4940 /www/nrich/html/content/id/4940/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> Here are some dominoes taken out of the full set:<br></br> <div><mdo:image height="233" width="550" src="dominoes.gif" alt="dominoes from 1, 0 to 3, 3"></mdo:image></div> <br></br> <div>Sort them into two groups - one group with an odd number of spots and one group with an even number of spots.</div> <div>Do you have any dominoes left over? Why or why not?</div> <br></br> <div>Now put the dominoes into pairs. The number of spots on each pair of dominoes must make a total of $5$.</div> <div>How many pairs can you make?</div> <div>Which dominoes are left over?</div> <br></br> <div>Can you pair them up in any different ways so that each pair adds to $5$?</div> <div>Which dominoes are left over now?</div> <br></br> <div>Are there any dominoes which are always left over?</div> <div>Can you explain why?</div> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p class="editorial">Natasha and Izzy from Finney Primary School say:</p> <br></br> Our solution was that we gave each domino a number going left to right <span class="editorial">(and top to bottom)</span> . The first domino 1, second 2 and so on. The first solution was the odd numbers (in order 1, 2 so on) were: <br></br> <div style="clear: both;">odd: 1,4,5,6. even: 2,3,7,8,9.</div> <div style="clear: both;"></div> <div style="clear: both;"></div> <p class="editorial" style="clear: both;">So, in other words the dominoes which have an odd number of spots are:</p> <div style="clear: both;"></div> <div style="clear: both;"><mdo:image width="505" height="68" alt="odd spotted dominoes" src="sol1.gif"></mdo:image></div> <p class="editorial">And the rest have an even number of spots. Natasha and Izzy say that none of the dominoes was left over. Rachel from Brynteg School explains that:</p> <p>All whole numbers are even or odd. The numbers on the dominoes are even or odd so there are no numbers left.</p> <p class="editorial">Jack and Peter answered the second part of the problem. They wrote:</p> <div>For the pairs equalling five you have the 0:3 + 0:2, 0:1 + 1:3, 1:1 + 1:2. The ones left over are, 2:2, 2:3 and 3:3.</div> <div>Other ways include, 2:2 + 0:1, 1:1 + 0:3, 0:2 + 1:2 then all the others are left over.</div> <div>There are always some dominoes left over but there are two that are always left over: 3:3 and 3:2 because they both equal five or over.</div> <p class="editorial">Well done, Jack and Peter. There are other ways to make pairs which total 5 and Thomas from Reading explains why:</p> <div>You could pair the dominoes up to total a number of five spots in three ways at any one time by pairing a three with a two or a one with a four. There are two dominoes which add up to two and they could each go with either one of the dominoes which add up to three. There are also two dominoes with a total of four spots and either one of those could go with the domino with only one spot on it.</div> <br></br> <div class="editorial">So here are all the possible ways of grouping the dominoes to make 5:</div> <br></br> <div>2,2 and 0,1 0,3 and 0,2 1,2 and 1,1 or</div> <div>3,1 and 0,1 0,3 and 0,2 1,2 and 1,1 or</div> <div>2,2 and 0,1 0,3 and 1,1 1,2 and 0,2 or</div> <div>3,1 and 0,1 0,3 and 1,1 1,2 and 0,2</div> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Domino Sorting</h2> <br></br> Here are some dominoes taken out of the full set:<br></br> <div><mdo:image alt="dominoes from 1, 0 to 3, 3" height="233" src="dominoes.gif" width="550"></mdo:image></div> <br></br> <div>Sort them into two groups - one group with an odd number of spots and one group with an even number of spots.</div> <div>Do you have any dominoes left over? Why or why not?</div> <br></br> <div>Now put the dominoes into pairs. The number of spots on each pair of dominoes must make a total of $5$.</div> <div>How many pairs can you make?</div> <div>Which dominoes are left over?</div> <br></br> <div>Can you pair them up in any different ways so that each pair adds to $5$?</div> <div>Which dominoes are left over now?</div> <br></br> <div>Are there any dominoes which are always left over?</div> <div>Can you explain why?</div> </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=4940&amp;part=index">This problem</a> will help learners become more familiar with odd and even numbers, and number bonds to five. It will also challenge them to justify findings.</div> <br></br> <h3>Possible approach</h3> <div>If you have an interactive whiteboard, you may find our <a href="http://nrich.maths.org/public/viewer.php?obj_id=6361&amp;part=index">Dominoes Environment</a> useful for this problem.</div> <br></br> <div>You might like to start by giving pairs of children a whole set of dominoes to explore and ask them some open-ended questions such as:</div> <ul> <li>How can you sort them?</li> <li>Can you make a pattern?</li> <li>Can you make a snake?</li> <li>What did you notice?</li> </ul> <div>Learners can then find the subset of dominoes that they need for this task and tackle it in pairs. It will provoke a lot of meaningful discussion and will give pupils the experience of having to argue mathematically.</div> <br></br> <div>In a plenary, focus on the "can you explain why?"- the beginning of an understanding of proof. The sooner we start children justifying their conclusions, the better mathematicians they&#39;ll turn out to be!</div> <br></br> <h3>Key questions</h3> <div>Can you think of some pairs of numbers that add to five?</div> <div>Have you added up or counted the spots on each domino?</div> <div>Which domino could you pair with this one so that there are five spots altogether?</div> <br></br> <h3>Possible extension</h3> <div>You might want some children to find all the different ways of making pairs that add to $5$. This could be by picking two and then replacing them, or by finding all the different combinations which could be made at the same time (the problem as written focuses on the latter). Whichever way, part of their task should be to convince you that they have not missed any pairs out.</div> <br></br> <h3>Possible support</h3> <div>Children would really benefit from having sets of dominoes to manipulate as this allows them to change their mind easily, so giving them more confidence to begin the task, and also prevents them from using any domino twice.</div> <br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">You might find it useful to find some real dominoes to use.<br></br> Have you added up or counted the spots on each domino?<br></br> Can you think of some pairs of numbers that add to five?<br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> Even: 1,1 2,0 2,2 3,1 3,3<br></br> Odd: 1,0 2,1 3,0 3,2<br></br> None left over because all numbers either even or odd<br></br> <br></br> Pairs to make 5:<br></br> <br></br> 2,2 and 0,1 0,3 and 0,2 1,2 and 1,1<br></br> or 3,1and 0,1 0,3 and 0,2 1,2 and 1,1<br></br> or 2,2 and 0,1 0,3 and 1,1 and 1,2 and 0,2<br></br> or 3,1 and 0,1 0,3 and 1,1 1,2 and 0,2<br></br> <br></br> Description:Try grouping the dominoes in the ways described. Are there any left over each time? Can you explain why?<br></br></mdoxml> 2 3 1 0 0 0 0 Try grouping the dominoes in the ways described. Are there any left over each time? Can you explain why? Mathematics Tools Dominoes Numbers and the Number System Odd and even numbers Calculations and Numerical Methods Addition & subtraction Decision Mathematics and Combinatorics Combinations Handling, Processing and Representing Data Sorting data Admin Lower primary mapping document