HexPentas
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
Age
5 to 11
Challenge level
Problem
You've maybe come across a challenge of putting five squares together in different ways - often called pentominoes.
So here are five squares and four ways of putting them together - there are obviously more ways. (You could try the Penta Place problem if you haven't seen this before.)
Image
You may even have done a similar
challenge using triangles (if not have a go at our
Tri-five problem). Like this:
Image
But the challenge for today is to consider the same idea using
hexagons.
Image
So, the challenge is:
What different shapes can you make using five hexagons?
Be careful that you do not have any the same as each other.
For example, these two look different but are in fact
identical:
Image
Please send in all the different shapes you can get and let us know how you made sure you had found them all.
Perhaps you also have some ideas about how you may go further with the question "I wonder what would happen if I ...?"
Getting Started
You could cut out hexagons from this sheet or you may like to colour in your arrangements on this sheet of smaller shapes.
How will you check that all your shapes are different from each other?
How will you know you have found all the different shapes?
Student Solutions
Fergus and Nathan from Rutherglen Primary sent in all these. They are good. Once you've got these you can have a look to see if any are the same. I've circled in a reddish colour those that end up the same by simple turning. I've circled in a blue-ish/greenish colour those that if they were flipped over would be the same. I'm very pleased with the efforts these boys have put in.
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Hannah from Herts and Essex High School sent in this really colourful solution. I have again taken this further and joined together with reddish circles those that - ignoring colour - would be the same through turning. I do think that colouring in makes them so different!
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Four other pupils from Rutherglen Primary School: Hannah, Laura, Sarah and Aleisha, found out that hexpentas started off quite easy but at some stages it got a bit challenging. They sent in the following:
First we started off with setting
out the straight ones and then moving one every time. The hardest
part was figuring out if we had them all and trying to make a
system.
After that we counted all the
hexagons to see if we had five in each. The easiest part was
counting them and seeing if there was five hexagons in each shape.
Then we counted all the shapes and we got about twenty, but I think
we could have created more!
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Kate from St. John Bosco Primary School sent in the following useful suggestion:
How about children can make their
own pattern and learn by that.
I agree that it is very good if pupils make their own versions of the challenges.
Well done to all those who sent in solutions. I hope you enjoyed the activity and maybe can take it further.
Teachers' Resources
Hexpentas
You've maybe come across a challenge of putting five squares together in different ways - often called pentominoes.
So here are five squares and four ways of putting them together - there are obviously more ways. (You could try the Penta Place problem if you haven't seen this before.)
Image
You may even have done a similar challenge using triangles (if not have a go at our Tri-five problem). Like this:
Image
But the challenge for today is to consider the same idea using hexagons.
Image
So, the challenge is:
What different shapes can you make using five hexagons?
Be careful that you do not have any the same as each other. For example, these two look different but are in fact identical:
Image
Please send in all the different shapes you can get and let us know how you made sure you had found them all.
Perhaps you also have some ideas about how you may go further with the question "I wonder what would happen if I ...?"
Why do this problem?
This activity is accessible to a wide variety of pupils. At one level, learners can explore ways of putting the five shapes together by 'playing' with hexagon pieces. At a higher level, more experienced or older pupils can be expected to go about the activity in an organised, systematic way. In addition, this
problem encourages learners to use visualisation, and to apply their understanding of reflective and rotational symmetry.
Possible approach
You may like to start by giving pupils the experience of this problem using five squares (pentominoes). Rather than asking them to make the different shapes, you could give them the set of twelve pentominoes and arrange them in different ways so that they are convinced there aren't any missing. This will help
them to develop a system for working on the hexagon version.
You could introduce the challenge by creating some hexagons to be dragged around the interactive whiteboard screen, or you could cut large shapes out of card and fix them to the board or wall. Encourage a few children to make their own arrangements for all to see so that everyone is clear about the task before giving time for pairs to investigate further. Children will find it useful to cut
out hexagons from this sheet (larger shapes) or perhaps to colour in arrangements on this sheet (smaller shapes).
You may need to have a mini-plenary at an appropriate moment to discuss shapes that may be reflections or rotations of each other. At this point, having the interactive whiteboard would be helpful so that you can demonstrate rotations or flips easily (although this can be done by cutting out shapes or joining hexagons together so they can be manipulated physically).
In the plenary, you could pool all the different shapes that the group has found. You could then either invite some pairs up to explain their own ways of working systematically, or you could ask the class to order the shapes that are displayed so that any missing ones can be identified.
Key questions
Are any of your shapes the same?
Have you done this in some way so that you know what you are going to try next?
How are you making sure you do not have the same one twice?
How are you recording these?
Possible extension
One possibility is for pupils to explore what happens with more hexagons. Alternatively, you could encourage them to ask "I wonder what would happen if I ...?" and to go down their own line of enquiry.
Possible support
An appropriate drawing program on the computer may be a useful tool for some pupils. Others will need to have hexagons to arrange.