Domino Join Up

245 /www/nrich/html/content/03/05/letme1/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> You will need a double-six set of dominoes for this problem (a standard set of dominoes) but without the seven doubles (double one, double two etc). That is twenty-one domino pieces in all. <mdo:image height="63" width="402" src="245A.png" alt=""></mdo:image> Can you arrange fifteen of the pieces round a track like this so that all the touching domino pieces add to $6$ and the ends join up? <mdo:image height="285" width="398" src="245.png" alt=""></mdo:image> You will have six pieces left over. Using the same twenty-one pieces, can you now make all the joins add to $7$? Which six pieces will you have to leave out? Can you now make all the joins add to $5$ using the twenty-one pieces? Which six pieces will you have to leave out now? Now can you make all the joins add to make an even number? What about all the joins adding to make an odd number? You may find our <a href="http://nrich.maths.org/public/viewer.php?obj_id=6361&part=index"> Dominoes Environment</a> useful for working on this problem. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Sarah, Aleisha, Samuel and Jesse from Rutherglen Primary wrote to say: We had fun doing this, and it was fairly easy the way we understood it. The hardest part was making sure we had the right twenty one pieces in the first place, because we used some old mixed sets from school. We worked out a great system to check we had what we needed though! Joining to $6$. This was really easy, we all had lots of fun doing this we all liked it. <mdo:image height="187" width="248" alt="" src="rutherglen6.gif"></mdo:image> Joining to $7$. This was a bit tricky but we all worked as a team and got it done. We also worked out that the blank ones had to be left out. <mdo:image height="198" width="264" alt="" src="rutherglen7.gif"></mdo:image> Joining to $5$. This one was easy because we knew that we had to get the sixes out of the road so it can work. <mdo:image height="193" width="259" alt="" src="rutherglen5.gif"></mdo:image> Joining even. This was easy because we just had to put two even numbers or two odd numbers together and it will make an even number. <mdo:image height="186" width="245" alt="" src="ruthergleneven.gif"></mdo:image> Joining odd. This was easy because we got help off the even one a bit. Then we just kept on working really hard as a team and got it done. <mdo:image height="214" width="285" alt="" src="rutherglenodd.gif"></mdo:image> Joining even then odd. We did this one for a bit of fun and we had lots of fun. We had to work out what will fit together and what won't. <mdo:image height="222" width="295" alt="" src="rutherglenoddeven.gif"></mdo:image> Well done all of you. Tom from Crawley Down Village C of E School told us: Dominoes adding up to $6$ (which also solves the even number problem): Leave out: $1/4, 5/6, 2/0, 4/3, 2/5, 5/1$ I just tried different combinations. Dominoes adding up to $7$ (which also solves the odd number problem): Leave out: $0/1, 0/2, 0/3, 0/4, 0/5, 0/6$ I worked out that you couldn't have dominoes with zeros because it wouldn't add up to $7$. Dominoes adding up to $5$: Leave out: $6/0, 6/1, 6/2, 6/3, 6/4, 6/5$ I worked out that you couldn't have dominoes with six because it would be too much. Good reasoning, thank you Tom. Alex from Heathfield sent in some different solutions: Adding to $6$: <mdo:image height="267" width="400" alt="" src="alex6.gif"></mdo:image> Adding to $7$: <mdo:image height="300" width="400" alt="" src="alex7.gif"></mdo:image> Adding to $5$: <mdo:image height="300" width="400" alt="" src="alex5.gif"></mdo:image> Adding to an even number: <mdo:image height="311" width="400" alt="" src="alexeven.gif"></mdo:image> Adding to an odd number: <mdo:image height="307" width="400" alt="" src="alexodd.gif"></mdo:image> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <h3>Why do this problem?</h3> <a href="http://nrich.maths.org/public/viewer.php?obj_id=245&part=index">This problem</a> is useful for providing an interesting way of practising simple addition and subtraction. It can also be used to learn more about the rules for adding odd and even numbers. The problem requires a trial and improvement approach, and it is worthwile encouraging learners to articulate their method so that it can be compared with someone else's. <h3>Possible approach</h3> <div>You could start by joining together various pairs of dominoes so that the touching ends make a total of six. If you display several pairs on the board at the same time, you could ask children to talk about what they notice and this will make a nice lead into the problem itself. (Using the <a href="http://nrich.maths.org/public/viewer.php?obj_id=6361&part=index">Dominoes Environment</a> for this purpose may be useful.) Depending on their experience of using dominoes, you may need to give the group time to handle a full set of dominoes (one between two) first of all, before asking them to talk about how it is made up. You can then introduce the language of 'doubles' and pairs can remove the doubles from their own set.</div> <div>After this, the children could work on the problem with their partner so that they are able to talk through their ideas. <a href="/content/03/05/letme1/Domino%20Join%20Up.pdf">These sheets</a> have a board with spaces for the fifteen domino pieces. The second page has the twenty-one pieces from the reduced set. If you want to use the board with a real set in which the pieces are a different size from the one given, you may be able to use a photocopier that can enlarge or reduce to change the size of the board to fit.</div> <div>At the end of the lesson an interesting discussion could arise about which six pieces were left at the end of each activity and why this was so. You may also choose to focus on asking the children what they notice about adding odd and even numbers.</div> <h3>Key questions</h3> <div>How much more do you need to make six?</div> <div>What goes with one to make six?</div> <div>If you take four from six what does it leave?</div> <div>Can you find another way to do it?</div> <div>Which domino pieces will you have to leave out if you are making seven? Why can't you use them?</div> <div>Which domino pieces will you have to leave out if you are making five?</div> <div>Is this number odd or even?</div> <h3>Possible extension</h3> Children who find this problem straightforward could use a full set of dominoes and, following the same rules, try to make them join up. <h3>Possible support</h3> It might be helpful for children to make a pictorial list of the dots on the dominoes that make six. You might find <a href="http://nrich.maths.org/public/viewer.php?obj_id=4940&part=index">this problem</a> (Domino Sorting) useful as an alternative. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> You might find <a href="/content/03/05/letme1/DomJU.pdf">these sheets</a> with a picture of the track and twenty-one domino pieces useful if you don't have a set of dominoes to use. Don't be afraid to just have a go and to see what happens. You may be able to make some final 'tweaks' later on. </mdoxml> 2 4 1 0 0 0 0 Domino Join Up Can you arrange fifteen dominoes so that all the touching domino pieces add to 6 and the ends join up? Can you make all the joins add to 7? Using, Applying and Reasoning about Mathematics Trial and improvement Mathematics Tools Dominoes Calculations and Numerical Methods Addition & subtraction