8102
/www/nrich/html/content/id/8102/
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2011-02-01T00:00:01
<mdoxml version="1.0"><br></br>
<div style="color:#800080">
<h3>You've probably come across number patterns before, like;</h3>
</div>
<div style="color:#0000FF">
<p>$2, 4, 6, 8, 10, 12 . . .$</p>
<p>$512, 256, 128, 64, 32 . . .$</p>
<p>$220, 210, 200, 190, 180, 170 . . .$</p>
<p>$11, 14, 17, 20, 23, 26 . . .$</p>
</div>
<div style="color:#800080">
<p>Work out the rules that produced each of the patterns.</p>
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<p> </p>
<p>Well, let's form some patterns of our own. But this time we will use two maths operations [ 'operations' mean adding, multiplying, dividing, subtracting etc.] to produce each new number in the pattern.</p>
<p>You will use multiplication with either adding or subtraction. BUT you can have the multiplication last if you like.</p>
<p>So let's get started, you will need to do $5$ things for this work:</p>
<p>$1$) You need to choose a STARTING NUMBER (In my example suppose it's $7$)<br></br>
$2$) You need to decide which OPERATION to do first [addition, subtraction or multiplication] (In my example suppose it's multiplication )<br></br>
$3$) You need to decide what NUMBER to use with the operation you chose in b (In my example suppose it's $3$ )<br></br>
$4$) You need to decide on your next OPERATION (I chose to use multiplication before so it's got to be addition or subtraction, suppose it's subtraction)<br></br>
$5$) You need to decide what NUMBER to use with the operation you chose in d. (In my example suppose it's $3$)</p>
<p>Let's run through this a bit.</p>
<p>Starting number $7 [ x 3 - 3 ]$ gives $18$</p>
<p>Now we use $18$.....$ [ x 3 - 3 ]$ gives $51$</p>
<p>Now we use $51$.....$ [ x 3 - 3 ]$ gives $150$</p>
<p>I think it would be better to write it down without the brackets bit, although we do have to do both operations before we write the result down and it would be helpful to write it all out in columns going down:-</p>
<p><iframe allowfullscreen="" frameborder="0" height="315" scrolling="no" src="http://http://www.youtube.com/embedded/5mg8aZXolCY" width="420"></iframe></p>
<div class="c1"> </div>
<div class="c1">
<p> </p>
<p>I now write down ALL THE THINGS THAT I NOTICE ABOUT THIS PATTERN.</p>
<p>Things like:-</p>
<p>The answers go odd, even, odd, even . . . . .</p>
<p>The units figures go $7, 8, 1, 0, 7, 8, 1, 0 ,$ . . . etc.</p>
<p>You might be able to see lots more, I've just written down a few quick ones.</p>
<p>When I've had a good long, hard look and talked with others, perhaps, I do the next stage; which is:-</p>
<p>Take a look at a, b, c, d, e, that I used and make a SMALL CHANGE</p>
<p>SUPPOSE I decide to change e, to subtracting $4$ instead of $3$ BUT everything else stays the same so:-</p>
<p>Starting Number $7 [ x 3 - 4 ]$ gives ......</p>
<p><mdo:image alt="" src="b_2.gif"></mdo:image></p>
<p>Like before I write down all the things that I notice :-</p>
<p>LIKE :-</p>
<p>The answers all end in a $7$</p>
<p>The tens figures go $4, 3, 0, 1, 4, 3, 0, 1,$ . . .</p>
<p>AND maybe many more things.</p>
<p>Then I COMPARE what I noticed this time with last time, similar things and rather different things.</p>
<p>SO I might write something like:-</p>
<p>In the first pattern the units went in a pattern of $4$ repeating different figures, in the second pattern it's the tens that do that. Both patterns end with a $7$ every $4th$ one. I also notice that in the units of the first you could say that the $8 - 1$ gives $7$!</p>
<p>WELL after that L O N G introduction it's time for you to have a go!</p>
<p>$1$) Choose you five things to start off with a, b, c, d, e.<br></br>
$2$) Produce at least $8$ answers underneath each other in good columns [it helps to see tens patterns etc.].<br></br>
$3$) Write about the things you notice.<br></br>
$4$) Make one small change in the $5$ starting things a, b, c, d, e.<br></br>
$5$) Do $2$) and $3$) again.<br></br>
$6$) Compare what's happened this time with the first time.</p>
<p>Then CHOOSE :-</p>
<p>i) It might be good to make a similar change again and see what happens [like subtracting $5$ in our example]<br></br>
ii) Make a very different change [so it might be the starting number this time changed to $10$ in our example]<br></br>
iii) Make a totally fresh new start [for example $3 ( x 11 + 1)]$<br></br>
iv) Follow something through that it interesting you when you compare what you notice.</p>
<p>If you notice something very interesting happen then you may be able to do some predicting and then asking yourself WHY does this particular thing happen.</p>
<p><br></br>
</p>
</div>
<p> </p></mdoxml>
<mdoxml version="1.0"><br></br>
<p class="editorial">Here is a little journey taken by one creative
explorer:-</p>
<p class="editorial">They decided on an easy start by taking:</p>
<br></br>
<br></br>
<strong>3 x 8 + 4<br></br></strong><br></br>
<p class="editorial">The pattern went as follows:-</p>
<br></br>
<br></br>
<table border="0">
<tbody>
<tr>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td>2</td>
<td>8</td>
</tr>
<tr>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td>2</td>
<td>2</td>
<td>8</td>
</tr>
<tr>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td>1</td>
<td>8</td>
<td>2</td>
<td>8</td>
</tr>
<tr>
<td></td>
<td></td>
<td></td>
<td></td>
<td>1</td>
<td>4</td>
<td>6</td>
<td>2</td>
<td>8</td>
</tr>
<tr>
<td></td>
<td></td>
<td></td>
<td>1</td>
<td>1</td>
<td>7</td>
<td>0</td>
<td>2</td>
<td>8</td>
</tr>
<tr>
<td></td>
<td></td>
<td></td>
<td>9</td>
<td>3</td>
<td>6</td>
<td>2</td>
<td>2</td>
<td>8</td>
</tr>
<tr>
<td></td>
<td></td>
<td>7</td>
<td>4</td>
<td>8</td>
<td>9</td>
<td>8</td>
<td>2</td>
<td>8</td>
</tr>
<tr>
<td></td>
<td>5</td>
<td>9</td>
<td>9</td>
<td>1</td>
<td>8</td>
<td>6</td>
<td>2</td>
<td>8</td>
</tr>
<tr>
<td>4</td>
<td>7</td>
<td>9</td>
<td>3</td>
<td>4</td>
<td>9</td>
<td>0</td>
<td>2</td>
<td>8</td>
</tr>
</tbody>
</table>
<p class="editorial">This was quite nice, after the first number
[the chosen one] all the digits in the units column are 8, the tens
are 2 and the digital roots go:- 1, 3, 1, 3, 1, 3, 1, 3, 1.</p>
<p class="editorial">So what happened when the starting number is
changed?</p>
<br></br>
<br></br>
<strong>4 x 8 + 4<br></br></strong><br></br>
<br></br>
<pre>
</pre>
<table border="0">
<tbody>
<tr>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td>4</td>
</tr>
<tr>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td>3</td>
<td>6</td>
</tr>
<tr>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td>2</td>
<td>9</td>
<td>2</td>
</tr>
<tr>
<td></td>
<td></td>
<td></td>
<td></td>
<td>2</td>
<td>3</td>
<td>4</td>
<td>0</td>
</tr>
<tr>
<td></td>
<td></td>
<td></td>
<td>1</td>
<td>8</td>
<td>7</td>
<td>2</td>
<td>4</td>
</tr>
<tr>
<td></td>
<td></td>
<td>1</td>
<td>4</td>
<td>9</td>
<td>7</td>
<td>9</td>
<td>6</td>
</tr>
<tr>
<td></td>
<td>1</td>
<td>1</td>
<td>9</td>
<td>8</td>
<td>3</td>
<td>7</td>
<td>2</td>
</tr>
<tr>
<td></td>
<td>9</td>
<td>5</td>
<td>8</td>
<td>6</td>
<td>9</td>
<td>8</td>
<td>0</td>
</tr>
</tbody>
</table>
<p class="editorial">And the digital roots now go:- 9, 4, 9, 4, 9,
4, 9, 4</p>
<p class="editorial">Whoa! That's quite something. The units now
go, after the first chosen number, 6204 and repeat.</p>
<p class="editorial">Well let's summarize what the rest of the
exploration gave :-</p>
<table border="1" width="100%">
<tbody>
<tr>
<th width="15%">RULE</th>
<th width="15%">UNITS</th>
<th width="15%">DIG.ROOTS</th>
<th width="55%">OTHER COMMENTS</th>
</tr>
<tr>
<td width="15%">3 x 8 + 4</td>
<td width="15%">8....</td>
<td width="15%">1 3 1 3 1 3 ...</td>
<td width="55%">Tens > 2 Hunds > 2860....</td>
</tr>
<tr>
<td width="15%">4 x 8 + 4</td>
<td width="15%">6204 ...</td>
<td width="15%">9 4 9 4 9 4 ...</td>
<td width="55%"></td>
</tr>
<tr>
<td width="15%">5 x 8 + 4</td>
<td width="15%">4260 ...</td>
<td width="15%">8 5 8 5 8 5 ...</td>
<td width="55%"></td>
</tr>
<tr>
<td width="15%">6 x 8 + 4</td>
<td width="15%">2046 ...</td>
<td width="15%">7 6 7 6 7 6 ...</td>
<td width="55%"></td>
</tr>
<tr>
<td width="15%">7 x 8 + 4</td>
<td width="15%">0462 ...</td>
<td width="15%">6 7 6 7 6 7 ...</td>
<td width="55%"></td>
</tr>
<tr>
<td width="15%">8 x 8 + 4</td>
<td width="15%">8...</td>
<td width="15%">5 8 5 8 5 8 ...</td>
<td width="55%">Tens > 6480</td>
</tr>
<tr>
<td width="15%">9 x 8 + 4</td>
<td width="15%">6204 ...</td>
<td width="15%">4 9 4 9 4 9 ...</td>
<td width="55%"></td>
</tr>
<tr>
<td width="15%">10 x 8 + 4</td>
<td width="15%">4620 ...</td>
<td width="15%">3 1 3 1 3 1 ...</td>
<td width="55%">So we could predict that the next ones go:-</td>
</tr>
<tr>
<td width="15%">11 x 8 + 4</td>
<td width="15%">2 ...</td>
<td width="15%">2 ...</td>
<td width="55%">Dig. Roots should be 2, 11 but 11 is 2 so all are
2!</td>
</tr>
<tr>
<td width="15%">12 x 8 + 4</td>
<td width="15%">0462 ...</td>
<td width="15%">1 3 1 3 1 3 ...</td>
<td width="55%"></td>
</tr>
<tr>
<td width="15%">13 x 8 + 4</td>
<td width="15%">8 ...</td>
<td width="15%">9 4 9 4 9 4 ...</td>
<td width="55%">Checked, and Tens > 0648....</td>
</tr>
</tbody>
</table>
<p class="editorial">Well reading this has been very exciting and
led Bernard to make a further change, using 7 instead of 8.</p>
<table border="1" width="100%">
<tbody>
<tr>
<th width="15%">RULE</th>
<th width="15%">UNITS</th>
<th width="15%">DIG.ROOTS</th>
<th width="55%">OTHER COMMENTS</th>
</tr>
<tr>
<td width="15%">2 x 7 + 4</td>
<td width="15%">8024...</td>
<td width="15%">945378612 ..</td>
<td width="55%">Tens > 1310 ...</td>
</tr>
<tr>
<td width="15%">3 x 7 + 4</td>
<td width="15%">5973...</td>
<td width="15%">786129453 ..</td>
<td width="55%">Tens > 2750 ...</td>
</tr>
<tr>
<td width="15%">4 x 7 + 4</td>
<td width="15%">2804...</td>
<td width="15%">537861294 ..</td>
<td width="55%">Tens > 3200 ... Hunds > All even</td>
</tr>
<tr>
<td width="15%">5 x 7 + 4</td>
<td width="15%">9735...</td>
<td width="15%">378612945 ..</td>
<td width="55%">Tens > 3740 ...</td>
</tr>
<tr>
<td width="15%">6 x 7 + 4</td>
<td width="15%">6...</td>
<td width="15%">129453786 ..</td>
<td width="55%">Tens > 4280 ... Hunds > 3200...</td>
</tr>
<tr>
<td width="15%">7 x 7 + 4</td>
<td width="15%">3597...</td>
<td width="15%">861294537 ..</td>
<td width="55%">Tens > 5720 ...</td>
</tr>
<tr>
<td width="15%">8 x 7 + 4</td>
<td width="15%">0428...</td>
<td width="15%">612945378 ..</td>
<td width="55%">Tens > 6270 ...</td>
</tr>
<tr>
<td width="15%">9 x 7 + 4</td>
<td width="15%">7359...</td>
<td width="15%">453786129 ..</td>
<td width="55%">Tens > 6710 ...</td>
</tr>
<tr>
<td width="15%">10 x 7 + 4</td>
<td width="15%">4280...</td>
<td width="15%">294537861 ..</td>
<td width="55%">Tens > 7251 ...</td>
</tr>
<tr>
<td width="15%">11 x 7 + 4</td>
<td width="15%">1...</td>
<td width="15%">945378612 ..</td>
<td width="55%">Tens > 8701 ... Hunds > 5000 Thous >
Even</td>
</tr>
</tbody>
</table>
<p class="editorial">There is so much here that is fascinating.</p>
<p class="editorial">I hope that your explorations have gone
well!</p>
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<h3>Why do this problem?</h3>
<a href="http://nrich.maths.org/public/viewer.php?obj_id=33&part=">
This activity</a> , I have found to be very enjoyable for pupils,
because it opens up a new world for most of them - that they can
create their own number patterns and explore them! It's a healthy
change for many of them to feel that they are not just being handed
something that the teacher already knows an awful lot about. So if
these thoughts encourage you then present it to your pupils.<br></br>
<br></br>
<h3>Possible approach</h3>
<div>I've found it valuable to
use this with the whole class and focus on an introduction where
they're meeting the usual number patterns. As they look at some
familiar patterns, I note down for all to see the comments that
pupils are making. I've usually numbered their findings and got to
at least number six for each one! This then leads into the idea of
them creating their own, to explore in similar ways.</div>
<br></br>
<h3>Key questions</h3>
<div>Tell me about your
rules.</div>
<div>Do you notice anything
that you want to tell me about?</div>
<br></br>
<h3>Possible extension</h3>
<div>See the 'Then choose'
suggestions at the end of <a href="http://nrich.maths.org/public/viewer.php?obj_id=33&part=">
the problem</a> itself.</div>
<br></br>
<h3>Possible support</h3>
<div>Calculators are useful
here so that the pupls are free to explore rather than getting tied
down by the calculations.</div>
<br></br>
<h3>For the
highest-attaining</h3>
<h4 style="font-weight: 400;">
The pupils could go to <a href="http://nrich.maths.org/6928&part=">Become a Maths
Detective</a> which is an interactive version of this activity.
They can then explore much further and do some powerful comparisons
of results. Further ideas relating to that later activity can be
found by following the link in Become a Maths Detective for the
NRICH Projects site where other pupils' ideas can be viewed and
commented on once you have registered.</h4>
<h3>Extra</h3>
The patterns that are generated can be very exciting. I find it
useful if the children have already met things like the patterns
that are evident in the ninetimes table to take things a bit go
further and investigate <a href="http://nrich.maths.org/public/viewer.php?obj_id=5529&part=">
Digital Roots</a> . I have also found that following the a, b, c,
d, e parts as suggested in this activity, writing what they notice,
changing something slightly and repeating etc. to be a very good
investigational process for the youngsters to get used to. Caleb
Gattegno in the 1960's said; "Mathematics is the study of the
invariances under a set of transformation". Or if you prefer it, in
my words now; "Doing mathematics is taking something, changing it
in some way and observing what is the same and what is
different."<br></br>
<div>BE WARNED it may be hard
to stop some children once they get going!</div>
<br></br>
<div>You may also find the
article <a href="http://nrich.maths.org/public/viewer.php?obj_id=1308&part=">
Divisibility Tests</a> useful when with older pupils.</div>
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Clone of Wonderful Number Patterns
EWWNP means Exploring Wild and Wonderful Number Patterns Created by Yourself! Investigate what happens if we create number patterns using some simple rules.
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