8343
/www/nrich/html/content/id/8343/
1
0000-00-00T00:00:00
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><p><mdo:image alt="pencil cases" src="pencilcase-3.png" style="float: right; width: 250px; height: 250px;"></mdo:image>I want a new pencil case for my 15 coloured pencils. I don't want one that will take up too much room in my bag, but it does need to be big enough for my pencils.</p>
<p>Alison thinks I should get the blue triangular one. Charlie thinks I should get the green cylindrical one. Steve prefers the red hexagonal one.</p>
<p>I like them all, so I'm going to get the one which takes all 15 pencils comfortably, but takes up the least room in my bag.</p>
<p>When I measured them I discovered that they were all 20cm long, with a maximum width of 4cm. They are all prisms and their cross-sections are an equilateral triangle, a circle and a regular hexagon respectively.</p>
<p> </p>
<p>Try making three 'pencil cases' out of card, to see which would be the best fit.</p>
<ul>
<li>How many pieces of card do you need for each?</li>
<li>What shape would they be, and how many of each shape do you need?</li>
<li>What lengths do you already know, what might you need to find out? Is there a simple practical way to find any missing information, or could you calculate any missing lengths?</li>
</ul>
<p>Cut out the three nets then stick them together with sellotape to make the 'pencil cases', leaving an opening of some kind. Looking at them, do you have any sense which would be best? Which appears to have most space inside, which least? Which will take up most room in my bag, which least?</p>
<p>Find 15 pencils and see how they stack in the 'pencil cases'. Which is best now?</p></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><h3>Why do this problem?</h3>
<p><a href="http://nrich.maths.org/8343">This problem</a> provides students with an opportunity to engage in mathematical modelling, using practical activity as a way of investigating a problem which focuses on nets and prisms. Many students find it difficult to relate the net of a solid to its 3-d appearance or to mentally unpack a solid to visualise its net, and the modelling approach
will help them with this, without getting bogged down in calculation. This problem could be linked with the Design Technology curriculum, and used to support approaches to design covered in DT.</p>
<h3>Possible approach</h3>
<h4>Equipment required:</h4>
<ul>
<li>Three demonstration prisms - one cylindrical, one triangular and one hexagonal - as close to length 20cm and maximum width 4cm as possible. Perhaps the DT department could help!</li>
<li>Other prisms for students to take apart to see what their nets look like.</li>
<li>Lots of pencils (ordinary or coloured).</li>
<li>Elastic bands suitable to hold 15 pencils.</li>
<li>Sellotape or parcel tape.</li>
<li>Lots of used A4 paper.</li>
<li>A4 card.</li>
<li>Isometric paper.</li>
<li>Scissors, compasses and rulers.</li>
</ul>
<p>Start by showing the students the three prisms, then explain the problem: they need to investigate which shape will on the one hand provide enough room for 15 pencils, but on the other hand not take up any more room in a bag than necessary. Invite them to suggest which they think would be best. This is also an opportunity to rehearse the use of appropriate words, such as
prism and net, and the difference between a three-dimensional object and its two dimensional cross-section (a triangular prism is not a triangle!).</p>
<p>Students should then think about what the net of each prism looks like. If it is to be made in card, how many separate pieces does each require, what shape are they, what lengths do they know, how will they find out any they don't. </p>
<p>Rather than wasting card at this stage, it might be a good idea to investigate the nets with paper. There is no need to worry about tabs, as the nets can be stuck together with sellotape or parcel tape.</p>
<p>Sticking them together will test the accuracy of their nets, and will help reinforce that sides which are to be stuck together need to be the same length. If students have not yet learnt how to calculate the circumference of a circle, they can draw a circle with diameter 4cm then measure its circumference with string. Isometric paper will help with drawing an equilateral triangle
and a hexagon with a maximum width of 4cm, and will also facilitate comparing their areas to see which prism has most space inside it (ie. the greatest cross-sectional area, since the volumes/capacities of prisms are dependent on the areas for a given length). If students have not yet learnt how to calculate the area of a circle, they can compare their paper circle with their paper
equilateral triangle and hexagon and see which fits inside which.</p>
<p>Once students are happy with their paper nets, they can make card nets to be made up into card pencil cases. They can then experiment with 15 pencils to see which pencil case best fits the constraints given. To start with it might be best to bunch the pencils together in an elastic band</p>
<h3>Key questions</h3>
<ul>
<li>What shape is the body of the cylinder?</li>
<li>How can we make sure that edges which need to match in a net are the same length?</li>
<li>How can we compare the areas of a triangle, a hexagon and a circle - and if they all have a maximum width of 4cm, which is going to have the smallest area, which the largest?</li>
<li>Why is it sufficient to compare the cross-sectional areas of prisms of a given length, if we want to know how their volume/capacity compares?</li>
<li>What kind of stacking arrangement of the pencils is best for a given shape? (The number of pencils suggested is 15 because they can be stacked in a triangular array as well as other ways).</li>
</ul>
<h3>Possible extension</h3>
<p><a href="https://nrich.maths.org/8342">Design and make your own pencil case</a> for a particular number of pencils.</p>
<h3>Possible support</h3>
<p><a href="http://nrich.maths.org/6307">Air Nets</a> may help students who have difficulty in visualising how a 2-d net will fold up to form a 3-d solid.</p></mdoxml>
2
3
0
0
1
0
0
Packing Pencils
Will my pencils fit into these pencil cases?
Applications
Maths in STEM
Applications
Design
3D Geometry, Shape and Space
Solids
3D Geometry, Shape and Space
Prisms
3D Geometry, Shape and Space
Nets
3D Geometry, Shape and Space
Cylinders
2D Geometry, Shape and Space
Circles
Using, Applying and Reasoning about Mathematics
Mathematical modelling
Using, Applying and Reasoning about Mathematics
Real world
Measures and Mensuration
Perimeters
Measures and Mensuration
Area
Using, Applying and Reasoning about Mathematics
Practical Activity