7756 /www/nrich/html/content/id/7756/ 1 0000-00-00T00:00:00 <mdoxml version="1.0"> <p style="text-align: center;"> </p> <p style="text-align: center;"><em>This activity has been particularly created for the most able. (The pupils that you come across in many classrooms just once every few years.)</em></p> <p style="text-align: center;"> </p> <p style="text-align: center;"><mdo:image src="diag%20cuts%20example.jpg"></mdo:image></p> <p style="text-align: center;"> </p> <p>Imagine the central square in a big city and its paved with large square tiles. It may be rectangular rather than squrea! You are going to go straight from one corner, diagonally across to the other corner. You may be walking, cycling, skate boarding or using roller blades. Which ever way you travel you will go absolutely straight.</p> <p>In the picture above showing a very, very small example (a $4$ by $3$ rectangle) you see that the blue line of travel goes through six of the square tiles. Maybe there are other small rectangles that would need you to cross six square tiles.</p> <p>Your challenge is to find what different sizes of rectangles would mean you travelled across $36$ tiles?</p> <p>Can you find a generalization that would enable you to find solutions more easy?</p> <p> </p> <p> </p> <p> </p> </mdoxml> <mdoxml version="1.0"> <h3><span style="font-weight: bold;">Why do this problem?</span></h3> <div>This <a href="http://nrich.maths.org/7756">activity</a> challenges the most able pupils in their abilities to link the numerical with the spatial.</div> <br></br> <h3>Possible approach</h3> <p>As this is intended for the most able I would suggest printing out the activity and discussing together first of all, then let them produce their creations.</p> <h3>Key questions</h3> <div>Tell me about how you arrive at these solutions.</div> <div>So what relationships do you think are involved here?</div> <div>What further questions can you ask?</div> <p><br></br>  </p> </mdoxml> <mdoxml version="1.0"> <p>It may be helpful it some circumstances to look at some other ways of seeing the challenge. Imagine you are looking through a small strange window and can only see part of the town square ( which maybe a rectangle). You may see the path of a friend going across from corner to corner in a straight line. So what size can the rectangle be and how many tiles are cut through?</p> <p style="text-align: center;"><mdo:image src="funny%20window%201.jpg"></mdo:image></p> <p> </p> <p>or maybe in another town through another strnage window you see the path is like this;</p> <p style="text-align: center;"><mdo:image src="funny%20window%202.jpg"></mdo:image></p> <p style="text-align: center;"> </p> <p>So what size can the rectangle be and how many tiles are cut through?</p> <p> </p> </mdoxml> <mdoxml version="1.0"> <p>To cut across 36 tiles the rectangles are,</p> <p>2,35     2,36    3,34      3,36   4,33       4,34    4,36     5,32    6,31     6,32   6,33   6,36</p> <p>7,30     8,29    8,30     9,28   9,30       9,36   10,27   10,28   11,6  </p> <p>12,25   12,26   12,27   12,28   12,30   12,36   13,24   14,23   14,24</p> <p>15,22   15,24   16,21   16,22   17,20   18,19    18,20   18,21   18,24   18,36   36,36 </p> </mdoxml> 2 5 0 1 1 0 0