This problem 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.
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!).
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.
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.
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.
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
Design and make your own pencil case for a particular number of pencils.
Air Nets may help students who have difficulty in visualising how a 2-d net will fold up to form a 3-d solid.