4869 /www/nrich/html/content/id/4869/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> I can join two equilateral triangles together along their edges like this:<br></br> <br></br> <div><mdo:image alt="two triangles" height="82" src="twotri.gif" width="138"></mdo:image></div> <br></br> <div>How many different ways are there to join three equilateral triangles together? (You must match an edge to an edge.)</div> <div>How do you know you have got them all?</div> <br></br> <div>How many different ways are there to join four equilateral triangles together?</div> <div>Have you got a system for checking that you have them all?</div> <br></br> <div>You might like to use this interactivity to try out your ideas. Click the red triangle to make a new triangle appear. You can move the triangles around by dragging with your mouse. If you hover your mouse over a corner, you will see a white circle where you can click to rotate the triangle.</div> <br></br> <div><a href="/content/id/4869/polygons.swf">Full Screen Version</a></div> <mdo:flash height="400" id="/content/id/4869/polygons.swf" width="400"><param name="allowfullscreen" value="true"></param><param name="allowfullscreen" value="true"></param> <param name="movie" value="/content/id/4869/polygons.swf"></param> <param name="flashplayerversion" value="7"></param> <param name="height" value="400"></param> <param name="width" value="400"></param> </mdo:flash><br></br> <br></br> For a further challenge, you might like to look at <a href="http://nrich.maths.org/public/viewer.php?obj_id=1022&amp;part=index">Tri-Five</a>.<br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p class="editorial">Grace and Brittany from Stradbroke Primary both made the problem simpler to start with (this is often a good way of starting a problem, so good thinking!). Grace wrote:</p> <br></br> $1$ triangle makes $1$ shape<br></br> <div style="clear: both;">$2$ triangles makes $1$ shape<br></br> <div style="clear: both;">$3$ triangles makes $1$ shape<br></br> <div style="clear: both;">$4$ triangles makes $3$ shapes<br></br> <div style="clear: both;">It was hard to spot the ones that were the same. I drew them on a piece of paper to make it easier to check them.</div> <p class="editorial" style="clear: both;">Jonathan from Brick Street Public School explained how he found the three different ways for 4 triangles, starting with the shape made from three triangles:</p> <div style="clear: both;">There are five sides to put the fourth triangle in, but there are only three different sides to put the fourth triangle in because two sides make the same shape.</div> <p class="editorial" style="clear: both;">Here is a picture of the three shapes that Jonathan sent:</p> </div> </div> </div> <br></br> <div style="text-align: center;"><mdo:image width="235" height="235" alt="three different ways to put four triangles together" src="trianisolgif.gif"></mdo:image></div> <br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Triangle Animals</h2> <br></br> I can join two equilateral triangles together along their edges like this:<br></br> <div><mdo:image alt="two triangles" height="82" src="twotri.gif" width="138"></mdo:image></div> <br></br> <div>How many different ways are there to join three equilateral triangles together? (You must match an edge to an edge.)</div> <div>How do you know you have got them all?</div> <br></br> <div>How many different ways are there to join four equilateral triangles together?</div> <div>Have you got a system for checking that you have them all?</div> <br></br> <div>You might like to use this interactivity to try out your ideas. Click the red triangle to make a new triangle appear. You can move the triangles around by dragging with your mouse. If you hover your mouse over a corner, you will see a white circle where you can click to rotate the triangle.</div> <br></br> <div><a href="/content/id/4869/polygons.swf">Full Screen Version</a></div> <mdo:flash height="400" id="/content/id/4869/polygons.swf" width="400"><param name="allowfullscreen" value="true"></param><param name="allowfullscreen" value="true"></param> <param name="movie" value="/content/id/4869/polygons.swf"></param> <param name="flashplayerversion" value="7"></param> <param name="height" value="400"></param> <param name="width" value="400"></param> </mdo:flash><br></br> <br></br> For a further challenge, you might like to look at <a href="http://nrich.maths.org/public/viewer.php?obj_id=1022&amp;part=index">Tri-Five</a> .</div> <br></br> <h3><span style="font-weight: bold;">Why do this problem?</span></h3> <div>There are two main reasons for using <a href="http://nrich.maths.org/4869&amp;part=">this problem</a> in the classroom. The first is to encourage children to develop a systematic approach to solving problems and the second is to enable them to understand that rotating a shape does not change the shape itself, it just puts it in a different place (or orientation). Both these ideas can be touched upon as you introduce the problem.</div> <div> </div> <h3>Possible approach</h3> <div>To begin with, explore just two triangles. This might be best done by having cut out triangles on an OHP, or by using the interactivity. Suggest to the group that one of the triangles could be kept still while you look at the different positions of the second triangle. This will encourage them to move the second triangle around the first in a particular direction i.e. having a system so that no possibilities are left out. Then, by looking at each of the different shapes and rotating them, children can be asked what they notice. (They are all the same.)</div> <div>In this way, the class can conclude that in fact there is only one way to put two triangles together. This initial exploration and discussion will equip them for tackling the problem itself.</div> <br></br> <h3>Key questions</h3> <div>Have you decided what is the same and what is different?</div> <div>How do you know that you have them all?</div> <br></br> <h3>Possible extension</h3> Children could have a go at <a href="http://nrich.maths.org/1022&amp;part=">Tri-Five</a>. They could also see what happens if they use two or three isoceles triangles. Are their results different? Why? Again the focus should be on developing the use of accruate descriptive language and beginning to introduce the correct mathematical vocabulary of vertices, sides and angles.<br></br> <br></br> <h3>Possible support</h3> Having lots of cut-out triangles for children to stick down will help them access this challenge.  You could use <a href="/content/id/4869/EquilateralTris.pdf">this sheet</a> of equilateral triangles for printing and cutting (and possibly laminating too).<br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> What is the same?<br></br> What is different?<br></br> How do you know that you have them all?<br></br> If you are working away from the computer, you might like to print off  <a href="/content/id/4869/EquilateralTris.pdf">this sheet</a> of equilateral triangles and cut them out.<br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <br></br> 3 triangles - 1 way<br></br> 4 triangles - 3 ways<br></br></mdoxml> 5 4 1 0 0 0 0 How many different ways can you find to join three equilateral triangles together? Can you convince us that you have found them all? Mathematics Tools Sets of shapes Information and Communications Technology Interactivities Decision Mathematics and Combinatorics Combinations Using, Applying and Reasoning about Mathematics Working systematically 2D Geometry, Shape and Space Equilateral triangles