# Unexpected Ordering

Watch the video below:*If you can't access YouTube, here is a direct link to the video: UnexpectedOrdering.mp4. You can view closed captions by selecting 'CC' in the bottom menu of the video once it is playing.*

What do you notice?

What do you want to ask?

Watch it again.

Perhaps some of your questions have been answered. Or you might have thought of new questions.

Try the task for yourself. What happens?

What happens if you use a different order of 1s and 2s in dealing out the cards?

Can you explain why this is maths and not a magic trick?

*You may be interested in the other problems in our Using Wonder to Promote Curiosity Feature.*

*This problem featured in an NRICH video in June 2020.*

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?

The first time she dealt the cards out there was one card left the second time there were no cards left.

Lucia added:

I tried the trick and I figured that firstly, I had 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and then I flipped it over just like the video did and ten was at the back. So then I flipped ten with nine so the ten was on the ground and nine on top. Then I flipped the eight and seven around so that eight was nearest to the ground and seven on top . I repeated this pattern so that I could get 1 on top of 2, 2 on top of 3, 3 on top of 4, and etc. That is why I would have the numbers backwards when I look at the cards/numbers.

Israel from Lark Rise Academy wrote:

Because when you collect the cards up, the bigger ones end up on the smaller ones so if you swap them around it would make it in order so that is how that trick is made.

### 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:

What would happen if you could deal in threes as well?