183 /www/nrich/html/content/00/11/letme2/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p>In a certain city houses had to be built in a particular way.</p> <p>There had to be two rooms on the ground floor and all other rooms had to be built on top of these.</p> <p>Families were allowed to build just one room for each person living in the house.</p> <p>So a house for two people would look like this:</p> <div style="text-align: center;"><mdo:image height="67" width="132" alt="" src="lmt1.gif"></mdo:image></div> <br></br> <p>but a house for three people could look like one of these:</p> <p style="text-align: center;"><mdo:image height="126" width="376" alt="" src="lmt2.gif"></mdo:image></p> <p>There are some families of seven people living in the town.<br></br> <br></br> In how many different ways can they build their houses?</p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p><span class="editorial">Thank you for the solutions we received for this problem. You agreed that there are six different ways of building a house for seven people. Chris from Holyport C of E Primary, Yasi from Canberra Grammar and James from St Woolos, Newport all sent in pictures of the houses. Here is James':</span></p> <br></br> <mdo:image height="165" width="550" src="JamesSol.gif" alt=""></mdo:image><br></br> <span class="editorial"> </span><br></br> <p><span class="editorial"><span class="editorial">I like the way you have done this in a very logical order, James, moving one 'room' from the left to the right each time. Jordan also wrote to us and described the same method as James', but in words:</span></span></p> I started with seven cubes because I had to house seven people.<br></br> I drew one stack and reversed it, just moving the odd cube across. <br></br> I used the same method two more times, changing the height of the stack and moving it from left to right. <br></br> My answer is six different types of house. <br></br> <br></br> <p class="editorial">Alice from Henrietta Barnett and Patricia from Chongfu Primary School both described the six solutions too. </p> <p><span class="editorial">However, Callum from Canberra Grammar School went a step further and decided to look at different combinations of rooms if they were different colours. So, for example, you may decide you have two red rooms, two yellow rooms and three blue rooms. You could then investigate the number of different ways you could arrange the coloured rooms for each of James' pictures. This makes a nice extension to the problem, Callum - well done!</span></p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>A City of Towers</h2> <br></br> <p>In a certain city houses had to be built in a particular way.</p> <p>There had to be two rooms on the ground floor and all other rooms had to be built on top of these.</p> <p>Families were allowed to build just one room for each person living in the house.</p> <p>So a house for two people would look like this:</p> <div style="text-align: center;"><mdo:image alt="" height="67" src="lmt1.gif" width="132"></mdo:image></div> <br></br> <p>but a house for three people could look like one of these:</p> <p style="text-align: center;"><mdo:image alt="" height="126" src="lmt2.gif" width="376"></mdo:image></p> <p>There are some families of seven people living in the town.<br></br> <br></br> In how many different ways can they build their houses?</p> </div> <br></br> <h3><span style="font-weight: bold;">Why do this problem?</span></h3> <a href="http://nrich.maths.org/public/viewer.php?obj_id=183&amp;part=index">This problem</a> is an investigation into combinations of a number of cubes. It is a practical activity which involves visualising and relating $3$D shapes to their representation on paper. Young children are often introduced to sets of regular polyhedra and similar sorts of shapes, less often do they systematically explore shapes made up from cubes.<br></br> <br></br> <h3>Possible approach</h3> <div>You could start with <a href="/content/00/11/letme2/City%20of%20Towers.doc">this story</a> as an introduction to the problem. Alternatively you could simply talk through the problem as it is written. Ideally, it would be good to supply interlocking cubes or other cube bricks and $2$ cm squared paper or plain paper for recording. (You could use <a href="/content/00/11/letme2/183.pdf">this sheet</a>.) It might help to begin the challenge all together before asking children to work in pairs on the problem so that they are able to talk through their ideas and compare their results with a partner.</div> <br></br> <div>Some children may need help recording their models and you could demonstrate this on the interactive whiteboard. If $2$ cm cubes have been used then they can lay their shape on the paper and see how it fits into the squares. Alternatively, children might just sketch their models on plain paper or, if you have enough cubes, they can keep each model.</div> <br></br> <div>In the plenary, as well as comparing results, it would be good to spend time talking about how the children approached the problem. Some might have started straight away with seven cubes, others might have tried four cubes, then five, etc. Some children might have made the models, some might have been able to picture the houses and draw them without using cubes. It can be useful to discuss the advantages and disadvantages of each different method. Depending on the children&#39;s experience, you can also draw attention to those that have used a systematic way of finding all the houses. If most of the children have not developed a system, you could line up models in a particular order for all to see so that they notice the system themselves. This way, they may be able to spot any that are missing.</div> <br></br> <h3>Key questions</h3> <div>How many cubes are there in this one? Would it be a good idea to count them?</div> <div>Are all your houses different from each other?</div> <div>Could you put this cube in a different place?</div> <div>How will you draw your houses?</div> <br></br> <h3>Possible extension</h3> Some children could investigate other numbers of cubes or create their own rules for building houses.<br></br> <h3>Possible support</h3> <div>You may like to suggest that some children start by finding all the houses for four people, then five etc.</div></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> You could draw the houses on squared paper or make them with cube bricks.<br></br> How will you know you have found all the different houses?<br></br> You could start by finding all the houses for four people, then five people etc.<br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p>Robin from Moorfield Junior, Stockport says:<br></br> The answer is :- There are six ways of building a tower for seven people.</p> <p><mdo:image alt="" src="fig1.gif"></mdo:image></p> <br></br></mdoxml> 2 4 1 0 0 0 0 In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses? Decision Mathematics and Combinatorics Combinations Using, Applying and Reasoning about Mathematics Visualising Using, Applying and Reasoning about Mathematics Working systematically Using, Applying and Reasoning about Mathematics Recording mathematics Using, Applying and Reasoning about Mathematics Practical Activity 3D Geometry, Shape and Space 2D representations of 3D shapes Admin Lower primary mapping document