It's a Tie

5516 /www/nrich/html/content/id/5516/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">Kaia's father always wore a white shirt and coloured tie for work. He didn't like deciding which tie to wear so he would put his hand in the drawer and wear the first tie that he pulled out. Each evening the tie would go back in the drawer.<br></br> <br></br> One Friday morning, Kaia said, "Do you like that tie? You usually wear it at least twice a week." <br></br> "I do not," replied her father, "I have eight ties and I only work five days a week. I certainly don't wear any tie more than once a week." <br></br> "You have worn that tie twice a week in at least five of the ten weeks you have been taking me to school," Kaia insisted. <br></br> "Impossible! Prove it!" challenged her father.<br></br> <br></br> Do you think Kaia or her father was right? Why?<br></br> <br></br> That weekend Kaia did an experiment. She made the following drawings of his ties and created the chart shown below. <br></br> <br></br> <mdo:image width="600" height="305" alt="eight different ties labelled A-H" src="ties.gif"></mdo:image><br></br> <table border="1"> <tbody> <tr> <td>Week no.</td> <td>Monday</td> <td>Tuesday</td> <td>Wednesday</td> <td>Thursday</td> <td>Friday</td> </tr> <tr> <td style="text-align: center;">1</td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> </tr> <tr> <td style="text-align: center;">2</td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> </tr> <tr> <td style="text-align: center;">3</td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> </tr> <tr> <td style="text-align: center;">4</td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> </tr> <tr> <td style="text-align: center;">5</td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> </tr> <tr> <td style="text-align: center;">6</td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> </tr> <tr> <td style="text-align: center;">7</td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> </tr> <tr> <td style="text-align: center;">8</td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> </tr> <tr> <td style="text-align: center;">9</td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> </tr> <tr> <td style="text-align: center;">10</td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> </tr> <tr> <td style="text-align: center;">11</td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> </tr> <tr> <td style="text-align: center;">12</td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> </tr> <tr> <td style="text-align: center;">13</td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> </tr> <tr> <td style="text-align: center;">14</td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> </tr> <tr> <td style="text-align: center;">15</td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> </tr> <tr> <td style="text-align: center;">16</td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> </tr> <tr> <td style="text-align: center;">17</td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> </tr> <tr> <td style="text-align: center;">18</td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> </tr> <tr> <td style="text-align: center;">19</td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> <td style="text-align: center;"></td> </tr> <tr> <td style="text-align: center;">20</td> <td></td> <td></td> <td></td> <td></td> <td></td> </tr> </tbody> </table> <br></br> How might Kaia use the drawings and chart to prove her father wrong? <br></br> Try doing an experiment yourself to find out which ties might be chosen. <br></br> What did the result of your experiment show? <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p class="editorial">Jannis from Northcross Intermediate said:</p> "I think Kaia was right. The ties come out of the closet randomly. We know each tie has 1 eighth of a chance to be pulled out, so its quite likely that the father will not wear 1 tie twices in the week. But its random so there is a chance of him wearing a tie twice in the week." <p class="editorial">Making the point that they're probabilities so you can't be sure.</p> <p class="editorial">Matthew and James from Stradbroke Primary did an experiment, and found that they picked the same tie twice in 9 out of 10 weeks. Ege and Onur from FMV Ozel Erenkoy Isik Primary School, Turkey, also did experiments, and you can find their results <a href="/content/id/5516/OPV1LZ-its_a_tie_ege.doc">here</a> and <a href="/content/id/5516/bvc4fN-its_a_tie_onur.doc">here</a> .</p> <p class="editorial">Some people have tried to calculate the probability of picking the same tie twice, but none correctly yet. Here is what happened when we tried the experiment:</p> <p class="editorial"><mdo:image width="500" height="450" src="Its%20a%20tie%20results%20table.gif" alt=""></mdo:image></p> <div>We found that there were:</div> <div>6 weeks out of 20 when he wore a different tie every day (green), so the probability of this is 6/20,</div> <div>12 weeks out of 20 when he wore the same tie twice in a week (blue), so the probability is 12/20, and</div> <div>2 weeks out of 20 when he wore the same tie more than twice, so the probability of this is 2/20.</div> <br></br> <div>These are only approximate probabilities, and each time you do the experiment you will get a different answer. You can improve the approximation by increasing the number of times you do the experiment and adding up all the results as this will be more accurate.</div> <br></br> <div>Here are some some photos of children working on this problem:</div> <br></br> <div style="text-align: center;"><mdo:image width="500" height="499" src="preparing%20materials.jpg" alt="preparing materials"></mdo:image></div> <div style="text-align: center; font-style: italic;">Preparing materials</div> <div style="text-align: center; font-style: italic;"> </div> <div style="text-align: center; font-style: italic;"> <mdo:image width="600" height="355" src="Drawing%20ties.jpg" alt="drawing ties"></mdo:image></div> <div style="text-align: center; font-style: italic;">Drawing ties</div> <div style="text-align: center; font-style: italic;"> </div> <div style="text-align: center; font-style: italic;"> <mdo:image width="500" height="502" src="completing%20table.jpg" alt="completing a table"></mdo:image></div> <div style="text-align: center; font-style: italic;">Completing the table</div> <div style="text-align: center; font-style: italic;"> </div> <div style="text-align: center; font-style: italic;"> <mdo:image width="500" height="532" src="collating.jpg" alt="collating results"></mdo:image></div> <div style="text-align: center; font-style: italic;">Collating results</div> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>It's a Tie</h2> Kaia's father always wore a white shirt and coloured tie for work. He didn't like deciding which tie to wear so he would put his hand in the drawer and wear the first tie that he pulled out. Each evening the tie would go back in the drawer.<br></br> <br></br> One Friday morning, Kaia said, "Do you like that tie? You usually wear it at least twice a week."<br></br> "I do not," replied her father, "I have eight ties and I only work five days a week. I certainly don't wear any tie more than once a week."<br></br> "You have worn that tie twice a week in at least five of the ten weeks you have been taking me to school," Kaia insisted.<br></br> "Impossible! Prove it!" challenged her father.<br></br> <br></br> Do you think Kaia or her father was right? Why?<br></br> <br></br> That weekend Kaia did an experiment. She made the following drawings of his ties and created the chart shown below.<br></br> <br></br> <mdo:image alt="eight different ties labelled A-H" height="305" src="ties.gif" width="600"></mdo:image><br></br> <table style="" border="1"> <tbody> <tr> <td>Week no.</td> <td>Monday</td> <td>Tuesday</td> <td>Wednesday</td> <td>Thursday</td> <td>Friday</td> </tr> <tr> <td style="text-align: center;">1</td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> </tr> <tr> <td style="text-align: center;">2</td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> </tr> <tr> <td style="text-align: center;">3</td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> </tr> <tr> <td style="text-align: center;">4</td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> </tr> <tr> <td style="text-align: center;">5</td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> </tr> <tr> <td style="text-align: center;">6</td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> </tr> <tr> <td style="text-align: center;">7</td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> </tr> <tr> <td style="text-align: center;">8</td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> </tr> <tr> <td style="text-align: center;">9</td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> </tr> <tr> <td style="text-align: center;">10</td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> </tr> <tr> <td style="text-align: center;">11</td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> </tr> <tr> <td style="text-align: center;">12</td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> </tr> <tr> <td style="text-align: center;">13</td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> </tr> <tr> <td style="text-align: center;">14</td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> </tr> <tr> <td style="text-align: center;">15</td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> </tr> <tr> <td style="text-align: center;">16</td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> </tr> <tr> <td style="text-align: center;">17</td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> </tr> <tr> <td style="text-align: center;">18</td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> </tr> <tr> <td style="text-align: center;">19</td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> <td style="text-align: center;"> </td> </tr> <tr> <td style="text-align: center;">20</td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> </tr> </tbody> </table> <br></br> How might Kaia use the drawings and chart to prove her father wrong?<br></br> Try doing an experiment yourself to find out which ties might be chosen.<br></br> What did the result of your experiment show?</div> <br></br> <h3><span style="font-weight: bold;">Why do this problem?</span></h3> <div><a href="http://nrich.maths.org/public/viewer.php?obj_id=5516&part=index">This problem</a> allows children to simulate a probability experiment; to be involved, be creative and construct their own meanings and explanations for the results they come up with. The problem fits a wide variety of skill and age groups because the design of the experiment and the analysis and interpretation of the data can be stretched to the children's level of understanding. The familiar setting of the problem also makes it meaningful for children. Children have intuitive ideas about probability, which are capitalised on, and their understanding of generating random events is developed by problems like this.</div> <br></br> <h3>Possible approach</h3> <div>Having presented the task orally to the group, children can decide how best to approach it. They could either cut out and randomly draw ties from a bag with replacement (you could print off <a href="/content/id/5516/TiestoCut.doc">this sheet</a> ), represent the ties with letters, or use a computer simulation, depending on their level.</div> <br></br> <div>Organising the class into groups, or pairing children, provides the opportunity to communicate mathematically about the design of the experiment, discuss and revise estimates as more data become available and describe the results in terms of experimental probability.</div> <br></br> <div>When discussing their results, this is a good opportunity to talk about whether the data they collected was what they expected. Why or why not? This could lead into introducing theoretical probability. The results from the various groups could be compared and then combined to provide a more accurate estimate of the theoretical probability.</div> <br></br> <h3>Key questions</h3> <div>How many weeks were there when Kaia's father wore the same tie more than once?</div> <div>Is that what you expected?</div> <div>What is the likelihood that he wears the same tie three times in a week?</div> <div>What is the chance that he will wear a different tie every day in the week?</div> <br></br> <h3>Possible extension</h3> <div>The problem can be extended by changing the number of ties and/or the number of days in the rotation. Once this problem had been modelled and different experimental designs and solutions shared with the class, children should be encouraged to pose their own questions and design simulations that will solve their own problems.</div> <div>An extra challenge can be to devise an experiment that will show how many days in a row it is likely he will wear a different tie. This requires a different approach: Total number of days divided by the total number of trial.</div> <div>If the pupils have enough prior experience with probability experiments, they may be able to compute the theoretical probability to compare with the experimental results.</div> <br></br> <h3>Possible support</h3> <div>Physically performing an experiment by pulling ties out of a "hat" should help all children participate in this task.</div> <br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> Why not create your own experiment? You could draw and cut out ties (or print off <a href="/content/id/5516/TiestoCut.doc">this sheet</a>) and perhaps put them in a bag to be picked out. <br></br> Maybe you could create a simulation on the computer?<br></br> It might be a good idea to work with some friends in different groups so that you have more results to look at.<br></br> Look at the results of your experiment. How many times did you pull out the same tie in a week?<br></br> How does this compare with your friends' experiments?<br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> Preparing materials<br></br> Drawing ties<br></br> Filling in the table<br></br> Collating results<br></br> <br></br> Description:Kaia is sure that her father has worn a particular tie twice a week in at least five of the last ten weeks, but her father disagrees. Who do you think is right?<br></br></mdoxml> 2 4 0 1 0 0 0 It's a Tie Kaia is sure that her father has worn a particular tie twice a week in at least five of the last ten weeks, but her father disagrees. Who do you think is right? Probability Experimental probability Using, Applying and Reasoning about Mathematics Practical Activity Probability Theoretical probability Admin Upper primary mapping document