Number Round Up

188 /www/nrich/html/content/01/02/letme1/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> If you are a teacher, click <a href="/188&part=note">here</a> for a version of the problem suitable for classroom use, together with supporting materials. Otherwise, read on ... Arrange the numbers $1$ to $6$ in each set of circles below. The sum of each side of the triangle should equal the number in the centre of the triangular shape. <mdo:image alt="9" src="triangle1.gif" style="width: 194px; height: 168px;"></mdo:image> <mdo:image alt="" height="175" src="spacer.gif" width="75"></mdo:image> <mdo:image alt="10" src="triangle2.gif" style="width: 194px; height: 168px;"></mdo:image> <mdo:image alt="11" src="triangle3.gif" style="width: 194px; height: 168px;"></mdo:image> <mdo:image alt="" height="168" src="spacer.gif" width="75"></mdo:image> <mdo:image alt="12" src="triangle4.gif" style="width: 194px; height: 168px;"></mdo:image> Once you've had a chance to think about it, click below to see how three different pupils began working on the task. Dan said: <div class="toggle"> "I used counters which had $1$ to $6$ on them. I put the counters in a triangle in any old way, then I added up the sides. Then I moved the counters around to try and get the right total on each side." </div> Emma said: <div class="toggle"> "I noticed that three of the numbers are odd ($1, 3$ and $5$) and three of the numbers are even ($2, 4$ and $6$). I thought this might help. I know that $9$ is an odd number so it can be made using odd + odd + odd or using even + even + odd." </div> Farah said: <div class="toggle"> "If I want a small total on each side, I'll need small numbers in the corners of the triangle." </div> Can you take each of these starting ideas and develop it into a solution? A practical version of this activity is included in the Year 3/4 Brain Buster Maths Box which contains hands-on challenges developed by members of NRICH and produced by BEAM. For more details and ordering information, please scroll down <a href="http://nrich.maths.org/public/viewer.php?obj_id=4833&part=index" style="font-style: italic;">this page</a> . </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> We had some good solutions sent in with some diagrams to support them. From Isseya St Andrews School Sukhumvit inThailand we had this sent in; <mdo:image src="Isseya%22s%20solution.jpg"></mdo:image> We had correct solutions from Ethan at the Good Shepherd Lutheran School in Australia, Rebecca, Ryan and Georgia from Falcon Junior School in England. Emerald and Brooke from Myland School in England sent in the following interesting account of their work; Emerald First I tried putting any numbers into the circles. After a few goes I made this triangle where the sides add up to $11$, $12$ and $13$. Because $11$, $12$, $13$ was so close, I swapped the $3$ for the $2$ to make $13$ one smaller and $11$ one higher. Then all the sides total $12$. Brooke I tried to add up to nine lots of different ways only using numbers $1$ to $6$. There were only $3$ different ways $6 + 1 + 2 = 9$, $1 +5 + 3 = 9$ and $3 + 4 + 2 = 9$. I found that I couldn?t have $5$ and $2$ on the same side because you would need another $2$ to make it add up to $9$. I put the three sets of numbers into the triangle to see if they match up to make nine. I found that I needed to hide the numbers $4, 5$ and $6$ on a middle, not a corner because they were only used once in the ways of making $9$. For the $10$ triangle the ways to make $10$ are $5 + 3 + 2 = 10$, $5 + 1 + 4 = 10$, $6 + 1 + 3 = 10$. You can't have a $4, 2$ or $6$ on a corner because they are only in one way of making $10$. Emerald To make the $11$ triangle I started using Brooke's method. I made $11$ by adding $6 + 4 + 1$ and $5 + 2 + 4$. I put those sides on a triangle with the $4$ at the top because it was in both sentences. The only number missing was $3$, so that had to go in the empty circle. That made the bottom side total $14$. I swapped the $5$ and the $2$ to make the bottom side less. All the sides equal $11$. Brooke I guessed that we could make other triangles with the numbers $1$ to $6$. I tried to make triangle totals of $3, 5, 8, 13$ and $15$, but none of them could be done. For the $8$ triangle there were not two numbers to go with $6$ that would make $8$. For the $13$ and $15$ triangles there were not two numbers to go with $1$ to make $13, 15$ or higher numbers. From Valley Road Primary School the following came from Lauren; Well in school I was quite scared when I saw it but then we had to work with our talk partners and then I really enjoyed it I didn't want to stop. My solution is: I had counters $1$ to $6$ then I put them in any old order but then I then knew that I had to have them all equal to $11$ or $10$ so then I got them all. Thank you for being so honest Lauren and thank you for all the emails that were sent in, on this occasion no one sent any ideas into the <a class="blogbutton" href="http://nrich.maths.org/z/infinities">blog</a> . </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Number Round Up</h2> If you are a teacher, click <a href="/188&part=note">here</a> for a version of the problem suitable for classroom use, together with supporting materials. Otherwise, read on ... Arrange the numbers $1$ to $6$ in each set of circles below. The sum of each side of the triangle should equal the number in the centre of the triangular shape. <mdo:image alt="9" src="triangle1.gif" style="width: 194px; height: 168px;"></mdo:image> <mdo:image alt="" height="175" src="spacer.gif" width="75"></mdo:image> <mdo:image alt="10" src="triangle2.gif" style="width: 194px; height: 168px;"></mdo:image> <mdo:image alt="11" src="triangle3.gif" style="width: 194px; height: 168px;"></mdo:image> <mdo:image alt="" height="168" src="spacer.gif" width="75"></mdo:image> <mdo:image alt="12" src="triangle4.gif" style="width: 194px; height: 168px;"></mdo:image> Once you've had a chance to think about it, click below to see how three different pupils began working on the task. Dan said: <div class="toggle"> "I used counters which had $1$ to $6$ on them. I put the counters in a triangle in any old way, then I added up the sides. Then I moved the counters around to try and get the right total on each side." </div> Emma said: <div class="toggle"> "I noticed that three of the numbers are odd ($1, 3$ and $5$) and three of the numbers are even ($2, 4$ and $6$). I thought this might help. I know that $9$ is an odd number so it can be made using odd + odd + odd or using even + even + odd." </div> Farah said: <div class="toggle"> "If I want a small total on each side, I'll need small numbers in the corners of the triangle." </div> Can you take each of these starting ideas and develop it into a solution? A practical version of this activity is included in the Year 3/4 Brain Buster Maths Box which contains hands-on challenges developed by members of NRICH and produced by BEAM. For more details and ordering information, please scroll down <a href="http://nrich.maths.org/public/viewer.php?obj_id=4833&part=index" style="font-style: italic;">this page</a> . </div> <h3>Why do this problem?</h3> <div><a href="http://nrich.maths.org/public/viewer.php?obj_id=188&part=index">This problem</a> is an interesting context in which children can practise addition and subtraction. It can be solved in many different ways and the sample approaches offer a basis for discussion of possible different methods. You might spend a couple of lessons on this activity.</div> <h3>Possible approach</h3> <div>You could introduce this challenge by having one of the triangles on the interactive whiteboard with numbers $1$-$6$ to drag into the circles. Begin by dragging the numbers into any places and ask the children to talk about what they notice. Some may mention the numbers used but direct them to look at totals along each "side" if it doesn't come up naturally. This will mean they understand the set-up and you can then leave them to try the problem itself. Try not to direct learners too much at this stage but make sure they understand that they can use any resources or equipment that they might find helpful. (You might want to have these sheets of blank triangles available should anyone request them: <a class="doclink" href="/content/01/02/letme1/NumberRoundUpSheet.doc">Word document</a> or <a class="pdflink" href="/content/01/02/letme1/NumberRoundUpSheet.pdf">pdf</a>). Once you feel that most children have made progress and understand the problem well (this does not necessarily mean that they have found all the solutions), give out this (<a class="doclink" href="/content/01/02/letme1/BBNumber%20round%20up%20new%20sheet.doc">doc</a> <a class="pdflink" href="/content/01/02/letme1/BBNumber%20round%20up%20new%20sheet.pdf">pdf</a> ). Suggest to the class that when they've finished or can't make any further progress, they should look at the sheet showing three approaches used by children working on this task. Pose the question, "What might each do next? Can you take each of their starting ideas and develop them into a solution?". It might be appropriate to read through each method as a whole class before giving pairs time to work on each one. Alternatively, you may prefer to allocate a particular starting point to each pair. Allow at least fifteen minutes for a final discussion. Invite some pairs to explain how the three different methods might be continued. You may find that some members of the class used completely different approaches when they worked on the task to begin with, so ask them to share their methods too. You can then facilitate a discussion about the advantages and disadvantages of each. Which way would they choose to use if they were presented with a similar task in the future? Why? (You may find that conversations arise about the number of different solutions for each total. Encourage children to articulate what they think is the same and what is different in this context. They might not all agree!) </div> <div> </div> <h3>Key questions</h3> <div>What will you try first? Why did you put that number there? Tell me about how you found that solution. Is there only one solution? Tell me about this approach. What do you think s/he was doing? How do you think this will help to solve the problem? What do you think s/he would have done next? </div> <h3>Possible extension</h3> <div>Children could investigate whether other totals are possible. Why or why not?</div> <h3>Possible support</h3> <div>Learners might request a range of different resources to help them tackle this challenge, for example numbered counters, mini-whiteboards, number lines. Try not to pre-empt their requests by placing equipment out on tables at the start, but do make sure these kind of resources are easily accessible to the children, should they want to use them and do your best to accommodate any requests which you hadn't anticipated!</div> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> What will you try first? Why did you put that number there? Tell me about how you found that solution. Is there only one solution? Tell me about this approach. What do you think s/he was doing? How do you think this will help to solve the problem? What do you think s/he would have done next? </mdoxml> 5 5 1 0 0 0 0 Number Round Up Arrange the numbers 1 to 6 in each set of circles below. The sum of each side of the triangle should equal the number in its centre. Calculations and Numerical Methods Addition & subtraction Decision Mathematics and Combinatorics Combinations Using, Applying and Reasoning about Mathematics Trial and improvement Numbers and the Number System Odd and even numbers Admin Lower primary mapping document