Let's Investigate Triangles

93 /www/nrich/html/content/01/09/bbprob2/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. <mdo:image alt="" height="112" src="93A.png" width="587"></mdo:image> There are: <ul> <li>yellow strips with $3$ holes</li> <li>black strips with $4$ holes,</li> <li>red strips with $5$ holes and</li> <li>green strips with $6$ holes.</li> </ul> There are plenty of strips of each colour. Vincent makes a triangle with a green strip, a yellow strip and a red strip. <mdo:image alt="" height="136" src="93B.png" width="244"></mdo:image> Tara makes a triangle with two green strips and one red strip and another triangle with three black strips. <mdo:image alt="two triangles." src="pic3.gif"></mdo:image> How many different triangles can you make with these lengths? Can you find three strips which cannot be made into a triangle when you use them together? </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> We had a few suggestions as to what sould be done with these strips. I've chosen three from pupils of very different ages. Matthew Solution: $1$ green and $2$ yellow strips Explanation: because the green strips have $6$ holes and the yellow strips have $3$ holes and so the green strips don't have enough holes for two yellow strips together to make a triangle. Zareah Well, the three strips that can't make a triangle are the green, the yellow and the black because the green strip is too long to connect the yellow and black. Furthermore, you can also make a triangle with it if you space it out properly. Oleg In total, there $4 \times 4 \times 4 = 64$ possible combinations of strips. We picked one of the four strips, then pick again two times and make a triangle of them. But Green + Yellow + Yellow makes a degenerate triangle, that looks like a line. This is an interesting argument but I think Oleg has counted lots of triangles more than once when they are essentially the same. Can you see how he has done that? Perhaps you can offer us a different solution? </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Let's Investigate Triangles</h2> Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. <mdo:image alt="" height="112" src="93A.png" width="587"></mdo:image> There are: <ul> <li>yellow strips with $3$ holes</li> <li>black strips with $4$ holes,</li> <li>red strips with $5$ holes and</li> <li>green strips with $6$ holes.</li> </ul> There are plenty of strips of each colour. Vincent makes a triangle with a green strip, a yellow strip and a red strip. <mdo:image alt="" height="136" src="93B.png" width="244"></mdo:image> Tara makes a triangle with two green strips and one red strip and another triangle with three black strips. <mdo:image alt="two triangles." src="pic3.gif"></mdo:image> How many different triangles can you make with these lengths? Can you find three strips which cannot be made into a triangle when you use them together? </div> <h3>Why do this problem?</h3> <div><a href="http://nrich.maths.org/public/viewer.php?obj_id=93&part=index">This problem</a> allows children to make important discoveries about triangles for themselves through representing the sides of triangles with strips from a construction kit. They will have the opportunity to find out which combinations of lengths cannot be made into triangles and why this is so. They should also discover how rigid triangles are and that three fixed lengths can only be arranged in one way (unlike other polygons such as quadrilaterals where the angles can be changed).</div> <div>This investigation requires plenty of equipment and space, so is probably best done in a small group.</div> <h3>Possible approach</h3> <div>You could start by demonstrating with the interactivity in <a href="http://nrich.maths.org/public/viewer.php?obj_id=2342&part=index">this problem</a> to show the children what can be done with three rods or sticks.</div> After they have been introduced to the problem, encourage the children to work on making different triangles with three lengths of strips. If they are mature enough to work in pairs, the children will gain from talking through their ideas with a partner. If they are working on a suitable plastic table the length of the rods (as numbers) can be written on the table with whiteboard markers to check if any of the triangles made are repeats. Go on to using all four lengths when you judge the children are ready. <div>If you do not have access to a construction set or suitable rods, <a href="/content/01/09/bbprob2/LetsInvestigateTrianglesColour.pdf">this sheet</a> will give you four copies of each strip in the colours in the problem. If they are printed out on card and laminated they can be laid flat on a table to make the triangles. <a href="/content/01/09/bbprob2/LetsInvestigateTrianglesB%26W.pdf">This sheet</a> has uncoloured strips that can be photocopied. Paper copies can be pasted onto a backing sheet but make sure that the children have made a triangle before they have access to any glue or paste as they will probably be frustrated on finding that the third side does not fit!</div> <div>In a plenary, children can show the triangles they have made and any three lengths that will not make a triangle. Invite them to explain why the three lengths cannot make a triangle, and look out for those learners who begin to make generalisations.</div> <h3>Key questions</h3> <div>Have you made any triangles using this length of strip?</div> <div>Have you made any triangles with all the sides the same colour/length?</div> <div>Have you made any triangles with all the sides different colours/lengths?</div> <div>Are you sure you haven't made one like that already?</div> <div>Why do you think you can't make a triangle with those three sticks/rods?</div> <h3>Possible extension</h3> Children who work quickly could be encouraged to find a way to show that they have discovered all of the possible triangles using four different lengths. They can be encouraged to compare the different triangles they have made and describe them to one another looking carefully at their characteristics. <h3>Possible support</h3> Some learners might find it helpful to use just two lengths to start with. (Make sure that the longest one is shorter than two of the others put together lengthways.) You can make four different triangles with these. You can then add a third length. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Have you made triangles with all the sides the same colour? Have you made triangles with all the sides different colours? Have you found three sticks you can't make a triangle with? <a href="/content/01/09/bbprob2/LetsInvestigateTrianglesColour.pdf">This sheet</a> will give you four copies of each strip in the colours in the problem, which you could print off and cut out. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">Equilateral triangles: 4 Isosceles triangles: 3,3,4 - 3,3,5 - (3,3,6 gives a straight line); 4,4,3 - 4,4,5 - 4,4,6; 5,5,3 - 5,5,4 - 5,5,6; 6,6,3 - 6,6,4 - 6,6,5 Scalene triangles: 3,4,5 - 4,5,6 - 3,5,6 - I make that 18 altogether. Solutions prior to Dec 2012 We had a few suggestions as to what sould be done with these strips. I've chosen three from pupils of very different ages. Matthew Solution: $1$ green and $2$ yellow strips Explanation: because the green strips have $6$ holes and the yellow strips have $3$ holes and so the green strips don't have enough holes for two yellow strips together to make a triangle. Zareah Well, the three strips that can't make a triangle are the green, the yellow and the black because the green strip is too long to connect the yellow and black. Furthermore, you can also make a triangle with it if you space it out properly. Oleg In total, there $4 \times 4 \times 4 = 64$ possible combinations of strips. We pick one of $4$ strips, then do it again $2$ times and make a triangle of them. But Green + Yellow + Yellow make degenerate triangle, that looks like a line. This is an interesting argument but Oleg has counted lots of triangles more than once when they are essentially the same. Can you see how he has done that and offer us a correct solution?</mdoxml> 2 3 1 0 0 0 0 Let's Investigate Triangles Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make? Using, Applying and Reasoning about Mathematics Working systematically Decision Mathematics and Combinatorics Combinations 2D Geometry, Shape and Space Mixed triangles Mathematics Tools Construction kits Using, Applying and Reasoning about Mathematics Investigations Admin Lower primary mapping document