Triangle Pin-Down

2851 /www/nrich/html/content/id/2851/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <mdo:image alt="a right angled triangle drawn on a four-pin board" height="187" src="triangles.gif" width="550"></mdo:image> A right-angled triangle has been drawn on the four-pin board above left. Can you draw the same type of triangle on a three-pin board? How many pins could there be on the board for you to be able to draw the same type of triangle? Do you notice anything about the number of pins for which this is possible? What kind of triangle is drawn on the six-pin board above? How many pins could there be on the board for you to be able to draw the same type of triangle? Do you notice anything about the number of pins for which this is possible? Can you name the type of triangle drawn on the nine-pin board above? On what size board could you draw the same type of triangle? Do you notice anything about the number of pins for which this is possible? You might like to use the interactivity to try out your ideas. <a href="/content/id/2851/circleAngles.swf">Full Screen Version</a> <mdo:flash height="400" id="/content/id/2851/circleAngles.swf" width="550"><param name="allowfullscreen" value="true"></param><param name="movie" value="/content/id/2851/circleAngles.swf"></param> <param name="flashplayerversion" value="7"></param> <param name="height" value="400"></param> <param name="width" value="550"></param> </mdo:flash> For printable sets of circle templates for use with this activity, please see <a href="http://nrich.maths.org/public/viewer.php?obj_id=6676&part=index">Printable Resources page</a>. Many thanks to Geoff Faux who introduced us to the merits of the 9 pin circular geo-board. The boards, moulded in crystal clear ABS that can be used on an OHP (185 mm in diameter), together with a teacher's guide, are available from Geoff at <a href="http://www.education-initiatives.co.uk/">Education Initiatives</a> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">A lot of students interpreted the "same type of triangle" as a right-angled triangle that is also isosceles. Emma, aged 13 wrote : To get an exact copy of the right-angled triangle you need the number of points to be a multiple of 4. If you interpret "same type of triangle" to mean any right-angled triangle, your answer would be different. What would it be? Bob has an answer for the six-pin board : <div>For the equilateral triangle the number of pegs must be a multiple of 3.</div> Can you give an explanation for these observations? Thank you to Kyle from New Zealand who correctly identified certain circles and states the type of triangle as <div>Isosceles.</div> If you consider that an isosceles triangle can also be a right-angled triangle, then many answers are possible. Do you think that we can consider an equilateral triangle to be a special form of isosceles triangle? <div class="editorial">If you consider that some triangles can be more than one type of triangle at the same time, can you work out the answer to the nine-pin part of the question?</div> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Triangle Pin-Down</h2> <mdo:image alt="a right angled triangle drawn on a four-pin board" height="187" src="triangles.gif" width="550"></mdo:image> A right-angled triangle has been drawn on the four-pin board. Can you draw the same type of triangle on a three-pin board? How many pins could there be on the board for you to be able to draw the same type of triangle? Do you notice anything about the number of pins for which this is possible? What kind of triangle is drawn on the six-pin board? How many pins could there be on the board for you to be able to draw the same type of triangle? Do you notice anything about the number of pins for which this is possible? Can you name the type of triangle drawn on the nine-pin board? On what size board could you draw the same type of triangle? Do you notice anything about the number of pins for which this is possible? You might like to use the interactivity to try out your ideas. <a href="/content/id/2851/circleAngles.swf">Full Screen Version</a> <mdo:flash height="400" id="/content/id/2851/circleAngles.swf" width="550"><param name="allowfullscreen" value="true"></param><param name="allowfullscreen" value="true"></param> <param name="movie" value="/content/id/2851/circleAngles.swf"></param> <param name="flashplayerversion" value="7"></param> <param name="height" value="400"></param> <param name="width" value="550"></param> </mdo:flash> For printable sets of circle templates for use with this activity, please see <a href="http://nrich.maths.org/public/viewer.php?obj_id=6676&part=index">Printable Resources page</a>. Many thanks to Geoff Faux who introduced us to the merits of the 9 pin circular geo-board. The boards, moulded in crystal clear ABS that can be used on an OHP (185 mm in diameter), together with a teacher's guide, are available from Geoff at <a href="http://www.education-initiatives.co.uk/">Education Initiatives</a></div> <h3>Why use this activity?</h3> This activity encourages children to explore the angles and relative lengths of sides of different triangles that can be made on pinboards. All the angles can be worked out using angles at the centre of a circle and the fact that the three angles in a triangle sum to $180^{\circ}$. No more complex geometric knowledge is required. It might help for learners to have copies of blank peg boards to record their work on. These can be downloaded here: Sheet of <a href="/content/id/2851/threepeg.doc">three-peg</a> boards Sheet of <a href="/content/id/2851/fourpeg.doc">four-peg</a> boards Sheet of <a href="/content/id/2851/fivepeg.doc">five-peg</a> boards Sheet of <a href="/content/id/2851/sixpeg.doc">six-peg</a> boards Sheet of <a href="/content/id/2851/sevenpeg.doc">seven-peg</a> boards Sheet of <a href="/content/id/2851/eightpeg.doc">eight-peg</a> boards Sheet of <a href="/content/id/2851/ninepeg.doc">nine-peg</a> boards Teachers may find the article <a href="http://nrich.maths.org/public/viewer.php?obj_id=2855&part=">Angle Measurement: An Opportunity for Equity</a> to be of interest. <h3>Possible approach</h3> The task can be introduced to the whole class using an interactive whiteboard and involving the children in contributing to decisions about their answers. The children could then try the task for themselves either on computers or using the sheets of geoboard pictures. Once they have had a chance to explore, support them in feeding back their solutions to the rest of the class. Different groups might work on the different number of pegs. <h3>Key questions</h3> The key questions are given in the text of the problem but here they are again: Can you draw the same type of triangle on a three-pin board? How many pins could there be on the board for you to be able to draw the same type of triangle? Do you notice anything about the number of pins for which this is possible? What kind of triangle is drawn on the six-pin board? How many pins could there be on the board for you to be able to draw the same type of triangle? Do you notice anything about the number of pins for which this is possible? Can you name the type of triangle drawn on the nine-pin board? On what size board could you draw the same type of triangle? Do you notice anything about the number of pins for which this is possible? <h3>Possible extension</h3> You could extend the problem to pinboards with more pins on them and learners could explore the relationship between the number of pins, the shapes and angles involved. <h3>Possible support</h3> Children will find this easier if they have the opportunity to engage with the task on real pinboards. They may also find it difficult to record their results. Encorange them to draw or mark the lines from the centre to the edge of the triangles.</mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">Have you looked at the angles in the triangles drawn? It might help to draw in some radii and work out angles from the centre. How will you record the triangles you can make? You might like to print off these sheets if you're not using the interactivity: Sheet of <a href="/content/id/2851/threepeg.doc">three-peg</a> boards Sheet of <a href="/content/id/2851/fourpeg.doc">four-peg</a> boards Sheet of <a href="/content/id/2851/fivepeg.doc">five-peg</a> boards Sheet of <a href="/content/id/2851/sixpeg.doc">six-peg</a> boards Sheet of <a href="/content/id/2851/sevenpeg.doc">seven-peg</a> boards Sheet of <a href="/content/id/2851/eightpeg.doc">eight-peg</a> boards Sheet of <a href="/content/id/2851/ninepeg.doc">nine-peg</a> boards </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">Right angled triangles can be drawn on 4 peg, 6 peg, 8 peg etc. Even number of pegs needed. Equilateral triangles can be drawn on 3 pegs, 6 pegs, 9 pegs etc. Number of pegs must be a multiple of 3. Isosceles triangles can be drawn on any number of pegs except 3 peg board (sometimes the isosceles are right angled too). </mdoxml> 2 5 0 1 0 0 0 Triangle Pin-Down Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs. Using, Applying and Reasoning about Mathematics Investigations Using, Applying and Reasoning about Mathematics Generalising 2D Geometry, Shape and Space Mixed triangles 2D Geometry, Shape and Space Angle properties of shapes Information and Communications Technology Interactivities Mathematics Tools Pinboard/geoboard