1175
/www/nrich/html/content/03/06/penta2/
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2011-02-01T00:00:01
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<p>How many different sets of numbers with at least four members can you find in the numbers in this box?</p>
<p class="c1" style="text-align: center;"><mdo:image alt="" height="179" src="4sets.png" width="344"></mdo:image></p>
<p class="c2"> </p>
<p class="c2">For example, one set could be multiples of $4$ {$8, 36 ...$}, another could be odd numbers {$3, 13 ...$}.</p>
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<div><a href="http://nrich.maths.org/public/viewer.php?obj_id=5906&part=">Click here for a poster of this problem</a>.</div>
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<p class="editorial">Children from Beaumaris North Primary School,
Melbourne told us:</p>
In our Year 3 Enrichment group we discussed all the possible sets
that could be found. Here is what we found!<br></br>
even numbers {$2, 8, 36, 56, 64, 136$}<br></br>
odd numbers {$3, 13, 17, 27, 39, 49, 51, 91, 119, 121, 125,
143$}<br></br>
multiples of one {all of them!}<br></br>
multiples of $2$ {$2, 8, 36, 56, 64, 136$ - the same as even
numbers!!}<br></br>
multiples of $3$ {$3, 27, 36, 39, 51$}<br></br>
multiples of $4$ {$8, 36, 56, 64, 136$}<br></br>
multiples of $7$ {$49, 56, 91, 119$}<br></br>
multiples of $8$ {$8, 56, 64, 136$}<br></br>
numbers starting with $1$ {$13, 17, 119, 121, 125, 136, 143$}<br></br>
square numbers {$36, 49, 64, 121$}<br></br>
numbers containing $2$ {$2, 27, 121, 125$}<br></br>
square roots {all of them are square roots !}<br></br>
numbers containing $1$ {$13, 17, 51, 91, 119, 121, 125, 136,
143$}<br></br>
numbers containing $6$ {$36, 56, 64, 136$}<br></br>
cube numbers {$8, 27, 64, 125$}<br></br>
numbers containing $3$ {$3, 13, 36, 39, 136, 143$}<br></br>
prime numbers {$2, 3, 13, 17$}<br></br>
numbers containing $9$ {$39, 49, 91, 119$}<br></br>
numbers that you can subtract one from {all of them again!}<br></br>
numbers between one and one hundred and forty four {all of them
AGAIN!}<br></br>
numbers between one and one hundred {$2, 8, 13, 17, 27, 36, 39, 49,
51, 56, 64, 91$}<br></br>
whole numbers {all of them}<br></br>
triple digit numbers {$119, 121, 125, 136, 143$}<br></br>
numbers with double digits {$13, 17, 27, 36, 39, 49, 51, 56, 64,
91$}<br></br>
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We enjoyed this because we liked picking out different
possibilities and seeing if they worked out. We could have
submitted more but the bell rang! Thanks for the great
challenge.<br></br>
<br></br>
<p><span class="editorial">Olly from North Molton Primary also
found:</span></p>
multiples of $13$: {$13, 39, 91, 143$}<br></br>
multiples of $17$: {$17, 51, 119, 136$}<br></br>
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... and multiples of $1$ which of course is all of the
numbers.<br></br>
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Jin and Anushkawho go to Dubai International Academy had one more
set:<br></br>
<br></br>
digits add to $8${$8,17,125,143$}<br></br>
<p class="editorial">Can you find any more? How about
triangle numbers and tetrahedral numbers?</p>
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<h3><span style="font-weight: bold;">Why do this
problem?</span></h3>
<a href="http://nrich.maths.org/public/viewer.php?obj_id=1175&part=index">
This problem</a> gives plenty of opportunity for both discussion
and exploring patterns, properties and relationships involving
numbers. It also helps learners to make general statements about
properties of numbers such as squares, factors, multiples, odd and
even, and to identify examples for which a statement is true or
false.<br></br>
<h3>Possible approach</h3>
<div>You could also use <a href="/content/03/06/penta2/like100.swf">this interactivity</a> as a
starter to this problem where you drag numbers you "like" (i.e. are
part of a set) to one side and numbers you "don't like" (i.e. are
not in your set) to the other. The children then have to ask
questions with yes/no answers to determine the name of your set.
You could invite some learners to take the lead on this themselves,
with the rest of the group asking questions.</div>
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<div>You could then introduce the problem itself and learners could
work in pairs from <a href="/content/03/06/penta2/1175.pdf">this
sheet</a> so that they are able to talk through their ideas with a
partner. (The sheet has two copies of the box of numbers on it.)
You could also challenge pairs to find two other possible members
for their sets from the numbers under $150$.</div>
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<div>At the end learners could be asked for the titles of the sets
they have found and then others could suggest members for these
sets. You could make statements such as "$9$ is a member of this
set" and ask learners to identify whether the statement is true or
false. When sets of multiples of $2$, $4$ and $8$ have been found
and discussed, learners could be asked to explain why no other sets
of multiples of even numbers are possible from the numbers in the
box.</div>
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<h3>Key questions</h3>
<div>What can you tell me about this number?</div>
<div>Is this odd or even? What else do you know about it?</div>
<div>Which multiplication tables will you find this number
in?</div>
<div>Can you see any other multiples of that number?</div>
<div>Can you suggest other numbers that could go together in this
set?</div>
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<h3>Possible extension</h3>
Learners could make their own list of numbers to make into as many
sets as possible. You could suggest using triangular and cube
numbers as well as various multiples.<br></br>
<h3>Possible support</h3>
Some children might find it useful to list all the properties of
each number in the box.<br></br>
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You could list all the properties of each number as a
start.<br></br>
Can you see some numbers which have the same property?<br></br></mdoxml>
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1
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Sets of Numbers
How many different sets of numbers with at least four members can
you find in the numbers in this box?
Using, Applying and Reasoning about Mathematics
Investigations
Sequences, Functions and Graphs
Sequences
Numbers and the Number System
Factors and multiples
Numbers and the Number System
Odd and even numbers