2293
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2011-02-01T00:00:01
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It's easy to work out the areas of squares drawn on a grid if
they are oriented in the usual way:<br></br>
<br></br>
<div style="text-align: center;"> <mdo:image height="106" width="376" src="tilted1.jpg" alt="squares with area 1,2,3 and 4"></mdo:image></div>
<br></br>
<br></br>
<span style="font-weight: bold;">I wonder if you can find a quick
and easy method to work out the areas of tilted
squares.</span><br></br>
<br></br>
Here are some squares with a tilt of 1:<br></br>
<div style="text-align: center;"><mdo:image height="131" width="395" src="tilted2.jpg" alt="squares with tilt 1"></mdo:image></div>
<div style="text-align: center;"><span style="font-style: italic;">See the</span> <a style="font-style: italic;" href="http://nrich.maths.org/2293&part=clue">hint</a> <span style="font-style: italic;">for suggested ways to calculate their
areas.</span></div>
<br></br>
Notice anything special about their areas?<br></br>
Can you predict the areas of other squares with a tilt of 1?<br></br>
<br></br>
What about squares with a tilt of 2? Or 3? Or 4? Or...?<br></br>
Notice anything interesting?<br></br>
<br></br>
<span style="font-weight: bold;">Can you make any conjectures about
the areas of tilted squares?</span><br></br>
<span style="font-weight: bold;">Can you prove your
conjectures?</span><br></br>
<br></br>
<span style="font-style: italic;">You might like to use the
interactivity below to help you to draw tilted
squares.</span><br></br>
<br></br>
<div><a href="/content/id/2293/squareAnimation3.swf">Full Screen
version</a> [Click inside the diagram first.]</div>
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<span class="editorial">We received three excellent solutions from
Jonathan, Bryn and Marissa of Madras College. Marissa's solution
follows:</span><br></br>
<br></br>
<mdo:image width="600" height="2249" alt="" src="marissa.jpg"></mdo:image><br></br>
<br></br>
<span class="editorial">The results above</span> <span class="editorial" style="font-weight: bold;">suggested</span> <span class="editorial">a relationship between the area of tilted squares
and the distance travelled along and up to get from one vertex to
an adjacent vertex.</span><br></br>
<span class="editorial"> </span> <br></br>
<span class="editorial">Alex from Llandovery College was able to
prove the relationship and sent us</span> <a class="editorial" href="/content/id/2293/Alex.doc">this proof</a> <span class="editorial">that the formula will always work.</span><br></br></mdoxml>
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<h3>Why do this problem?</h3>
<div>This problem offers an opportunity to spot patterns, make generalisations and eventually discover Pythagoras's Theorem, while giving students the chance to practise working out areas of squares and right-angled triangles.</div>
<br></br>
<h3>Possible approach</h3>
<div><span style="font-style: italic;">The video below shows two members of the NRICH team introducing the problem to a group of Year 9 students who came to visit the maths department at Cambridge. Video footage of the later stages of the lesson appear further down the page.</span></div>
<div> </div>
<br></br>
<div><mdo:flash data="http://downloads.sms.cam.ac.uk/mediaplayer.swf?file=http://downloads.sms.cam.ac.uk/1093714/1093728.flv" height="360" type="application/x-shockwave-flash" width="480"><param name="allowfullscreen" value="true"></param><param name="movie" value="http://downloads.sms.cam.ac.uk/mediaplayer.swf?file=http://downloads.sms.cam.ac.uk/1093714/1093728.flv&allowfullscreen=true&usefullscreen=true&allowscriptaccess=sameDomain"></param>
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<div><span style="font-style: italic;">Here is a description based on the approach used by a teacher with a Year 8 class, interspersed with some more video footage from the lesson in Cambridge:</span></div>
<div>I established that everyone could work out the area of squares when they were in the usual orientation:</div>
<div> </div>
<div>"4 by 4?"</div>
<div>"16"</div>
<div> </div>
<div>"5 by 5?"</div>
<div>"25"</div>
<div>etc.</div>
<div> </div>
<div>"But what if we had to work out the area of a tilted square?"</div>
<div> </div>
<div>I drew a point on the board, then moved 3 units to the right and 1 unit up and drew another point there.</div>
<div> </div>
<div>I used this as the base of my square and then drew the other three sides.</div>
<div><span style="font-style: italic;">This</span> <a href="http://nrich.maths.org/content/id/2667/squareCoords.swf" style="font-style: italic;">interactivity</a> <span style="font-style: italic;">could be used to familiarise students with drawing tilted squares.</span></div>
<div> </div>
<div>"How might we work out the area of this square?"</div>
<div>One student suggested that the base was 3 cm long and that the area would be 9 square cm.</div>
<div> </div>
<div>One student suggested we measure the length with a ruler and then square the result.</div>
<div>We discussed the problems with the two approaches and then I drew a 4 by 4 square around the tilted square and suggested that if we box the square in, work out the area of the box (16 square cm) and subtract the area of the four unwanted triangles (6 square cm) we would have the area of our tilted square (10 square cm). <span style="font-style: italic;">See the</span> <a href="http://nrich.maths.org/2203&part=clue" style="font-style: italic;">hint</a> <span style="font-style: italic;">for a diagram and an alternative method.</span></div>
<div> </div>
<div>"It would be nice to be able to work out the areas of these tilted squares as fast as you worked out the areas of the original squares"</div>
<div> </div>
<div>Students in one row were asked to draw a square with a base that went 4 along and 1 up.</div>
<div>Students in the next row were asked to draw a square with a base that went 5 along and 1 up.</div>
<div>Students in the next row were asked to draw a square with a base that went 6 along and 1 up. Students in the next row were asked to draw a square with a base that went 7 along and 1 up.</div>
<div> </div>
<div><span style="font-style: italic;">Here is a second video clip, showing the results being collected (approximately 5 minutes later) in the lesson in Cambridge:</span></div>
<div> </div>
<br></br>
<div><mdo:flash data="http://downloads.sms.cam.ac.uk/mediaplayer.swf?file=http://downloads.sms.cam.ac.uk/1095000/1095009.flv" height="360" type="application/x-shockwave-flash" width="480"><param name="allowfullscreen" value="true"></param><param name="movie" value="http://downloads.sms.cam.ac.uk/mediaplayer.swf?file=http://downloads.sms.cam.ac.uk/1095000/1095009.flv&allowfullscreen=true&usefullscreen=true&allowscriptaccess=sameDomain"></param>
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<div> </div>
<div>Students were all asked to work out the area of their squares and I then collected their results:</div>
<div>4 along and 1 up: 17</div>
<div>5 along and 1 up: 26</div>
<div>6 along and 1 up: 37</div>
<div>7 along and 1 up: 50</div>
<div> </div>
<div>"Do you notice anything about the areas?"</div>
<div>"All 1 more than a square number"</div>
<div> </div>
<div>"If you drew a square with a base 8 along and 1 up, what would you expect the area to be?"</div>
<div>"65"</div>
<div> </div>
<div>"If you drew a square with a base x along and 1 up, what would you expect the area to be?"</div>
<div>"x squared + 1"</div>
<div style="text-align: center;"> </div>
<div>"Great, it looks like we can now work out the areas of these tilted squares very easily."</div>
<div> </div>
<div>But what would happen if they were more tilted, say, 3 along and 2 up, or 4 along and 2 up, or...?"</div>
<div>Someone suggested that the rule would be x squared + 2</div>
<div> </div>
<div style="font-style: italic;">Here is the final video clip, showing the last part of the lesson in Cambridge:</div>
<div> </div>
<br></br>
<div><mdo:flash data="http://downloads.sms.cam.ac.uk/mediaplayer.swf?file=http://downloads.sms.cam.ac.uk/1095199/1095229.flv" height="360" type="application/x-shockwave-flash" width="480"><param name="allowfullscreen" value="true"></param><param name="movie" value="http://downloads.sms.cam.ac.uk/mediaplayer.swf?file=http://downloads.sms.cam.ac.uk/1095199/1095229.flv&allowfullscreen=true&usefullscreen=true&allowscriptaccess=sameDomain"></param>
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<div> </div>
<div>Again the class was split up to work out the areas of these tilted squares and we then collected their results:</div>
<div>3 along and 2 up: 13</div>
<div>4 along and 2 up: 20</div>
<div>5 along and 2 up: 29</div>
<div>6 along and 2 up: 40</div>
<div> </div>
<div>"Was our conjecture (x squared + 2) correct?"</div>
<div>"It's x squared + 4"</div>
<div>"It's x squared + 2 squared"</div>
<div>"The first one should have been x squared + 1 squared"</div>
<div> </div>
<div><span style="font-style: italic;">At this point, it may be appropriate to work on some ways of justifying the $x^2 + 1^2$ and $x^2 + 2^2$ conjectures.</span></div>
<div><span style="font-style: italic;">For groups who have met the idea of expanding $(x+1)^2$ this can be done algebraically.</span></div>
<div><span style="font-style: italic;">Alternatively, a pictorial approach could be offered, and students could be asked to explain why this proves their conjecture:</span></div>
<div> </div>
<div style="text-align: center;"><mdo:image alt="proof of Pythag" height="157" src="pythag.jpg" width="318"></mdo:image></div>
<div> </div>
<div>"What do you think will happen if the squares are even more tilted, say 3, or 4, or 5 up?"</div>
<div>"x squared + 3 squared"</div>
<div>"x squared + 4 squared"</div>
<div>"x squared + 5 squared"</div>
<div> </div>
<div>Students were split into groups again in the following lesson to check these conjectures and report back.</div>
<br></br>
<p><span style="font-style: italic;">Finally, algebraic or pictorial approaches used to justify earlier conjectures can be adapted to prove Pythagoras's Theorem.</span><br></br>
<br></br>
Follow-up lessons could focus on working out the lengths of the sides of right-angled triangles when two lengths have been given.</p>
<h3>Key questions</h3>
<div>How could you work out the area of a tilted square?</div>
<div>If you drew a square with a base x along and 1 up, what would you expect the area to be?</div>
<div>If you drew a square with a base x along and 2 up, what would you expect the area to be?</div>
<div>If you drew a square with a base x along and y up, what would you expect the area to be?</div>
<br></br>
<h3>Possible extension</h3>
<div>It is possible to draw squares with areas of 1, 2, 4, 5, 8, 9... but not 3, 6, 7, 11, 12...</div>
<div>Students could explore some of the properties of numbers which are and are not possible areas of tilted squares.</div>
<div>Can they prove that numbers of the form 4n+3 are not possible areas of tilted squares?</div>
<div> </div>
<p>Another possible follow-up task is <a href="http://nrich.maths.org/693&part=">Of All the Areas</a>.</p>
<h3>Possible support</h3>
<p>Start with <a href="http://nrich.maths.org/2526&part=">Square It</a> or <a href="http://nrich.maths.org/2667&part=">Square Coordinates</a> to help students become confident at drawing tilted squares.</p>
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<div style="clear: both;"><br></br>
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<div class="framework"><br></br>
NOTES AND BACKGROUND Ken Nisbet, Mathematics teacher at Madras College in Fife, Scotland, has added:
<p>"I used tilted squares as the basis of individual/group work with a top set (age 14). They were given time to explore this as an open ended question in groups in a brain storming session. Write ups were to be done individually, partly in class but completed at home. This is an important stage in the pupils' mathematical development where the idea of "proof" is coming to the fore. This
excellent investigation allows algebra to come to the fore as the language of generalisation and the means of "proof" of patterns. At this stage algebra skills are limited but we have now used this investigation as a springboard to developing necessary algebra skills - e.g. double brackets, squares, expressions etc."</p>
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<p style="font-weight: 400; text-decoration: none;">Two standard
methods for working out the areas of shapes may help.</p>
<p style="font-weight: 400; text-decoration: none;">Boxing in the
tilted squares, working out the area of the box and subtracting the
areas of the unwanted right-angled triangles: </p>
<p style="font-weight: 400; text-decoration: none; text-align: center;">
<mdo:image width="455" height="290" alt="boxing in method" src="boxingin3.gif"></mdo:image></p>
<p style="font-weight: 400; text-decoration: none;"> </p>
<p style="font-weight: 400; text-decoration: none;">Splitting the
tilted squares into right-angled triangles and squares and adding
the areas of the different parts:</p>
<br></br>
<p style="text-align: center;"><mdo:image width="312" height="282" align="top" src="splittingup.gif" bgcolor="" alt="Diagram of tilted square split up into 4 right angled triangles and a square, with their corresponding areas."></mdo:image></p>
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<mdoxml version="1.0">We received three excellent solutions from Jonathan, Bryn and
Marissa of Madras College. Marissa's solution follows:<br></br>
<br></br>
<mdo:image height="2249" width="600" src="marissa.jpg" alt=""></mdo:image><br></br>
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Tilted Squares
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
Using, Applying and Reasoning about Mathematics
Generalising
2D Geometry, Shape and Space
Squares
Measures and Mensuration
Area
2D Geometry, Shape and Space
Pythagoras' theorem
Information and Communications Technology
Interactivities