Bryony's Triangle
Watch the video to see how to fold a square of paper to create a flower. What fraction of the piece of paper is the small triangle?
Age
7 to 11
Challenge level
Watch the video BryonyTriangle.mp4 in which Bryony demonstrates how to make a flower from a square of paper.
She then sets you a challenge: what fraction of the original square of paper is the shaded triangle?
We would love to hear how you approach this problem.
You may want to send us in pictures or photos along with your explanations.
Thank you to Bryony Black for giving us permission to use this problem.
How might you use the folds of your flower to help?
What fractions of the piece of paper can you see using the folds?
How could you split the paper in half using the fold lines? A quarter ...?
What fractions of the piece of paper can you see using the folds?
How could you split the paper in half using the fold lines? A quarter ...?
Izak, who is home schooled, told us how he went about tackling this problem:
After folding the flower and shading the triangle, I got another square of paper the same size as the first one, but left it unfolded. I traced the shaded triangle onto the corner of the unfolded square of paper.
Then, I divided the unfolded square into $16$ equal squares.
Then, I used a ruler to divide one of the $16$ squares into triangles the same size as the shaded one on the flower. I counted the number of triangles in the $16$th of the square. There were $32$ total triangles in that $16$th.
$32 \times 16 = 512$
So, there are $512$ triangles the size of the shaded one in the original square of paper, therefore the shaded triangle is $\frac{1}{512}$.
Well done also to Josh from Goonhavern whose solution was similar to Izak's. Josh sent us this photo:
Image
Nikki C and Dominic B from Woore Primary School, and AI and JB from Gledhow Primary, used a similar method. Here is what AI and JB had to say:
The solution to the question is $\frac{1}{512}$.The explanation is, after we created the flower we marked the triangle as we needed to find out what fraction this triangle was out of the original square.
We unfolded the flower so we had the original square with the marked triangle in the bottom corner.
We marked out $\frac{1}{4}$ of the square, we then quartered this which gave us $\frac{1}{16}$.
We quartered again and got $\frac{1}{64}$.
Yet again we quartered giving us $\frac{1}{256}$.
The marked triangle was half of this so this time we halved and got $\frac{1}{512}$.
Mark, also from Gledhow Primary, used the same approach and he sent us a picture to help us see what he did:
Image
Joe, Sam, Ollie and Matthew from Keer Mackie Primary used another slightly different method:
First we worked out there were $16$ squares in the paper.
Next we worked out that in the squares were $4$ triangles so we found the product of $4$ and $16$ which was $64$.
Our next step was to find out how many of the smallest triangles fitted in the biggest triangle. The answer was $8$ so we timesed $8$ by $64$ and our final answer was $512$.
Ellie from Wellesley Park had a good way of using trial and improvement to solve this challenge:
I made the flower then opened it up and counted all the triangles. I counted 16. I worked out what each triangle represented. I cut out a triangle and tried to fit it into the small triangle. I kept trying until the triangle fitted perfectly onto the sheet of paper. Finally, the triangle fitted. The fraction is $\frac{1}{512}$.
Thank you for your clear solutions. It's not easy explaining how you worked on a problem when it involves folding paper!
Why do this problem?
The practical aspect of this task will appeal to many children. It offers an opportunity for learners to share their understanding of fractions with each other as they work to solve the problem.
Possible approach
You could begin by watching the video straight through all together and then talking about the task as a whole group so that everyone feels confident to have a go. It might be worth playing it again all through before handing out squares of paper. You could then play the video a third time in short sections as learners fold their own paper.
Alternatively, you could demonstrate the folding yourself if you are not able to use the video in the classroom.
Once all learners have made the flower, give them a chance to talk in pairs about the fraction part of the challenge. It might be that at this point, more squares of paper are needed which could be folded again and annotated. After some time, you could ask pairs to join together to form groups of four so that each pair has chance to explain their thinking so far.
You might give each group or pair a piece of large paper for them to record their ideas, which could then be presented to the rest of the class.
Key questions
How might you use the folds of your flower to help?
What fractions of the piece of paper can you see using the folds?
How could you split the paper in half using the fold lines? A quarter ...?
Possible extension
Some children might like to have a go at Fraction Fascination.
Possible support
You could encourage children to open out the flower and draw lines on the square over the fold lines to mark halves, quarters etc. You may want to have scissors and coloured pencils/pens available too. It might be appropriate to offer a different paper-folding task which is more accessible. You could challenge children to find a fraction contained within that, which perhaps
requires fewer 'steps'.