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## Number Round Up

If you are a teacher, click here for a version of the problem suitable for classroom use, together with supporting materials. Otherwise, read on ...

Arrange the numbers $1$ to $6$ in each set of circles below.

The sum of each side of the triangle should equal the number in the centre of the triangular shape.

Once you've had a chance to think about it, click below to see how three different pupils began working on the task.

Dan said:

"I used counters which had $1$ to $6$ on them.

I put the counters in a triangle in any old way, then I added up the sides.

Then I moved the counters around to try and get the right total on each side."

Emma said:

"I noticed that three of the numbers are odd ($1, 3$ and $5$) and three of the numbers are even ($2, 4$ and $6$).  I thought this might help.

I know that $9$ is an odd number so it can be made using odd + odd + odd or using even + even + odd."

Farah said:

"If I want a small total on each side, I'll need small numbers in the corners of the triangle."

Can you take each of these starting ideas and develop it into a solution?

A practical version of this activity is included in the Year 3/4 Brain Buster Maths Box which contains hands-on challenges developed by members of NRICH and produced by BEAM. For more details and ordering information, please scroll down this page .

### Why do this problem?

This problem is an interesting context in which children can practise addition and subtraction.  It can be solved in many different ways and the sample approaches offer a basis for discussion of possible different methods.  You might spend a couple of lessons on this activity.

### Possible approach

You could introduce this challenge by having one of the triangles on the interactive whiteboard with numbers $1$-$6$ to drag into the circles. Begin by dragging the numbers into any places and ask the children to talk about what they notice. Some may mention the numbers used but direct them to look at totals along each "side" if it doesn't come up naturally. This will mean they understand the set-up and you can then leave them to try the problem itself.

Try not to direct learners too much at this stage but make sure they understand that they can use any resources or equipment that they might find helpful.  (You might want to have these sheets of blank triangles available should anyone request them: Word document or pdf).

Once you feel that most children have made progress and understand the problem well (this does not necessarily mean that they have found all the solutions), give out this (.doc  or pdf). Suggest to the class that when they've finished or can't make any further progress, they should look at the sheet showing three approaches used by children working on this task.  Pose the question, "What might each do next? Can you take each of their starting ideas and develop them into a solution?".  It might be appropriate to read through each method as a whole class before giving pairs time to work on each one.  Alternatively, you may prefer to allocate a particular starting point to each pair.

Allow at least fifteen minutes for a final discussion.  Invite some pairs to explain how the three different methods might be continued.  You may find that some members of the class used completely different approaches when they worked on the task to begin with, so ask them to share their methods too.  You can then facilitate a discussion about the advantages and disadvantages of each.  Which way would they choose to use if they were presented with a similar task in the future? Why?

(You may find that conversations arise about the number of different solutions for each total.  Encourage children to articulate what they think is the same and what is different in this context.  They might not all agree!)

### Key questions

What will you try first?
Why did you put that number there?
Tell me about how you found that solution.
Is there only one solution?

Tell me about this approach.  What do you think s/he was doing?
How do you think this will help to solve the problem?
What do you think s/he would have done next?

### Possible extension

Children could investigate whether other totals are possible. Why or why not?

### Possible support

Learners might request a range of different resources to help them tackle this challenge, for example numbered counters, mini-whiteboards, number lines.  Try not to pre-empt their requests by placing equipment out on tables at the start, but do make sure these kind of resources are easily accessible to the children, should they want to use them and do your best to accommodate any requests which you hadn't anticipated!