5429
/www/nrich/html/content/id/5429/
5
2011-02-01T00:00:01
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Here is a $100$ grid with some numbers shaded:<br></br>
<br></br>
<mdo:image height="416" width="450" src="grid1.gif" alt="multiples of 5 and 7"></mdo:image><br></br>
<br></br>
What do all the numbers shaded blue have in common?<br></br>
What do you notice about all the numbers shaded pink?<br></br>
Can you work out why two of the numbers are shaded in a maroon
colour?<br></br>
<br></br>
Now, here is part of a $100$ square shaded in a different
way:<br></br>
<br></br>
<mdo:image height="173" width="200" alt="shaded pink: 24, 26, 34, 36, 44, 46 shaded blue: 35" src="grid2.gif"></mdo:image><br></br>
<br></br>
Can you explain the shading this time?<br></br>
<br></br>
Here are some more parts of the $100$ square, each one shaded
according to different rules. Can you work out what the rules are
for each?<br></br>
Is there only one solution each time?<br></br>
<br></br>
<mdo:image height="173" width="200" src="grid3.gif" alt="shaded blue: 66 and 77; shaded maroon: 88"></mdo:image><br></br>
<br></br>
<mdo:image height="173" width="200" src="grid4.gif" alt="shaded blue: 45, shaded maroon: 36 and 54"></mdo:image><br></br>
<br></br>
<mdo:image height="173" width="200" src="grid5.gif" alt="shaded pink: 16; shaded blue: 27"></mdo:image><br></br>
<br></br>
<p><span style="font-style: italic;">This problem is featured in
Maths Trails - Excel in Problem Solving, one of the books in the
Maths Trails series written by members of the NRICH Team and
published by Cambridge University Press. For more details about the
other books in the series, please see our</span> <a href="http://nrich.maths.org/public/viewer.php?obj_id=4833&part=index" style="font-style: italic;">publications page</a> <span style="font-style: italic;">.</span></p>
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<p class="editorial">Lulu and Sophie from St. Nicolas C. of E.
Junior School, Newbury have sent us a solution to the Multiples
Grid problem. Thank you both! For the first large square, they
say:</p>
<br></br>
The blue numbers are all multiples of 7.<br></br>
<div style="clear: both;">The pink numbers are all multiples of
5.<br></br>
<div style="clear: both;">The maroon numbers are multiples of 5 and
7. (Both.)</div>
<div style="clear: both;"></div>
<p class="editorial" style="clear: both;">In this square:</p>
<div style="clear: both;"></div>
<div style="clear: both;"><mdo:image width="201" height="172" alt="" src="grid2.gif"></mdo:image></div>
<p class="editorial" style="clear: both;">they say:</p>
<div style="clear: both;">The pink numbers are multiples of
2.</div>
<div style="clear: both;">
<div>The blue numbers are multiples of 5 or 7.</div>
<br></br>
<div style="clear: both;"></div>
<p style="clear: both;" class="editorial">Are you sure the blue
could be multiples of 5? In this square:</p>
<div style="clear: both;"></div>
<div style="clear: both;"><mdo:image width="198" height="171" src="grid3.gif" alt=""></mdo:image></div>
<div style="clear: both;"></div>
<p style="clear: both;" class="editorial">they say:</p>
<div style="clear: both;"></div>
<div style="clear: both;">The blue numbers are multiples of
11.<br></br>
<div style="clear: both;">The pink numbers are supposed to be
multiples of 8.<br></br>
<div style="clear: both;">The maroons are multiples of 8 and
11.</div>
<div style="clear: both;"></div>
<p style="clear: both;" class="editorial">In the next square:</p>
<div style="clear: both;"></div>
<div style="clear: both;"><mdo:image width="198" height="171" src="grid4.gif" alt=""></mdo:image></div>
<div style="clear: both;"></div>
<p style="clear: both;" class="editorial">they explain:</p>
<div style="clear: both;"></div>
<div style="clear: both;">The blues are multiples of 9.<br></br>
<div style="clear: both;">The pinks are multiples of 6.<br></br>
<div style="clear: both;">The maroon numbers are multiples of 9 and
6.</div>
<div style="clear: both;"></div>
<p style="clear: both;" class="editorial">Finally, in this last
square:</p>
<div style="clear: both;"></div>
<div style="clear: both;"><mdo:image width="198" height="171" src="grid5.gif" alt=""></mdo:image></div>
<div style="clear: both;"></div>
<p style="clear: both;" class="editorial">Sophie and Lulu
wrote:</p>
<div style="clear: both;"></div>
<div style="clear: both;">The blues are multiples of 9.<br></br>
<div style="clear: both;">The pinks are multiples of 4 or 8.</div>
<div style="clear: both;"></div>
<p style="clear: both;" class="editorial">Thank you for sending us
your solution, girls. Well done!</p>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
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<h2>Multiples Grid</h2>
<br></br>
Here is a $100$ grid with some numbers shaded:<br></br>
<br></br>
<mdo:image alt="multiples of 5 and 7" height="416" src="grid1.gif" width="450"></mdo:image><br></br>
<br></br>
What do all the numbers shaded blue have in common?<br></br>
What do you notice about all the numbers shaded pink?<br></br>
Can you work out why two of the numbers are shaded in a maroon colour?<br></br>
<br></br>
Now, here is part of a $100$ square shaded in a different way:<br></br>
<br></br>
<mdo:image alt="shaded pink: 24, 26, 34, 36, 44, 46 shaded blue: 35" height="173" src="grid2.gif" width="200"></mdo:image><br></br>
<br></br>
Can you explain the shading this time?<br></br>
<br></br>
Here are some more parts of the $100$ square, each one shaded according to different rules. Can you work out what the rules are for each?<br></br>
Is there only one solution each time?<br></br>
<br></br>
<mdo:image alt="shaded blue: 66 and 77; shaded maroon: 88" height="173" src="grid3.gif" width="200"></mdo:image><br></br>
<br></br>
<mdo:image alt="shaded blue: 45, shaded maroon: 36 and 54" height="173" src="grid4.gif" width="200"></mdo:image><br></br>
<br></br>
<mdo:image alt="shaded pink: 16; shaded blue: 27" height="173" src="grid5.gif" width="200"></mdo:image><br></br>
<br></br>
<p><span style="font-style: italic;">This problem is featured in Maths Trails - Excel in Problem Solving, one of the books in the Maths Trails series written by members of the NRICH Team and published by Cambridge University Press. For more details about the other books in the series, please see our</span> <a href="http://nrich.maths.org/public/viewer.php?obj_id=4833&part=index" style="font-style: italic;">publications page</a> <span style="font-style: italic;">.</span></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=5429&part=index">This problem</a> is an interesting way of reinforcing understanding of factors and multiples.</div>
<br></br>
<h3>Possible approach</h3>
<div>To start with, ask children to talk in pairs about why the numbers in the first 100 square are shaded blue, pink and maroon. Invite them to share their ideas and encourage correct use of vocabulary.</div>
<br></br>
Learners could continue to work in pairs, perhaps using <a href="/content/id/5429/multiplesgrid.doc">this sheet</a> of the four parts of differently-shaded 100 squares. As they work on the problem, trying to find out which factors have been chosen in order to produce the shading, encourage them to justify their solutions to their partners, and perhaps then to the whole class. How are they going
about the task? It might be useful to discuss ways of working systematically so that no solutions are omitted.<br></br>
<br></br>
<a href="/content/id/5429/Multiples%20Grid.xls">This spreadsheet</a> ,which shades the squares according to the chosen factors, can be used to check their hypotheses. In a plenary session, you could use the second sheet of the spreadsheet to pre-prepare some shaded sections of the 100 grid without numbers. If you tell them which multiples have been shaded, can the class work out where the small
part of the 100 grid is, i.e. which numbers it contains?<br></br>
<br></br>
<h3>Key questions</h3>
<div>What do the numbers shaded blue have in common?</div>
<div>What do the pink numbers have in common?</div>
<div>Can you rule out some factors straight away? How?</div>
<div>How will you know you have found all the possible solutions?</div>
<br></br>
<h3>Possible extension</h3>
<div>Learners could explore the spreadsheet for themselves at a computer. Challenge them to make up their own questions to ask a friend.</div>
<br></br>
<h3>Possible support</h3>
<div>A multiplication square may be useful for those children who find instant recall of multiplication facts difficult.</div>
<br></br>
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There is a clue in the title of the problem!<br></br>
You may like to investigate <a href="/content/id/5429/Multiples%20Grid.xls">this spreadsheet</a> which
shades the numbers on the $100$ square. The pictures have been
taken from this spreadsheet. Can you see what is happening? <br></br>
Are there some answers which you can rule out straight away in each
case?<br></br>
How will you know you have all the solutions for each grid?<br></br></mdoxml>
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multiples of 2 and 7<br></br>
multiples of 8 and 11<br></br>
multiples of 6 and 9 or 3<br></br>
multiples of 4 and 9, 8 and 9, 16 and 9... (depending on how high
you go)<br></br>
<br></br>
This problem is an interesting way to reinforce understanding of
factors and multiples. You might like to use this spreadsheet with
the class, which shades the squares according to the chosen
factors. One way of introducing the problem might be to invite
students to investigate the spreadsheet in pairs so that they get a
good feel for what it is doing. As they work on the problem, trying
to find out which factors have been chosen in order to produce the
shading, encourage them to justify their solutions to their
partners, and perhaps then to the whole class. How are they going
about the task? It might be useful to discuss ways of working
systematically so that no solutions are omitted. The spreadsheet
can be used to check their hypotheses. <br></br></mdoxml>
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Multiples Grid
What do the numbers shaded in blue on this hundred square have in
common? What do you notice about the pink numbers? How about the
shaded numbers in the other squares?
Information and Communications Technology
Interactivities
Information and Communications Technology
Spreadsheets
Numbers and the Number System
Factors and multiples
Using, Applying and Reasoning about Mathematics
Working systematically
Admin
Upper primary mapping document