Surprising split
Does this 'trick' for calculating multiples of 11 always work? Why or why not?
Problem
Take a look at this silent video:
If you can't access YouTube, here is a direct link to the video: SurprisingSplit.mp4
Click below if you would like to read a description of what happens in the video.
54 x 11 is written.
Below that, the 5 and 4 are written again but separated so that there is a gap between them. A curved line is drawn linking the 5 and 4 and an addition sign is written below it. The number 9 is written between the 5 and 4.
Below that, 594 = 54 x 11 is written.
What do you notice?
What would you like to ask?
You might want to watch the video again.
Can you re-create this trick for yourself for 54 x 11?
Try the trick for a different multiple of 11. What happens?
Does the trick always work, for any multiple of 11?
This isn't really a trick, it's maths! Can you explain why it works?
Thank you to David Frankau and Mr Hadfield from the Pointer School in Greenwich who drew our attention to a trick for the 11 times table, which we have adapted for this task.
Getting Started
Here is another way of thinking about the trick...
For 54 x 11, set out a column addition with 54 as the first number to add.
Write 54 again, below the first 54, but this time line up the 4 in the ones column of the second 54 with the 5 in the tens column of the first 54.
Add them together:
What digit is missing from the ones column of the second number in the sum?
What do you know about numbers that end in that digit?
Why is this the same mathematically as the trick in the video?
Student Solutions
A large number of submissions described the mathematics behind the split and why it works.
Gold Class at Grayshott CE Primary sent in the following:
We tried it for lots of numbers and we found that it works for all double digit numbers.
We think it works because if you split 11 you get 10 and 1. So when you times the number 54 by 10, you get 540 and when you times the 54 by 1 then it's like you're ADDING another 54 onto the 540. So in the tens column you have 5 + 4 which is 9.
Isaac and Reuben from Garras C.P School wrote:
This works with all two-digit numbers.
If you do any three-digit number multiplied by eleven you can put the number you get (by adding your split numbers) either side of the the first two split numbers (and then you can get the correct answer) but you can also get it wrong if you put the number on the wrong side.
Crane Class from Larkrise Primary, Oxford sent their explanation and also considered decimals and three-digit numbers:
We tested it lots of times and found that the trick works for two-digit numbers where the digits add up to <10.
Paddy, Dai Dai, Zulekha and Ottilie worked together to work out that you can make it work for numbers where the digits sum >10. When you have a two digit total, use the 'ones' as the middle number and add the tens on to the left hand number. See the picture:
William wondered if it worked for three digit numbers... It didn't!
Mustafa wondered if it would work for decimal numbers. It did, as long as you put the decimal point back in to the right of the total in the middle.
Zulekha noticed that you could use the trick to multiply numbers by 11, then double them or quadruple them etc to find x22/33/44/55/66/77/88/99!
Lucas from The British School in the Netherlands used some algebra to generalise a solution. Here he shows his working:
Orphée, who is home schooled, sent in the following, which also includes some ideas about three-digit numbers:
Thank you for these solutions. if you come across this task again and have a different kind of solution then do send it in.
Teachers' Resources
Why do this problem?
Sharing this trick with learners will hook them in, firing their curiosity and thereby providing the motivation to explore, and understand, the underlying mathematics. This trick will help children gain a deeper understanding of place value and may also provide an opportunity to practise column addition in a meaningful context.
Possible approach
Without saying much at all, play the silent video for the class, asking them to consider what is happening. You could check the multiplication using a calculator and dramatically announce the answer! Give them chance to talk to a partner about what they notice in the video and any questions they would like to ask before playing it once more. At this point, open up discussion to the whole group, inviting comments and questions. You could write these up on the board as they are mentioned and, rather than answering or responding yourself, encourage other members of the class to do so.
Suggest that, as a class, you try a different multiple of 11 (you could invite a pupil to give you one). With the children's help, rehearse the process step by step, this time writing on the board, stating what you are doing as you go. Ask someone to check the result on a calculator.
Set pairs off on exploring further. Is the answer always going to be correct? You may wish to suggest that everyone sticks to a two-digit multiple of 11 for now (see the 'Possible extension' ideas below).
As they work, look/listen out for those children who are recording in a helpful way, or those who are beginning to explain what is happening. You may want to warn some pairs that you are going to ask them to share their work with the whole class during a mini plenary.
After a suitable length of time, bring everyone together again and ask whether anyone found an example of a calculation when the trick didn't work. If some pairs think they have come across one, encourage them to share it and invite the whole group to check. Will this trick always work or is it just that we need more time to find examples that don't? Allow more time for pairs to think about this, encouraging them to create a watertight argument that would convince you one way or the other.
If appropriate, you could share the approach in the 'Getting started' section with the class. This is a different way of presenting the same underlying mathematics, which may be more accessible even though it has slightly less of a 'wow' factor!
The final plenary can involve everyone in building an explanation together, drawing on their understanding of partitioning 11 into 10 and 1. You could suggest that children amaze their family with this new trick at home!
Key questions
What is the 9 worth in 594? So, rather than 5 and 4, which two numbers have been added to make that amount?
Where have these two numbers 'come from'?
If you were asked to multiply by 11 mentally, how might you do it?
How would you partition 11 to help with this calculation?
Possible support
There is a slightly different way of presenting this trick outlined in the 'Getting started' section, which some learners may find more accessible.
Possible extension
Does the trick work for single-digit multiples of 11? What about three-digit multiples? What happens with decimal multiples of 11? Can learners make a conjecture before exploring?