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Now, if they all finished with a $12$ what were their starting numbers?
Doing and Undoing
Age 5 to 11
Challenge Level 
Doing and Undoing
Maybe you are used to making knots and sometimes you undo them!
Let's look at undoing some maths that's been done.
Suppose we have a starting number and then we doubled it.
That's the bit we'll call "doing".
To "undo" the maths we start with just the answer and see if we can get back to our starting number.
When I doubled my starting number, I got $6$. What do I have to do, to "undo" the $6$ and get back to my starting number?
Suppose that I did it again with a new starting number so I doubled and got to $16$. What would you have to do to "undo" and get back to my new starting number?
That's the bit we'll call "doing".
To "undo" the maths we start with just the answer and see if we can get back to our starting number.
When I doubled my starting number, I got $6$. What do I have to do, to "undo" the $6$ and get back to my starting number?
Suppose that I did it again with a new starting number so I doubled and got to $16$. What would you have to do to "undo" and get back to my new starting number?
Can you think what you would have to do to "undo" these three children's maths?
Danesh says
"I took $4$ away, what should I do to get back to my starting number?"
Meg says
" I added $8$, what should I do to get back to my starting number?"
Chris says
"I halved, what should I do to get back to my starting number?"
Now, if they all finished with a $12$ what were their starting numbers?
Photograph acknowledgements;
http://www.instructables.com/image/FPRA3T8FZ8J4A39/How-to-tie-various
-knots.jpg
http://2.bp.blogspot.com/-Jg0te1K_F5g/TpFcEY8xT6I/AAAAAAAAHy4/vgkke
-0QQ98/s1600/knot.jpg
http://www.instructables.com/image/F0PMDJ8FZ8J4A36/Figure-8-Knot.jpg
http://www.instructables.com/image/FPRA3T8FZ8J4A39/How-to-tie-various
-knots.jpg
http://2.bp.blogspot.com/-Jg0te1K_F5g/TpFcEY8xT6I/AAAAAAAAHy4/vgkke
-0QQ98/s1600/knot.jpg
http://www.instructables.com/image/F0PMDJ8FZ8J4A36/Figure-8-Knot.jpg
Why do this problem?
This activity possibly presents a new way for many pupils to think about the arithmetic they do. The idea of inverse operations is core mathematical concept and this activity offers opportunities to explore them in a meaningful way. The task also presents the chance to allow pupils to have the freedom to be curious and explore all kinds of
mathematics further, as well as persevere in their attempts to "undo" various calculations.
Possible approach
One way would be to start by saying "Here's the number $4$. I'll double it", and inviting the pupils to say what the answer is and then how to get back to the $4$.
You could try some different numbers and repeat the process with doubling each time as the operation.
The operation can then be changed to an addition or subtraction one.
This can now lead to the bigger question about whether same inverse operation works for every starting number.
If your pupils are secure then use the same rule but choose a different, and this time mystery, starting number and tell them the finishing number. Invite them to think about what calculation they'd do to get back to your mystery starting number.
You could try some different numbers and repeat the process with doubling each time as the operation.
The operation can then be changed to an addition or subtraction one.
This can now lead to the bigger question about whether same inverse operation works for every starting number.
If your pupils are secure then use the same rule but choose a different, and this time mystery, starting number and tell them the finishing number. Invite them to think about what calculation they'd do to get back to your mystery starting number.
Key questions
What number do you get?
If you 'undo' the operation what do you get?
What do you have to do to 'undo' adding $6$?
What do you have to do to 'undo' doubling?
Tell me about your ideas.
Possible extension
For those who show fluency in the activity then a further challenge is to say that the starting number has gone through two operations. Can the children find out how to undo them each in turn? What order do they have to undo them in?
Possible support
Children may need to stick to small numbers so that they are not overloaded by trying to remember number facts. It may help to have a 'machine box' and think about the numbers going in and the numbers coming out, and what happens if you undo the action by putting in the numbers backwards.
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