhat you do when a problem has no rule: you decide, and you say why.
Groups of four, Grades 6 to 8. Every session is a whole problem, started and finished in ninety minutes, so a student who misses a week misses one problem, not a chapter.
A group opens when four students are ready.