Getting the Balance

5676 /www/nrich/html/content/id/5676/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> Here we have a balance for you to work on:<br></br> <br></br> <a href="/content/id/5676/balancer.swf" linkindex="8">Full Screen Version</a> <br></br> <mdo:flash height="400" width="550"><param value="/content/id/5676/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> <br></br> It is a number balance, sometimes it's called a "Balance Bar" and sometimes an "Equalizer".<br></br> It has weights like these;<br></br> <div style="text-align: center;"><mdo:image height="34" width="40" src="NosBal3.gif" alt="NosBal3"></mdo:image></div> <div>These weights are hung below the numerals. It balances equal amounts, for example, with $10$ on one side and $2$ and $8$ on the other we have;</div> <div style="text-align: center;"><mdo:image height="115" width="296" src="NosBal2.gif" alt="NosBal2"></mdo:image></div> If you like this idea try "<a linkindex="9" href="http://nrich.maths.org/public/viewer.php?obj_id=4725&part=index" set="yes">Number Balance</a> ", then return here. <div style="text-align: left;"></div> <div>Now this challenge is about getting the balance.</div> <br></br> <div>Rule : All the while you can only have one weight at each numeral on the balance.</div> <br></br> <div style="text-align: left;"></div> Let's start by saying that on one side of the balance, place two weights and keep them there. Make it balance by placing $3$ weights on the other side (remember only one at each numeral!). <div style="text-align: left;"></div> <div style="text-align: left;"></div> <div style="text-align: left;"></div> <br></br> So you might start with an $8$ and a $3$ on one side, and find you have to have something like these for it to balance:<br></br> <div style="text-align: center;"><mdo:image height="226" width="417" src="Bal1.jpg" alt="Bal1"></mdo:image></div> <br></br> <div style="text-align: center;"><mdo:image height="222" width="428" src="Bal2.jpg" alt="Bal2"></mdo:image></div> <div>So choose your two places on one side and find many different balance places on the other side.</div> <br></br> <div style="text-align: left;"></div> <div>When you've done all you can, it might be an idea to choose another (maybe higher) pair of numbers for one side and find all the ways of placing $3$ weights on the other side.</div> <br></br> <div style="text-align: left;"></div> Are you recording your results? If so, how? <div style="text-align: left;"></div> <div style="text-align: left;"></div> <div style="text-align: left;"></div> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p class="editorial">Daniel from Englefield Infant School had a go at this challenge. First he tried 8 and 3 on the left, as the question suggested:</p> 8+3=1+2+8<br></br> 8+3=1+3+7 <br></br> 8+3=1+4+6 <br></br> 8+3=2+3+6 <br></br> 8+3=2+4+5 <br></br> <br></br> <p class="editorial">I like the way you have written these in a very logical order, Daniel. Well done! Next, Daniel tried 10 and 9 on the left side of the balance:</p> 10+9=1+8+10<br></br> 10+9=2+7+10 <br></br> 10+9=2+8+9 <br></br> 10+9=3+6+10 <br></br> 10+9=3+7+9 <br></br> 10+9=4+5+10 <br></br> 10+9=4+6+9 <br></br> 10+9=4+7+8 <br></br> 10+9=5+6+8<br></br> <br></br> <p class="editorial">Fantastic! Izzy from Tonbridge Grammar School reminds us that:</p> Whatever the sum of numbers of weights on one side add up to has to be equal with the mass of weights on the other.<br></br> <br></br> <p class="editorial">Thank you to you both.</p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Getting the Balance</h2> <br></br> Here we have a balance for you to work on:<br></br> <br></br> <a href="/content/id/5676/balancer.swf" linkindex="8">Full Screen Version</a><br></br> <mdo:flash height="400" id="/content/id/5676/balancer.swf" width="550"><param name="allowfullscreen" value="true"></param><param name="movie" value="/content/id/5676/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> <br></br> It is a number balance, sometimes it's called a "Balance Bar" and sometimes an "Equalizer".<br></br> It has weights like these;<br></br> <div style="text-align: center;"><mdo:image alt="NosBal3" height="34" src="NosBal3.gif" width="40"></mdo:image></div> <div>These weights are hung below the numerals. It balances equal amounts, for example, with $10$ on one side and $2$ and $8$ on the other we have;</div> <div style="text-align: center;"><mdo:image alt="NosBal2" height="115" src="NosBal2.gif" width="296"></mdo:image></div> If you like this idea try "<a href="http://nrich.maths.org/public/viewer.php?obj_id=4725&part=index" linkindex="9" set="yes">Number Balance</a> ", then return here. <div style="text-align: left;"> </div> <div>Now this challenge is about getting the balance.</div> <br></br> <div>Rule : All the while you can only have one weight at each numeral on the balance.</div> <br></br> <div style="text-align: left;"> </div> Let's start by saying that on one side of the balance, place two weights and keep them there. Make it balance by placing $3$ weights on the other side (remember only one at each numeral!). <div style="text-align: left;"> </div> <div style="text-align: left;"> </div> <div style="text-align: left;"> </div> <br></br> So you might start with an $8$ and a $3$ on one side, and find you have to have something like these for it to balance:<br></br> <div style="text-align: center;"><mdo:image alt="Bal1" height="226" src="Bal1.jpg" width="417"></mdo:image></div> <br></br> <div style="text-align: center;"><mdo:image alt="Bal2" height="222" src="Bal2.jpg" width="428"></mdo:image></div> <div>So choose your two places on one side and find many different balance places on the other side.</div> <br></br> <div style="text-align: left;"> </div> <div>When you've done all you can, it might be an idea to choose another (maybe higher) pair of numbers for one side and find all the ways of placing $3$ weights on the other side.</div> <br></br> <div style="text-align: left;"> </div> Are you recording your results? If so, how? <div style="text-align: left;"> </div> <div style="text-align: left;"> </div> <div style="text-align: left;"> </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=5676&part=index">This activity</a> sets the scene for some important algebraic ideas - that if three unknowns have a particular total, there are potentially different possibilities for these unknowns. As well as giving children practice in addition and subtraction, this problem could be used as a focus for introducing the idea of working systematically.</div> <br></br> <h3>Possible approach</h3> <div>It would be good to have the balance up on the interactive whiteboard and, with the class watching, put a weight on the $10$ on the left-hand side and a weight on the $8$ on the right-hand side. Ask them to talk in pairs about what will happen when you put a weight on the $2$ on the right-hand side. Share some ideas, in particular their reasons for their predictions, then test them out on the interactivity. Repeat this a few times, with different numbers of weights on each side until you feel as if the children are happy with the set-up.</div> <br></br> <div>Challenge them to investigate the problem, preferably with pairs working at a computer. Alternatively, you may have "real" balances so that you do not need to use the computer for long periods of time. After inviting children to try out some ideas, it would be good to spend some time talking to the group about how they are finding solutions and what they are recording. You may want to draw attention to any systems that the children are using which help them find all possibilities. For example, they might keep one weight the same on the right hand side and find all combinations that can go with it; then increase this fixed weight to the next one along and find all combinations etc. You may find that some children are not using the balance itself and are just recording on paper.</div> <br></br> <h3>Key questions</h3> <div>Where are you going to hang the weights on the left-hand side?</div> <div>What will you need to hang on the right-hand side?</div> <div>Can you find another weight to balance them with the same weights on the left?</div> <div>How will you know you have found all the possibilities?</div> <div>How are you recording what you've done?</div> <br></br> <h3>Possible extension</h3> <div>Ask the pupils what they think the total of the two weights on the left would be to give the biggest number of possibilities for rearrangements on the right-hand side. How do they know and how could they prove this? Learners could be challenged to find all the different ways that two weights can balance two weights. Alternatively, children could investigate what happens if you are allowed to have three weights, but any two weights may be on one numeral.</div> <br></br> <h3>Possible support</h3> <div>It might help if children record each possibility on a different strip of paper. Once they have found a few for a particular pair of weights on the left, you could help them order the strips which would reveal any gaps.</div> <br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">Where are you going to hang the weights on the left-hand side? <br></br> What will you need to hang on the right-hand side? <br></br> Can you find another weight to balance them with the same weights on the left? <br></br> How will you know you have found all the possibilities? <br></br> How are you recording what you've done? <br></br> Try out your ideas using the interactive balance, or your school might have a real balance - ask your teacher.<br></br> <br></br></mdoxml> 2 5 1 0 0 0 0 Getting the Balance If you hang two weights on one side of this balance, in how many different ways can you hang three weights on the other side for it to be balanced? 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