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/www/nrich/html/content/02/10/penta5/
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2011-02-01T00:00:01
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<br></br>
Arrange the four number cards (1, 2, 3 and 21) on the grid above to make a diagonal, vertical or horizontal line.<br></br>
You can put a number card on a square with
<ul>
<li>the same number</li>
<li>a multiple of that number</li>
<li>a factor of that number.</li>
</ul>
<br></br>
In how many different ways can you do it?<br></br>
<br></br>
You can use the interactivity above or <a href="/content/02/10/penta5/Factor%20Lines.pdf">this sheet</a> of grids. The interactivity keeps track of the number of different lines you have created and you can view the lines you've already made by clicking 'Show lines'.<br></br>
<br></br>
This problem has been adapted from the book "Cards on the Table" by Fran Mosley, published by <a href="http://www.beam.co.uk">BEAM Education</a> .<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<p class="editorial">Joe and Sarah described their solutions very
well, although they didn't find all the possible lines of four.
Here is Joe's:</p>
<p><mdo:image height="648" width="477" alt="Joe's Block 4 Solution" src="block_4_sol.jpg"></mdo:image></p>
<p class="editorial">Clement Chan from Sha Tin College in Hong Kong
has found thirteen ways. He says:</p>
<p>I first tried to find all the possible lines I could make on the
grid than looked at each one carefully for example:</p>
<p>I want to make a vertical line from the $1$, I check whether the
$21$ can go in anywhere and if it can I check whether the $3$ can
go anywhere, and then the $2$ and put the $1$ on to finish the
line.</p>
<p>If I want to make a line diagonally from the $5$ then I check it
in the same way, and if I want to make a horizontal line from the
$15$ I check it the same way etc.</p>
<p></p>
<p class="editorial">Laquiesha and Chloe from Kingsmoor Primary
also thought they had found thirteen ways, but they didn't list
them all. They pointed out:</p>
<p>We also found out that you couldn't do diagonals because the no.
$21$ had to be placed on the four odd numbers $1$, $3$, $7$, $21$
and you couldn't put the number $2$ in the diagonal because $2$
needs to be on a even number but they were all odd in the
diagonal.</p>
<p class="editorial">(In fact, $2$ could go on the $1$ in the
diagonal, couldn't it? But I see what you mean, that you can't then
place the other two numbers in the diagonal.) Ben T and Ben H from
Brewood Middle School realised that diagonals are impossible
too.</p>
<p class="editorial">Mrs Rankin from Sandown Primary School wrote
to say:</p>
<div>A group of children worked together on this investigation
today and became very excited when they found fifteen solutions to
this problem by using a systematic approach!! (That'??s one more
than your solution suggests!)</div>
<br></br>
<p class="editorial">Well done all of you - you're right. We
thought that there were only fourteen solutions. We obviously
weren't being as systematic as we thought! Here is a table which
shows the group's solutions:</p>
<br></br>
<mdo:image height="492" width="307" src="sandownsol.gif" alt=""></mdo:image><br></br>
<br></br>
<p class="editorial">I like the way you have organised your
recording. I have highlighted the two $21$s in the
bottom of your table because I think in fact these should be $7$s.
I hope you agree with me.</p>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><div class="embed">
<h2>Factor Lines</h2>
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<param name="allowFullScreen" value="true"></param>
<param name="flashplayerversion" value="10"></param>
</mdo:flash></div>
<br></br>
Arrange the four number cards (1, 2, 3 and 21) on the grid above to make a diagonal, vertical or horizontal line.<br></br>
You can put a number card on a square with<br></br>
<ul>
<li>the same number</li>
<li>a multiple of that number</li>
<li>a factor of that number.</li>
</ul>
<br></br>
In how many different ways can you do it?<br></br>
<br></br>
You can use the interactivity above or <a href="/content/02/10/penta5/Factor%20Lines.pdf">this sheet</a> of grids. The interactivity keeps track of the number of different lines you have created and you can view the lines you've already made by clicking 'Show lines'.<br></br>
<br></br>
This problem has been adapted from the book "Cards on the Table" by Fran Mosley, published by <a href="http://www.beam.co.uk">BEAM Education</a> .</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=1138&part=index">This problem</a> will help children become more familiar with factors and multiples, and in order to find all the lines, a systematic approach is required. It will encourage learners to talk to each other and provides a good context in which to discuss different methods of approach. It has lots of scope for being
opened out and extended through 'what if ...?' questions.</div>
<br></br>
<h3>Possible approach</h3>
<div>You could introduce this problem by showing the group the interactivity in an edited form. Click on the 'View Editor' button and you can then choose the numbers of rows, columns and the length of a 'winning' line. If you choose to have a $4$ by $4$ grid of numbers $1$ to $16$. You can change the numbers to drag onto the grid by clicking in the corners of them and increasing or decreasing
them so choose to make lines of three and drag the numbers $3$, $5$ and $10$. Tell the children that you are going to make lines of three, but they have to work out the rules for making a line. Make one line (e.g. the $5$, $10$ and $3$ over the $1$, $2$ and $3$ respectively) and invite the children to talk to each other about what they think the rules are at this stage. Then give them another
line (e.g. the $5$, $10$ and $3$ over the $1$, $5$ and $9$ respectively) and ask them to talk further to clarify the rules. You could give them another line before having a whole-class discussion about the rules.</div>
<br></br>
<div>Once you have established the rules, you could set the challenge of finding lines of four with the $1$ to $25$ grid. Children could use the interactivity or a copy of <a href="/content/02/10/penta5/Factor%20Lines.pdf">this sheet</a> . Give them time to work in pairs without saying too much more and then bring them all together for a mini-plenary at a suitable moment. Give learners a chance
to share what they have done so far so that some different ways of approaching the task are revealed and different ways of recording are discussed. You may find some draw attention to the properties of some of the numbers in the grid, for example the way in which the prime numbers affect a line of four. If this does not come up naturally, you may like to bring it up later in the lesson.</div>
<br></br>
<div>After giving children more time to work on the problem, you could invite everyone to contribute to a whole-group solution, perhaps recording lines collectively on the board using the interactivity. At this point, encourage the pupils to scrutinise the grid carefully so that everyone is happy all the lines have been found. This will rely on having a systematic way of working, but there are
many systems that could be adopted.</div>
<br></br>
<div>You can extend this challenge in many different ways (see below) and you may like to choose some of the suggestions to leave for children to mull over during the following week. You could set up part of your wall as a place for learners to pin up contributions and then you could plan to review the work altogether at a future date.</div>
<br></br>
<div>Of course, the interactivity in this problem can be returned to time after time, as it can be tailored to suit your objectives.</div>
<br></br>
<h3>Key questions</h3>
<div>Which of the cards can be multiples?</div>
<div>Which can be factors?</div>
<div>Can any be both?</div>
<div>How will you remember which lines you have found?</div>
<div>Are there any other ways to make that line?</div>
<div>Are there any numbers on the grid which are impossible to cover? Why or why not?</div>
<div>Which of the four numbers will be the most difficult to place do you think? Why?</div>
<div>Which of the four numbers will be the easiest to place do you think? Why?</div>
<br></br>
<h3>Possible extension</h3>
<div>Children could investigate whether by rearranging the grid numbers, they could increase/decrease the number of lines. Could they pick four different dragging numbers so that more/fewer lines could be made? Could they choose different grid numbers to increase/decrease the number of lines? What would happen if, rather than a line, the aim was to get a block of four, or an 'L' shape? Of course
children will be able to come up with their own questions to investigate.</div>
<br></br>
<h3>Possible support</h3>
<div>You can use the interactivity to change the size of the grid and the numbers used. Some children might find it easier to start with a $4$ by $4$ grid and to identify lines of three, for example. Cutting out the numbers at the bottom of <a href="/content/02/10/penta5/Factor%20Lines.pdf">this sheet</a> might be useful so that they can be moved around the grid. Alternatively, transparent
counters with numbers stuck on them would be helpful.</div>
<br></br>
<div style="font-style: italic;">Many thanks to all the teachers participating in the <a href="http://nrich.maths.org/public/viewer.php?obj_id=6340&part=index">Enriching Mathematics in Devon Project</a> who provided lots of ideas and inspiration for these notes.</div>
<br></br>
<br></br></mdoxml>
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<mdoxml version="1.0"><br></br>
<div>Which of the cards can be multiples?</div>
<div>Which can be factors?</div>
<div>Can any be both?</div>
<div>How will you remember what lines you have found?</div>
<div>How will you know you have got them all?</div>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0">21, 22, 23, 24 using 21, 2, 1, 3<br></br>
21, 16, 11, 6 using 21, 2, 1, 3<br></br>
7, 12, 17, 22 using 21, 3, 1, 2<br></br>
7, 8, 9, 10 using 21, 2, 3, 1<br></br>
7, 8, 9, 10 using 21, 1, 3, 2 <br></br>
2, 7, 12, 17 using 2, 21, 3, 1<br></br>
6, 7, 8, 9 using 1, 21, 2, 3<br></br>
6, 7, 8, 9 using 3, 21, 2,
1 <br></br>
6, 7, 8, 9 using 2, 21, 1, 3<br></br>
3, 8, 13, 18 using 21, 2, 1, 3<br></br>
1, 2, 3, 4 using 21, 1, 3, 2<br></br>
1, 2, 3, 4 using 21, 2, 3, 1<br></br>
1, 2, 3, 4 using 3, 1, 21, 2<br></br>
1, 6, 11, 16 using 21, 3, 1, 2<br></br>
<br></br></mdoxml>
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Factor Lines
Arrange the four number cards on the grid, according to the rules,
to make a diagonal, vertical or horizontal line.
Using, Applying and Reasoning about Mathematics
Working systematically
Numbers and the Number System
Factors and multiples
Numbers and the Number System
Prime numbers
Information and Communications Technology
Interactivities
Admin
Upper primary mapping document