Halving
These pictures show squares split into halves. Can you find other ways?
Age
5 to 7
Challenge level
These images show squares split in half:
Image
Image
How might you check that each was correct?
Can you think of more ways to split a square into two halves?
Have you thought about using the squares on your squared paper to
help you check your drawings?
How about cutting out your square? How would this help to check that you had divided it in half?
How about cutting out your square? How would this help to check that you had divided it in half?
Aoibheann and Padriac from Cloghans Hill NS in Ireland, said:
You can check that the squares are properly halved by cutting out the pieces and assembling them back to back or on top of each other, manipulated so they are the starting in the same places.
Very well explained, thank you.
Other ways to do halves are included in the photo they sent:
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These ideas were sent in by Donna of Zion Heights.
To check that each way is correct, you can measure the area of each red/white piece and see if the combined total area of the white pieces is equal to the total area of the red pieces. You can also cut out the white and red pieces and see if you can rearrange them so that every white piece has a matching, congruent red piece.
Here are two more possible ways to cut a square in half:
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Krystof also sent in another way to cut a square in half:
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Thank you to all of you.
Yusuke and Alex from the United World College of South East Asia East Campus wrote
Dylan from St Louis de Monfort in Australia wrote
To start with, a straight line down the middle can easily be seen as even however when there are harder shapes like when the amount of red and the amount of white are in totally different spots then it can get challenging. By simply checking that the shape is symmetrical then you can safely assumethat they are even
Please don't worry that your solution is not "complete" - we'd like to hear about anything you have tried. Teachers - you might like to send in a summary of your children's work.
Thank you for those of you who sent in your ideas and thoughts.
Halving
These images show squares split in half:
Image
Image
How might you check that each was correct?
Can you think of more ways to split a square into two halves?
You might like to use squared paper to draw your ideas on or to print out the uncoloured squares above.
Why do this problem?
This exploratory problem is a fantastic way to consolidate children's understanding of halving and halves. It also gives learners experience of mathematical proof.
Possible approach
To introduce the problem, you could use this PowerPoint presentation. It shows squares being halved in different ways and would provoke interesting discussion. (It loops so starts again when it reaches the end.) You may want to show the presentation (or at least some of it) and invite the
children to watch in silence. What is happening? Give them time to talk to a partner about what they've seen and then bring the whole group together. This will allow children to clarify their understanding of halving before going on to the main task.
It would be a good idea to give pairs of children a print-out of these sheets, the first of which contains the smaller squares, and the second the larger squares in the problem. Pupils will have many ideas of their own when it comes to checking or proving that the squares in
the problem really are split in half. Some will be able to explain orally, in terms of looking for red and white parts that are exactly the same shape and size. Others will prefer to draw them on squared paper and count squares. Some may want to cut out the squares and physically lay the halves on top of each other. Alternatively, it might be possible in some instances to cut out the square and
fold it to show that it has been split in half. Learners can use the uncoloured squares on the sheet to investigate different ways of shading halves for themselves.
In the plenary, as well as giving the class a chance to share their solutions and ways of working, you might like to distinguish between the squares that have been divided into two pieces (one red, one white) from those where there are more than two pieces. In all cases, we are comparing the total amount of red with the total amount of white.
Key questions
Have you thought about drawing the squares on squared paper and then using the small squares to help you check that they are split in half?
How about cutting out your square? How would this help to check that you had divided it in half?
Possible extension
Some children might enjoy investigating how to split other shapes in half. They could try Happy Halving which involves doing just that.Possible support
This sheet has the larger squares drawn on a grid of squares which may be useful for many children.