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/www/nrich/html/content/98/04/bbprob1/
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2011-02-01T00:00:01
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<p><mdo:image align="right" src="worm.gif" alt=""></mdo:image></p>
<p>Imagine a square patch of ground in a garden, and because it's a
good garden there are worms around, and they feed from the soil.
Well, I've taken an imaginary square and given it the numbers 1 to
100.</p>
<br></br>
<p style="text-align: center;"><mdo:image height="373" width="372" alt="100" src="%2100%204%20Worms.jpg"></mdo:image></p>
<p>You've probably come across number squares like this in maths
sometimes at school. So each part of this square piece of ground
has a number. But now, suppose that the centre piece of this square
is what the worm likes to eat most of all. So I've taken the centre
16 squares and made them a little darker.</p>
<br></br>
<p style="text-align: center;"><mdo:image height="369" width="368" alt="pic1" src="Picture%201.jpg"></mdo:image></p>
<p>You will probably notice that the lowest number in this square
is 34 and the highest is 67. Well, that's enough about the ground.
What about the worm? The worm will be 4 squares long and to make
sure we know which is the head and which is the tail I've coloured
the head part differently.</p>
<p style="text-align: center;"><mdo:image height="152" width="46" alt="pic5" src="Picture%205.jpg"></mdo:image></p>
<br></br>
<p>Of course the worm can bend itself in different ways. But I'm
going to suggest that on this occasion it has 4 square parts [like
4 multilink] which have to be joined edge to edge in the "normal''
way. Here are three examples that would be O.K., and one which is
not because the head is not at the end of the body!</p>
<p style="text-align: center;"><mdo:image height="276" width="332" alt="pic6" src="Picture%206.jpg"></mdo:image></p>
<br></br>
<p>Now, to set up the challenge. With a square in front of you,
with the numbers 1 to 100 marked on as shown and the centre 16
squares marked in some special way, place your worm, perhaps made
out of multilink, on the square. [Multilink are 2 cm long so get
hold of some 2 cm squared paper; the backing paper for some
"taky-back'' film is just the right size.] When you place your worm
you have to make sure that the head is on one of the centre squares
that are shaded differently. The head stays there, but while the
worm is eating it wriggles and changes the shape of its body, in as
many different ways as it can. Here are just three of the many ways
in which the worm can be, and I've placed its head on the number
34.</p>
<p style="text-align: center;"><mdo:image height="370" width="372" alt="pic2" src="Picture%202.jpg"></mdo:image><mdo:image height="370" width="371" alt="pic3" src="Picture%203.jpg"></mdo:image></p>
<br></br>
<div style="text-align: center;"><mdo:image height="372" width="370" alt="pic4" src="Picture%204.jpg"></mdo:image></div>
<br></br>
<br></br>
<p>The challenge is to:-</p>
<ol type="a">
<li>Find all the shapes and positions that the worm can be with its
head fixed in the one place [in my example, on number 34, but you
can choose any of those 16 squares]. Don't forget about the worm's
body going North, South, East and West.</li>
<li>Add up the numbers that the worm is on, that is the numbers
that the head and the rest of the body are resting on. [In my first
example 34 + 33 + 43 + 53 = 163; 2 <sup>nd</sup> example gives 34 +
33 + 23 + 13 = 103; 3 <sup>rd</sup> example gives 34 + 33 + 23 + 22
=112.]</li>
<li>Find a good way of recording your results
<ol>
<li>so that you can make sure you do not do the same one twice
and</li>
<li>so that you can compare the results easily.</li>
</ol>
</li>
</ol>
<p>Well, that's really it! Of course at the end you have to say "I
wonder what would happen if I ...?''.</p>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<p><span class="editorial">Here are three solutions to this problem
from Anna, Oliver and Lucy.</span></p>
<p class="editorial">From Anna we had the following which was very
well presented:</p>
I tried out two numbers and compared them. The numbers were 34 and
67.<br></br>
These were my results:- 34 is the head of the worm<br></br>
<br></br>
<table border="0">
<tbody>
<tr>
<td>34 + 24 + 23 + 33 = 114
<div style="text-align: center;"><mdo:image width="149" height="149" alt="" src="Anna1.gif"></mdo:image></div>
</td>
<td><br></br>
<div>34 + 24 + 25 + 35 = 118</div>
<div><mdo:image width="149" height="149" src="Anna2.gif" alt=""></mdo:image></div>
<br></br>
<br></br></td>
</tr>
<tr>
<td>
<div>34 + 44 + 33 + 43 = 154</div>
<div><mdo:image width="149" height="149" src="Anna3.gif" alt=""></mdo:image></div>
</td>
<td>
<div>34 + 35 + 45 + 44 = 158</div>
<div><mdo:image width="149" height="149" src="Anna4.gif" alt=""></mdo:image></div>
</td>
</tr>
<tr>
<td>
<div>34 + 24 + 14 + 4 = 76</div>
<div><mdo:image width="78" height="290" src="Anna5.gif" alt=""></mdo:image></div>
</td>
<td>
<div>34 + 35 + 36 + 37 = 142</div>
<div><mdo:image width="290" height="78" src="Anna6.gif" alt=""></mdo:image></div>
</td>
</tr>
<tr>
<td>
<div>34 + 44 + 54 + 64 = 196</div>
<div><mdo:image width="78" height="290" src="Anna7.gif" alt=""></mdo:image></div>
</td>
<td>
<div>34 + 33 + 32 + 31 = 130</div>
<div><mdo:image width="290" height="78" src="Anna8.gif" alt=""></mdo:image></div>
</td>
</tr>
<tr>
<td>
<div>34 + 24 + 14 + 15 = 87</div>
<div><mdo:image width="149" height="220" src="Anna9.gif" alt=""></mdo:image></div>
</td>
<td>
<div>34 + 35 + 36 + 46 = 151</div>
<div><mdo:image width="220" height="149" src="Anna10.gif" alt=""></mdo:image></div>
</td>
</tr>
<tr>
<td>
<div>34 + 44 + 54 + 53 = 185</div>
<div><mdo:image width="149" height="219" src="Anna11.gif" alt=""></mdo:image></div>
</td>
<td>
<div>34 + 33 + 32 + 22 = 121</div>
<div><mdo:image width="219" height="149" src="Anna12.gif" alt=""></mdo:image></div>
<br></br></td>
</tr>
</tbody>
</table>
<br></br>
<p>I found 18 more but the highest score that I found was 196, and
it was a vertical worm, crossing through 34, 44, 54 and 64. These
are some of the worms where the head starts at 67:</p>
<br></br>
<table border="0">
<tbody>
<tr>
<td>
<div>67 + 77 + 87 + 97 = 328</div>
<div><mdo:image width="78" height="290" alt="" src="Anna13.gif"></mdo:image></div>
</td>
<td>
<div>67 + 57 + 47 + 37 = 208</div>
<div><mdo:image width="78" height="290" alt="" src="Anna14.gif"></mdo:image></div>
</td>
</tr>
<tr>
<td>
<div>67 + 68 + 69 + 70 = 274</div>
<div><mdo:image width="290" height="78" alt="" src="Anna15.gif"></mdo:image></div>
</td>
<td>
<div>67 + 66 + 65 + 64 = 262</div>
<div><mdo:image width="290" height="78" alt="" src="Anna16.gif"></mdo:image></div>
<br></br></td>
</tr>
<tr>
<td>
<div>67 + 57 + 56 + 66 = 246</div>
<div><mdo:image width="149" height="149" alt="" src="Anna17.gif"></mdo:image></div>
</td>
<td>
<div>67 + 68 + 58 + 59 = 250</div>
<div><mdo:image width="149" height="149" alt="" src="Anna18.gif"></mdo:image></div>
</td>
</tr>
</tbody>
</table>
<br></br>
I found 18 more of these, but the highest score was 328.<br></br>
<br></br>
I wonder what would happen if I could use any numbers in the
square?<br></br>
I wonder what would happen if I could draw the worm diagonally?
Would the score be higher?<br></br>
I wonder what would happen if there were more numbers in the
square?<br></br>
I wonder what would happen if the square was a triangle or a
different shape? I wonder?<br></br>
<br></br>
<p><span class="editorial">Well Anna, thank you so much for your
thoughts. I liked the picture of you thinking these
thoughts.</span></p>
<p class="editorial">Stephen in the same school [West Flegg in
Yarmouth] did some similar work and his thoughts were:-</p>
I wonder what would happen if there were two worms and not
one?<br></br>
<br></br>
<p><span class="editorial">Oliver, from a little village near
Somerton in Somerset,drew the starting field of 100 in the way
suggested.</span></p>
<br></br>
<br></br>
<mdo:image width="361" height="361" src="Oliver1.gif" alt=""></mdo:image><br></br>
<br></br>
<p><span class="editorial">Then he drew, quite systematically the
following worms and showed their additions with the head on
43.</span></p>
<br></br>
<br></br>
<mdo:image width="498" height="583" src="Answer.jpg" alt=""></mdo:image><br></br>
<br></br>
<p><span class="editorial">Well done Oliver!</span></p>
<p class="editorial">Lucy, from Wells in Somerset, started off with
this different arrangement of numbers:</p>
<br></br>
<br></br>
<mdo:image width="221" height="221" src="lucy1.gif" alt=""></mdo:image><br></br>
<br></br>
<p><span class="editorial">She then coloured in the worms as
solutions and I've used the same colours to colour in the ones that
she drew together with the totals.</span></p>
<br></br>
<br></br>
<mdo:image width="643" height="296" src="lucy2.gif" alt=""></mdo:image><br></br>
<br></br>
<p><span class="editorial">She then numbered the square in a very
interesting way. See if you can see what she has done and then
perhaps go one to do a very different one for yourself.</span></p>
<br></br>
<br></br>
<mdo:image width="598" height="457" alt="" src="Lucy3.gif"></mdo:image><br></br>
<span class="editorial"></span><br></br>
<p><span class="editorial">Unfortunately she did not have time to
do very many, but I got caught up with this idea so I did the last
twenty showing the kind of shape I was using. Then I began to
notice things!!!</span></p>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><div class="embed">
<h2>Worms</h2>
<br></br>
<br></br>
<p><mdo:image align="right" alt="" src="worm.gif"></mdo:image></p>
<p>Imagine a square patch of ground in a garden, and because it's a good garden there are worms around, and they feed from the soil. Well, I've taken an imaginary square and given it the numbers 1 to 100.</p>
<br></br>
<p style="text-align: center;"><mdo:image alt="100" height="373" src="%2100%204%20Worms.jpg" width="372"></mdo:image></p>
<p>You've probably come across number squares like this in maths sometimes at school. So each part of this square piece of ground has a number. But now, suppose that the centre piece of this square is what the worm likes to eat most of all. So I've taken the centre 16 squares and made them a little darker.</p>
<br></br>
<p style="text-align: center;"><mdo:image alt="pic1" height="369" src="Picture%201.jpg" width="368"></mdo:image></p>
<p>You will probably notice that the lowest number in this square is 34 and the highest is 67. Well, that's enough about the ground. What about the worm? The worm will be 4 squares long and to make sure we know which is the head and which is the tail I've coloured the head part differently.</p>
<p style="text-align: center;"><mdo:image alt="pic5" height="152" src="Picture%205.jpg" width="46"></mdo:image></p>
<br></br>
<p>Of course the worm can bend itself in different ways. But I'm going to suggest that on this occasion it has 4 square parts [like 4 multilink] which have to be joined edge to edge in the "normal'' way. Here are three examples that would be O.K., and one which is not because the head is not at the end of the body!</p>
<p style="text-align: center;"><mdo:image alt="pic6" height="276" src="Picture%206.jpg" width="332"></mdo:image></p>
<br></br>
<p>Now, to set up the challenge. With a square in front of you, with the numbers 1 to 100 marked on as shown and the centre 16 squares marked in some special way, place your worm, perhaps made out of multilink, on the square. [Multilink are 2 cm long so get hold of some 2 cm squared paper; the backing paper for some "taky-back'' film is just the right size.] When you place your worm you have to
make sure that the head is on one of the centre squares that are shaded differently. The head stays there, but while the worm is eating it wriggles and changes the shape of its body, in as many different ways as it can. Here are just three of the many ways in which the worm can be, and I've placed its head on the number 34.</p>
<p style="text-align: center;"><mdo:image alt="pic2" height="370" src="Picture%202.jpg" width="372"></mdo:image><mdo:image alt="pic3" height="370" src="Picture%203.jpg" width="371"></mdo:image></p>
<br></br>
<div style="text-align: center;"><mdo:image alt="pic4" height="372" src="Picture%204.jpg" width="370"></mdo:image></div>
<br></br>
<br></br>
<p>The challenge is to:-</p>
<ol type="a">
<li>Find all the shapes and positions that the worm can be with its head fixed in the one place [in my example, on number 34, but you can choose any of those 16 squares]. Don't forget about the worm's body going North, South, East and West.</li>
<li>Add up the numbers that the worm is on, that is the numbers that the head and the rest of the body are resting on. [In my first example 34 + 33 + 43 + 53 = 163; 2 <sup>nd</sup> example gives 34 + 33 + 23 + 13 = 103; 3 <sup>rd</sup> example gives 34 + 33 + 23 + 22 =112.]</li>
<li>Find a good way of recording your results
<ol>
<li>so that you can make sure you do not do the same one twice and</li>
<li>so that you can compare the results easily.</li>
</ol>
</li>
</ol>
<p>Well, that's really it! Of course at the end you have to say "I wonder what would happen if I ...?''.</p>
</div>
<h3>Why do this problem?</h3>
<div>This <a href="http://nrich.maths.org/public/viewer.php?obj_id=40&part=">activity</a> is a fun activity that involves addition as well as spacial awareness. It allows pupils to extend their knowledge of both in an accessible environment.</div>
<br></br>
<h3>Possible approach</h3>
<div>I have used this activity with whole classes and small groups of children. I prepared the grid sheets for them and asked them to get four multilink each, three of one colour and one of a different colour. (<a href="/content/98/04/bbprob1/Worms%20100%20Square.pdf">Here</a> is a $100$ square that you can print off.)</div>
<br></br>
<div>It was good to see the invented methods the children used to record their results. Some used loads of different coloured crayons and drew them on the sheet. Others created little pictures or diagrams and the sums that went with them.</div>
<br></br>
<div>There are a few rather surprising things that occur and it is good for the children to find out why. It shows a lot about children's number awareness with numbers going up and down in tens, and up and down in units. When they had finished with the head in one square they compared their results with others who had started somewhere different.</div>
<br></br>
<div>Finally I gave them the option of numbering the $100$ square in a different manner. I do recommend this as a very interesting thing to do. There are lots of possibilities and they result in all kinds of answers when the same rules apply as they had at the start.</div>
<br></br>
<h3>Key questions</h3>
<div>Tell me about this worm you've got.</div>
<div>What's the largest total you've seen so far?</div>
<div>Tell me about your recording.</div>
<br></br>
<h3>Possible extension</h3>
<div>Using different sizes of worms and/or different sizes of field and then compare. One child once came up with the idea of allowing the worms to bend upwards so that you had two or more parts over the same number. So we'd see something like this:</div>
<div style="text-align: center;"><mdo:image alt="3d" height="367" src="3d.jpg" width="367"></mdo:image></div>
<div>So it had its head on $64$ and three parts all on $63$: $64 + 63 + 63 + 63 = 253$</div>
<br></br>
<h4> </h4>
<br></br>
<h3>For the exceptionally mathematically able</h3>
<div>What is the effect of numbering the $100$ square in a different way. Examine the $4x4$ square in the middle. Which way of numbering gets the largest/smallest numbers in this central square?</div>
<div>Consider other shapes to the field and new ways of numbering the smaller areas inside. eg. a hexagonal/octagonal field.</div>
<h3>Possible support</h3>
<div>Some support is sometimes necessary for some pupils and separating the two elements out may help. So we make all the different shapes of worms with cubes first and then consider where the head will go.</div>
<br></br>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
You could try keeping the second and third body parts in one place
to begin with. How many different places are there for the fourth
part?<br></br>
What happens if you still keep the second part in the same place
and investigate different arrangements of the third and fourth
now?<br></br>
You might find <a href="/content/98/04/bbprob1/100Square.doc">this
100 square</a> useful.<br></br></mdoxml>
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Worms
Place this "worm" on the 100 square and find the total of the four
squares it covers. Keeping its head in the same place, what other
totals can you make?
Using, Applying and Reasoning about Mathematics
Investigations
Numbers and the Number System
Comparing and Ordering numbers
Using, Applying and Reasoning about Mathematics
Working systematically
Calculations and Numerical Methods
Addition & subtraction
Mathematics Tools
100 square