942 /www/nrich/html/content/99/04/penta4/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><mdo:image height="111" width="138" src="pyramid.gif" alt=""></mdo:image><br></br> <br></br> How many trapeziums, of various sizes, are hidden in this picture? <br></br> <br></br> (By the way, some of us here at NRICH think they should be called trapezia . What do you think?)<br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <comment> soln.html </comment> <p>Mark thinks it could be 33 trapeziums. Try counting the trapeziums and challenge Mark's answer if you disagree.</p> <p><comment> /soln.html </comment></p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <h3><span style="font-weight: bold;">Why do this problem?</span></h3> <div>It&#39;s quite easy to hazard a guess about how many trapezia there are in <a href="http://nrich.maths.org/public/viewer.php?obj_id=942&amp;part=index">this problem</a> but to be absolutely certain (and convince someone else that you have all the possible solutions) requires some really systematic work.</div> <br></br> <h3>Possible approach</h3> <div style="float: right;"><mdo:image alt="" height="98" src="hexagon.gif" width="114"></mdo:image></div> <div>You could begin with a whole class challenge of a similar but simpler kind - for example how many trapezia in this shape?</div> <br></br> <div>Check that all the children know what a trapezium is and ask for a system for finding all possibilities in this diagram. Emphasise working systematically and what this means in practice - for example starting at the top and working clockwise.</div> <br></br> <div>Offer the <a href="/content/99/04/penta4/TrapeziaTemplate.doc">recording sheet</a> for those who want it - and scissors for them to cut out the variations and re-order them to check for missing diagrams. Working systematically does not come naturally to young children so being able to impose a structure onto randomly generated pictures can be a valuable step in learning how to be systematic.</div> <br></br> <h3>Key questions</h3> <div>Where will we start?</div> <div>What story can we tell which will convince a friend that we have them all? How do we know?</div> <div>Is there another way of arranging them in a pattern?</div> <br></br> <h3>Possible extension</h3> <div>Adding another layer of triangles to the bottom of the diagram increases the complexity, but for children who are already working systematically this will be only slightly more challenging.</div> <br></br> <div>There is a collection of similar style problems <a href="/content/99/04/penta4/counting%20shapes.doc">here</a>.</div> <br></br> <h3>Possible support</h3> <div>Reducing the picture by one layer can be helpful for children who find visualising difficult. Provide a <a href="/content/99/04/penta4/TrapeziaSupport.doc">recording sheet</a> so that they can model the same ordering strategy as in the main activity.</div> <br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><mdo:image height="142" width="136" src="trap-eg.gif" alt=""></mdo:image><br></br></mdoxml> 2 5 0 1 0 0 0 How many trapeziums, of various sizes, are hidden in this picture? 2D Geometry, Shape and Space Trapezia Using, Applying and Reasoning about Mathematics Working systematically