2132
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2011-02-01T00:00:01
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<td>Seven squares are set inside each other. The centre points of each side of the outer square are joined to make a smaller square inside it and so on.</td>
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<div class="c1" style="text-align: center;"><mdo:image alt="seven squares set inside each other" height="226" src="7squares.gif" width="234"></mdo:image></div>
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<td>The centre square has the area of $1$ (one) square unit.</td>
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<div>What is the total area of the four outside triangles which are outlined in red?</div>
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This problem is also available in French: <a href="http://nrich.maths.org/7125">Les 7 carrés</a><br></br>
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<div><a href="http://nrich.maths.org/public/viewer.php?obj_id=6811&part=">Click here for a poster of this problem</a>.</div>
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<span class="editorial">You have found several different ways of
answering this problem which is great. Many of you, including Fiona
at Tattingstone School and Douglas at Burgoyne Middle School,
decided to draw the squares. However, your approaches weren't
necessarily the same</span>.<br></br>
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<p><span class="editorial">This is what Fiona says:</span></p>
The first thing I did was to get some squared paper to draw the
figure on. I found out I couldn't start the middle square as one
square because I couldn't draw the other squares accurately so I
did the middle square as a four and that gave me points for the
rest of the figure.<br></br>
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<td><mdo:image height="345" width="366" src="Fiona1.gif" alt="Fiona's diagram"></mdo:image></td>
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<div>To find the total area of the numbered triangles:</div>
<div>I noticed that $1$ and $4$ were two halves of a square and
also $2$ and $3$ are two halves of a square.</div>
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<td><mdo:image height="337" width="317" alt="Fiona's second diagram" src="Fiona2.gif"></mdo:image></td>
<td>Now I just have to count every square in this top rectangle as
one unit: My answer is $32$ square units.</td>
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<p class="editorial">Courtney, Charlotte and Tyler from Gateway
Primary also used this approach.</p>
<p class="editorial">Here is what Douglas did:</p>
<div>1) I drew the squares starting with the inside square that
measured $1$ cm by $1$ cm.</div>
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<div>2) I measured the sides and calculated the area of each of the
outside triangles: Height $4$ cm width $4$ cm Area of $1$ triangle
= $8$ cm</div>
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<div class="editorial">Sally also sent us her solution. She spotted
some interesting patterns: good work!</div>
<div class="editorial"> </div>
<div>When I drew the squares, I knew the first one had side length
$1$, then the third one had side length $2$, the fifth had side
length $4$, and the seventh had side length $8$ (they kept
doubling). I drew the picture on squared paper, and when I counted
the squares, each square (all of them, not just the odd ones) had
area twice as big as the one before. So the first one had area $1$,
then the next one had area $2$, then $4$, then $8$, then $16$, then
$32$, then $64$. The area of the triangles is the area of the
biggest one take away the area of the next one down, so that's
$64-32=32$.</div>
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<p class="editorial">Gemma and Nathalie from City of London
Freemen's School, Roger, Mark and Sam from Spalding Grammar
School and George from Strand on the Green Juniors also
noticed this pattern and so they didn't need to do any drawing to
find the solution. Nathalie wrote:</p>
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<div>As each square is half of the square surrounding it the total
area of the outside square is $64$. You then have to take away the
square directly inside it to find the area of just the outside
triangles. As the square directly inside it is simply half the
outside square then we are left with $32$ as the area for the
outside triangle. </div>
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<p class="editorial">Well done too to Callum from Arthur Mellows
Village College and children from Truscott Street Public
School.</p>
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<h3><span style="font-weight: bold;">Why do this
problem?</span></h3>
<a href="http://nrich.maths.org/public/viewer.php?obj_id=2132&part=index">
This problem</a> provides a challenge when you are focusing on
drawing lines and shapes accurately or finding areas either by
counting squares or calculation. The problem calls on learners'
understanding of squares and right-angled triangles, and requires
working systematically and visualisation as well as reasonably
accurate drawing.<br></br>
<h3>Possible approach</h3>
<div>You could introduce this problem by making the beginnings of
the design using paper folding. Each learner will need a square of
paper. (This could be made in class by folding the end of the sheet
diagonally, thus forming a square and cutting off the
excess.)</div>
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<div>Take the square of paper and fold it both ways diagonally so
finding the centre. Fold each corner to the centre and crease it.
Thus forming the second square (in red).</div>
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<div style="text-align: center;"><mdo:image height="116" width="116" alt="" src="2132A.png"></mdo:image></div>
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<div>After this, fold each side to the centre and crease, forming
the third square (in green).</div>
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<div style="text-align: center;"><mdo:image height="115" width="117" alt="" src="2132B.png"></mdo:image></div>
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<div>Learners could be challenged to continue this process which
becomes increasingly difficult and prone to inaccuracy. This
introduction will show learners where the problem "is coming
from".</div>
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<div>Provide $1$ cm$^2$ paper, or even better, square dotty paper,
sharp pencils and rulers for working on the actual problem.
Learners would benefit from working in pairs so that they are able
to talk through their ideas with a partner but should do their own
drawing. <a href="/content/02/12/penta1/2132.pdf">This sheet</a>,
showing the diagram, could prove useful.</div>
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<div>Learners could be left on their own to discover the best place
to start the drawing and the best size to choose, or the whole
group could discuss this together before they begin working
individually.</div>
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<div>At the end of the lesson learners can discuss both their
methods for tackling the drawing and for finding the area of the
four triangles. There are several different ways of answering the
problem and it can be done without any drawing at all, so there
should be plenty to talk about. It might be useful, if no one has
suggested it, to cut out the four triangles and put them together
as a square.</div>
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<h3>Key questions</h3>
<div>What would be a good way to start, with the first, smallest
square or the largest one?</div>
<div>How big should you draw the centre square to make it easier to
draw the others?</div>
<div>Is that a $45^o$ angle?</div>
<div>Now you have done the drawing, how are you going to find the
area of the four triangles?</div>
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<h3>Possible extension</h3>
Learners who find this problem straightforward could find the area
of each of the successive squares in the diagram and predict the
size of the next squares, or try the problem <a href="http://nrich.maths.org/public/viewer.php?obj_id=1854&part=index">
Fitted</a>, and/or <a href="http://nrich.maths.org/public/viewer.php?obj_id=6522&part=index">
Baravelle</a>.<a href="http://nrich.maths.org/public/viewer.php?obj_id=1854&part=index">
</a><br></br>
<h3>Possible support</h3>
Children could start with a $4 \times 4$ square and then draw a
second square at $45^o$ to this with each side centred
on a corner of the first square. Each side of this
second square will be placed diagonally across two squares. A third
and successive squares can be drawn in the same way.<br></br>
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<p>You could draw it out on squared or dotty paper starting with
the centre square.<br></br>
Think about the scale - what size could you draw the centre square
to make it easier to draw the others?<br></br>
How does the second square compare in size to the centre
square?</p>
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Inside Seven Squares
What is the total area of the four outside triangles which are
outlined in red in this arrangement of squares inside each other?
2D Geometry, Shape and Space
Squares
2D Geometry, Shape and Space
Right angled triangles
Measures and Mensuration
Area
Using, Applying and Reasoning about Mathematics
Visualising