5864
/www/nrich/html/content/id/5864/
1
2011-02-01T00:00:01
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Play the game <a href="http://nrich.maths.org/public/viewer.php?obj_id=5911&part=">Connect Three</a> below, and then take a look at the questions that follow.<br></br>
<br></br>
This is a mathematical version of noughts and crosses that uses differently numbered dice (one with the numbers $1$, $2$, $3$, $-4$, $-5$, $-6$ and the other with the numbers $-1$, $-2$, $-3$, $4$, $5$, $6$). To place your counter, roll the dice and decide whether to add or subtract the numbers.<br></br>
<br></br>
You can use the interactive version below or print <a href="/content/id/5864/Connect%20Three.pdf">this board</a> to play away from the computer.<br></br>
<br></br>
<a href="/content/id/5864/ConnectThreePM.swf">Full Screen Version</a><br></br>
<br></br>
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<param name="movie" value="/content/id/5864/ConnectThreePM.swf"></param>
<param name="flashplayerversion" value="8"></param>
</mdo:flash><br></br>
<span style="font-weight: bold;">Are some squares easier to land on than others?</span><br></br>
<br></br>
<div style="margin-left: 40px;">I can only land on $12$ if I get a $6$ and a $-6$:</div>
<div style="margin-left: 40px;">$6 - (-6) =12$</div>
<div style="margin-left: 40px;"> </div>
<p style="margin-left: 40px;">but there are many ways of landing on $4$:<br></br>
$5 - 1 = 4$<br></br>
$2 - (-2) = 4$<br></br>
$(-2) - (-6) = 4$<br></br>
.....</p>
<br></br>
<span style="font-weight: bold;">Can you work out the number of different ways of achieving each of the different totals?</span><br></br>
<br></br>
<span style="font-weight: bold;">Can these results help you work out a strategy for improving your chances of winning the game?</span><br></br></mdoxml>
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<p><span class="editorial">Daniel from Gateway Primary School,
noted that:</span></p>
<p>It is hardest to get $12$ and negative $12$. You are most likely
to get the low positive and negative numbers. You are least likely
to get the high positive and negative numbers.</p>
<p class="editorial">Jeremy from Longston School wrote:</p>
<p>I wrote a table of all the numbers you can get, and then looked
at how many ways of making each number there are:</p>
<p><mdo:image height="446" width="151" src="Numbers.jpg" alt=""></mdo:image></p>
<p>Then you can see that getting a $3$ or $-3$ are most likely to
happen. So you should place a counter near the $3$ or $-3$ because
then you are most likely to get it next turn, and then you will
have two in a row.</p>
<p></p>
<p></p>
<p></p>
<p><span class="editorial">Aidan commented</span>:</p>
You should always aim to get $0$, as is it is in the centre of the
grid, and offers the most possibilities. And if the other person
has two in a row, you should block them like noughts and crosses if
you can.<br></br>
<p class="editorial">Very well noticed by everyone. Can anyone add
further comments?</p>
<br></br>
<br></br></mdoxml>
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<h3><span style="font-weight: bold;">Why do this
problem?</span></h3>
<div><a href="http://nrich.maths.org/public/viewer.php?obj_id=5864">This
problem</a> offers students a chance to analyse a game which
involves adding and subtracting positive and negative numbers, and
requires them to work out the probability of the different possible
outcomes. Teachers could explain that by knowing about the
mathematics behind a game we can sometimes determine a winning
strategy (or more realistically, a strategy that improves our
chances of winning).</div>
<br></br>
<div>Teachers may find it interesting to read the NCETM
Mathemapedia Entry:</div>
<div><a href="http://www.ncetm.org.uk/mathemapedia/Games%20as%20challenges">Games
as challenges to stimulate curiosity and support learning</a></div>
<br></br>
<h3>Possible approach</h3>
The game could be introduced using an interactive whiteboard by
asking for two volunteers to play, each linked to one of the
coloured counters, or the class could be divided into two teams.
<br></br>
<br></br>
After the introduction students could play in pairs at individual
computers, or <a href="/content/id/5864/Connect%20Three.pdf">this
board</a> could be printed for students to play away from the
computers. <a href="/content/id/5864/ConnectThreeDice.xls">This
spreadsheet</a> can be used to simulate throwing the two
dice.<br></br>
<br></br>
To switch attention from consolidation of number skills to the
mathematics behind the game, this needs to be followed by a whole
class discussion that focuses on emerging strategies, observations,
explanations and justifications.<br></br>
<br></br>
Students can then go back to working in pairs to establish the
numbers of ways of achieving the different totals. At the end of
the lesson a plenary discussion can offer students a chance to
present their findings. The discussion can compare the merits of
the different approaches used (eg listing possibilities vs sample
space diagrams).<br></br>
<br></br>
Following on from this problem, students could take a look at:
<br></br>
<a href="http://nrich.maths.org/public/viewer.php?obj_id=5958&part=">Weights</a>
<br></br>
<a href="http://nrich.maths.org/public/viewer.php?obj_id=5868&part=">Consecutive
Negative Numbers</a> <br></br>
<h3>Key questions</h3>
<div>Are there some numbers that we should be aiming for?
Why?</div>
<br></br>
<div>Are certain numbers easier to 'cover' than others? Why?</div>
<br></br>
<div>"Have you got all the solutions?" "How do you know?"</div>
<br></br>
<h3>Possible extension</h3>
<div>If appropriate, teachers could set up an open ended activity
for the rest of the week in which students investigate changes they
could make to the game.</div>
<div>For example:</div>
<div style="margin-left: 40px;"></div>
<div style="margin-left: 40px;">* Different shaped boards</div>
<div style="margin-left: 40px;">* Boards where some numbers appear
more than once</div>
<div style="margin-left: 40px;">* Different dice (e.g.
dodecahedral) or different numbers on the dice</div>
<div style="margin-left: 40px;">* Allowing multiplication and
division and changing the board accordingly</div>
<br></br>
<h3>Possible support</h3>
<div>Students who struggle with adding and subtracting negative
numbers can play <a href="http://nrich.maths.org/public/viewer.php?obj_id=5865&part=">First
Connect Three</a> , a simpler version of the game that just
requires dice with positive numbers.</div>
<br></br>
<br></br>
Teachers may like to take a look at the article on <a href="http://nrich.maths.org/public/viewer.php?obj_id=5947&part=">Adding
and Subtracting Negative Numbers</a> <br></br></mdoxml>
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<div><span style="font-weight: bold;">You need to consider all the
different possible outcomes when you throw the dice.</span></div>
<br></br>
You could get:<br></br>
<br></br>
<div style="margin-left: 40px;">$^-5$ on the Yellow dice &
$^-3$ on the Red dice, or</div>
<div style="margin-left: 40px;">$3$ on the Yellow dice & $^-1$
on the Red dice, or...</div>
<br></br>
How many different possible outcomes are there?<br></br>
<br></br>
<br></br>
<span style="font-weight: bold;">You can make different totals with
each combination.</span> <br></br>
<br></br>
If you get $3$ & $^-1$ you can make totals of $2$ or $4$ or
$^-4$:<br></br>
<br></br>
<div style="margin-left: 40px;">Yellow dice + Red dice<br></br>
$3 + (^-1) = 2$<br></br>
<br></br>
Yellow dice - Red dice<br></br>
$3 - (^-1) = 4$<br></br>
<br></br>
Red dice - Yellow dice<br></br>
$^-1 - 3 = ^-4$</div>
<br></br>
How many combinations are there for each throw of the dice?<br></br>
<br></br>
How many combinations are there altogether?<br></br>
<br></br>
<br></br></mdoxml>
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<mdoxml version="1.0">Int<a href="/content/id/5864/ConnectThree.swf">eractiv</a>ity<br></br>
<br></br>
There are 36 possible dice combinations:<br></br>
1 & -1<br></br>
1 & -2<br></br>
1 & -3<br></br>
etc., etc...<br></br>
<br></br>
Each combination allows for 3 different totals:<br></br>
A + B<br></br>
A - B<br></br>
B - A<br></br>
<br></br>
So there are 108 different arrangements to be considered.<br></br>
<br></br>
Number of ways of achieving each total:<br></br>
<br></br>
<table border="1">
<tbody>
<tr>
<td style="text-align: center;">possible totals</td>
<td style="text-align: center;">number of ways</td>
</tr>
<tr>
<td style="text-align: center;">12</td>
<td style="text-align: center;">1</td>
</tr>
<tr>
<td style="text-align: center;">11</td>
<td style="text-align: center;">2</td>
</tr>
<tr>
<td style="text-align: center;">10</td>
<td style="text-align: center;">3</td>
</tr>
<tr>
<td style="text-align: center;">9</td>
<td style="text-align: center;">3</td>
</tr>
<tr>
<td style="text-align: center;">8</td>
<td style="text-align: center;">3</td>
</tr>
<tr>
<td style="text-align: center;">7</td>
<td style="text-align: center;">3</td>
</tr>
<tr>
<td style="text-align: center;">6</td>
<td style="text-align: center;">3</td>
</tr>
<tr>
<td style="text-align: center;">5</td>
<td style="text-align: center;">5</td>
</tr>
<tr>
<td style="text-align: center;">4</td>
<td style="text-align: center;">7</td>
</tr>
<tr>
<td style="text-align: center;">3</td>
<td style="text-align: center;">8</td>
</tr>
<tr>
<td style="text-align: center;">2</td>
<td style="text-align: center;">7</td>
</tr>
<tr>
<td style="text-align: center;">1</td>
<td style="text-align: center;">6</td>
</tr>
<tr>
<td style="text-align: center;">0</td>
<td style="text-align: center;">6</td>
</tr>
<tr>
<td style="text-align: center;">-1</td>
<td style="text-align: center;">6</td>
</tr>
<tr>
<td style="text-align: center;">-2</td>
<td style="text-align: center;">7</td>
</tr>
<tr>
<td style="text-align: center;">-3</td>
<td style="text-align: center;">8</td>
</tr>
<tr>
<td style="text-align: center;">-4</td>
<td style="text-align: center;">7</td>
</tr>
<tr>
<td style="text-align: center;">-5</td>
<td style="text-align: center;">5</td>
</tr>
<tr>
<td style="text-align: center;">-6</td>
<td style="text-align: center;">3</td>
</tr>
<tr>
<td style="text-align: center;">-7</td>
<td style="text-align: center;">3</td>
</tr>
<tr>
<td style="text-align: center;">-8</td>
<td style="text-align: center;">3</td>
</tr>
<tr>
<td style="text-align: center;">-9</td>
<td style="text-align: center;">3</td>
</tr>
<tr>
<td style="text-align: center;">-10</td>
<td style="text-align: center;">3</td>
</tr>
<tr>
<td style="text-align: center;">-11</td>
<td style="text-align: center;">2</td>
</tr>
<tr>
<td style="text-align: center;">-12</td>
<td style="text-align: center;">1</td>
</tr>
</tbody>
</table>
<br></br></mdoxml>
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Playing Connect Three
In this game the winner is the first to complete a row of three.
Are some squares easier to land on than others?
Probability
Equally likely outcomes
Calculations and Numerical Methods
Addition & subtraction
Numbers and the Number System
Positive-negative numbers
Secondary Mapping Document
Place value, integers, ordering & rounding - Stage 3
Secondary Mapping Document
DisplayCabinet