5865 /www/nrich/html/content/id/5865/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> In this game the winner is the first to complete a row of three, either horizontally, vertically or diagonally. <br></br> <br></br> Roll the dice, place each dice in one of the squares and decide whether you want to add or subtract to produce a total shown on the board. Your total will then be covered with a counter. <br></br> <br></br> You cannot cover a number which has already been covered. <br></br> If you are unable to find a total which has not been covered you must Pass.<br></br> <br></br> You can use the interactive version below or print <a href="/content/id/5865/gameboard.pdf">this board</a> to play away from the computer.<br></br> <br></br> <a href="/content/id/5865/PrimaryConnectThree.swf">Full Screen Version</a> <br></br> <mdo:flash height="370" width="600"><param value="/content/id/5865/PrimaryConnectThree.swf" name="movie" ></param><param value="8" name="flashplayerversion" ></param></mdo:flash><br></br> <br></br> <br></br> Are there some numbers that we should be aiming for? Why?<br></br> <br></br> Which number on the grid is the easiest to get? Why?<br></br> <br></br> Which number is the most difficult to get? Why?<br></br> <br></br> For a more challenging version of this game, you could look at <a href="http://nrich.maths.org/public/viewer.php?obj_id=5911&amp;part=index"> Connect Three</a>.<br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p class="editorial">In answer to what numbers we should be aiming for, Tom and Chester from Hotwells Primary School said:</p> <p>It is better to get the numbers in the middle of the board because then you have more choice and it&#39;s easer to get three in a row.</p> <div><span class="editorial">Jeremy from Longston School wrote about what numbers are easiest to get:</span></div> <p>I wrote a table of all the pairs the dice can throw, and then the numbers you can add and subtract to get</p> <p> </p> <mdo:image alt="" height="350" src="Grid.jpg" width="555"></mdo:image><br></br> I then made a table of all the totals you can make, and how many ways of making them there are:<br></br> <br></br> <table style="" border="1"> <tbody> <tr> <td>Result</td> <td>How</td> <td>No. ways</td> </tr> <tr> <td>-5</td> <td>(1-6) twice</td> <td>2</td> </tr> <tr> <td>-4</td> <td>(1-5) twice, (2-6)twice</td> <td>4</td> </tr> <tr> <td>-3</td> <td>(1-4) twice, (2-5) twice (3-6) twice</td> <td>6</td> </tr> <tr> <td>-2</td> <td>(1-3) twice, (2-4) twice, (3-5) twice, (4-6) twice</td> <td>8</td> </tr> <tr> <td>-1</td> <td>(1-2) twice, (2-3) twice, (3-4) twice, (4-5) twice, (5-6) twice</td> <td>10</td> </tr> <tr> <td>0</td> <td>(1-1), (2-2), (3-3), (4-4), (5-5), (6-6)</td> <td>6</td> </tr> <tr> <td>1</td> <td>(6-5) twice, (5-4) twice, (4-3) twice, (3-2) twice, (2-1) twice</td> <td>10</td> </tr> <tr> <td>2</td> <td>(6-4) twice, (5-3) twice, (4-2) twice, (3-1) twice, (1+1)</td> <td>9</td> </tr> <tr> <td>3</td> <td>(6-3) twice, (5-2) twice, (4-1) twice, (1+2) twice</td> <td>8</td> </tr> <tr> <td>4</td> <td>(6-2) twice, (5-1) twice, (1+3) twice, (2+2)</td> <td>7</td> </tr> <tr> <td>5</td> <td>(6-1) twice, (1+4) twice, (2+3) twice</td> <td>6</td> </tr> <tr> <td>6</td> <td>(1+5) twice, (2+4) twice, (3+3)</td> <td>5</td> </tr> <tr> <td>7</td> <td>(1+6) twice, (2+5) twice, (3+4) twice</td> <td>6</td> </tr> <tr> <td>8</td> <td>(2+6) twice, (3+5) twice, (4+4)</td> <td>5</td> </tr> <tr> <td>9</td> <td>(3+6) twice, (4+5) twice</td> <td>4</td> </tr> <tr> <td>10</td> <td>(4+6) twice, (5+5)</td> <td>3</td> </tr> <tr> <td>11</td> <td>(5+6) twice</td> <td>2</td> </tr> <tr> <td>12</td> <td>(6+6)</td> <td>1</td> </tr> </tbody> </table> <br></br> <br></br> <p>So it is easiest to make $-1$ and $1$.</p> <p>It is hardest to make $12$ as there is only one way to make it.</p> <p class="editorial">Well done Jeremy, we like your logical and well planned answer.</p> <p class="editorial">Tables are a great way to write down all your information in a game, so you can discover new things.</p> <p><span class="editorial">Try playing it against the computer and have a think yourself. We&#39;d love to hear from you</span>.</p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>First Connect Three</h2> <br></br> In this game the winner is the first to complete a row of three, either horizontally, vertically or diagonally.<br></br> <br></br> Roll the dice, place each dice in one of the squares and decide whether you want to add or subtract to produce a total shown on the board. Your total will then be covered with a counter.<br></br> <br></br> You cannot cover a number which has already been covered.<br></br> If you are unable to find a total which has not been covered you must Pass.<br></br> <br></br> You can use the interactive version below or print <a href="/content/id/5865/gameboard.pdf">this board</a> to play away from the computer.<br></br> <br></br> <a href="/content/id/5865/PrimaryConnectThree.swf">Full Screen Version</a><br></br> <mdo:flash height="370" id="/content/id/5865/PrimaryConnectThree.swf" width="600"><param name="allowfullscreen" value="true"></param><param name="allowfullscreen" value="true"></param> <param name="movie" value="/content/id/5865/PrimaryConnectThree.swf"></param> <param name="flashplayerversion" value="8"></param> </mdo:flash><br></br> <br></br> <br></br> Are there some numbers that we should be aiming for? Why?<br></br> <br></br> Which number on the grid is the easiest to get? Why?<br></br> <br></br> Which number is the most difficult to get? Why?<br></br> <br></br> For a more challenging version of this game, you could look at <a href="http://nrich.maths.org/public/viewer.php?obj_id=5911&amp;part=index">Connect Three</a>.</div> <br></br> <h3><span style="font-weight: bold;">Why do</span> <span style="font-weight: bold;">this problem</span><span style="font-weight: bold;">?</span></h3> <div><a href="http://nrich.maths.org/public/viewer.php?obj_id=5865&amp;part=">This problem</a> is a great way for students to take responsibility for their own learning. They can avoid negative numbers if they are not confident or they can push themselves to calculate negative answers. In analysing the game more fully, rather than just playing it, the idea is for learners to develop a system for finding all the possible ways of making each number on the grid so they can justify which are the easiest to get.</div> <br></br> <h3>Possible approach</h3> <div>You could introduce the game by playing against the class, or by splitting the class into two teams to play against each other, or with the class playing against the computer. Students can play against each other in pairs to get more of an idea of the game. You can print off <a href="/content/id/5865/gameboard.pdf">this board</a> if the students are not playing at a computer.</div> <br></br> <div>After a suitable length of time, ask the suggested questions in a whole-class discussion that focuses on emerging strategies, observations, explanations and justifications. Students can then go back to working in pairs to establish the numbers of ways of achieving the different totals.</div> <br></br> <div>At the end of the lesson a plenary discussion can offer a chance to present findings and you can draw attention to those methods which were particularly efficient. This would then lead to a discussion about how their findings might affect the way they play the game to win.</div> <br></br> <h3>Key questions</h3> <div>Are there some numbers that we should be aiming for? Why?</div> <div>Which numbers on the grid are the easiest to get? Why?</div> <div>Which numbers are most difficult to get? Why?</div> <br></br> <h3>Possible extension</h3> <div>Further challenges could be provided by asking what would happen if:</div> <ul> <li>there was a differently shaped board</li> <li>numbers appeared more than once on the board and you could place more than one counter in a turn</li> <li>you could use dodecahedral dice or, for example, $1-12$ spinners</li> <li>you wanted to design a board for a game where you allowed multiplication and division</li> </ul> For students who are able to add and subtract both positive and negative numbers, the game <a href="http://nrich.maths.org/public/viewer.php?obj_id=5911&amp;part=">Connect Three</a> and the problem <a href="http://nrich.maths.org/public/viewer.php?obj_id=5864&amp;part=">Playing Connect Three</a> are suitable extensions.<br></br> <h3>Possible support</h3> <div>If some pupils are struggling, you could adapt the board so that it only contains the numbers $1-12$.</div> <div> </div> <div>Handouts for teachers are available here (<a href="/content/id/5865/First%20Connect%20Three.doc">word document</a>, <a href="/content/id/5865/First%20Connect%20Three.pdf">pdf document</a>), with the problem on one side and the notes on the other.  </div> <br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> Are some numbers part of more lines of three than others? This might affect whether we aim for them or not. <br></br> How can you get $1$ by throwing the two dice? How many ways are there altogether?<br></br> How can you get $12$ by throwing two dice?<br></br> It might be useful to look at all the numbers on the grid in turn in this way.<br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <table border="1"> <tbody> <tr> <td>Number on grid</td> <td>Number of lines it could be in</td> </tr> <tr> <td>-5</td> <td>3</td> </tr> <tr> <td>-4</td> <td>5</td> </tr> <tr> <td>-3</td> <td>4</td> </tr> <tr> <td>-2</td> <td>3</td> </tr> <tr> <td>-1</td> <td>3</td> </tr> <tr> <td>0</td> <td>5</td> </tr> <tr> <td>1</td> <td>8</td> </tr> <tr> <td>2</td> <td>7</td> </tr> <tr> <td>3</td> <td>3</td> </tr> <tr> <td>4</td> <td>3</td> </tr> <tr> <td>5</td> <td>7</td> </tr> <tr> <td>6</td> <td>8</td> </tr> <tr> <td>7</td> <td>6</td> </tr> <tr> <td>8</td> <td>3</td> </tr> <tr> <td>9</td> <td>3</td> </tr> <tr> <td>10</td> <td>4</td> </tr> <tr> <td>11</td> <td>5</td> </tr> <tr> <td>12</td> <td> <div>3</div> </td> </tr> </tbody> </table> <br></br> <div>Therefore, might want to aim for the 1 and the 6 as they're in the most lines.</div> <br></br> <table border="1"> <tbody> <tr> <td>Answer</td> <td>How</td> <td>No. ways</td> </tr> <tr> <td>-5</td> <td>(1-6) twice</td> <td>2</td> </tr> <tr> <td>-4</td> <td>(1-5) twice, (2-6)twice</td> <td>4</td> </tr> <tr> <td>-3</td> <td>(1-4) twice, (2-5) twice (3-6) twice</td> <td>6</td> </tr> <tr> <td>-2</td> <td>(1-3) twice, (2-4) twice, (3-5) twice, (4-6) twice</td> <td>8</td> </tr> <tr> <td>-1</td> <td>(1-2) twice, (2-3) twice, (3-4) twice, (4-5) twice, (5-6) twice</td> <td>10</td> </tr> <tr> <td>0</td> <td>1-1, 2-2, 3-3, 4-4, 5-5, 6-6 all twice</td> <td>12</td> </tr> <tr> <td>1</td> <td>(6-5) twice, (5-4) twice, (4-3) twice, (3-2) twice, (2-1) twice</td> <td>10</td> </tr> <tr> <td>2</td> <td>(6-4) twice, (5-3) twice, (4-2) twice, (3-1) twice, 1+1</td> <td>9</td> </tr> <tr> <td>3</td> <td>(6-3) twice, (5-2) twice, (4-1) twice, (1+2) twice</td> <td>8</td> </tr> <tr> <td>4</td> <td>(6-2) twice, (5-1) twice, (1+3) twice, 2+2</td> <td>7</td> </tr> <tr> <td>5</td> <td>(6-1) twice, (1+4) twice, (2+3) twice</td> <td>6</td> </tr> <tr> <td>6</td> <td>(1+5) twice, (2+4) twice, 3+3</td> <td>5</td> </tr> <tr> <td>7</td> <td>(1+6) twice, (2+5) twice, (3+4) twice</td> <td>6</td> </tr> <tr> <td>8</td> <td>(2+6) twice, (3+5) twice, 4+4</td> <td>5</td> </tr> <tr> <td>9</td> <td>(3+6) twice, (4+5) twice</td> <td>4</td> </tr> <tr> <td>10</td> <td>(4+6) twice, 5+5</td> <td>3</td> </tr> <tr> <td>11</td> <td>(5+6) twice</td> <td>2</td> </tr> <tr> <td>12</td> <td>6+6</td> <td>1</td> </tr> </tbody> </table> <br></br> 0 is the easiest number to get as there are more ways of getting it than any other. <br></br> <div>12 is the hardest as there's only one way to make it.</div> <br></br> <div><a href="/content/id/5865/PrimaryConnectThree.zip">flash files</a></div> <br></br> <br></br> <br></br></mdoxml> 2 4 0 1 1 0 0 The idea of this game is to add or subtract the two numbers on the dice and cover the result on the grid, trying to get a line of three. Are there some numbers that are good to aim for? Using, Applying and Reasoning about Mathematics Working systematically Information and Communications Technology Interactivities Using, Applying and Reasoning about Mathematics Games Numbers and the Number System Positive-negative numbers Calculations and Numerical Methods Addition & subtraction Secondary Mapping Document Place value, integers, ordering & rounding - Stage 3 Secondary Mapping Document DisplayCabinet Admin Upper primary mapping document