1110 /www/nrich/html/content/02/05/penta2/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> The eight sets of coordinates below define various quadrilaterals (four-sided shapes). In each case one coordinate is missing. <ol> <li>$(2,11), \; (0,9),\; (2,7),\; (?,?)$</li> <li>$(3,7),\; (3,4),\; (8,4),\; (?,?)$</li> <li>$(18,3),\; (16,5), \;(12,5),\; (?,?)$</li> <li>$(13,12),\; (15,14),\; (12,17),\; (?,?)$</li> <li>$(7,14),\; (6,11),\; (7,8),\; (?,?)$</li> <li>$(15,9),\; (19,9),\; (16,11),\; (?,?)$</li> <li>$(11,3),\; (15,2),\; (16,6),\; (?,?)$</li> <li>$(9,16),\; (2,9),\; (9,2),\; (?,?)$</li> </ol> <p>These eight quadrilaterals are all symmetrical. This may be rotational or line symmetry or both. The shapes are all in the first quadrant.</p> <p>The set of eight missing coordinates define another shape - this time with eight sides which has both rotational and line symmetry.</p> <p>Can you draw this new eight-sided shape on a graph like the one below?</p> <p><mdo:image alt="Grid." src="grid.gif"></mdo:image></p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p class="editorial">We have had the following solution from Matthew, David and Jack at St. Nicolas School, Newbury. They thought there was a problem with number 6 but there isn't!</p> <p>Our solutions to the missing coordinates are:</p> <ol> <li>(2,11), (0,9), (2,7) ..............(4,9)<br></br> which had both rotational and line symmetry.</li> <li>(3,7), (3,4), (8,4) ..............(8,7)<br></br> which had both rotational and line symmetry.</li> <li>(18,3), (16,5), (12,5) ..............(10,3)<br></br> which had line symmetry.</li> <li>(13,12), (15,14), (12,17) ..............(10,15)<br></br> which had both rotational and line symmetry.</li> <li>(7,14), (6,11), (7,8) ..............(8,11)<br></br> which had both rotational and line symmetry.</li> <li>(15,9), (19,9), (16,11) ..............(12,11)<br></br> which had rotational symmetry.</li> <li>(11,3), (15,2), (16,6) ..............(12,7)<br></br> which had both rotational and line symmetry.</li> <li>(9,16), (2,9), (9,2) ..............(16,9)<br></br> which had both rotational and line symmetry.</li> </ol> <p>We plotted these 8 sets of coordinates, which made a symmetrical star.</p> <p><mdo:image src="star.gif" alt="Symmetical 4-pointed star on graph paper."></mdo:image></p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>A Cartesian Puzzle</h2> <br></br> The eight sets of coordinates below define various quadrilaterals (four-sided shapes). In each case one coordinate is missing. <ol> <li>$(2,11), \; (0,9),\; (2,7),\; (?,?)$</li> <li>$(3,7),\; (3,4),\; (8,4),\; (?,?)$</li> <li>$(18,3),\; (16,5), \;(12,5),\; (?,?)$</li> <li>$(13,12),\; (15,14),\; (12,17),\; (?,?)$</li> <li>$(7,14),\; (6,11),\; (7,8),\; (?,?)$</li> <li>$(15,9),\; (19,9),\; (16,11),\; (?,?)$</li> <li>$(11,3),\; (15,2),\; (16,6),\; (?,?)$</li> <li>$(9,16),\; (2,9),\; (9,2),\; (?,?)$</li> </ol> <p>These eight quadrilaterals are all symmetrical. This may be rotational or line symmetry or both. The shapes are all in the first quadrant.</p> <p>The set of eight missing coordinates define another shape - this time with eight sides which has both rotational and line symmetry.</p> <p>Can you draw this new eight-sided shape on a graph like the one below?</p> <p><mdo:image alt="Grid." src="grid.gif"></mdo:image></p> </div> <br></br> <h3><span style="font-weight: bold;">Why do this problem?</span></h3> <a href="http://nrich.maths.org/public/viewer.php?obj_id=1110&amp;part=index">This problem</a> is one that requires some understanding of coordinates in the first quadrant. It will also call on knowledge of both rotational and line symmetry, and the properties of various quadrilaterals.<br></br> <h3>Possible approach</h3> <div>You could play a game of &#39;twenty questions&#39; to begin with so that pupils get a chance to familiarise themselves with properties of shapes. Choose a quadrilateral and write the name of it on a piece of paper. Invite the class to ask questions to guess what your quadrilateral is, but you can only answer yes or no. Keep a tally of the number of questions asked - if they get it in less than twenty, they win, otherwise you win. You could repeat this a few times with pupils choosing shapes.</div> <br></br> <div>You could start on the problem itself by showing it to the group on an interactive whiteboard or data projector. Alternatively, if they are already very familiar with coordinates in the first quadrant, you could get them to work in pairs from a printed sheet of the problem from the beginning. It is important that they are able to talk through their ideas with a partner while doing the problem. <a href="/content/02/05/penta2/1110.pdf">This sheet</a> of the first quadrant could be used for both rough working and the final results. Otherwise supply plenty of squared paper! It might help learners to know that the coordinates of each quadrilateral are given going round in an anti-clockwise direction.</div> <br></br> <div>One of the nice things about this problem is that learners will know that they have solved it correctly. In the plenary, therefore, you can concentrate on asking some pairs to explain the way they tackled the problem, rather than focusing on the answer. Were some ways more efficient than others?</div> <br></br> <h3>Key questions</h3> <div>What kind of quadrilateral do you think this one is?</div> <div>Where is its fourth vertex?</div> What kind of symmetry do you think this quadrilateral has?<br></br> <h3>Possible extension</h3> Learners could plot their own quadrilaterals with one vertex of each forming a hexagon and so make a similar problem for a friend to try.<br></br> <h3>Possible support</h3> <div>You might want to tell some children that the shapes include one parallelogram, one trapezium and one rhombus, and are otherwise squares and rectangles.</div> <br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p>The coordinates are given going round each quadrilateral in an anti-clockwise direction.</p> <p>The shapes include one parallelogram, one trapezium and one rhombus and are otherwise squares and rectangles.</p> <br></br></mdoxml> 2 3 0 1 0 0 0 Find the missing coordinates which will form these eight quadrilaterals. These coordinates themselves will then form a shape with rotational and line symmetry. Transformations and their Properties Symmetry Coordinates and Coordinate Geometry Coordinates 2D Geometry, Shape and Space Quadrilaterals Admin Upper primary mapping document