1156 /www/nrich/html/content/03/02/penta3/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p>Skeleton shapes are made with balls of modelling clay and straws.</p> <p>This shows a cube and a skeleton cube:</p> <div style="text-align: center;"><mdo:image width="336" height="163" src="skeleton1B.gif" alt=""></mdo:image></div> <p>How many balls of modelling clay and how many straws does it take to make the cube?</p> <p>Here are some piles of modelling clay balls and straws:</p> <p style="text-align: center;"><mdo:image width="400" height="161" alt="piles of balls and straws: 6 straws with 4 balls, 9 straws with 6 balls, 8 straws with 5 balls, 12 straws with 8 balls, 18 straws with 12 balls" src="Skeleton2.gif"></mdo:image></p> <p>Look at the shapes below and decide which piles are needed to make a skeleton of each shape.</p> <p style="text-align: center;"><mdo:image width="400" height="160" alt="solid triangular prism, tetrahedron, cuboid, square pyramid and hexagonal prism" src="Skeleton3.gif"></mdo:image></p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p>We had lots of correct solutions to this problem and you used many different ways of helping yourselves to reach the answers. Some of you made the shapes from straws and modelling clay, like Rachel, Abigail and Alistair from Histon and Impington Infant School. They sent in a photo of their models:</p> <p><mdo:image width="300" height="228" alt="" src="photo.gif"></mdo:image></p> <p>Others of you decided to make the shapes from Polydron, while some of you drew the shapes carefully. Alice from Tattingstone, and Scarlett and Sam from Cupernham, were among those who opted for drawing, but William, also from Cupernham sent in particularly careful sketches using isometric paper:</p> <p><mdo:image width="200" height="200" src="cuboid.gif" alt="cuboid"></mdo:image> Cuboid : 8 balls, 12 straws</p> <p><mdo:image width="200" height="152" src="hexpris.gif" alt="hexagonal prism"></mdo:image> Hexagonal prism : 12 balls, 18 straws</p> <p><mdo:image width="200" height="160" src="sqpyr.gif" alt="square pyramid"></mdo:image> Square pyramid : 5 balls, 8 straws</p> <p><mdo:image width="200" height="193" src="tetra.gif" alt="tetrahedron"></mdo:image> Tetrahedron : 4 balls, 6 straws</p> <p><mdo:image width="200" height="140" src="triangpris.gif" alt="triangular prism"></mdo:image> Triangular prism : 6 balls, 9 straws</p> <p>Chris and Michael from Moorfield Junior School also sent some excellent drawings and took the problem a step further by looking at the relationship between the number of faces, edges and vertices of different prisms. They enclosed this table:</p> <table cellspacing="0" cellpadding="5" border="1"> <tbody> <tr> <th>Type of prism</th> <th>Faces</th> <th>Vertices</th> <th>Edges</th> </tr> <tr> <td>triangular</td> <td>5</td> <td>6</td> <td>9</td> </tr> <tr> <td>cube or cuboid</td> <td>6</td> <td>8</td> <td>12</td> </tr> <tr> <td>pentagonal</td> <td>7</td> <td>10</td> <td>15</td> </tr> <tr> <td>hexagonal</td> <td>8</td> <td> <div class="c1">12</div> </td> <td>18</td> </tr> <tr> <td>heptagonal</td> <td>9</td> <td>14</td> <td>21</td> </tr> <tr> <td>octagonal</td> <td>10</td> <td>16</td> <td>24</td> </tr> <tr> <td>nonagonal</td> <td>11</td> <td>18</td> <td>27</td> </tr> <tr> <td>decagonal</td> <td>12</td> <td>20</td> <td>30</td> </tr> </tbody> </table> <p>Chris and Michael say:</p> <p>There are patterns you can see in the vertices and edges columns. The number of vertices is double the amount of sides on the 2D shape at each end. In the edges column it's three times the amount of sides on the 2D shape.</p> <p>Perhaps you can spot some more patterns too. Let us know if so.</p> <p>Thank you also to these people who sent us correct answers too:</p> <p>Hugo, Emile and Benny from Wesley College Prahran Preparatory School in Melbourne, Australia.<br></br> Charlotte and Thomas.<br></br> Al from Dudley.<br></br> Ryo, Jake and Charlie from Moorfield Junior School.<br></br> Samuel from Bispham Drive Junior School in Nottingham.<br></br> David from Tithe Barn Primary School.</p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Skeleton Shapes</h2> <br></br> <p>Skeleton shapes are made with balls of modelling clay and straws.</p> <p>This shows a cube and a skeleton cube:</p> <div style="text-align: center;"><mdo:image alt="" height="163" src="skeleton1B.gif" width="336"></mdo:image></div> <p>How many balls of modelling clay and how many straws does it take to make the cube?</p> <p>Here are some piles of modelling clay balls and straws:</p> <p style="text-align: center;"><mdo:image alt="piles of balls and straws: 6 straws with 4 balls, 9 straws with 6 balls, 8 straws with 5 balls, 12 straws with 8 balls, 18 straws with 12 balls" height="161" src="Skeleton2.gif" width="400"></mdo:image></p> <p>Look at the shapes below and decide which piles are needed to make a skeleton of each shape.</p> <p style="text-align: center;"><mdo:image alt="solid triangular prism, tetrahedron, cuboid, square pyramid and hexagonal prism" height="160" src="Skeleton3.gif" width="400"></mdo:image></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=1156&amp;part=index">This problem</a> helps children begin to understand the various properties of common geometric solid shapes, concentrating on edges and vertices. It also helps in promoting discussion and experimentation. Naming the shapes should be a help during discussion and description of what has been done, rather than being an exercise in its own right.<br></br> <h3>Possible approach</h3> <div>Before doing this problem children should have had plenty of free play, building with sets of solid shapes so that they begin to get a feel for their properties. They should also have chance to experiment with building skeleton shapes either with a kit such as "Constructo Straws" or with drinking straws and modelling clay (such as plasticine).</div> <br></br> <div>You could start on the problem by asking the group to tell you what they know about cubes. Using a large cube, ask them to count the faces, the edges and the vertices (corners). (Avoid the word "side" which can be confusing when discussing 3D shapes and use instead "face" and "edge".)</div> <br></br> <div>After this you could encourage the group to work in pairs on the actual problem from a printed sheet so that they are able to talk through their ideas with a partner. It is essential that children have real 3D shapes to handle and count as they work and if at all possible they should have opportunity to experiment by making skeleton shapes as well. <a href="/content/03/02/penta3/1156.pdf">This sheet</a> might be useful for recording for those children who would find making their own table for results difficult.</div> <br></br> <h3>Key questions</h3> <div>How many edges did you count? What does this tell you about the number of straws we need?</div> <div>How many corners did you count? What does this tell you about the number of balls of modelling clay we need?</div> How many edges meet at this corner?<br></br> <h3>Possible extension</h3> <div>Children could find other solid shapes and continue the activity. They could also record by drawing the shapes they have used on isometric paper although this is rather tricky!</div> <h3>Possible support</h3> Start by counting the faces on a cube - a large dice might be useful - and then the edges and finally the vertices. A non-permanent pen could be used to mark a real shape if children are having difficulty keeping track of their counting.<br></br> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> It might help to use real shapes so that you can count the number of edges and vertices (corners) they have.<br></br> How many edges does the first shape have? So how many straws would be needed? <br></br> How many vertices does it have? So how many balls of modelling clay would we need to make it? <br></br></mdoxml> 2 4 1 0 0 0 0 How many balls of modelling clay and how many straws does it take to make these skeleton shapes? 3D Geometry, Shape and Space Polyhedra Using, Applying and Reasoning about Mathematics Visualising Mathematics Tools Sets of shapes Admin Lower primary mapping document