2852
/www/nrich/html/content/id/2852/
1
2011-02-01T00:00:01
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
How many different triangles can you make on a circular pegboard
that has nine pegs?<br></br>
<br></br>
You may like to use the interactivity to try out your ideas. If you
prefer to work on paper, you might find this sheet of <a href="http://nrich.maths.org/content/id/6676/9-Dot_noCentralPoint.pdf">
nine-peg boards</a> useful.<br></br>
<br></br>
<a href="/content/id/2852/circleAngles.swf">Full Size
Version</a><br></br>
<mdo:flash height="400" width="550"><param value="/content/id/2852/circleAngles.swf" name="movie" ></param><param value="7" name="flashplayerversion" ></param><param value="400" name="height" ></param><param value="550" name="width" ></param></mdo:flash><br></br>
<br></br>
<br></br>
<span class="editorial">Many thanks to Geoff Faux who introduced us
to the merits of the nine-pin circular geoboard.</span><br></br>
<span class="editorial">The boards, moulded in crystal clear ABS
that can be used on an OHP (185 cm in diameter), together with a
teacher's guide, are available from Geoff at</span> <a href="http://www.education-initiatives.co.uk/">Education
Initiatives</a><br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<p class="editorial">Cong who goes to St. Peter's RC Primary, Aberdeen, sent in a correct solution to this problem. The key to answering it is to be sure you know what you mean by "different" triangles. Cong found 7 different triangles could be drawn on the nine-pin board which he drew:</p>
<br></br>
<br></br>
<div style="text-align: center;"><mdo:image width="500" height="321" src="9pinsol.gif" alt="nine different triangles"></mdo:image></div>
<br></br>
<p class="editorial">
He also sent in a table which gave some more information about each triangle:</p><table border="1"><tr><td>Number</td><td>Colour</td><td>Type</td></tr><tr><td>1</td><td>Green</td><td>Isosceles</td></tr><tr><td>2</td><td>Light blue</td><td>Scalene</td></tr><tr><td>3</td><td>Purple</td><td>Scalene</td></tr><tr><td>4</td><td>Orange</td><td>Isosceles</td></tr><tr><td>5</td><td>Pink</td><td>Scalene</td></tr><tr><td>6</td><td>Blue</td><td>Isosceles</td></tr><tr><td>7</td><td>Red</td><td>Equilateral</td></tr></table><br></br>
<p class="editorial">Well answered, Cong, thank you.</p>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><div class="embed">
<h2>Nine-pin Triangles</h2>
<br></br>
How many different triangles can you make on a circular pegboard that has nine pegs?<br></br>
<br></br>
You may like to use the interactivity to try out your ideas. If you prefer to work on paper, you might find this sheet of <a href="http://nrich.maths.org/content/id/6676/9-Dot_noCentralPoint.pdf">nine-peg boards</a> useful.<br></br>
<br></br>
<a href="/content/id/2852/circleAngles.swf">Full Size Version</a><br></br>
<mdo:flash height="400" id="/content/id/2852/circleAngles.swf" width="550"><param name="allowfullscreen" value="true"></param><param name="movie" value="/content/id/2852/circleAngles.swf"></param>
<param name="flashplayerversion" value="7"></param>
<param name="height" value="400"></param>
<param name="width" value="550"></param>
</mdo:flash><br></br>
<br></br>
<br></br>
<span class="editorial">Many thanks to Geoff Faux who introduced us to the merits of the nine-pin circular geoboard.</span><br></br>
<span class="editorial">The boards, moulded in crystal clear ABS that can be used on an OHP (185 cm in diameter), together with a teacher's guide, are available from Geoff at</span> <a href="http://www.education-initiatives.co.uk/">Education Initiatives</a></div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div><a href="http://nrich.maths.org/public/viewer.php?obj_id=2852&part=index">This problem</a> will help learners extend their knowledge of properties of triangles. It requires visualisation, a systematic approach and is a good context for generalisation and symbolic representation of findings.</div>
<br></br>
<h3>Possible approach</h3>
<div>To start with, you could pose the problem orally, asking children to imagine a circle with nine equally spaced dots placed on its circumference. How many triangles do they think it might be possible to draw by joining three of the dots? Take a few suggestions and then ask how they think they could go about finding out.</div>
<br></br>
<div>Show the interactivity, or draw a nine-point circle on the board. Invite them each to imagine a triangle on this circle. How would they describe their triangle to someone else? Let the class offer some suggestions e.g. by numbering the dots and describing a triangle by the numbers at its vertices, and then return to the problem of the number of different triangles. Discuss ways in which they
will be able to keep track of the triangles and how they will know they have them all. Some children may wish to draw triangles in a particular order, for example those with a side of 1 first (i.e. adjacent pegs joined), then 2 etc. Others may feel happy just to list the triangles as numbers. <a href="/content/id/2852/ninepin.doc">This sheet of blank nine-point circles</a> may be useful.
Encourage children to work in small groups to find the total number.</div>
<br></br>
<div>Bring them together to share findings and systems, using the interactivity to aid visualisation.</div>
<br></br>
<div><a href="http://nrich.maths.org/public/viewer.php?obj_id=962&part=index">Quadrilaterals</a> is a similar problem which pupils could try next.</div>
<br></br>
<h3>Key questions</h3>
<div>How do you know your triangles are all different?</div>
<div>How do you know you have got all the different triangles?</div>
<br></br>
<h3>Possible extension</h3>
<div>You could challenge pupils to think about whether they could predict the number of different triangles which are possible for different point circles. How would they do about finding out? It may be useful to have sheets of other point circles available: <a href="/content/id/2852/fourpeg.doc">four-point</a> , <a href="/content/id/2852/fivepeg.doc">five-point</a> , <a href="/content/id/2852/sixpeg.doc">six-point</a> , <a href="/content/id/2852/eightpeg.doc">eight-point</a> . Are there any similarities between all the circles with an odd numbers of points? How about those with an even number?</div>
<br></br>
<h3>Possible support</h3>
<div>Chidlren could begin by investigating the <a href="/content/id/2852/sevenpeg.doc">seven-point circle</a> .</div>
<br></br>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
How will you record the triangles you've made? You might like to print off this <a href="/content/id/2852/ninepin.doc">sheet</a>
to use. <br></br>
You could try drawing triangles which have a side made by joining two pegs next to each other. How many different triangles can you make in this way?<br></br>
How will you know when you've got them all?<br></br>
<br></br>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
Seven possible in total. If number pegs as 1-9, then can describe
them:<br></br>
<br></br>
Four with "base" of 1 i.e. joining adjacent dots: 1, 2, 3; 1, 2, 4;
1, 2, 5; 1, 2, 6<br></br>
Two more different ones with "base" of 2: 1, 3, 5; 1, 3, 1, 3,
6<br></br>
One more different with "base" of 3: 1, 4, 7<br></br>
<br></br>
This problem offers an opportunity for pupils to work in a
systematic way, using their knowledge of the properties of
triangles. A useful discussion about which triangles are the same
and which are different could be encouraged. If working on paper
rather than using the interactivity, pupils may find it helpful to
print this sheet off. <br></br></mdoxml>
2
5
0
1
0
0
0
Nine-Pin Triangles
How many different triangles can you make on a circular pegboard
that has nine pegs?
Using, Applying and Reasoning about Mathematics
Visualising
2D Geometry, Shape and Space
Mixed triangles
Using, Applying and Reasoning about Mathematics
Working systematically
Information and Communications Technology
Interactivities
Mathematics Tools
Pinboard/geoboard
Admin
Upper primary mapping document