1885
/www/nrich/html/content/03/10/penta1/
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2011-02-01T00:00:01
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<mdoxml version="1.0"><br></br>
<p>Four bags contain a large number of $1$s, $3$s, $5$s and $7$s.</p>
<p align="center"><mdo:image alt="four bags containing 1s, 3s, 5s and 7s" height="117" src="1_3_5_7.gif" width="391"></mdo:image></p>
<p align="left">Pick any ten numbers from the bags above so that their total is $37$.</p>
<br></br>
<a href="http://nrich.maths.org/7205&part=">Click here for a poster of this problem</a>.
<p> </p></mdoxml>
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<mdoxml version="1.0"><br></br>
<p class="editorial">I hope you enjoyed tackling this problem - a
bit of a teaser! Many of you spotted that it is impossible to make
37 from ten odd numbers. Adam and Ryan from Moorfield Junior School
wrote:</p>
<blockquote>
<p>This problem is not possible because with an even number of odd
numbers you cannot make an odd number. You can make 36 and 38 using
10 numbers but not 37. You can make 37, but by using 9 numbers.
Here are some examples:</p>
<p>36 (10 numbers): 5 + 5 + 5 + 5 + 5 + 3 + 3 + 3 + 1 + 1<br></br>
38 (10 numbers): 1 + 1 + 1 + 3 + 3 + 5 + 5 + 5 + 7 + 7<br></br>
37 (9 numbers): 5 + 5 + 5 + 5 + 5 + 5 + 5 + 1 + 1</p>
</blockquote>
<p class="editorial">Alice from Tattingstone School sent a similar
explanation and demonstrated other ways of making 37 with nine,
eleven and thirteen numbers:</p>
<blockquote>
<p>37 (9 numbers): 5 + 5 + 5 + 5 + 5 + 5 + 3 + 3 + 1<br></br>
37 (9 numbers): 5 + 5 + 5 + 5 + 5 + 7 + 1 + 1 + 3<br></br>
37 (11 numbers): 5 + 5 + 5 + 5 + 5 + 3 + 3 + 3 + 1 + 1 + 1<br></br>
37 (13 numbers): 7 + 7 + 7 + 7 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 +
1</p>
</blockquote>
<p class="editorial">Joshua from Tattingstone School explained very
clearly why the problem was impossible:</p>
<blockquote>
<p>I looked at the numbers in the bag and discovered that they were
all odd. I know that it is a mathematical fact that if you take any
two odd numbers and add them together, you will always get an even
number as the answer e.g. 9 + 7 = 16. Therefore any even
combination of odd numbers will also always give an even number as
the answer e.g. 7 + 1 + 5 + 9 = 22. The question asks me to make
37, which is an odd number, out of 10 odd numbers which due to the
facts above is impossible.</p>
</blockquote>
<p class="editorial">Joanna from Tattingstone suggested more ways
of making 36 or 38 using ten numbers and mentioned that she could
make 37 with seven, nine, eleven etc numbers. Interestingly, Joanna
pointed out that you couldn't make 37 using five of the numbers
because 7 is the biggest number you could use and 5 x 7 = 35 which
is no good.</p>
<p class="editorial">Alistair who goes to Histon and Impington
Junior School looked at it in a slightly different way:</p>
<blockquote>
<p>You can make 37 using a combination of 9 numbers, but to use 10
numbers, you need to split one of the 9 numbers into two others
(that are in the bags).<br></br>
Since all these numbers are odd, it is impossible to split them up
into exactly two whole numbers from other bags.</p>
</blockquote>
<p class="editorial">Aryeh and Gitty from Bournemouth Jewish Day
School found it was impossible and suggested that you could do it
if you could multiply then add or subtract:</p>
<blockquote>
<p>5 x 5 + 5 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 37<br></br>
5 x 5 + 3 + 1+ 1 + 3 +1 + 1 + 1 + 1 = 37<br></br>
5 x 7 + 1 + 1 + 1 + 1 - 1 - 1 - 1 + 1 =37</p>
</blockquote>
<p class="editorial">Julie from Hillside Community Primary School
also noticed that if multiplying were allowed, it would have
worked:</p>
<blockquote>
<p>1 x 1 x 1 x 1 x 1 x 1 x 5 x 7 + 5 - 3 = 37</p>
</blockquote>
<p class="editorial">Well done also to all these people who noticed
that you couldn't make 37 in this way:</p>
<blockquote>
<ul>
<li><span class="editorial">Charlotte and Lauren from Hillside
Community Primary School</span></li>
<li><span class="editorial">Helen, Katie, Pauline, Ryo and Jake
from Moorfield Junior School</span></li>
<li><span class="editorial">Deep, Chad, Henri, and Olof at the
Canadian Academy in Kobe, Japan</span></li>
<li><span class="editorial">Pupils at Balgowan Primary
School</span></li>
</ul>
</blockquote>
<br></br></mdoxml>
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<mdoxml version="1.0"><div class="embed">
<h2>Make 37</h2>
<br></br>
<p>Four bags contain a large number of $1$s, $3$s, $5$s and $7$s.</p>
<p align="center"><mdo:image alt="four bags containing 1s, 3s, 5s and 7s" height="117" src="1_3_5_7.gif" width="391"></mdo:image></p>
<p align="left">Pick any ten numbers from the bags above so that their total is $37$.</p>
<br></br>
<a href="http://nrich.maths.org/7205&part=">Click here for a poster of this problem</a>.
<p> </p>
</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=1885&part=index">Make 37</a> offers learners the chance to apply basic number properties and operations, and will challenge the idea of mathematical problems having one right answer. It is an opportunity to encourage children to generalise and, even at this level, they can be asked to prove these generalisations.</div>
<br></br>
<h3>Possible approach</h3>
<div>One way to start is to ask pupils to make $16$ using any four of the numbers. Make sure they have paper or mini whiteboards for working on. They will quickly find a variety of answers.</div>
<br></br>
<div>Declare this is too easy and set them on the task of making $37$ using ten of the numbers. Then give the pupils time to "play", trying out different numbers for themselves. It would be a good idea to encourage them to check their results with a neighbour! Refrain from asking if it is possible; wait until a pupil suggests it can't be done, to which the response could be: "Why can't it be
done?"</div>
<br></br>
<div>It might be appropriate to ask pupils what totals they <span style="font-style: italic;">have</span> been able to make and what they notice about these totals. Turning pupils' attention to the properties of the totals that <span style="font-style: italic;">are</span> possible will encourage them to suggest conclusions. Have they noticed that all the totals are even? This may alert them to
the fact that making $37$ is much harder than they thought or does not seem to be possible.</div>
<br></br>
<div>Having given the group more time to work in pairs or small groups, draw their ideas together using the key questions below. Once they have drawn conclusions based on odd and even numbers, challenge them to convince you this is the case. Drawing <a href="/content/03/10/penta1/odd%2Bodd.doc">diagrams</a> of odd and even numbers may help pupils to visualise the mathematics.</div>
<br></br>
<div>A follow-up question along the lines of "How could you change the problem so that we <span style="font-style: italic;">can</span> make $37$ from ten numbers?" or "How can we change the problem so that it <span style="font-style: italic;">is</span> possible?" provides a useful opportunity to gauge how fully the children have understood the ideas.</div>
<br></br>
<h3>Key questions</h3>
<div>What do you notice about the numbers in the bags? What do they have in common?</div>
<div>What totals have you managed to make using ten numbers? Do they have anything in common?</div>
<div>Can you make $37$ with a different number of numbers? How many?</div>
<div>Can you explain why?</div>
<br></br>
<h3>Possible extension</h3>
<div>This problem could be used as a starter before tackling the much more challenging <a href="http://nrich.maths.org/public/viewer.php?obj_id=1866&part=index">Take Three from Five</a> .</div>
<br></br>
<h3>Possible support</h3>
<div>You can use a lower total and fewer numbers to make the arithmetic easier without losing the mathematical thinking and reasoning. For example, the challenge could be to make $15$ using four numbers.</div>
<br></br>
<br></br></mdoxml>
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<mdoxml version="1.0"><br></br>
<p>What numbers <span style="font-style: italic;">can</span> you
make? Do they have anything in common?<br></br>
Have you made $37$ with a different number of numbers? How many? Do
these numbers have anything in common?<br></br>
What do you notice about the numbers in the bags?</p>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0">One way to start is to ask pupils to make 16 using any 4 of the
numbers. They will quickly find a variety of answers. Declare this
is too easy and set them on the task of making 37 using 10 of the
numbers. Make sure they have paper for working on. Then give the
pupils time to "play", trying out different numbers for themselves.
It would be a good idea to encourage them to check their results
with a neighbour! Refrain from asking if it is possible; wait till
a pupil suggests it can't be done. To which the response could be:
"Why can't it be done?" It might be appropriate to ask pupils what
numbers they HAVE been able to make and what they notice about
these totals. Turning pupils' attention to the properties of the
numbers that are possible will encourage them to suggest
conclusions. Have they noticed that all the totals are even? This
may alert them to the fact that making 37 is much harder than they
thought or does not seem to be possible. You could invite them to
share their thoughts using the following prompts: * Can you make 37
with a different number of numbers? How many? * Can you explain
why? Drawing diagrams of odd and even numbers may help pupils to
visualise the mathematics. A follow-up question along the lines of
"How could you change the problem so that we can make 37 from ten
numbers?" provides a useful opportunity to gauge how fully the
children have understood the ideas. This problem could be used as a
starter before tackling the more challenging Take Three from Five
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Make 37
Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten
numbers from the bags above so that their total is 37.
Numbers and the Number System
Odd and even numbers
Calculations and Numerical Methods
Addition & subtraction
Using, Applying and Reasoning about Mathematics
Generalising
Information and Communications Technology
smartphone
Admin
Upper primary mapping document