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Add Infinitum – Teaching Times Tables

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Somerset Times Edition

Week 1,
Term One, 2016

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The perennial debate over Times Tables teaching has been re-energised in England with the recent announcement that all Year 6 students will be expected to know their tables up to 12 x 12 and will be tested, against the clock, using an online test.

Critics argue that rote learning will be encouraged at the expense of understanding. The dispute is also whether we should encourage arithmetic skill over mathematical reasoning.

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“Maths is a non-negotiable of a good education,” explained Education secretary Nicky Morgan. Ironically, when asked on a live morning TV interview last year, she refused to give an answer to 7 x 8. This was the same problem that the former Labor school’s minister got wrong in 1988 (he gave the answer as 54). Unfortunately he didn’t know the trick; 56 = 7 x 8 (so it’s just 5678). The ironically named Christine Blower, leader of the National Union of Teachers, pointed out that many children could find the answers to such calculations on their phones.

The Australian Curriculum stipulates that for a Year 4 Achievement Standard, students should recall Multiplication Facts up to 10 x 10, so by Year 6 they should certainly be proficient up to 12 x 12.

Let us dissect what is involved in learning tables. Ostensibly, it seems that these poor little children will be forced to absorb 144 pieces of information in an old fashioned class room situation, chanting meaningless equations with the threat of failure, punishment and disgrace hanging over them - known by some as the ‘good old days’, “which never did me any harm”.

The one times table is trivial but illustrates an important calibrating tool that any number multiplied by 1 retains its value. At this stage it might also be pertinent to think about multiplying by zero!

Multiplying by 10 is also straight forward, so from 144 multiplication facts we are down to 124.

The 11 times table is also simple (up to 9 x 11) and for two digit numbers multiplied by 11 there is a simple trick to ‘split them and add them’. For 11 x 11, we split the digits of 11 to get 1 ... 1. Then add 1+1 = 2, and place the result in the middle to give 1 2 1. This works for any two digit numbers (although you may need to carry numbers over) and the same process works for 3, 4 or more digit numbers multiplied by 11 so you can see how 134 x 11 = 1474. You might also be surprised to know that 111111111 x 111111111 = 12345678987654321 for the same reason!

When we examine the nine times tables we notice that the result of each one; 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108 reveal a pattern in that the sum of the digits also equal 9. If that is too daunting there is also the “fingers trick” which can make the nine times table fun for all ages.

The five times table has its own pattern, with the second digit alternating between 5 and 0. The two times table is an easy one as the numbers are relatively low and indeed, by Year 3, students are supposed to know the two, three, five and ten times tables anyway. The three times table has another interesting property in that the sum of the digits in the answer is always divisible by 3.

To summarise the situation so far, it looks like there are patterns evident in the 1, 2, 5, 9, 10 and 11 times tables so we are now left to analyse the 3, 4, 6, 7, 8 and 12 times tables. That makes it 72 multiplication facts to learn, but when you consider the commutative property of multiplication (that 3 x 4 = 4 x 3), this halves all the rote learning. Now I reckon the number of multiplication facts that need committing to memory has been reduced to around 21 (shaded in the diagram below) which does not seem an unreasonable number of facts to recall.

Figure 1

By the way, the calculations that are most often worked out incorrectly are those in the middle of the grid with 6 x 8 (and 8 x 6) found to be the hardest of them all.
One type of number that has no factors apart from one and itself is a prime number and earlier this month there was great excitement when a new one was discovered. It contains 22,338,618 digits and has a value of 274,207,281 – 1. There are an infinite number of primes and contrary to the belief that prime numbers have no real life application, they actually help protect phone passwords and credit card transactions.

In the final analysis, why should we need to know the 12 times tables at all, when decimalisation of our currency took place over 40 years ago and feet and inches have given way to the metric system? Well, for a start, there are 12 months in a year so if you are quoted a monthly phone plan, you should be able to convert that to an annual amount - and you won’t need to use your phone to help you!

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