Teaching multiplication tables up to 12×12
16 October 2013Add to My Folder
My six year old daughter is very fond of telling me what 12×12 is. She knows the answer is 144 and she loves telling anyone she meets. Although she might not understand what she is saying, I get the feeling that she won’t forget it. Her understanding will come later. She has simply enjoyed learning what she considers to be ‘grown-up’ maths.
Multiplication tables in the 2014 curriculum
When you start drilling into the new maths curriculum for lower Key Stage 2, you will find that the Year 4 programme of study states that pupils should be ‘taught to recall multiplication and division facts for multiplication tables up to 12×12’. The ‘old’ curriculum stated that children needed to know up to the 10 times table by the end of primary school. Many feel that learning up to 12×12 by the age of nine is a tall order, but it certainly shouldn’t phase us. In fact, it’s a great opportunity to learn some interesting concepts and patterns.
What order should you teach tables?The times tables facts are important, and make a significant contribution to numeracy. However, there is no research indicating an ideal order for teaching the times tables from 0 to 12. The usual consensus about learning tables is as follows:
- Start with children building up a table using physical apparatus such as cubes and rods,
- Move on to pictorial representation of tables,
- Symbolise the two types of table – for example, the table of 2s and the two times table,
- Practise the tables in both written and oral forms.
1, 2, 10 and 5 times tables
Children need to know how to double and how to halve. Teaching the 1, 2, 10 and 5 tables emphasises how effective doubling and halving strategies can be – so they seem to be a good place to start.
Rather than 12×12, I’d stick with thinking about learning up to 10 by 10 first, using a a blank 0-10 square. If the facts for 0, 1, 2, 5 and 10 are completed, there are only 36 facts left blank. Six of these facts are square numbers (3 squared, 4 squared and so on). The remaining 30 can be halved to 15 because of the commutative property of ab = ba. So, there are 21 remaining facts beyond the easy tables. Using rules based on the understanding of relationships, together with the derivation of facts from known facts, decreases the memory demand of the multiplication facts.9 times table
Deriving the nine times table from the ten times table is easy enough if the distributive law of multiplication over subtraction is applied. Basically, this is the 10 times table minus the 1 times table. The digits of the answer will always add up to 9.
9×1 = 10 – 1 = 9
9×2 = 20 – 2 = 18
9×3 = 30 – 3 = 27
9×4 = 40 – 4 = 36
Another strategy and a handy way to multiply by 9 is to use your fingers by holding up your hands with the fingers outstretched and doing the following:
For example, for 3×9:
- Start with your left hand and count along three fingers from the left hand side. Bend down this third finger.
- Count the number of fingers to the left of your bent finger (2). Each of these fingers count as ten so we have a value of 20.
- Then count the remaining fingers to the right of the bent finger. Each of these fingers count as units so we have a value of 7.
- Add the tens to the units and you have the answer: 27
4 times table
For multiplying by four simply double twice.
For example, for 4×6 start with 2×6 = 12
Then double again: 2×12 = 24
As with the ten strategy for 9, this ‘double double’ technique works with numbers beyond the table square collection.
Using square number facts
If the numbers you are multiplying are separated by 2 (for example, 7 and 5) them multiply the number in the middle by itself (in other words, square it) and subtract one. For example:
6×4 = 5 squared (25) subtract 1 = 25
7×5 = 6 squared (36) subtract 1 = 35
9×7 = 8 squared (64) subtract 1 = 63
Top tips and tricks
Use these clever tricks when multiplying by the following numbers:
0* – Multiplying by zero always produces zero
1* – The answer is always the same as the question – the number remains unchanged
2* – Double the number so add the number to itself (for example, 2×8 = 8 + 8)
3* – Double the number and then add the number to the double
4* – Double and double again
5* – The last digit always goes 5, 0, 5, 0, 5 and so on. The answer is always half of 10x (for example, 5×6 = half of 10×6). The answer is half the number times 10 (e.g. 5×6 = 10×3 = 30)
6* – If you multiply 6 by an even number they both end in the same digit (for example, 6×6 = 36)
7* – Calculate 6 times the number plus the number (or 8 times the number minus the number).
8* – Multiplying by 8 can be made easier by doubling three times
9* – Calculate 10x the number minus the number (for example, 9×6 = 10×6 – 6 = 60 – 6 = 54)
10 – Put a zero after it (except for decimal numbers)
11 – For up to 9×11, just repeat the digit (for example, 3×11 = 33).
12 – Calculate 10x plus 2x
Finger mathsThe hardest times tables to learn seem to be 6, 7, 8 and 9. What can we do to conquer these? By far the most successful strategy I have ever used and taught with children and teachers for multiplying from 6 through to 10 is using my hands. It is a very effective method and offers a neat and clever way of solving a multiplication problem when the going gets tough. For example, let’s think about 8×9:
- Start with your palms facing you.
- Assign each digit of your hands with a number. Write the number 10 on your thumb, then 9 on your index finger, 8 on your middle finger, 7 on your ring finger and 6 on your little finger. Do this on the other hand.
- Join the 8 finger on the left hand with the 9 finger on the right hand.
- The two fingers that have joined and any fingers beneath them are worth ‘ten’. So, in this example, you have a value of 70.
- The fingers above those joined are now multiplied together. So, on your left hand you have two fingers above the join and on your right hand you have one finger above the join making 2×1.
- Now add 70 and 2 to make 72.
Finer computation requires practise but it may be a method that children come to use when they ‘get stuck’ or use as the strategy of choice.
Alternatives to rote learning
Whilst there is a lot of brouhaha about the place for rote learning of tables, rote learning certainly has its place. Children need a fluent knowledge of number facts and without this being accessible and automatic they cannot demonstrate enough facility with numbers to enable them to solve problems. But rote learning cannot dominate. What is clearly needed alongside this is a range of fun strategies and games that addresses a range of learning styles, such as these ones from the Child Education resource bank: