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Fifteen Cards

Age 7 to 11
Challenge Level

Fifteen Cards

I have fifteen cards numbered $1 - 15$.

I put down seven of them on the table in a row.

7 cards
The numbers on the first two cards add to $15$.
The numbers on the second and third cards add to $20$.
The numbers on the third and fourth cards add to $23$.
The numbers on the fourth and fifth cards add to $16$.
The numbers on the fifth and sixth cards add to $18$.
The numbers on the sixth and seventh cards add to $21$.

What are my cards?

Can you find any other solutions?

How do you know you've found all the different solutions?

Why do this problem?

This problem requires no mathematical ideas beyond simple addition and possibly subtraction, but it does require the perseverence to stick with a trial and improvement approach, combined with some systematic working.

Possible approach

You could start by giving pairs of learners a set of cards numbered from $1$ to $15$.  (Numbered cards can be downloaded here. If they are printed onto thin card they will be easier to use and if covered with plastic film they should last a long time.)  Suggest they put seven cards face down in a row at random and find out the sum of the numbers on the first two cards.  You could find out whether any pairs had the same total and ask the rest of the class to predict whether they had used the same numbers.  Invite children to justify their responses, for example by listing possible combinations to make that sum.  
Having enabled learners to get a feel for the situation in this way, introduce the problem itself, perhaps by giving out copies of this printed sheet or by displaying the challenge on the board.  Try not to say anything else at this stage and give them chance to work in their pairs so that they are able to talk through their ideas with their partner.
Observe the way that pairs work and, once they have made some headway, bring the group back together to discuss strategies so far.  What have they been doing to try and solve the problem?  Some may have adopted a trial and improvement approach.  Others may be working more systematically, perhaps by listing possible pairs of numbers for a particular total.  Can anyone suggest a good place to start the problem?  Has anyone tried something that didn't work?  After this discussion, give more time for pairs to complete the task.  You may find that some adopt a strategy suggested by another pair rather than pursuing their original method.
In the final plenary, come together again to reflect on the process.  You could share final answers, but also encourage some pairs to explain their methods from start to finish.  This could be an opportunity to discuss which method/s were particularly elegant or efficient.

Key questions

How might you start this problem? 
How could this number be made with two cards?  Is there another way?  And another ...?
How will you remember which combinations you have tried?
I wonder whether the order of these two cards might make a difference?

Possible extension

After demonstrating that they have found all the possible solutions, learners could make up a similar problem for others to try. Remind them that you will expect them to know the solutions to their own problem before giving it to others to try out!


Possible support

Using digit cards will encourage learners to try out different combinations without having to commit anything to paper at first.  They may need reminding that, for example,  that $12$ followed by $3$ will give a different order from $3$ followed by $12$.  You could suggest that they focus on just one pair to begin with and consider all possible combinations, then try to work out what the other cards could be based on each of those possibilities.