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Difference of squares

When two numbers being multiplied sit the same distance from the same number, the product comes out of a single square and a single subtraction. Take their midpoint, square it, and subtract the square of the distance:

18 × 22 = 20 × 20 − 2 × 2 = 400 − 4 = 396

18 and 22 are both two away from twenty. Twenty squared is 400, two squared is 4, and the answer is 396.

47 × 53 = 2500 − 9 = 2491

How to spot it. Check whether the midpoint of the two factors is round. The quick way: if one is as far below a round number as the other is above it, the technique fits. 18 and 22, 47 and 53, 96 and 104.

It is at its best when the distance is small, because then the square being subtracted is tiny. At a distance of two there are only four to take off.

The midpoint does not have to be a multiple of ten, only one whose square you know. 23 × 27 is 625 − 4, which is 621, because 25 squared is 625.

Practise

The exercise always gives a pair whose midpoint is a multiple of ten and asks for the three steps separately. In a real calculation you have to spot the pair yourself, and that is where all the work in this technique lies.

Mental arithmetic techniques

Addition and subtraction

• Number bonds to tenThe foundation of mental arithmetic. Learn the five pairs and practise them right here.
• Bridging through tenWhen crossing ten, always step through ten. Fill the first number up to ten and add the rest.
• Adding from the leftIn your head you add from the larger place values, not from the smaller ones the way you do on paper. Start with the tens.
• Subtracting in partsTake the tens off, then the units. The same move as in addition, only in the other direction.
• Subtracting by counting upInstead of borrowing, count up from the smaller number. The first step always uses a number bond to ten.
• Round and adjustWhen one number sits just below a round one, compute with the round one and fix it at the end. Take away when adding, add back when subtracting.
• Constant differenceShift both numbers by the same amount until the number being taken away is round. The difference does not change and no correction is needed.

Multiplication

• Multiplying by eightThree doublings in a row, and you get the twos and the fours for free along the way.
• Squares as anchorsLearn the nine squares by heart and use them to reach the neighbouring products. The hardest corner of the times table sits right next to the squares.
• Multiplying by nineReally just multiplying by ten and subtracting once.
• Multiplying in partsSplit the bigger factor into its tens and its units, multiply each separately, then add.
• Multiplying two-digit numbersSplit one factor and multiply the other factor by each part. Nothing new here, just one layer more.
• Rounding in multiplicationMultiply by the nearby round number and correct afterwards. The correction is the gap times the other factor.
• Halving and doublingMultiplying by five is multiplying by ten and halving. Halving one factor while doubling the other leaves the product unchanged.
• Multiplying by elevenPull the digits apart and put their sum in the middle. When the sum goes past nine, carry one to the left-hand digit.
• Squaring any two-digit numberChoose the nearest multiple of ten. Move one factor to it and the other the same distance in the opposite direction, multiply the new factors, and add the square of the shift.
• Difference of squaresWhen two factors sit the same distance from a round number, square the number between them and subtract the square of the distance.
• Squaring numbers ending in fiveDrop the final five, multiply what is left by the next number up, and put 25 on the end. 35 squared is 3 × 4 and 25, which is 1225.

Division

• Dividing in partsDivision is a multiplication question asked backwards. Split a bigger dividend into parts that divide easily.
• Division with a remainderWhen a division does not come out exactly, the answer is a quotient and a remainder. The same move also tells you roughly how big it is.
• Halving in divisionDividing by four is two halvings, by eight it is three. Dividing by five is doubling and then dividing by ten.
• Factoring the divisorWhen the divisor breaks into factors, divide twice in a row with smaller numbers. Dividing by six is dividing by two and then by three.
• Simplifying divisionDivide both numbers by the same factor. The quotient stays the same and you carry on with smaller numbers.
• Divisibility rulesBefore dividing, you can almost see at a glance whether the division will come out exactly.

Checking

• Estimating the sizeRound, get a rough answer, and only then work it out exactly. That way a big mistake does not slip through.
• Casting out ninesReduce every number to a single digit by repeatedly adding its digits. If the operation is correct, it must also hold for the reduced forms.

Mental arithmetic