### Pebbles

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

### It Figures

Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?

### Bracelets

Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?

# Surprising Split

## Surprising Split

Take a look at this video, which has no sound:

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.

### 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?