One Big Triangle

192 /www/nrich/html/content/01/04/letme1/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <div class="framework"> Drag and drop these triangles so that the numbers that touch add up to $10$ Try to use all the small triangles and move them to fill in all the spaces in the big triangle. </div> <a href="/content/01/04/letme1/1BTri.swf">Full screen version</a> <mdo:flash height="400" id="/content/01/04/letme1/1BTri.swf" width="550"><param name="allowfullscreen" value="true"></param><param name="allowfullscreen" value="true"></param> <param name="movie" value="/content/01/04/letme1/1BTri.swf"></param> <param name="flashplayerversion" value="8"></param> </mdo:flash> You can also print out <a href="/content/01/04/letme1/One%20Big%20Triangle%20Cards.pdf">these triangles,</a> cut them out and rearrange them to make the big triangle. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">Well done to two schools that sent in a picture of their solution, firstly class 4M at St. James' School in England and from Yunsil at Ballybay Elementary School in The Republic of Ireland. Congratulations and thanks for the picture. <div style="text-align: center;"><mdo:image src="triangle%20sol.png"></mdo:image></div> <div> Catherine sent us her solution. Is it the same as yours? </div> <div style="text-align: center;"><mdo:image src="big%20triangle.gif"></mdo:image></div> <div> Nicole from Mercy College told us: To work this out I just did trial and error and kept on working out different ways until finally after five minutes I got the answer. I like to think of 'trial and error' as 'trial and improvement'! This is certainly a good strategy for this problem. Their completed triangle was the same as the very first one above. Pupils at St Michael's Kirkham wrote: We looked at the numbers on the triangles and found the number bonds to $10$. We had to decide which triangle to start with so the rest of them would fit in. We used the colours to help us as well. Ryan, Josh and Jake from Marpool Primary School also spotted that the colours were useful: We have found a solution for One Big Triangle. Each colour must match when putting number bonds to ten together. You first have to find the triangle that has the numbers $8$, $3$ and $2$ and put it in the bottom left hand corner because the number bonds will not work in any other way. Then put the triangle with $9$, $9$ and $4$ and put it in the bottom right corner. Then it should be easier to work it out from then on. Cerys and Rachael from Ysgol Bryncrug, Tywyn, Gwynedd sent a photo of their completed triangle: <mdo:image src="photo.gif"></mdo:image> Using printed cards makes the problem a bit harder as each triangle can be rotated too. Jack from Birchwood, Suffolk also had a good strategy: The easiest way was to start from the bottom and work towards the top. I realised that certain numbers couldn't be paired to make $10$ so I put them at the bottom so they didn't have to link up with anything. Sion gave us a very detailed explanation, which is very clear: The first thing that you have to look for is the triangle that can only go in a limited amount of spaces or where the there are only a few numbers that add up to ten with it. In this case it is the triangle with the two nines and a four. I would put this in a corner. After you have found this you have to look for the triangles that match e.g. the two fives. They both have a one but only one of them has a one that could connect to a nine if the nine was in a corner. After you have solved one piece, look for the pieces that can add up to the remaining faces of the other triangles. The triangle with a seven in the red part and the triangle with the three in the red must go on the bottom because there aren't any of the other number to make ten with. There are now four empty spaces two for the pair with the two and eight in the red areas and two for the pair with the nine and one in it. Look where they could connect, the blue areas on the triangles with one and eight. Match them up and their respective partners. Finally, Jack, Kyle, Louis, George, and Mr. Phillips from Hennock School sent <a href="/content/01/04/letme1/SolvingOneBigTriangle.ppt">this</a> PowerPoint file which explains how to complete the triangle.</div></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>One Big Triangle</h2> <div class="framework">Notes for adults The small triangles can be moved to fill in the spaces in the big triangle. Just drag and drop. The numbers that touch need to add up to $10$. It is possible to make one big triangle so that all the the numbers on the sides that touch add to $10$.</div> <a href="/content/01/04/letme1/1BTri.swf">Full screen version</a> <mdo:flash height="400" id="/content/01/04/letme1/1BTri.swf" width="550"><param name="allowfullscreen" value="true"></param><param name="allowfullscreen" value="true"></param> <param name="movie" value="/content/01/04/letme1/1BTri.swf"></param> <param name="flashplayerversion" value="8"></param> </mdo:flash> You can also print out <a href="/content/01/04/letme1/One%20Big%20Triangle%20Cards.pdf">these triangles,</a> cut them out and rearrange them to make the big triangle. </div> <h3> Why do this problem?</h3> <a href="http://nrich.maths.org/public/viewer.php?obj_id=192&part=index">This problem</a> is useful for recalling and learning all the pairs of numbers with a total of 10 and working out the corresponding subtraction facts. The interactive version will help children to remain engaged with the problem as it is easy to use. However, for children who are fluent with the combinations to ten, this problem is still relevant because of the element of logic. If using the cards rather than the interactivity, learners will also need to visualise triangles in different orientations in order to solve the problem. <h3>Possible approach</h3> <div>You could start the lesson by asking the children to tell you pairs of numbers that add to $10$. Then you could introduce the interactivity with the whole group on the interactive whiteboard.</div> <div>After this, children can work in pairs on the problem with cards made from <a href="/content/01/04/letme1/One%20Big%20Triangle%20Cards.pdf">this sheet</a> so that they are able to talk through their ideas with a partner. (If these are printed onto thin card and laminated you will have a permanent set that can be used for other purposes as well.) The cards are more difficult to arrange than the interactivity because they can be rotated.</div> <div>During a plenary, encourage the children to talk about the strategies they have used. Did they guess and then check? Or did they have a more systematic approach to the problem? Did they imagine what a triangle would look like in a particular position before placing it there? You might find learners have used visualisation to plan which piece will go where. Discuss the solutions that have been found. Are they all the same? Do they look the same as the interactivity? If not, can the children suggest why this should be so?</div> <h3>Key questions</h3> <div>What do you need to put with $7$ ... to make $10$?</div> Can you find a different card with that number on it? <h3>Possible extension</h3> Children who finish quickly could use the cards to make a shape (not necessarily a triangle) where the touching numbers add to $9$ (or $8$ or $11$). Alternatively, they could add their own choice of numbers to blank triangular pieces to create their own activity. <h3>Possible support</h3> Children could use <a href="/content/01/04/letme1/One%20Big%20Triangle%20Cards.pdf">the cards</a> to make a different shape (not necessarily a triangle) where the touching numbers add to $10$. Alternatively, learners could use the interactivity which will be sure to hold attention and increase persistence and perseverance. Some children may find it difficult to cope with matching more than one pair at a time in which case using a domino type activity would be more accessible. A set of 9 spot dominoes would be useful for this and you can find one <a href="/content/01/04/letme1/9SpotDoms.pdf">here</a>. The task could be to join the dominoes together so that the 'match' adds to 10 or any other number of the children's choice. This will then give them plenty of practice in identifying and remembering number bonds. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">You could try putting one piece in to start with and then seeing whether you can place another piece next to it. Are you sure that triangle is the only possibility? </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Catherine sent us her solution. Is it the same as yours? <div style="text-align: center;"><mdo:image alt="Example of a solution" height="340" src="big%20triangle.gif" width="388"></mdo:image>Nicole from Mercy College told us: To work this out I just did trial and error and kept on working out different ways until finally after 5 minutes I got the answer. I like to think of 'trial and error' as 'trial and improvement'! This is certainly a good strategy for this problem. Here is a picture of the completed triangle: Pupils at St Michael's Kirkham wrote: We looked at the numbers on the triangles and found the number bonds to $10$. We had to decide which triangle to start with so the rest of them would fit in. We used the colours to help us as well. Ryan, Josh and Jake from Marpool Primary School also spotted that the colours were useful: We have found a solution for One Big Triangle. Each colour must match when putting number bonds to ten together. You first have to find the triangle that has the numbers 8, 3 and 2 and put it in the bottom left hand corner because the number bonds will not work in any other way. Then put the triangle with 9, 9 and 4 and put it in the bottom right corner. Then it should be easier to work it out from then on. Cerys and Rachael from Ysgol Bryncrug, Tywyn, Gwynedd sent a photo of their completed triangle: Using printed cards makes the problem a bit harder as each triangle can be rotated too. Jack from Birchwood, Suffolk also had a good strategy: The easiest way was to start from the bottom and work towards the top. I realised that certain numbers couldn't be paired to make $10$ so I put them at the bottom so they didn't have to link up with anything. Sion gave us a very detailed explanation, which is very clear: The first thing that you have to look for is the triangle that can only go in a limited amount of spaces or where the there are only a few numbers that add up to ten with it. In this case it is the triangle with the two nines and a four. I would put this in a corner. After you have found this you have to look for the triangles that match e.g. the two fives. They both have a one but only one of them has a one that could connect to a nine if the nine was in a corner. After you have solved one piece, look for the pieces that can add up to the remaining faces of the other triangles. The triangle with a seven in the red part and the triangle with the three in the red must go on the bottom because there aren't any of the other number to make ten with. There are now four empty spaces two for the pair with the two and eight in the red areas and two for the pair with the nine and one in it. Look where they could connect, the blue areas on the triangles with one and eight. Match them up and their respective partners. Finally, Jack, Kyle, Louis, George, and Mr. Phillips from Hennock School sent this PowerPoint file which explains how to complete the triangle.</div> <div> </div> </mdoxml> 2 3 1 0 0 0 0 One Big Triangle Make one big triangle so the numbers that touch on the small triangles add to 10. You could use the interactivity to help you. Calculations and Numerical Methods Addition & subtraction Using, Applying and Reasoning about Mathematics Trial and improvement Information and Communications Technology Interactivities Using, Applying and Reasoning about Mathematics Visualising Using, Applying and Reasoning about Mathematics Representing Admin Lower primary mapping document