Unexpected Ordering

Why do this problem?

Card tricks tap into children's natural curiosity and can provide the motivation for exploring the underlying mathematics in order to unpick how they are done.  The mathematics of this particular trick is very accessible and will help to reinforce the order of number names in the counting sequence.  The challenge, therefore, is being able to explain why the trick always works and so it provides a fantastic opportunity to encourage young children to articulate their reasoning.
 

Possible approach

This problem featured in an NRICH video in June 2020.

Play the video without interruption and invite the class to consider what they notice and what they want to ask.  Without saying any more at this stage, play the video a second time. Some of the children's questions might have been answered by seeing the clip for a second time, but you could now gather 'noticings' and questions, writing them up for all to see rather than commenting/answering yourself.  It may be that other learners can offer further insight so that some of the questions are answered.

Allow time for children to have a go for themselves in pairs.  You could suggest that they replicate exactly what Fran does in the video as a starting point. You can leave the video playing on loop so that children can tune in or out at any point.

Bring the whole group together again and invite contributions.  What have they noticed now, having had a go for themselves?  Why do they think that Fran was surprised in the video? Set pairs off again, this time to try out their own sequence of dealing out in ones and twos.  

As they work, look out for those pairs who seem to have found useful ways of recording and ways of working on the task. You could facilitate a mini plenary to give time for some pairs to share their approaches, which may help those who are struggling to make progress. 

After a suitable amount of time, draw them together again.  Hopefully, some pairs will conjecture that the cards always end up in numerical order, no matter how you deal them out. Challenge the class to prove or disprove this conjecture and emphasise that this trick isn't 'magic', it is maths.  Can they find a way of dealing in ones or twos, as in the video, which doesn't result in the cards being in numerical order? If they agree with the conjuecture, can they explain why this is the case?

The final plenary is a chance to create a chain of logical reasoning (a proof) as a whole class, with everyone chipping in as appropriate. 
 

Key questions

What happens if you use a different order of ones and twos in dealing out the cards?
How will you keep track of what you have tried?
How could you share what you think is going on with someone else?
 

Possible support

You could suggest that learners do this task with the cards facing upwards.  In this way they will easily be able to see how the order is being affected when the cards are dealt out again.
 

Possible extension

Invite learners to explore one or more of the following questions: