2421
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2011-02-01T00:00:01
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<p style="text-align: left;">Andy and his friend Sam were walking along the road together. Andy had a big bag of marbles.</p>
<p style="text-align: center;"> </p>
<p style="text-align: center;"><mdo:image align="top" alt="Bag of Marbles" bgcolor="" height="176" src="marbles1.gif" width="140"></mdo:image></p>
<br></br>
Unfortunately the bottom of the bag split and all the marbles spilled out. Poor Andy!<br></br>
<br></br>
<div style="text-align: center;"><mdo:image align="top" alt="Bag of Marbles with marbles falling out" bgcolor="" height="226" src="marbles2.gif" width="408"></mdo:image></div>
<br></br>
One third ($\frac{1}{3}$) of the marbles rolled down the slope too quickly for Andy to pick them up. One sixth ($\frac{1}{6}$) of all the marbles disappeared into the rain-water drain.<br></br>
<br></br>
Andy and Sam picked up all they could but half ($\frac{1}{2}$) of the marbles that remained nearby were picked up by other children who ran off with them.<br></br>
<br></br>
Andy counted all the marbles he and Sam had rescued.<br></br>
<br></br>
<div style="text-align: center;"><mdo:image align="top" alt="Pile of marbles" bgcolor="" height="71" src="marbles3.gif" width="151"></mdo:image></div>
<br></br>
<br></br>
He gave one third ($\frac{1}{3}$) of these to Sam for helping him pick them up. Andy put his remaining marbles into his pocket. There were $14$ of them.<br></br>
<br></br>
How many marbles were there in Andy's bag before the bottom split?<br></br>
<br></br>
What fraction of the total number that had been in the bag had he lost or given away?<br></br>
<br></br>
<br></br>
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<p class="editorial">Several very well reasoned solutions came in
for this problem about Andy's marbles. Katherine from Treffos,
North Wales, pupils from Wigginton Primary and Stephen from Frisby
School all went about the problem in a similar way. Stephen
said:</p>
<br></br>
I know that to solve this problem I have to work backwards,
beginning with the last part, and ending with the first.
<div style="clear: both;">So... Andy walks away with 14 marbles,
2/3 of what was recovered. His friend walks away with 1/3 of what
was recovered. 14 is 2/3 so to find out 1/3 I then half 14, getting
7. Sam had 7 and Andy had 14. A total rescued of 14 + 7 = 21<br></br>
<div style="clear: both;">21 is half (1/2) of what was lying around
on the ground, the other half having been taken by the children. To
find how many were on the ground nearby I have to double 21. I now
have an answer of 42.<br></br>
<div style="clear: both;">42 must be doubled to get the final
answer, as 1/3 and 1/6 went down drains or hills (1/3 + 1/6 = 1/6 +
2/6 =1/2).<br></br>
<div style="clear: both;">This gives me a final answer of 84
marbles in the bag.<br></br>
<div style="clear: both;">To find out what fraction of the marbles
Andy had given away or lost you have to find the fraction of
marbles he was left with. He was left with 14 out of 84 marbles
(14/84). You then work out the fraction in its lowest form. Firstly
divide 14 and 84 by 2 to get 7/42 then divide them by 7 to get an
answer of 1/6. However 1/6 is what he is left with so he must have
lost or given away 5/6 of his marbles!</div>
</div>
</div>
</div>
</div>
<br></br>
<p class="editorial">Michael from King Henry VIII found a different
method for solving the problem:</p>
Andy has x marbles<br></br>
<div style="clear: both;">1/6 go down the drain, and 1/3 roll down
the slope<br></br>
<div style="clear: both;">That comes to 3/6 or 1/2<br></br>
<div style="clear: both;">x/2<br></br>
<div style="clear: both;">1/2 the remaining 1/2 are stolen, leaving
1/4<br></br>
<div style="clear: both;">x/4<br></br>
<div style="clear: both;">He gave 1/3 to his friend, leaving
1/4*2/3 0r 2/12 or 1/6<br></br>
<div style="clear: both;">x/6<br></br>
<div style="clear: both;">That left 14<br></br>
<div style="clear: both;">x/6=14<br></br>
<div style="clear: both;">Times both sides by 6 and you get
x=84<br></br>
<div style="clear: both;">He had 84 marbles, and he had lost or
given away 5/6</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<br></br>
<p class="editorial">These are both very clear solutions, well
done. Working from the end of a problem can be a useful way of
tackling it - something to remember!</p>
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<h2>Andy's Marbles</h2>
<br></br>
<p style="text-align: left;">Andy and his friend Sam were walking along the road together. Andy had a big bag of marbles.</p>
<p style="text-align: center;"> </p>
<p style="text-align: center;"><mdo:image align="top" alt="Bag of Marbles" bgcolor="" height="176" src="marbles1.gif" width="140"></mdo:image></p>
<br></br>
Unfortunately the bottom of the bag split and all the marbles spilled out. Poor Andy!<br></br>
<br></br>
<div style="text-align: center;"><mdo:image align="top" alt="Bag of Marbles with marbles falling out" bgcolor="" height="226" src="marbles2.gif" width="408"></mdo:image></div>
<br></br>
One third ($\frac{1}{3}$) of the marbles rolled down the slope too quickly for Andy to pick them up. One sixth ($\frac{1}{6}$) of all the marbles disappeared into the rain-water drain.<br></br>
<br></br>
Andy and Sam picked up all they could but half ($\frac{1}{2}$) of the marbles that remained nearby were picked up by other children who ran off with them.<br></br>
<br></br>
Andy counted all the marbles he and Sam had rescued.<br></br>
<br></br>
<div style="text-align: center;"><mdo:image align="top" alt="Pile of marbles" bgcolor="" height="71" src="marbles3.gif" width="151"></mdo:image></div>
<br></br>
<br></br>
He gave one third ($\frac{1}{3}$) of these to Sam for helping him pick them up. Andy put his remaining marbles into his pocket. There were $14$ of them.<br></br>
<br></br>
How many marbles were there in Andy's bag before the bottom split?<br></br>
<br></br>
What fraction of the total number that had been in the bag had he lost or given away?<br></br>
<br></br>
<br></br>
</div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<a href="http://nrich.maths.org/public/viewer.php?obj_id=2421&part=index">This problem</a> involves complicated reasoning about fractions that challenges children's understandings of the concepts involved. It is a good example of how fractions relate to multiplication and division.<br></br>
<h3>Possible approach</h3>
You could start a lesson with some oral challenges, such as:
<div>"I bought some apples at the market. After I had given half of them to my sister I had $7$ left. How many did I buy?"</div>
<div>"Tom gave a quarter of his bag of sweets to Ben and ate half of them himself. He had $6$ left. How many sweets were there in the bag to begin with?"</div>
<div>Encourage learners to talk to each other about how they solved each of the above - they may not have used exactly the same method.</div>
<br></br>
You could introduce the problem verbally, as a <a href="/content/id/2421/Andy%27sMarbles.pdf">printed sheet</a> or on an interactive whiteboard. Ask learners to spend a bit of time working on it in pairs and then share ideas on how to get started amongst the whole group. It would be good to encourage children to jot down whatever they find helpful as they tackle the problem.
<h3>Key questions</h3>
<div>Where shall we start?</div>
<div>How many marbles did Andy and Sam rescue?</div>
<div>How might this help you to work out the number of marbles Andy had before the bag split?</div>
<br></br>
<h3>Possible extension</h3>
Children could be encouraged to make up their own version of a similar problem for a friend to solve.<br></br>
<h3>Possible support</h3>
<div>It might help some learners to start slowly from the end asking questions such as:</div>
<div>"How many marbles did Andy have at the end?"</div>
<div>"How many did he give to Sam? And how many did he keep for himself?"</div>
<div>"Can you work out from that how many Andy and Sam picked up altogether?"</div>
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How many marbles did Andy and Sam rescue?<br></br>
How might this help you to work out the number of marbles which
Andy had before the bag split?<br></br></mdoxml>
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Andy's Marbles
Andy had a big bag of marbles but unfortunately the bottom of it
split and all the marbles spilled out. Use the information to find
out how many there were in the bag originally.
Fractions, Decimals, Percentages, Ratio and Proportion
Calculating with fractions
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Upper primary mapping document