Number Round Up

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 its centre.

Triangle Animals

How many different ways can you find to join three equilateral triangles together? Can you convince us that you have found them all?

Button-up

My coat has three buttons. How many ways can you find to do up all the buttons?

Plants

Plants

Uncle Raj has three children. Next year, when they've had their birthdays, Naomi will be $5$, Alex will be $6$ and Chris will be $7$. The family has decided on something rather unusual for part of their presents.

All three children have their birthday in the late spring and since they are keen on gardening they are going to buy some plants for the garden, one for each year they have been alive.

Here is the plan of their house and garden:

You notice that there are three circular paths that cross over each other. Each child is to have a circle but there will be some bits that are shared, around the middle.

When the time comes, the four of them go off to the garden centre to choose the plants. They do not have a lot of money so they're looking for special offers. They find a very special offer which gives a good discount if you buy ten plants altogether. The three children say that that is no good because they need more than ten. But Uncle Raj realises they can manage with only ten.

They go to the cafe and have some cool drinks, and Uncle Raj draws a plan of the three paths and puts little marks to show the plants.

Here is his idea:

The children are fascinated to see that Naomi has $1$ and shares $4$, Alex has $2$ and shares $4$ and Chris has $3$ and shares $4$. They think that's rather cool and it saves them a lot of money. So they finish their drinks and off they go to buy their ten plants.

Well now it's your turn to have a go and find some different solutions.
REMEMBER:- You must use exactly ten plants (no more, no less)
REMEMBER:- The circles must contain $5$, $6$ and $7$ plants (no more, no less).

As you try, you may find that you are developing a system for getting the next one. If so, we'd love to hear about it.  You might like to try to find them all, and write about all the things you notice about each solution.

You could print off this sheet to help you record them, if you like. However, you might find a different way of recording them altogether. Some people find it's easiest to do it quite large and have ten objects to move around in different places.

As with most of these challenges you can and should ask "I wonder what would happen if ...?" Well you might try a different number of objects (plants). You could try different numbers for each circle, as if the children were different ages.

Good luck and don't forget about sending in any results you have.

Why do this problem?

This activity is a good one to choose when wanting to encourage children to think creatively and 'have a go' at problem solving. It is also accessible to a wide range of attainment levels. This activity can also be a platform from which to give opportunities for children to talk freely about their thinking and to encourage them really to listen to each other.

Possible approach

Before you embark upon this activity with pupils, I do suggest that you play around with it a bit on your own or with colleagues. This is often a good idea, but in this case I think it is particularly valuable as you find that interesting strategies come into play that you probably would not have thought about had you not indulged yourself.

You could, with some youngsters, just introduce three hoops and a number of counters (to represent plants) and, having labelled the hoops with the quantities that need to be inside, ask the youngsters to see if they can find a solution. With the suggested $5$, $6$ and $7$ for each of the three circles, you could explore the activity using between $7$ and $18$ counters. It would be quite something for some of your pupils to explore all the different possibilities with each of the numbers from $7$ through to $18$, particularly with the emphasis on talking, listening and discussion.

Key questions

How many are there now in this part?
How could we make this part have a correct number in it?

Possible extension

a.  This idea actually came from a pupil doing this investigation. She labelled each area and then produced a table to show the number of plants in each.

Pupils might consider the areas D, E, F and G as "worth" more than $1$, (D, E, F being $2$ and G $3$) and record results in a table.  There's a lot to explore in such tables, and it's interesting at the start to find out how the pupils explore to get the table. Some may be using a spreadsheet, or mental calculations, or just looking at the picture of three overlapping circles, whilst others may use something practical to check that all is well with their ideas. Some interesting discussions may arise from some pupils who work very arithmetically and come up with a system but unfortunately ignore the maximum number allowed in each circle.

b.  Explore other groups of numbers instead of just $5$, $6$ and $7$ - what about numbers going up in $2$s, $4$s, $6$s and $8$s or any other number?

c.  If pupils have happily constructed tables in which every possibility is discovered, you might explore the number of possibilities according to the difference between the total for the three circles if they were no overlaps, ($5+6+7=18$) and the number of items used. For example there were seven solutions for a difference of two:

Further exploration will reveal the number of solutions when $3$, $4$, $5$ etc. extra ones are needed (e.g. when $13$ items are used with $5$, $6$ and $7$ circles then there an extra ($18 - 13$) five items needed.

For the exceptionally mathematically able

Obvious extension work can be looked at by considering four or more areas - so take yourselves to More Children and Plants.

Possible support

I feel that the best support is to be alongside the children being the 'more mature thinker' in that you can more easily help them to keep track of where they are and prompt them to suggest the next move.