7392
/www/nrich/html/content/id/7392/
1
2011-02-01T00:00:01
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<p>Watch the video below in which Bryony demonstrates how to make a flower from a square of paper. (If you can't see the control bar, zoom out in your browser.)<br></br>
She then sets you a challenge: what fraction of the original square of paper is the shaded triangle?</p>
<br></br>
<video controls="controls" height="315" id="BryonyTriangle.mp4" src="BryonyTriangle.mp4" type="video/mp4" width="420"></video>
<br></br>
<br></br>
We would love to hear how you approach this problem.<br></br>
You may want to send us in pictures or photos along with your explanations.<br></br>
<p><span style="font-style: italic;">Thank you to Bryony Black for giving us permission to use this problem</span>.</p>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<p><span class="editorial">Izak, who is home schooled, told us how
he went about tackling this problem:</span></p>
<br></br>
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.<br></br>
Then, I divided the unfolded square into $16$ equal squares.<br></br>
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.<br></br>
<br></br>
$32 \times 16 = 512$<br></br>
<br></br>
So, there are $512$ triangles the size of the shaded one in the
original square of paper, therefore the shaded triangle is
$\frac{1}{512}$.<br></br>
<br></br>
<p><span class="editorial">AI and JB from Gledhow Primary used a
similar method:</span></p>
The solution to the question is $\frac{1}{512}$.<br></br>
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.<br></br>
We unfolded the flower so we had the original square with the
marked triangle in the bottom corner.<br></br>
We marked out $\frac{1}{4}$ of the square, we then quartered this
which gave us $\frac{1}{16}$.<br></br>
We quartered again and got $\frac{1}{64}$.<br></br>
Yet again we quartered giving us $\frac{1}{256}$.<br></br>
The marked triangle was half of this so this time we halved and got
$\frac{1}{512}$.<br></br>
<br></br>
<p><span class="editorial">Mark, also from Gledhow Primary, used
the same approach and he sent us a picture to help us see what he
did</span>:</p>
<br></br>
<br></br>
<mdo:image height="345" width="400" src="Mark.png" alt=""></mdo:image><br></br>
<br></br>
<p><span class="editorial">Joe, Sam, Ollie and Matthew from Keer
Mackie Primary used another slightly different method:</span></p>
<br></br>
First we worked out there were $16$ squares in the paper.<br></br>
Next we worked out that in the squares were $4$ triangles so we
found the product of $4$ and $16$ which was $64$.<br></br>
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 $512$.<br></br>
<br></br>
<p class="editorial">Thank you for your clear solutions. It's
not easy explaining how you worked on a problem when it involves
folding paper!</p>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<h3><span style="font-weight: bold;">Why do this
problem?</span></h3>
<div>The practical aspect of <a href="http://nrich.maths.org/7392&part=">this task</a> 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.</div>
<br></br>
<h3>Possible approach</h3>
<div>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. </div>
<div> </div>
<div>Alternatively, you could demonstrate the folding yourself if
you are not able to use the video in the classroom.</div>
<div> </div>
<div>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.</div>
<div> </div>
<div>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. </div>
<br></br>
<h3>Key questions</h3>
<div>How might you use the folds of your flower to help?</div>
<div>What fractions of the piece of paper can you see using the
folds? </div>
<div>How could you split the paper in half using the fold
lines? A quarter ...? </div>
<br></br>
<h3>Possible extension</h3>
<div>Some children might like to have a go at <a href="http://nrich.maths.org/5061&part=">Fraction
Fascination</a>.</div>
<br></br>
<h3>Possible support</h3>
<div>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.</div>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0">How might you use the folds of your flower to help? <br></br>
What fractions of the piece of paper can you see using the
folds?<br></br>
How could you split the paper in half using the fold lines? A
quarter ...? <br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0">$\frac{1}{512}$<br></br></mdoxml>
5
3
0
1
0
0
0
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?
Using, Applying and Reasoning about Mathematics
Practical Activity
Fractions, Decimals, Percentages, Ratio and Proportion
Fractions
Information and Communications Technology
Video