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2011-02-01T00:00:01
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Some years ago I suddenly had to do some maths with some boys who
were a bit turned off about it. If it had been today in England
they would have said, "It's not cool!"<br></br>
<br></br>
There were two small PE hoops nearby and some small bean bags. I
put down the hoops as you see:<br></br>
<div style="text-align: center;"><mdo:image alt="" src="2a.real.gif"></mdo:image></div>
<br></br>
<p>I collected eight of the bean bags. "Do they really have beans
in?" I asked. They did not know and neither did I. Never mind.</p>
<p>I suggested that we put them in the hoops. Four ended up being
in the blue hoop, six in the red hoop so that two were in the
overlap.</p>
<p>We went on to talk about how many were in the blue and how many
were in the red and how the ones in the middle seemed to be counted
twice. Try this for yourself.</p>
We tried putting the bean bags in the hoops in a different way and
each time we counted how many were in each of the two hoops.
<p>Well it was time to use the yellow hoop that had been
around:</p>
<div style="text-align: center;"><mdo:image alt="" src="3a.real.gif"></mdo:image></div>
<br></br>
<p>I suggested we made sure that there were four in the blue, five
in the red and six in the yellow. So we all tried and then ...?</p>
<p>Well have a go at this one.</p>
<br></br>
<div>Now the investigation is to take this much further. Try to
find as many ways as you can for having those numbers $4$, $5$ and
$6$ using just eight objects. I guess you'll need to record your
results somehow so that you do not do the same ones twice!</div>
<br></br>
<div>Have you found yourself using some kind of 'system' or
'method' for going from one arrangement to the next? Try to explain
it if you have.</div>
<br></br>
<div>When you're pretty sure you cannot find any more, check yours
with a friend and see if there are any new ones!</div>
<p>As always we then have to ask "I wonder what would happen if
...?"</p>
<p>This month it's very easy to invent new ideas, for example, "I
wonder what would happen if I used a different number of objects?"
You could go about this in order and try six objects and then
seven, you've done eight so move on to nine ...</p>
<p>Any other ideas?</p>
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<p class="editorial">We did not have any solutions sent in for this
activity, maybe it was done very practically and not much was
recorded. The recording is not always the most important aspect of
a piece of mathematics.</p>
<p><span class="editorial">However, we would still love to hear
from you if you have worked on this problem. Teachers, you may like
to send in a summary of your pupils' work. Please email us:</span>
<a class="editorial" href="mailto:primary.nrich@maths.org">primary.nrich@maths.org</a></p>
<p class="editorial">But a little time on and we got this reponse -
thank you.</p>
<p>We are Dominic and Sam (year 5) from St Nicolas CE Junior
School, Newbury.</p>
<p>We noticed that you had no solutions last month to this problem,
so here is our solution.</p>
<p>We started with 2 hoops, red and blue. We had 6 bean bags in the
red hoop and 4 in the blue. We worked out that we had to add the
number of bags in each together and subtract the number of bags to
get 2, which is the number of bags that go in the middle (where the
hoops overlap). Then if there are 6 in the red hoop, 2 are in the
middle so 4 are not in the middle.</p>
<p>On the other way, if there are 4 in the blue, 2 are in the
middle and 2 are not. For 3 hoops, we had 0 in the middle first and
then 1 in the middle, next 2 and lastly 3. This way we didn't have
two solutions the same. We worked out 14 solutions and we think we
found them all.</p>
<p><mdo:image width="399" height="596" alt="beans" src="Beans.jpg"></mdo:image></p>
<p> <mdo:image width="302" height="465" src="2ndBeans.jpg" alt="2ndbeans"></mdo:image></p>
<p> </p>
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<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div><a href="http://nrich.maths.org/public/viewer.php?obj_id=81&part=index">This activity</a> is especially good to do with a small number of pupils to give them confidence to talk with each other and discuss their ideas. It is amazing how many different ways of approaching the problem they will find.</div>
<div> </div>
<div>One teacher commented that "ALL children loved this! It was good for developing understanding of Venn diagrams and recording in a systematic way".</div>
<br></br>
<h3>Possible approach</h3>
<div>Introducing this activity as described in the problem with hoops and bean bags would be a good idea, if possible. This might mean going outside, taking over the school hall, or pushing all the tables to the edge of the classroom to create floor space. You could start with just two hoops and eight bean bags, and ask one child to put the bean bags in the hoops however they like. Ask questions
about this distribution of bean bags, such as the number in each hoop and the number in the overlap. It's important at this stage to listen to the pupils so that you can assess their understanding of the overlapping area.</div>
<br></br>
<div>Introduce a third hoop and set up the challenge. Invite children to work in pairs or groups to find at least one solution. Give them the choice as to what they use in terms of equipment: some may want hoops, some may be happy to draw representations. However they work at the problem, encourage them to record their solution/s somehow so they don't forget.</div>
<br></br>
<div>Once some solutions have been found, bring the group together again to share (and check!) the answers they've got so far. At this point, you can challenge them to find ALL the solutions. Give them a chance to talk in their pairs about how they might do this before sharing some suggestions. Draw attention to the methods which use some sort of system, so that there is less chance that
solutions will be missed out.</div>
<br></br>
<div>This investigation would make an engaging and attractive display, and you could encourage learners to describe the system they devised so that anyone looking at the display would be able to follow it.</div>
<br></br>
<div>(This activity would also be worth doing as staff CPD.)</div>
<br></br>
<h3>Key questions</h3>
<div>How many bean bags are in <span style="font-style: italic;">this</span> space?</div>
<div>Are there the correct number in the red hoop etc.?</div>
<br></br>
<h3>Possible extension</h3>
<div>Simply ask the pupils to suggest extensions by asking "I wonder what would happen if we ...?"</div>
<h3><br></br>
Possible further work which leads to material for the exceptionally mathematically able</h3>
Go to <a href="https://nrich.maths.org/36?part=note">Plants teachers' notes</a><br></br>
<h3>Possible support</h3>
<div>You could provide <a href="/content/00/09/bbprob1/BeanBags.pdf">this recording sheet</a> for those having difficulty in recording. It contains the three hoops printed six times.</div>
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<mdoxml version="1.0">You might find <a href="/content/00/09/bbprob1/BeanBags.pdf">this
recording sheet</a> useful.<br></br>
How will you know that you have found all the ways?
Have you got a good system?<br></br>
You could start by focusing on the blue hoop, for
example. Where <span style="font-style: italic;">could</span> the bean bags go? <br></br></mdoxml>
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<mdoxml version="1.0">There's not a lot to say about this. The children were in the
1980's and were the lower achievers and found it quite captivating.
I'm quite convinced that with a helper 6 year olds up wards can get
into this even if they only use two hoops.<br></br>
<br></br>
If you use this as some staff inservice, it shows how important it
is to do this kind of activity for real and not just do it on
paper. The discussions are also very valuable - people do this
activity in so many different ways.<br></br>
<br></br>
I'm pretty certain that the following results are correct for
producing 4, 5 & 6 in the hoopswith a certain
number of bean bags:- <br></br>
using 6 bags- 2 ways <br></br>
using 7 bags- 6 ways <br></br>
using 8 bags- 16 ways <br></br>
using 9 bags- 21 ways <br></br>
using 10 bags- 28 ways<br></br>
using 11 bags- 18 ways <br></br>
using 12 bags- 8 ways <br></br>
using 13 bags- 5 ways <br></br>
using 14 bags- 3 ways <br></br>
using 15 bags- 1 way <br></br>
<br></br>
Now there's always something about numbers produced by doing an
activity like this. I'm convinced of that and pupils can catch the
excitement as you all find different things. I've not explored the
four hoops situation.<br></br></mdoxml>
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Bean Bags for Bernard's Bag
How could you put eight beanbags in the hoops so that there are
four in the blue hoop, five in the red and six in the yellow? Can
you find all the ways of doing this?
Using, Applying and Reasoning about Mathematics
Investigations
Calculations and Numerical Methods
Addition & subtraction
Decision Mathematics and Combinatorics
Combinations
Using, Applying and Reasoning about Mathematics
Working systematically