5578
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2011-02-01T00:00:01
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Here is an example of a factor-multiple chain of four
numbers:<br></br>
<br></br>
<mdo:image height="91" width="550" alt="3-6-30-90" src="egchain.gif"></mdo:image><br></br>
<br></br>
Can you see how it works? Perhaps you could make some statements
about some of the numbers in the chain using the words "factor" and
"multiple".<br></br>
<br></br>
In these chains, each blue number can range from $2$ up to $100$
and must be a whole number.<br></br>
<br></br>
You may like to experiment with <a href="/content/id/5578/Chains.xls">this spreadsheet</a> which allows you
to enter numbers in each box. Perhaps you can make some more chains
for yourself.<br></br>
<br></br>
What are the smallest blue numbers that will make a complete chain?
<br></br>
What are the largest blue numbers that will make a complete chain?
<br></br>
What numbers cannot appear in any chain? <br></br>
What is the biggest difference possible between two adjacent blue
numbers? <br></br>
What is the largest and the smallest possible range of a complete
chain? (The range is the difference between the largest and
smallest values.) <br></br>
<br></br>
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<p class="editorial">Joel, Callum and Bethan from William Harding
School each sent in a solution to this problem. They all agreed
that the smallest blue numbers that make a complete chain are
2-4-8-16 and the largest blue numbers that will make a complete
chain are 5-25-50-100. (Although what is the largest number
possible in position one?)</p>
<p class="editorial">Tiberiu wrote to us to say:</p>
All prime numbers between 13 and 99 cannot be used in a
chain.<br></br>
<br></br>
<p class="editorial">I wonder if you can explain why? Are there
some other numbers which can't appear?</p>
<p class="editorial">Tiberiu continued:</p>
Three numbers closest to 2 and the last one closest to 100 will
give you the largest difference <span class="editorial">(between
adjacent numbers)</span> : 2,4,8,96. The largest difference in this
case is 88.<br></br>
The largest range: 2,10,50,100 range: 98 <br></br>
The smallest range: 2,4,8,16 range: 14<br></br>
<br></br>
<p class="editorial">The "Southville Sizzlers" at Southville
Primary School also worked hard on this problem. Well done to all
of you.</p>
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<h2>Factor-multiple Chains</h2>
<br></br>
Here is an example of a factor-multiple chain of four numbers:<br></br>
<br></br>
<mdo:image alt="3-6-30-90" height="91" src="egchain.gif" width="550"></mdo:image><br></br>
<br></br>
Can you see how it works? Perhaps you could make some statements about some of the numbers in the chain using the words "factor" and "multiple".<br></br>
<br></br>
In these chains, each blue number can range from $2$ up to $100$ and must be a whole number.<br></br>
<br></br>
You may like to experiment with <a href="/content/id/5578/Chains.xls">this spreadsheet</a> which allows you to enter numbers in each box. Perhaps you can make some more chains for yourself.<br></br>
<br></br>
What are the smallest blue numbers that will make a complete chain?<br></br>
What are the largest blue numbers that will make a complete chain?<br></br>
What numbers cannot appear in any chain?<br></br>
What is the biggest difference possible between two adjacent blue numbers?<br></br>
What is the largest and the smallest possible range of a complete chain? (The range is the difference between the largest and smallest values.)<br></br>
<br></br>
</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=5578&part=index">This problem</a> offers opportunities for pupils to reinforce their understanding of factors and multiples, and to become confident in using this vocabulary. It also gives them the chance to justify their solutions and to be creative by making their own chains.</div>
<br></br>
<h3>Possible approach</h3>
<div>You may wish to show <a href="/content/id/5578/Chains.xls">this spreadsheet</a> to the whole class to introduce the problem. It is an interactive chain, which allows you to enter numbers in each box, giving feedback as to when factors and multiples occur. As you make different chains, ask the children to explain what is happening so that everyone fully understands the environment.</div>
<br></br>
<div>Encourage pupils to work on the different questions, ideally in pairs so that they have somone with whom to discuss their ideas. They could work on paper or mini-whiteboards, and access to calculators might be useful. You might like the class to be at computers so they can manipulate the spreadsheet themselves and check their solutions. You may have to discuss what is meant by 'largest' and
'smallest', and come to an agreement. You could ask two questions, for example 'What are the largest blue numbers that will make a complete chain?' and 'What is the largest possible first number in a chain?'.</div>
<br></br>
<div>When sharing solutions, encourage learners to justify their answers - how do they know that their chain contains the smallest/largest numbers etc.? Some children will be using trial and improvement, some will have developed a system for trying numbers in turn, and others may have been able to combine these with their knowledge of number properties.</div>
<br></br>
<h3>Key questions</h3>
<div>Can any number start the chain? Why/why not?</div>
<div>Can any number be at the end of a chain? Why/why not?</div>
<div>What can you say about the numbers in the second and third positions in a chain?</div>
How will you keep track of what you have tried?<br></br>
<h3>Possible extension</h3>
<div>Children could investigate what would happen if the chain was made up of more than four numbers, or made up from a different range of numbers.</div>
<br></br>
<h3>Possible support</h3>
<div>Enabling learners to manipulate the spreadsheet for themselves will help them access this problem and you can focus on justifying the solutions rather than pupils worrying about obtaining them. There is still a great deal of mathematical thinking taking place as children interact with the spreadsheet.</div>
<br></br>
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If you haven't already had a look, it might be worth investigating
<a href="/content/id/5578/Chains.xls">this
spreadsheet</a> which you can put different numbers into to make
chains.<br></br>
For the smallest chain, what number might it be a good idea to
start with on the left?<br></br>
What could you try first to make the largest chain? You could
choose to start with a particular number on the right this
time.<br></br>
How will you know which numbers you have already tried? <br></br></mdoxml>
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smallest: 2-4-8-16<br></br>
largest: 5-25-50-100 produces the largest possible number in the
last three positions but 12-24-48-96 contains the largest number
possible in position one<br></br>
for example, 26 cannot be in a chain. The latest it can appear is
position two with either 2 or 13 in position one but position four
is limited to a number up to 100 and so cannot offer a value to
make a chain. Also prime numbers must occupy position one in any
chain in which they appear<br></br>
88 is the maximum difference between adjacent numbers in a chain 2-
4-8-96<br></br>
For a greatest range 2-4-8-96 looks promising but 5-25-50-100 is
greater still, 95. The minimum range is 14 produced by
2-4-8-16<br></br>
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Factor-Multiple Chains
Can you see how these factor-multiple chains work? Find the chain
which contains the smallest possible numbers. How about the largest
possible numbers?
Information and Communications Technology
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Numbers and the Number System
Comparing and Ordering numbers
Using, Applying and Reasoning about Mathematics
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Numbers and the Number System
Factors and multiples
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