Fitted

1854 /www/nrich/html/content/03/09/six3/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><p><em><strong>If you are a teacher click <a href="/1854&part=note">here</a> for a version of the problem suitable for classroom use, together with supporting materials. Otherwise, read on ...</strong></em></p> <p><mdo:image align="right" alt="Picture of some squares" height="230" src="Fitt.gif" width="230"></mdo:image></p> <p>Nine squares with side lengths $1, 4, 7, 8, 9, 10, 14, 15$ and $18$ cm can be fitted together with no gaps and no overlaps, to form a rectangle.</p> <p>What are the dimensions of the rectangle?</p> <p><br></br> </p> <p>Once you've had a chance to think about it, click below to see how three different pupils began working on the task.</p> <p> </p> <p>This is how Anna started:</p> <div class="toggle"><mdo:image src="Fitted1.jpg"></mdo:image></div> <p> </p> <p>Here is what Brendan tried:</p> <div class="toggle"><mdo:image src="Fitted3.jpg"></mdo:image></div> <p> </p> <p>Here is Chandra's initial approach to the problem:</p> <div class="toggle"><mdo:image src="Fitted2.jpg"></mdo:image></div> <p style="margin: 0.1pt 0cm;"> </p> <p>Can you take each of these starting ideas and develop them into a solution?</p> <p> </p></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p> </p> <p><span class="editorial">We had $60$ solutions submitted and a bit of response in the <a href="http://www.nrich.maths.org/z/infinities" class="blogbutton">blog </a> Here are a selection.</span></p> <p><span class="editorial">This solution was created by students at Primary and Secondary</span> <span class="editorial">Master classes at the British School Al Khubairat in Abu Dhabi:</span></p> <p>Solutions submitted by:<br></br> Olaf and James (Yr $6$), (Primary) $20$ mins including cutting time.<br></br> Thomas (Yr $7$), Joanna (Yr $9$) & Madeleine (Yr $9$), (Secondary) . $15$ mins including cutting time.</p> <p style="text-align: center;"><mdo:image src="fitted%20pic.jpg"></mdo:image></p> <p style="text-align: center;">Dimensions of the Rectangle: 33 cm x 32 cm</p> <p> </p> <p><span class="editorial">Joe who studied this at home sent in the following:</span><br></br> Put nine, eight and fifteen along the top. Place the $1$ under the bottom of the left corner on the eight. Underneath the $9$ put the ten, next to the ten (and under the eight) put $7$. This will line up the $7$ under the eight creating a place for $15$. So fifteen is forming the top right hand corner of the rectangle. Under the $7$ place $4$ so it is next to the bottom part of $10$. Under the ten put fourteen , and next to the fourteen put eighteen. There you have it you have made a succseful rectangle with $33$cm long and $32$cm wide!!!!</p> <p> </p> <p><span class="editorial">From Kent College Prep School we had contributions from Megan, Polly and Amy, Sophia and Lottie, Olivia and Isabel - and Isabella and Victoria's solution was: </span></p> <p>Our first solution was we drew squares $1x1, 4x4, 7x7, 8x8, 9x9, 10x10, 14x14, 15x15$ and $18x18$. Then we cut them out and started to find a solution.<br></br> It took us a couple of goes to find our solution; we found our solution because we started to put numbers together to make another number e.g. $8x8$ and $10x10$ made $18x18$.<br></br> When we had finished we counted the width and the length and it equalled $32x33$.</p> <p><span class="editorial">At the last moment we had a solution in from John at</span> <span class="editorial">37th of Heraklion School in<br></br> Greece:</span></p> <p style="text-align: center;"><mdo:image src="John%27s%20New%20Fitted.jpg"></mdo:image></p> <p><br></br> <span class="editorial">We also had a solution from the British Manilla School in the Philippines sent in by Rishi.</span></p> <p><span class="editorial">We had a substantial contribution from year $6$ at East Ward School in Bury England. This can be seen</span> <a href="/content/03/09/six3/Bury%20School.pdf">here </a> <span class="editorial">.</span></p> <p><span class="editorial">A big WELL DONE to all those who sent in many other excellent correct solutions.</span></p> <p> </p> <p><br></br> <br></br> </p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Fitted</h2> <p><em><strong>If you are a teacher click <a href="/1854&part=note">here</a> for a version of the problem suitable for classroom use, together with supporting materials. Otherwise, read on ...</strong></em></p> <p><mdo:image align="right" alt="Picture of some squares" height="230" src="Fitt.gif" width="230"></mdo:image></p> <p>Nine squares with side lengths $1, 4, 7, 8, 9, 10, 14, 15$ and $18$ cm can be fitted together with no gaps and no overlaps, to form a rectangle.</p> <p>What are the dimensions of the rectangle?</p> <p><br></br> </p> <p>Once you've had a chance to think about it, click below to see how three different pupils began working on the task.</p> <p> </p> <p>This is how Anna started:</p> <div class="toggle"><mdo:image src="Fitted1.jpg"></mdo:image></div> <p> </p> <p>Here is what Brendan tried:</p> <div class="toggle"><mdo:image src="Fitted3.jpg"></mdo:image></div> <p> </p> <p>Here is Chandra's initial approach to the problem:</p> <div class="toggle"><mdo:image src="Fitted2.jpg"></mdo:image></div> <p style="margin: 0.1pt 0cm;"> </p> <p>Can you take each of these starting ideas and develop them into a solution?</p> <p> </p> </div> <br></br> <h3><span style="font-weight: bold;">Why do this problem?</span></h3> <div><a href="http://nrich.maths.org/public/viewer.php?obj_id=1854&part=index">This problem</a> is a challenging context in which to explore area. It can be solved in many different ways and reflection on different methods is encouraged through inclusion of the three sample approaches. You may well need more than one lesson for this activity.</div> <br></br> <h3>Possible approach</h3> <div>Pose the problem orally, or project the text onto a screen, without mentioning the examples of how some children started. Give the class a few minutes to consider, individually, how they might go about tackling the problem, then pair them up and suggest that they talk to their partner about their ideas so far. Try to stand back and observe, and resist the temptation to make helpful suggestions! <p>Allow pairs to work on the task so that you feel they have made some progress, but do not worry if they have not completed it or if they report being stuck. The aim at this stage is for everyone to 'get into' the problem and work hard on trying to solve it, but not necessarily to achieve a final solution. Make sure that the children have easy access to any resources that they are likely to need but don't put anything out on tables already otherwise this may lead them down particular routes.</p> <p>At a suitable time, hand out <a href="/content/03/09/six3/Fitted.pdf">this sheet</a> to pairs. Suggest to the class that when they've finished or can't make any further progress, they should look at the sheet showing three approaches used by children working on this task. Pose the question, "What might each do next? Can you take each of their starting ideas and develop them into a solution?". You may like pairs to record their work on large sheets of paper, which might be more easily shared with the rest of the class in the plenary. </p> <p>Allow at least fifteen minutes for a final discussion. Invite some pairs to explain how the three different methods might be continued. You may find that some members of the class used completely different approaches when they worked on the task to begin with, so ask them to share their methods too. You can then facilitate a discussion about the advantages and disadvantages of each. Which way would they choose to use if they were presented with a similar task in the future? Why?</p> </div> <div> </div> <h3>Key questions</h3> <div>Tell me about this approach. What do you think he/she was doing?<br></br> How do you think this will help to solve the problem?<br></br> What do you think he/she would have done next?</div> <div> </div> <h3>Possible extension</h3> <div><a href="http://nrich.maths.org/public/viewer.php?obj_id=35&part=index">Tiles on a Patio</a> is an investigation based on similar ideas which would make a good follow-up activity.</div> <br></br> <h3>Possible support</h3> <div>It might help some pupils to have sets of squares cut out ready for use. However, only give them out if children explicitly decide to approach the task in this way at the start, or if they are continuing Brendan's solution, and you think that they would spend too long accurately drawing and cutting the squares themselves.</div> <p> </p></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <p>What do you know about the rectangle?</p> <p>Have a look at each approach. What do you think he/she was doing?</p> <p>How do you think this will help to solve the problem?</p> <p>What do you think he/she would have done next?</p> </mdoxml> 5 5 0 1 0 0 0 Fitted Nine squares with side lengths 1, 4, 7, 8, 9, 10, 14, 15, and 18 cm can be fitted together to form a rectangle. What are the dimensions of the rectangle? 2D Geometry, Shape and Space Squares Using, Applying and Reasoning about Mathematics Trial and improvement 2D Geometry, Shape and Space Rectangles Numbers and the Number System Factors and multiples Measures and Mensuration Area Admin Upper primary mapping document