4734 /www/nrich/html/content/id/4734/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> You might like to have a look at <a href="http://nrich.maths.org/public/viewer.php?obj_id=4725&amp;part=index"> Number Balance</a> before trying this problem.<br></br> <br></br> Here is a number balance, or equaliser:<br></br> <br></br> <a href="/content/id/4734/balancer.swf">Full Screen Version</a> <br></br> <mdo:flash height="400" width="550"><param value="/content/id/4734/balancer.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> <div>It has weights hanging off it to make it balance.</div> <div>We can hang more than one weight from each hook.</div> <div>I hang one weight on the $10$. I want to use just one hook on the other side of the equaliser.</div> <div style="margin-left: 40px;">Where could I put weights and how many would I need to put on?</div> <div style="margin-left: 40px;">Is there only one answer?</div> <div>Now I hang two weights on the $10$ on one side of the equaliser.</div> <div style="margin-left: 40px;">Using only the $2$ hook, how could I add weights on the other side to make it balance?</div> <div style="margin-left: 40px;">Using only the $5$ hook, how could I add weights on the other side to make it balance?</div> <div style="margin-left: 40px;">How could you make it balance using both the $2$ and the $5$ hooks? Is there only one solution?</div> <div>Now I hang one weight on the $10$ hook on one side of the equaliser.</div> <div style="margin-left: 40px;">If I use the $3$ hook and one other, how can I make it balance?</div> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p class="editorial">Lottie and Adele sent us a great solution to this problem. They answer each part in turn and explain clearly as they go along:</p> 1. To make it balance with 10 on one side, using only one hook, you can put two on the 5 or one on the 10 or five on number 2 or ten on 1.<br></br> 2. Two weights on the 10=20. So put ten on number 2 hook to make it balance. <br></br> 3. You could put four weights on the 5 to make it balance with 20 (2 10s). <br></br> 4. You can put two on the 5 or put five weights on the 2. There is only one solution. <br></br> 5. It balances by putting one on the 7 (with the 3) and the 10 on the other side.<br></br> <p class="editorial">Thank you to you both. I wonder if there are other ways of making the equaliser balance if you have one weight on the 10 hook on one side and can use the 3 hook and one other hook on the other side? Don't forget that you can have more than one weight on a hook.</p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Are You Well Balanced?</h2> <br></br> You might like to have a look at <a href="http://nrich.maths.org/public/viewer.php?obj_id=4725&amp;part=index">Number Balance</a> before trying this problem.<br></br> <br></br> Here is a number balance, or equaliser:<br></br> <br></br> <a href="/content/id/4734/balancer.swf">Full Screen Version</a><br></br> <mdo:flash height="400" id="/content/id/4734/balancer.swf" width="550"><param name="allowfullscreen" value="true"></param><param name="movie" value="/content/id/4734/balancer.swf"></param> <param name="flashplayerversion" value="7"></param> <param name="height" value="400"></param> <param name="width" value="550"></param> </mdo:flash><br></br> <div>It has weights hanging off it to make it balance.</div> <div>We can hang more than one weight from each hook.</div> <div>I hang one weight on the $10$. I want to use just one hook on the other side of the equaliser.</div> <div style="margin-left: 40px;">Where could I put weights and how many would I need to put on?</div> <div style="margin-left: 40px;">Is there only one answer?</div> <div>Now I hang two weights on the $10$ on one side of the equaliser.</div> <div style="margin-left: 40px;">Using only the $2$ hook, how could I add weights on the other side to make it balance?</div> <div style="margin-left: 40px;">Using only the $5$ hook, how could I add weights on the other side to make it balance?</div> <div style="margin-left: 40px;">How could you make it balance using both the $2$ and the $5$ hooks? Is there only one solution?</div> <div>Now I hang one weight on the $10$ hook on one side of the equaliser.</div> <div style="margin-left: 40px;">If I use the $3$ hook and one other, how can I make it balance?</div> </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=4734&amp;part=index">This problem</a> can be used to introduce repeated addition and therefore concepts of multiplication. It could also be used to challenge children to work systematically to find all possible solutions.</div> <br></br> <h3>Possible approach</h3> <div>Before tackling this problem, pupils will need practical experience of working with balances, if possible. It would also be a good idea to have a look at <a href="http://nrich.maths.org/public/viewer.php?obj_id=4725&amp;part=index">Number Balance</a> which uses the same interactivity, but focuses on number bonds.</div> <br></br> To introduce the problem, show the class the balance on the interactive whiteboard and, without saying anything, hang weights to make it balance, perhaps one on the left and two on the right. Take the weights off and put on a different combination of weights on each side to balance the equaliser. Repeat this a few times and invite learners to talk to each other in pairs about what they think is happening. You could then try the first question as a whole group, asking the children to talk to each other about what to do before sharing ideas and checking using the interactivity. During this discussion, you can specifically introduce language to help children talk about what they are doing, for example "$2$ add $2$ add $2$ add $2$ add $2$ equals $10$ or five lots of $2$ equal $10$".<br></br> <br></br> For the other parts of the problem, you may want learners to be at a computer in pairs or you may prefer them to work on mini-whiteboards/paper. After leaving time for them to work on this, bring them together and discuss their solutions. When it comes to considering balancing a total of $20$ on one side with weights on both the $2$ hook and the $5$ hook, you may want to ask pairs to write up some solutions on individual strips of paper which can be stuck on the board. You could look for patterns in the solutions and order them to help decide whether any have been missed out. Having seen this, children could try to use a similar system to find all the solutions to the final part of the problem.<br></br> <h3>Key questions</h3> <div>Where have you tried hanging weights so far?</div> <div>How will you know that you have got all the ways of making the equaliser balanced?</div> <br></br> <h3>Possible extension</h3> <div>Children could investigate ways of balancing the equaliser if, for example, you can only put weights on the $2$ hook on one side and the $10$ hook on the other.</div> <br></br> <h3>Possible support</h3> <div>Having access to the interactivity, either on the whiteboard, or on individual computers, will help some children gain in confidence as they will be able to try out their ideas without the anxiety of getting things wrong.</div> <br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> Try starting off by putting weights on the $1$ hook. How many will you need to balance with $10$?<br></br> <div>What about trying the $2$ hook? How many will you need this time?</div> <div>Can you keep going in this way?</div> <div>If I had two weights on the $10$ hook, what would this be the same as in total?<br></br> If I then hang two weights on the other side, what would that balance with? So how many will I need on the $2$ hook to balance with the two on the $10$?</div> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> 10 = 1+1+1+1+1+1+1+1+1+1<br></br> 10 = 2+2+2+2+2<br></br> 10 = 5+5<br></br> 10 = 10<br></br> 20 = ten on the 2 hook<br></br> 20 = four on the 5 hook<br></br> 20 = five on the 2 and two on the 5<br></br> 10 = one on the 3 and one on the 7<br></br> 10 = one on the 3 and seven on the 1<br></br> 10 = two on the 3 and one on the 4<br></br> 10 = two on the 3 and two on the 2<br></br> 10 = two on the 3 and four on the 1<br></br> 10 = three on the 3 and one on the 1<br></br> <br></br> This problem is an extension of Number Balance . It builds on the idea of number bonds by introducing repeated addition and therefore concepts of multiplication. Inviting children to explore for themselves how the balance works by adding and taking off weights is a vital part of their learning. This could then be followed up by a whole class discussion before asking them to tackle the questions.<br></br></mdoxml> 2 5 1 0 0 0 0 Can you work out how to balance this equaliser? You can put more than one weight on a hook. Numbers and the Number System Factors and multiples Information and Communications Technology Interactivities Using, Applying and Reasoning about Mathematics Working systematically Admin Lower primary mapping document