Fractional Triangles
Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.
Age
7 to 11
Challenge level
Use the lines on this figure to show how the pattern of triangles can be used to divide the square into two halves, three thirds, six sixths and nine ninths.
Image
More lines are needed to divide it into four quarters.
What is the least amount of line needed to do this if the quarters are in one piece and all the same shape?
How many ways can you divide it into halves using just the lines given?
Click on a dot to choose a colour, then click on a triangle.
What fraction of the design are the small squares?
How many sixths are there in a third?
You may like to print off this sheet for you to work on.
Chris and Jenny sent us some colourful diagrams to show their answers:
Image
Image
Image
Image
Julia drew on the line she'd draw to get quarters. Can you see that there are now four identical triangles? (Concentrate on the thickest lines.)
Image
However, Anna wrote to say:
The least amount of line needed is one line dividing the central two triangles of the square in half.
I knew that the diagram had 18 triangles which is a number not divisible by 4. So I divided two triangles in half to have a total of 20 triangles which is a number divisible by 4. At first I drew the smaller triangles at the side of the square but this did not work. Then I divided all of the triangles in half using 9 lines giving me 36 triangles. I successfully divided the diagram into equally sized quarters of the same shape, but this did not give me the smallest amount of line. Once more I split two triangles in half but this time the halved triangle was in the middle.
Image
Image
The Maths Challenge Group at St Aidan's VC School also sent us some different ways of dividing the shape into two halves. Here are the images they sent:
Image
They told us:
We all compared our answers and came up with these as our final results. We looked very carefully at 5 and 7, 3 and 6 in order to be sure that they were different answers not merely a rotation. We are happy with our results after our discussion.
Thank you for sharing that with us and well done everyone.
Fractional Triangles
Use the lines on this figure to show how the pattern of triangles can be used to divide the square into two halves, three thirds, six sixths and nine ninths.
Image
More lines are needed to divide it into four quarters.
What is the least amount of line needed to do this if the quarters are in one piece and all the same shape?
How many ways can you divide it into halves using just the lines given?
Why do this problem?
This problem could be used as part of a lesson on finding fractions of various shapes. It should help develop an understanding of the relationship between the part and the whole. It allows children to explore fractions in a non-threatening, open-ended way and yet it does contain some real challenge.
Possible approach
The problem could be introduced by showing the design to all the group. You could either draw it on the board or display this image on an interactive whiteboard. Alternatively, you could print out this
sheet (enlarged to A3 if required). Whichever way you decide, it would be good if the image could be annotated by children during the lesson.
Ask the children what they can see and invite them to talk about it - think, pair, share. You could steer the conversation towards numbers and fractions if the children do not naturally bring it up. Asking general questions about the numbers and fractions of different shapes in the design will give children the confidence to tackle the problem.
This sheet (which contains six copies of the image) could be used both to work on and record answers to the problem. Coloured pencils would be useful to emphasise the different shapes within the design.
The final question in the problem: "How many ways can you divide it into halves using just the lines given?" could provide a useful starting point for a plenary as even those who have had difficulty with some of the questions could join in usefully.
Key questions
What fraction of the design are the small squares?
How many sixths are there in a third?