5677
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2011-02-01T00:00:01
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Before you try this activity, you might like to have a go at <a href="http://nrich.maths.org/public/viewer.php?obj_id=4725&part=index">Number Balance</a> and <a href="http://nrich.maths.org/public/viewer.php?obj_id=5676&part=index">Getting the Balance</a>, if you haven't already.<br></br>
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Here is a balance, or equaliser, which you might like to investigate:<br></br>
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<a href="/content/id/5677/balancer.swf" linkindex="8" set="yes">Full Screen Version</a><br></br>
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Now, what about a balance that has halves on it?<br></br>
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<mdo:image alt="balance with halves marked" height="159" src="BalHalvesDone.jpg" width="550"></mdo:image><br></br>
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Imagine we hung a weight on the number 7 on the left-hand side. Let's say we can only use the "halves" on the right. Which two numbers on the right-hand side could you hang weights from so that it would balance?<br></br>
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For example, you might have a weight on the $\frac{1}{2}$ and the $6\frac{1}{2}$.<br></br>
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Where else could you hang two weights on the right-hand side to make it balanced?<br></br>
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You can set yourselves all kinds of problems with this new set-up.<br></br>
What about asking, "I wonder what would happen if I had quarters marked up as well?"<br></br>
Or, "I wonder what it'd be like to have three weights balancing two using halves wherever we wanted?"<br></br>
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Let us know what you find out.<br></br>
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<p class="editorial">Boyang from Mountfields Lodge and Caroline
from St Jude's C of E School looked at this problem very logically.
To balance 7 on the left using two "half" weights on the right,
Boyang told us:</p>
<div>The two numbers on the right hand side could be:</div>
<div>$\frac{1}{2}$ + $6\frac{1}{2}$</div>
<div>$1\frac{1}{2}$ + $5\frac{1}{2}$</div>
<div>$2\frac{1}{2}$ + $4\frac{1}{2}$</div>
<div>$3\frac{1}{2}$ + $3\frac{1}{2}$</div>
<div>$4\frac{1}{2}$ + $2\frac{1}{2}$</div>
<div>$5\frac{1}{2}$ + $1\frac{1}{2}$</div>
<div>and finally, $6\frac{1}{2}$ + $\frac{1}{2}$.</div>
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<p class="editorial">I wonder whether the first solution is
different from the last one? What do you think?</p>
<p class="editorial">Caroline wasn't sure whether you are allowed
two weights on the same hook, but I think that's fine.</p>
<p class="editorial">Nobody has looked at hanging three weights on
the right-hand side, or having quarter divisions on the balance. If
you do investigate these situations, let us know! Please don't
worry that your solution is not "complete" - we'd like to hear
about anything you have tried.Teachers -
you might like to send a summary of your children's work.</p>
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<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
Whether <a href="http://nrich.maths.org/5677&part=">this problem</a> is
done with a balance or with pencil and paper, there is more to it
than just doing addition. Although addition and subtraction are
involved, you are likely to find the children using all sorts of
language associated with these two operations and one of the main
points here is to establish a balance or equivalence as well as
"equals". Some pupils are not so readily used to coming across
situations like $3 + 5 = 6 + 2$, for example. You could use this
activity to introduce some algebraic ideas, by encouraging the
class to articulate what they are trying to find in a general
sense. For example "$7$ balances two other numbers", or "$7$ =
'something' add 'something'"or "$7$ = ? $+$ ?". This will help them
to get the idea that they are finding different numbers which fit
this criterion - not just one answer.<br></br>
<h3>Key questions</h3>
<div>How might you
start?</div>
<div>How might you record your
thinking?</div>
<div>How do you know you've
got all the possibilities?</div>
<br></br>
<h3>Possible extension</h3>
As the question suggests, some learners might like to look at a
balance involving other fractions.<br></br>
<h3>Possible support</h3>
Children might like to try <a href="http://nrich.maths.org/5676&part=">Getting the
Balance</a> instead.<br></br>
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<mdoxml version="1.0">We've had one way of doing it using a weight on the $\frac{1}{2}$
peg. How could you balance 7 using $1\frac{1}{2}$? Can you carry on
going along the balance like this trying the "half numbers" in
turn?<br></br>
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How will you make a note of what you have done?<br></br>
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How will you know that you have got all the different ways?<br></br>
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<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0">Whether these activities are done with a balance or with pencil and
paper, there is more to it than just doing addition. Although
addition and subtraction are involved, you are likely to find the
children using all sorts of language associated with these two
operations and one of the main points here is to establish a
balance or equivalence as well as "equals". It's good to ask pupils
how they are thinking about the activity when there are two on one
side and more than one on the other, and how they might record
their thinking. Some pupils are not so readily used to coming
across situations like 3 + 5 = 6 + 2, for example. You could use
this activity to introduce some algebraic ideas, byencouraging the
class to articulate what they are trying to find in a general
sense. For example "7 balances two other numbers", or "7 =
'something' add 'something'"or "7 = ? + ?". This will help them to
get the idea that they are finding different numbers which fit this
criterion - not just one answer.<br></br></mdoxml>
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Balance of Halves
Investigate this balance which is marked in halves. If you had a
weight on the left-hand 7, where could you hang two weights on the
right to make it balance?
Algebra
Introducing algebra
Using, Applying and Reasoning about Mathematics
Investigations
Fractions, Decimals, Percentages, Ratio and Proportion
Calculating with fractions
Calculations and Numerical Methods
Addition & subtraction