7381
/www/nrich/html/content/id/7381/
1
2011-02-01T00:00:01
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<p>Charlie has been exploring squares with vertices drawn on the points of a <a href="http://nrich.maths.org/content/id/6676/10mm%20Dots%20square.pdf">square dotty grid</a>.<br></br>
Unfortunately, he rubbed out some of his work and only left behind one side of each square.<br></br>
</p>
<div style="text-align: center;"> <mdo:image alt="missing squares" height="249" src="charlie.jpg" width="500"></mdo:image></div>
<br></br>
<p>Can you recreate the squares he drew?<br></br>
Is there more than one possibility?<br></br>
<br></br>
Could <span style="font-weight: bold;">any</span> line joining two points be the side of a square whose vertices lie on grid points?<br></br>
How can you be sure?<br></br>
<br></br>
<br></br>
Alison has been drawing squares and their diagonals. Here are some of the diagonals she drew:</p>
<div style="text-align: center;"> <mdo:image alt="diagonals of squares" height="187" src="alison.jpg" width="500"></mdo:image></div>
<p>Can you recreate the squares she drew from her diagonals?<br></br>
Is there more than one possibility?<br></br>
Can you find a method to draw a square when you are just given the diagonal?<br></br>
<br></br>
Could <span style="font-weight: bold;">any</span> line joining two points be the diagonal of a square whose vertices lie on grid points?<br></br>
<br></br>
Can you find a way to help Alison decide whether a given line could be the diagonal of such a square?<br></br>
<br></br>
<br></br>
<span style="font-weight: bold;">Charlie and Alison played around with rhombuses next.</span><br></br>
<br></br>
Charlie said "Whenever I join two points to make a line, I can use my line as a side of several different rhombuses".</p>
<div style="text-align: center;"> <mdo:image alt="rhombuses" height="151" src="charlie2.jpg" width="407"></mdo:image></div>
<div style="text-align: left;">Do you agree with him?</div>
<div>When you are given a line, is there a quick way to work out how many rhombuses can be drawn using that line as one of the sides?</div>
<br></br>
<p> <br></br>
Alison said "When I draw a rhombus, it shares its diagonal with infinitely many other rhombuses."</p>
<div style="text-align: center;"> <mdo:image alt="rhombuses with same diagonal" height="173" src="alison2.jpg" width="315"></mdo:image></div>
<p>Do you agree with her?<br></br>
<br></br>
Not all lines can be the diagonal of a rhombus. Is there a quick way to decide which lines could be the diagonal of a rhombus?<br></br>
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<a href="/content/id/7381/EmilySolution.doc"><span class="editorial">Here</span></a> <span class="editorial">is how
Emily from Beeston Rylands Junior School started on the
problem.</span><br></br>
<br></br>
<span class="editorial">Damini from Newstead Wood School For
Girls discovered that not all diagonal lines that join two grid
points make a square:</span><br></br>
<br></br>
You find the centre of your diagonal line and go from there away
from the line in a straight line (at right angles) until you find
the right point. Then join the point to one end of the diagonal
line and you have a side! How easy was that. Not all diagonal lines
that join two grid points make a square. If you take a piece of
square dotty paper and join up any two dots it will be hard to find
the square, as there might not even be a square.<br></br>
<br></br>
I agree with the fact that a straight line can be the side of many
different rhombuses but to me I see no real way of determining how
many rhombuses go to one line.<br></br>
<br></br>
I also agree that the diagonal of one rhombus can have the same
diagonal line as infinitely many other rhombuses as if you take one
line you can keep editing the size of the rhombus without actually
changing the length of the first diagonal. <br></br>
<br></br>
<span class="editorial">Alexander from Wilson's School suggested a
strategy for finding the number of possible rhombuses that could be
made when he is given a side:</span><br></br>
<br></br>
To find how many different rhombuses can be found with a single
line that is between two dots, you need to choose one end
of the line. You then look at "one side" of that point, and
see how many different ways the line can be "twisted". You then
multiply it by four because the same number of "twisted" lines can
be found on the "other side" of that point, and on the "two sides"
of the other end of the line. The answer to this operation
determines how many rhombuses can be made from one line on a dotted
piece of paper. <br></br>
<span class="editorial">(Editor's comment: I think that Alexander
has double counted here; each of the "twisted lines" found at one
end will make rhombuses with each of the "twisted lines" found at
the other end of the line.)</span><br></br>
<br></br>
<span class="editorial">Niharika from Leicester High School for
Girls explained that if the vector $\pmatrix{a\cr b}$
represents the diagonal line, the square will have vertices at grid
points if a+b (and therefore a-b) is even.</span><br></br>
<br></br>
<span class="editorial">She also suggested a strategy for finding
the number of rhombuses that could be drawn if one of the sides is
given. Her solution is</span> <a href="/content/id/7381/NiharikaSolution.pdf" class="editorial">here</a><span class="editorial">.</span><br></br>
<br></br>
<span class="editorial">Niharika also contributed a solution
to </span> <a href="http://nrich.maths.org/7453&part=solution" class="editorial">Vector Journeys</a><span class="editorial">,
which uses vector notation so solve a related problem.</span><br></br>
<br></br>
<span class="editorial">We also received good responses from the
Maths Galaxy Explorers from North Walsham Junior
School.</span><br></br></mdoxml>
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<h3>Why do this problem?</h3>
<div>Many students think they are very familiar with squares and
rhombuses...</div>
<div>This problem requires a real appreciation of the properties of
these quadrilaterals in order to make and justify some interesting
generalisations.</div>
<div> </div>
<div><a href="http://nrich.maths.org/7472&part=">Here is an
article</a> that describes some of the background thinking that
informed the creation of this problem.</div>
<h3>Possible approach</h3>
<div>This problem requires students to draw tilted squares
reliably. <a href="http://nrich.maths.org/content/id/2293/squareAnimation3.swf">This
interactivity</a> might be helpful to demonstrate to students what
a tilted square looks like. Students could play <a href="http://nrich.maths.org/2526&part=">Square It</a> until
they can reliably spot tilted squares on a dotty grid.</div>
<div> </div>
<div><span style="font-weight: bold;">A possible start</span> which
involves the minimum of teacher input is to display the line below
and say:</div>
<div style="text-align: center;"> <mdo:image height="331" width="329" src="opposite.jpg" alt="line joining two points"></mdo:image></div>
<div style="text-align: center;"> </div>
<div>"Imagine we are drawing squares and rhombuses with vertices on
the dots of a square grid. This line can be</div>
<ul>
<li>the side of at least one square,</li>
<li>the diagonal of at least one square,</li>
<li>the side of at least one rhombus,</li>
<li>the diagonal of at least one rhombus.<br></br>
</li>
</ul>
<div>Work out how many different squares and rhombuses can include
this line, as either a side or a diagonal." </div>
<div> </div>
<div>This leads on to the challenge "In a while, I am going to ask
you to work out how many squares and rhombuses could be drawn using
a different line. The challenge will be to answer without doing any
drawing."</div>
<br></br>
<br></br>
<span style="font-weight: bold;">Alternatively,</span> <span style="font-weight: bold;">start</span> by showing the image of
Charlie's rubbed out squares, and give students some time to
recreate the squares on <a href="http://nrich.maths.org/content/id/6676/10mm%20Dots%20square.pdf">
dotty paper</a>. Once they have finished, ask them to compare in
pairs - have they always drawn the same square? Are there any other
possibilities?<br></br>
<br></br>
Bring the class together and share the techniques students were
using to complete the squares.<br></br>
"Draw any line you like on your dotty paper, and then try to
complete a square using your line as one of the sides."<br></br>
...<br></br>
"Has anyone found a line that they can't use as a side of a
square?" (If someone has found one, display it on the board and ask
the class to help.)<br></br>
"Do you think we can <span style="font-weight: bold;">always</span>
draw a square using <span style="font-weight: bold;">any</span>
line?"<br></br>
Give students some time to discuss this and come up with
justifications for their answer.<br></br>
<br></br>
<span style="font-weight: bold;">Next</span> show the image of
Alison's diagonals and give students time to recreate the
squares.<br></br>
Again, ask them to share approaches.<br></br>
"Draw any line you like on your dotty paper, and then try to
complete a square using your line as the diagonal."<br></br>
This time, it can't always be done. If students are struggling to
work out when it can and can't be done, suggest that they draw some
squares and see what the diagonals have in common.<br></br>
<br></br>
<span style="font-weight: bold;">Then move on to
rhombuses.</span><br></br>
"Draw any line you like on your dotty paper, and then try to
complete a rhombus using your line as one of the sides. How many
different rhombuses can you draw using your line?"<br></br>
Give students time to do this for a number of different lines of
their own choosing, and then bring the class together to share
their findings.<br></br>
<br></br>
"When we worked on squares, there were some lines that we couldn't
use as a diagonal. Will the same be true for rhombuses?"<br></br>
Once again, give students time to explore.<br></br>
<br></br>
Finally, bring the class together to share their ideas and justify
their findings.<br></br>
<br></br>
One technique for testing ideas at the end is to set a specific
challenge, for example, to draw some lines and ask them to
determine how many rhombuses could be drawn using each line as a
diagonal.<br></br>
<h3>Key questions</h3>
<div>Is there a method for completing a square given one of the
sides?</div>
<div>Is there a method for completing a square given its
diagonal?</div>
<div>Is there a quick way to determine whether a given line could
be the side or diagonal of a square with points on the dots of a
square grid?</div>
<div> </div>
<div>Is there a method for completing a rhombus given one of the
sides?</div>
<div>Is there a method for completing a rhombus given its
diagonal?</div>
<div>Is there a quick way to determine whether a given line could
be the side or diagonal of a rhombus with points on the dots of a
square grid?</div>
<br></br>
<h3>Possible extension</h3>
<a href="http://nrich.maths.org/6572&part="></a><a href="http://nrich.maths.org/7453&part=">Vector Journeys</a>
challenges students to explore similar relationships using vector
algebra.<br></br>
<h3>Possible support </h3>
<div>The interactivity in <a href="http://nrich.maths.org/2667&part=">Square Coordinates</a>
helps students to visualise tilted squares.</div>
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<mdoxml version="1.0">The interactivity in <a href="http://nrich.maths.org/2667&part=">Square Coordinates</a>
may help you to visualise tilted squares. <br></br>
You can challenge yourself by playing <a href="http://nrich.maths.org/2526&part=">Square It</a> against
the computer.<br></br></mdoxml>
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Opposite vertices
Can you recreate squares and rhombuses if you are only given a side
or a diagonal?
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