Find the mistake

Your approach to each question

1. Read the title: it says what is being worked out, and so which kind of slip to expect — a sum chain, an expression to simplify, or a price that goes up then down.
2. Check every line on its own first. Is the arithmetic written on it true, whatever came before? In the sum chains, the wrong line is always false on its own.
3. Then check that each line follows from the one before. In the algebra and price workings a line can be true arithmetic and still not follow from the line above.
4. When two neighbouring lines disagree, the later lines decide which is wrong: they are written from the correct working, so the line they agree with is the right one.

Techniques and tricks

• Last-digit check

  The last digit of a sum or product depends only on the last digits. A line whose last digit is impossible is wrong without doing the rest.

  e.g. 73 × 6 = 444 is wrong: 3 × 6 ends in 8, so the product ends in 8 (it is 438).

• Casting out nines

  Replace each number by its digit sum, repeated until one digit. The digit sums must obey the same sum or product. It catches the ten-out slips the last digit misses.

  e.g. 47 + 38 = 95? 47 → 2, 38 → 2, and 2 + 2 = 4, but 95 → 5. Wrong; 85 → 4 fits.

• Put in a number

  To test an algebra step, put a simple value such as x = 1 into both lines. If they give different numbers, the step is wrong.

  e.g. 6(x + 9) − 9(x − 6) at x = 1 is 105; “6x + 54 − 9x − 54” at x = 1 is −3, so that expansion lost a sign.

• Two minuses make a plus

  Opening −b(x − a) gives −bx + ab. Look straight at the sign in front of the second constant; it should be +.

• Collect the x terms by their signs

  After expanding a(x + b) − b(x − a), the x terms are ax and −bx, so the coefficient is a − b, and the constants are ab twice, so 2ab.

  e.g. 7(x + 3) − 3(x − 7) = 7x + 21 − 3x + 21 = 4x + 42, not 10x + 42.

• Percentages multiply

  A rise of u% is × (1 + u/100) and a fall of d% is × (1 − d/100). Multiply the two factors to get the overall change.

  e.g. +20% then −10%: 1.2 × 0.9 = 1.08, a change of 8%, not 10%. +10% then −10% is 0.99, a 1% fall.

How to practise

• This game has no settings of its own; leave Time pressure Off while you build the checking habits, then try Relaxed once they are quick.
• Make yourself say why a line is wrong before you tap it. The working names the slip, so compare your reason with it, not just your answer.
• Use Feedback “At each judgement” for a block once you are accurate: answering straight through, then reviewing, is how checking is done for real.
• In the algebra and price workings, line 1 only restates the question, so the slip is always in a later line. Spend your time there.