Facts as thinking tools

Leading on from last week’s blog on Thinking Differently, my thoughts this week centre on using mathematical facts as tools to solve problems. It occurs to me that children are encouraged to learn many facts, including times tables (up to 12 x 12): but rarely do we focus on why such facts need to be learned. Yet children are more inclined to rote learn these facts if they know WHY they need to. What is the purpose? Let’s consider a few problems (out of many) that are made considerably easier by instant recall of times tables.

Inverse: although it seems obvious that division is the opposite of multiplication, using times tables to also derive matching division facts speeds up calculation. For example, if 7 x 9 = 63, then the inverse fact that 63 divided by 7 would equal 9, and 63 divided by 9 would equal 7, should also be instantly recalled. One way to encourage such recognition, is to focus on numbers as related ‘families’ – the three numbers above, 7, 9 and 63 are a related family. We might call them the ‘inverse’ family.

How can knowing times tables help with more complex calculation, using much higher numbers? Times tables combined with place value form a formidable team for working out zero-based calculations. For example, 30 x 4 or, more complex, 70 x 600. Simply adding the required number of zeros to the times table, makes light work of such operations. Even ‘almost’ zero problems can be solved in this way: 41 x 60 = 240 + 60 = 300. Mere seconds of calculation.

Of course, children must understand how many zeros to add onto the answer and why. But times tables and an understanding of zero as place value markers: multiplication by ten, hundred or thousand, enable speedier problem solving.

What about decimal points? Yes, children can perform written calculations, but using place value with times tables works the opposite way as well – to divide by ten, hundred or thousand. Example: 2.5 x 400 = 10 with two zeros on the end = 1000. Another example: 60 x 1.2 = 72. Lets consider this second example. Pupils need to realise that the zero on 60 represents times 10 – but the decimal point in 1.2 represents division by 10. So, as 6 x 12 = 72, the inverse operations cancel out each other. One further example: 700 x 0.4. This needs thinking about. We know that 4 x 7 = 28. Two zeros mean multiplication by 100, but the decimal means division by ten – so only one zero needs to be added: 280.

Just as a plumber needs specialist tools, mathematicians needs facts in order to do a quick and efficient job.

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