1813 /www/nrich/html/content/03/07/letme2/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> Someone using an elastic band and a pegboard used four pegs to make the blue square you see below. They challenged another person to double the area by just moving two of the pegs. You can see what they did here.<br></br> <p align="center"><mdo:image alt="2 by 2 square made into 2 by 4 rectangle" height="199" src="Peg1.gif" width="289"></mdo:image></p> Have a go at these;<br></br> <p align="left">Can you make this into a right-angled triangle by moving just one peg?</p> <p align="center"><mdo:image alt="traingle of base 6 units, perpendicular height 4" height="215" src="Peg2.gif" width="333"></mdo:image></p> <p align="left"><br></br> Can you enlarge this to the same shape with all the sides twice the length, moving just two pegs?</p> <p align="center"><mdo:image alt="4 by 2 rectangle" height="253" src="Peg3.gif" width="333"></mdo:image></p> <p>You could set up some similar challenges for your friends.<br></br>  </p> <p align="left"> </p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p class="editorial">Lucy, who is educated at home, sent in a very clear solution to this question. For the first part she wrote:</p> <p>You move the top peg to the right by one space. If you cut a square from all four corners, you end up with a quarter of it. In the middle of the square you get four right angles.<br></br>  </p> <p class="editorial">I think there is at least one other way to get a right-angled triangle. Can you see how?</p> <p class="editorial">Lucy continued:</p> <p>For the second problem you know that the new shape is going to have sides $4  \times  8$ because the sides are multiplied by $2$. One of the sides is already $4$ so you just move the two right pegs $6$ spaces to the right.<br></br>  </p> <p class="editorial">Very well described solutions Lucy, thank you.</p> <span class="editorial">Matthew from Beechwood Park School 
wrote:</span><br></br> <br></br> Yes:<br></br> To make the triangle a right angle triangle in one move you move the highest band to a perpendicular angle to one of the other angles, and to make the rectangle longer you move the two furthest right corners to the furthest right pegs (in line with them).</mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Transformations on a Pegboard</h2> <br></br> Someone using an elastic band and a pegboard used four pegs to make the blue square you see below. They challenged another person to double the area by just moving two of the pegs. You can see what they did here.<br></br> <p align="center"><mdo:image alt="2 by 2 square made into 2 by 4 rectangle" height="199" src="Peg1.gif" width="289"></mdo:image></p> Have a go at these;<br></br> <p align="left">Can you make this into a right-angled triangle by moving just one peg?</p> <p align="center"><mdo:image alt="traingle of base 6 units, perpendicular height 4" height="215" src="Peg2.gif" width="333"></mdo:image></p> <p align="left"><br></br> Can you enlarge this to the same shape with all the sides twice the length, moving just two pegs?</p> <p align="center"><mdo:image alt="4 by 2 rectangle" height="253" src="Peg3.gif" width="333"></mdo:image></p> <p>You could set up some similar challenges for your friends.<br></br>  </p> <p align="left"> </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=1813&amp;part=index">This problem</a> is a good way of consolidating properties of shapes and visualising changes in their properties.<br></br> <br></br> <h3>Possible approach</h3> <div>You could introduce this problem by giving pegboards and elastic bands to pairs of children. If they have not used pegboards recently a few minutes of free play helps concentration later! Alternatively, learners could use the <a href="http://nrich.maths.org/public/viewer.php?obj_id=2883&amp;part=index">interactive virtual geoboard</a> to explore the challenges given (click on the circle icon to create a square grid). If you have an interactive whiteboard, using the virtual geoboard would be a good way to share ideas with the whole class during the lesson.</div> <br></br> <div>Children will discover that there is more than one way to do the first part of the problem. How many ways can they find? You could talk about how they know they have got them all - perhaps by looking at each vertex in turn in a systematic way. The problem will encourage children to think hard about what makes a triangle a right-angled one. You could ask them to investigate the other changes that occur when the length of sides of the rectangle are doubled (for example, what about the area?).</div> <br></br> Learners could draw their answers on <a href="/content/03/07/letme2/1cmdotty.doc">square dotty paper</a> or write instructions in words (which is much harder!).<br></br> <br></br> <h3>Key questions</h3> <div>Which pegs have you tried to move?</div> <div>Can you make the shape by moving any other pegs instead?</div> <div>Are there any other ways to do it?</div> <br></br> <h3>Possible extension</h3> Learners could make up similar puzzles for others to do using the <a href="http://nrich.maths.org/public/viewer.php?obj_id=2883&amp;part=index">virtual geoboard</a> or paper.<br></br> <br></br> <h3>Possible support</h3> Using a real pegboard with elastic bands will make this more accessible for many children. They could use two bands in different colours so that one can be left in the original place all the time.<br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> Using a pegboard would be helpful, or if you don't have one, try using <a href="/content/03/07/letme2/1cmdotty.doc">squared paper</a> or this <a href="http://nrich.maths.org/public/viewer.php?obj_id=2883&amp;part=index"> interactive geoboard</a>.<br></br> What are the properties of a right-angled triangle?<br></br> Which peg have you tried to move? <br></br> Can you make the shape by moving a different peg instead? <br></br> Are there any other ways to do it?<br></br></mdoxml> 2 3 0 1 0 0 0 How would you move the bands on the pegboard to alter these shapes? 2D Geometry, Shape and Space Angle properties of shapes 2D Geometry, Shape and Space Right angled triangles Measures and Mensuration Area Mathematics Tools Pinboard/geoboard 2D Geometry, Shape and Space Rectangles Admin Upper primary mapping document