2398 /www/nrich/html/content/id/2398/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p>The square below has been cut into two pieces:</p> <p><mdo:image width="214" height="214" align="top" alt="square cut in two" src="tangram.gif" bgcolor=""></mdo:image></p> <p>It is possible to fit the pieces together again to make a new shape.</p> <p class="standard">If you must match whole sides to each other so that the corners meet, how many new shapes can you make?</p> <p class="standard">Watch out for shapes which are really the same but just turned round or flipped over.</p> <p class="standard">You could download <a href="/content/id/2398/TangramTangle.pdf">this sheet</a> which contains four copies of the square.</p> <p class="standard">Alternatively, experiment with the interactivity below. You can rotate a shape by holding the mouse over a corner. Click on the stars to flip the shapes over.</p> <p class="standard"><a href="/content/id/2398/polygons.swf">Full Screen Version</a></p> <mdo:flash height="400" width="400"><param value="/content/id/2398/polygons.swf" name="movie" ></param><param value="7" name="flashplayerversion" ></param><param value="400" name="height" ></param><param value="400" name="width" ></param></mdo:flash> <p class="standard"> </p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <br></br> <p class="editorial">Ruth from Swanborne House School sent in a well-explained solution and I have added some diagrams to it:</p> <p>When you take away the right-angled triangle, you are left with an irregular quadrilateral. This has one short side which is the same size as the short side of the triangle. If you put these two edges together you get a large triangle:<br></br> <br></br> <mdo:image width="74" height="144" src="sol1.gif" alt="first solution"></mdo:image><br></br> <br></br> There are two remaining sides on the quadrilateral which are both the same size. If you match the triangle up with these sides (using the side which forms the right angle with the shorter side), you get either a parallelogram or another irregular quadrilateral:<br></br>  </p> <table> <tbody> <tr> <td><mdo:image width="90" height="58" src="sol2.gif" alt="second solution"></mdo:image></td> <td><mdo:image width="74" height="110" src="sol3.gif" alt="third solution"></mdo:image></td> </tr> </tbody> </table> <br></br> <p>You can't use the fourth side of the irreguar quadrilateral again, because you get back to a square again. So we think you can only make three shapes, unless you are allowed to flip the pieces.<br></br>  </p> <p class="editorial">Thank you for a well thought out solution, Ruth.</p> <p class="editorial">A, from Zion Heights Junior High School, did look at what happens when you flip the pieces over. Here is what he or she said:</p> <div>The square has been cut into a trapezoid <span class="editorial">(or trapezium)</span> and a right-angled triangle and since only matching sides can be connected to form a new shape, we should start by identifying pairs of sides - one from each smaller shape - that are of the same length.</div> <div>The hypotenuse <span class="editorial">(longest side)</span> of the right-angle triangle is the same length as the edge of the trapezoid from which it had been detached. If you flip the triangle along a diagonal line of reflection (sloping down from left to right) and refit it to the trapezoid, a pentagon is formed:</div> <br></br> <div style="text-align: center;"><mdo:image width="163" height="146" src="sol4.gif" alt="fourth solution"></mdo:image></div> <br></br> <div>The shorter leg of the triangle matches the shortest side of the trapezoid. From this pairing, the outcome is two possible new shapes: the triangle can be rotated $180^\circ$ and lined up to form a big right-angled triangle <span class="editorial">(like Ruth's first shape above)</span>; or the mirror reflection (along a vertical line of reflection) of its rotated $180^\circ$ form can be matched up to form a sort of pentagon:</div> <div style="text-align: center;"><mdo:image width="145" height="238" src="sol5.gif" alt="pentagon"></mdo:image></div> <div>The longer leg of the triangle matches the uncut sides of the original square - the two second longest sides of the trapezoid <span class="editorial">(or trapezium)</span>. On the left edge, two shapes can be formed: a parallelogram by a simple translation of the small triangle <span class="editorial">(Ruth's second shape above)</span>; or a larger trapezoid with the mirror reflection (along a horizontal line of reflection) of the translated triangle:</div> <br></br> <div style="text-align: center;"><mdo:image width="176" height="114" src="sol6.gif" alt="trapezium"></mdo:image></div> <div>On the top edge, two more shapes can be formed with a rotated version of the original triangle <span class="editorial">(Ruth's third shape)</span> and the reflection (along a vertical line of reflection) of the rotated version:</div> <br></br> <div style="text-align: center;"><mdo:image width="139" height="178" src="sol7.gif" alt="pentagon"></mdo:image></div> <p><span class="editorial">So, we've got seven new shapes altogether.</span></p> <p><span class="editorial">Heather from Cottenham Primary school sent us a picture of the shapes she made.  She explained that her mummy helped with the names:</span></p> <p><mdo:image src="heathersol.png"></mdo:image></p> <p><span class="editorial">Well done to you all and thank you for sending us your work.</span><br></br>  </p> <br></br> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Tangram Tangle</h2> <br></br> <p>The square below has been cut into two pieces:</p> <p><mdo:image align="top" alt="square cut in two" bgcolor="" height="214" src="tangram.gif" width="214"></mdo:image></p> <p>It is possible to fit the pieces together again to make a new shape.</p> <p class="standard">If you must match whole sides to each other so that the corners meet, how many new shapes can you make?</p> <p class="standard">Watch out for shapes which are really the same but just turned round or flipped over.</p> <p class="standard">You could download <a href="/content/id/2398/TangramTangle.pdf">this sheet</a> which contains four copies of the square.</p> <p class="standard">Alternatively, experiment with the interactivity below. You can rotate a shape by holding the mouse over a corner. Click on the stars to flip the shapes over.</p> <p class="standard"><a href="/content/id/2398/polygons.swf">Full Screen Version</a></p> <mdo:flash height="400" id="/content/id/2398/polygons.swf" width="400"><param name="allowfullscreen" value="true"></param><param name="movie" value="/content/id/2398/polygons.swf"></param> <param name="flashplayerversion" value="7"></param> <param name="height" value="400"></param> <param name="width" value="400"></param> </mdo:flash> <p class="standard"> </p> </div> <h3>Why do this problem?</h3> <p><a href="http://nrich.maths.org/2398&amp;part=">This activity</a> invites discussion amongst pupils which will encourage them to use vocabulary associated with position and transformations.</p> <h3>Possible approach</h3> <p>The problem could be introduced using the interactivity so that it is easy for everyone to see clearly. However, it is also important for children to have paper or card copies of the shapes to work on the activity. <a href="/content/id/2398/TangramTangle.pdf">This sheet</a> contains four copies of the square or you could make your own on squared paper (the line dividing the square in two is drawn from one corner to the midpoint of the opposite side). It may be a good idea to use paper which is coloured on one side only and talk about whether the new shape should be the same colour all over.</p> <p>Encouraging learners to be systematic in their discovery of &#39;new shapes&#39; is important if they are going to be asked how they "know" they have found every solution. Look out for those children who have developed a system and ask them to share their method with the whole group. For example, they might keep one shape fixed and find all the ways of placing the second shape.</p> <h3>Key questions</h3> <p>How will you record your findings?</p> <p>How do you know you have found all the shapes?</p> <h3>Possible extension</h3> <p>Invite children to make another cut so that they have three pieces. How many new shapes can they make now? What cuts make the &#39;best&#39; new set of shapes?</p> <h3>Possible support</h3> <p>Having several copies of the square will mean that children can stick down each arrangement.</p></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p>Cut out the square and the two pieces and try to make some shapes.  (You could print off <a href="/content/id/2398/TangramTangle.pdf">this sheet</a> which contains four copies of the square.)</p> <p>How will you record your findings?</p> <p>How do you know you have found all the shapes?</p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> Three possible shapes:<br></br> <br></br> <mdo:image width="189" height="207" src="tangramsolution.png" alt="three shapes"></mdo:image><br></br> <br></br></mdoxml> 5 5 1 0 0 0 0 If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make? Using, Applying and Reasoning about Mathematics Practical Activity Using, Applying and Reasoning about Mathematics Working systematically Using, Applying and Reasoning about Mathematics Visualising Admin Lower primary mapping document