Knowing by heart decimal and percentage equivalents to quarters is rather useful. These might be taught in units of work on fractions, decimals and percentages but in such isolation, they are wasted.
There’ll be plenty of work to do with children on proportion with some wonderful pictorial representations to develop their understanding and of course lots of repetition to ensure that those facts can be instantly recalled. However, what I’m interested in here is the application of these facts to do some pretty impressive calculating.
First of all, mental division. Take 17 ÷ 4. It’s relatively straightforward to show children counting in 4s until you get to 17; 4 groups of 4 with one left over – 4 remainder 1:
When children are dividing numbers by 4, it is only a short step to giving answers as decimals. If there is one left over, that’s one out of a possible group of 4 which is 0.25. If there is a remainder of 2, the answer will end in 0.5 and if there is a remainder of 3 then the answer will end in 0.75. The same can be done with dividing by 8. Any remainder can be converted into a decimal using known eighths facts. This example shows 27 ÷ 8:
Where we can really get children doing impressive things though is by using the same idea to multiply mentally. Using the quarters facts, children can multiply any number by 25 as long as they are secure in the knowledge that four 25s make 100:
Using this pattern, 8 x 25 is 200 and 12 x 25 is 300. Children can simply count in 4s with each 4 representing 100. 20 x 25 = 500, 32 x 25 = 800 and 44 x 25 = 1100. Impressive!
The next layer of difficulty is multiplying numbers that are not multiples of 4 by 100, such as 9 x 25 or 17 x 25. The same principle as before can be used. A group of four 25s is 100 and each individual tile in this pictorial representation is 25. Let’s look at 13 x 25 and 15 x 25:
This strategy can be adapted so that any number multiplied by 2.5 is efficient – a group of four 2.5s is 10. We can do the same with any number multiplied by 0.25, 250 etc.
Using the same principle along with eights facts, we can quickly figure out a strategy for multiplying any number by 12.5. Eight 12.5s make 100 so grouping 12.5s into eights makes any multiplication like this manageable. Here’s 24 x 12.5:
By simply knowing the 8 times table and that 8 x 12.5 = 100, 32 x 12.5 = 400, 40 x 12.5 = 500 etc. Groups of 8 12.5s make 100 and each individual tile is worth 12.5. When we’re multiplying a number by 12.5 and that number is not a multiple of 8, we just need to fall back on eighths facts. Here’s 17 x 12.5:
The strategy can be applied to multiplying any number by 125 (groups of 8 make 1000) or by 1.25 (groups of 8 make 10) or by 0.125 (groups of 8 make 1).