53 /www/nrich/html/content/98/10/bbprob2/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p style="text-align: center;"><mdo:image src="dicepic.jpg"></mdo:image><br></br>  </p> <p style="text-align: center;">Two dice are RED and one is GREEN.</p> <br></br> <p>Roll the dice and add up the numbers on the two RED dice and then subtract the number on the GREEN.</p> <p>So if one RED is $4$ and the other RED is $5$ and the GREEN is $3$ we should add together $4$ and $5$ to make $9$ and then subtract the $3$ so that gives us a final answer of $6$.</p> <p>You&#39;ll need to roll these dice many times and see what numbers you make each time by doing the addition and subtraction.</p> <p>In this game it would be good to find out:-</p> <ul> <li>what are the final answers by doing the addition and subtraction each time?</li> <li>what are all the different possible numbers?</li> <li>is there a good way of making sure you find them all?</li> <li>how will you record what you&#39;ve found out?</li> </ul> <p>Now have a go!</p> <div>Look at your results and write down some questions that you could ask about them. For example, do any of them have the same answers? If so, why?</div> <br></br> Then you could ask yourself, "I wonder what would happen if, instead, I ...?&#39;&#39;<br></br> <p> </p> <br></br> <div style="text-align: center;"> </div></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><span class="editorial">Molly from Churchers College Junior School sent in these well thought out ideas about this activity.</span><br></br> <br></br> * I have found out that all the different possible answers are between $ -4$ and $11$ including $11$ and $-4$. It is not possible to get any answers over $11$ and below $-4$.<br></br> I recorded all of this data in a table.<br></br> * I have found out that if all $3$ dice are the same the total will become the value of one of the die ( Eg. $1+1=2-1=1$ ).<br></br> * this would be the same conclusion as above if any $2$ of the die were the same. (E.g. $2+1=3-1=2$)<br></br> * I know there is a $50/50$ chance of the answer being odd or even because;<br></br> odd + odd - odd = odd<br></br> even + even - even = even<br></br> odd + even - odd = even<br></br> odd + odd - even = even<br></br> even + even - odd = odd<br></br> odd + even - even = odd   <br></br> These are all the posible ways of adding the dice.<br></br> Thank you for reading my solution I hope all is correct.<br></br> Molly        ;-)<br></br> <span class="editorial">Indeed Molly it is very good and I am impressed that you did this and came to those conclusions. You could of course extend the exploration by wondering about using $4$ dice and deciding whether to subtract just $1$ of those or maybe $2$.</span><br></br> <br></br> <span class="editorial">Ben, Harry, Will and Lucas from Tarporley Church of England School also worked on this activity and this was their report:</span><br></br> <br></br> There are four of us so two of us wrote ALL of the combinations down [one from $6+6-1$ and one from $1+1-6$]. There were $216$ possible calculations. At the same time the other two of us worked out which is the most likely answer[which is $4$]. Once we did that we were done.<br></br> <br></br> <span class="editorial">Sion from the same school added this extra piece of information;</span><br></br> <br></br> There are $225$ ways and your answer is the numbers $3$ and $4$. By finding all the $225$ calculations you then make a tally chart to show the most popular number.  Finally you count up the number and then your answer should be $3$ and $4$.<br></br> <br></br> <span class="editorial">We also had a number of good ideas from North Molton, namely, Michael, Jack, Beth, James and Sam.</span><br></br> <span class="editorial">Bram
 from the British School of Bucharest
 in  Romania
, sent in what I think is the first from Romania, - well done and thanks - saying;</span><br></br> <br></br> There is a higher probability to get $6$ than $2$ eg. there are fewer ways to get $2$ because there are $13$:<br></br> <br></br> $1+2-1=2$ , $1+3-2=2$ , $1+4-3=2$ , $1+5-4=2$ , $1+6-5=2$ , $2+3-3=2$ , $2+4-4=2$, $2+5-5=2$ , $2+6-6=2$ , $3+3-4=2$ , $3+4-5=2$ , $3+5-6=2$ , $4+4-6=2$<br></br> <br></br> and for $6$ there are:<br></br> <br></br> $1+6-1=6$ , $2+5-1=6$ , $2+6-2=6$ , $3+4-1=6$ , $3+5-2=6$<br></br> <br></br> <span class="editorial">Thanks you all, a great effort.</span><br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Roll These Dice</h2> <br></br> <p> </p> <p style="text-align: center;"><mdo:image src="the%203%20dice.jpg"></mdo:image></p> <br></br> <p>Two dice are RED and one is GREEN.</p> <p>Add up the numbers on the two RED dice and then subtract the number on the GREEN.</p> <p>So if one RED is $4$ and the other RED is $5$ and the GREEN is $3$ we should add together $4$ and $5$ to make $9$ and then subtract the $3$ so that gives us a final answer of $6$.</p> <p>You&#39;ll need to roll these dice many times and see what numbers you make each time by doing the addition and subtraction.</p> <p>In this game it would be good to find out:-</p> <ul> <li>what are the final answers by doing the addition and subtraction each time?</li> <li>what are all the different possible numbers?</li> <li>is there a good way of making sure you find them all?</li> <li>how will you record what you&#39;ve found out?</li> </ul> <p>Now have a go!</p> <div>Look at your results and write down some questions that you could ask about them. For example, do any of them have the same answers? If so , why?</div> <br></br> Then you could ask yourself, "I wonder what would happen if, instead, I ...?&#39;&#39;<br></br> <p> </p> </div> <br></br> <h3><span style="font-weight: bold;">Why do this problem?</span></h3> <div>This <a href="http://nrich.maths.org/public/viewer.php?obj_id=53&amp;part=">activity</a> offers practice in addition and subtraction, including negative results, but the main aim is for pupils to concentrate on making sure that all the ways of rolling the three dice are reached. This will need some sort of system and you could focus on how this could be recorded.</div> <br></br> <h3>Possible approach</h3> <div>You could introduce the problem using real dice and modelling the calculation a few times so that pupils get a feel for it. Once a few results have been recorded on the board, invite the pupils to speculate on how many different results there might be.</div> <br></br> <div>Ask them to work in pairs or small groups on the problem, saying very little else at this stage, but after a short time, bring them together again to share insights so far. Discuss the range of answers that learners have found up to that point and make sure they are happy with subtracting one number from a smaller number. (Using a number line which includes negative numbers might be helpful at this point.) Invite some pairs to describe how they are working. Some may be throwing real dice, others may be listing numbers. Encourage some sort of system so that they can be sure no results are left out. You could ask children to suggest ways of recording which would help - this could be in the form of a table or chart, but allow pupils to choose a way that suits them.</div> <br></br> <h3>Key questions</h3> <div>What are all the different possible numbers?<br></br> What are the final answers by doing the addition and subtraction each time?<br></br> Is there a good way of making sure you find all the possibilities?<br></br> How will you record what you&#39;ve found out?</div> <h3><br></br> Possible extension</h3> <div>Those children who are beginning to use the probability scale from $0$ to $1$, could begin to quantify the likelihood of getting particular results. Asking and investigating &#39;What if ...?&#39; questions would be another good follow-up activity.</div> <br></br> <h3>For the exceptionally mathematically able</h3> <div>Pupils can be challenged to use multiplication as well as addition and subtraction.  After some experimentation, they could try to predict totals that will NOT be possible, then check these predictions out, as well as explaining the predictions.</div> <br></br> <h3>Possible support</h3> <div>Most children will need dice to begin this activity.  For some the dice themselves are an essential prop. Some pupils will benefit from an adult working alongside them and asking questions along each step of the way until their confidence has increased.</div> <br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">What are the possible results of adding the red dice?<br></br> Is there a good way of making sure you find all the possibilities? <br></br> What are the possible results of taking away the number on the green dice from the red totals?<br></br> How will you record what you've found out? <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p class="editorial">Arthur from Cambridge has sent what he found out about adding two dice and subtracting another. He does a really good job of clearly explaining what he did. He says:</p> <p>First I worked out all the totals that you can get from adding two dice (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12). Then I started doing the take-aways for the third one but started getting in a muddle, so I made a table. Down the side I wrote all the totals for the two dice. Across the top I put the numbers from the dice to take-away. Then I filled in all the answers.</p> <table border="1"> <tbody> <tr bgcolor="#FFFFFF"> <td width="15"> </td> <td>1</td> <td>2</td> <td>3</td> <td>4</td> <td>5</td> <td>6</td> </tr> <tr> <td bgcolor="#FFFFFF">2</td> <td>1</td> <td>0</td> <td>-1</td> <td>-2</td> <td>-3</td> <td>-4</td> </tr> <tr> <td bgcolor="#FFFFFF">3</td> <td>2</td> <td>1</td> <td>0</td> <td>-1</td> <td>-2</td> <td>-3</td> </tr> <tr> <td bgcolor="#FFFFFF">4</td> <td>3</td> <td>2</td> <td>1</td> <td>0</td> <td>-1</td> <td>-2</td> </tr> <tr> <td bgcolor="#FFFFFF">5</td> <td>4</td> <td>3</td> <td>2</td> <td>1</td> <td>0</td> <td>-1</td> </tr> <tr> <td bgcolor="#FFFFFF">6</td> <td>5</td> <td>4</td> <td>3</td> <td>2</td> <td>1</td> <td>0</td> </tr> <tr> <td bgcolor="#FFFFFF">7</td> <td>6</td> <td>5</td> <td>4</td> <td>3</td> <td>2</td> <td>1</td> </tr> <tr> <td bgcolor="#FFFFFF">8</td> <td>7</td> <td>6</td> <td>5</td> <td>4</td> <td>3</td> <td>2</td> </tr> <tr> <td bgcolor="#FFFFFF">9</td> <td>8</td> <td>7</td> <td>6</td> <td>5</td> <td>4</td> <td>3</td> </tr> <tr> <td bgcolor="#FFFFFF">10</td> <td>9</td> <td>8</td> <td>7</td> <td>6</td> <td>5</td> <td>4</td> </tr> <tr> <td bgcolor="#FFFFFF">11</td> <td>10</td> <td>9</td> <td>8</td> <td>7</td> <td>6</td> <td>5</td> </tr> <tr> <td bgcolor="#FFFFFF">12</td> <td>11</td> <td>10</td> <td>9</td> <td>8</td> <td>7</td> <td>6</td> </tr> </tbody> </table> <br></br> <p>There&#39;s a pattern of all the same answers in the diagonals. If you look sloping the other way there&#39;s a pattern on both sides of the zeros. There&#39;s odd and even numbers in a line.</p> <p>To find out which answer came up the most I counted. I wrote a list of all the answers and underneath them I wrote how many times you get that answer.</p> <table> <tbody> <tr> <td>Answers</td> <td>-4</td> <td>-3</td> <td>-2</td> <td>-1</td> <td>0</td> <td>1</td> <td>2</td> <td>3</td> <td>4</td> <td>5</td> <td>6</td> <td>7</td> <td>8</td> <td>9</td> <td>10</td> <td>11</td> </tr> <tr> <td>How many</td> <td>1</td> <td>2</td> <td>3</td> <td>4</td> <td>5</td> <td>6</td> <td>6</td> <td>6</td> <td>6</td> <td>6</td> <td>6</td> <td>5</td> <td>4</td> <td>3</td> <td>2</td> <td>1</td> </tr> </tbody> </table> <p>There&#39;s another pattern. The numbers from 1 to 6 all can happen 6 times. I wondered if this is because they are the numbers on the dice. Taking away one dice sort of cancels out one of the other dice so there&#39;s just one dice left. That seems to make sense but I&#39;m not sure.</p> <br></br></mdoxml> 2 4 0 1 0 0 0 Roll two red dice and a green dice. Add the two numbers on the red dice and take away the number on the green. What are all the different possibilities that could come up? Using, Applying and Reasoning about Mathematics Investigations Calculations and Numerical Methods Addition & subtraction Mathematics Tools Dice Probability Experimental probability Probability Theoretical probability Using, Applying and Reasoning about Mathematics Working systematically Admin Upper primary mapping document