Why do this problem?

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.  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. 

Although this problem is called 'Witch's Hat' and making conical hats is a great way to initiate a lot of mathematical discussion and investigation, making hats is ideal for groups where some students need a simpler task and others need to be extended.

This problem could be linked with the Design Technology curriculum, and used to support approaches to design covered in DT.

Possible approach

Equipment required:

• Some hats for students to take apart and put together again
• Lots of scrap paper, material and card
• Rulers, protractors, compasses, scissors
• Squared paper
• Sellotape and glue

Start by getting the class to brainstorm what shapes they think they might need to cut out for each hat.  Then suggest students divide into groups according to which type of hat they want to work on.

The top hat and the fez could be modelled as cylinders, while the wizard's hat and the witch's hat are cones.  The fez could also be modelled as a frustum.  A cylinder needs a rectangular section with length equal to the circumference of the hat.  A cone needs a sector of a circle - any angle will do provided the sector is less than a full circle (why?).  However, a semi-circle is probably a good place to start.  A frustum starts off life as a cone and then has its top removed.

Students need to remember that the point of a hat is to sit on someone's head!  They will probably need to measure some heads to find out what a sensible circumference ought to be.

Initial designs could be made with scrap paper and sellotape to see what works.  Once groups are happy that their nets do make up into viable hats, then it's time to make a real hat out of card or fabric - this could be an opportunity for the maths and DT departments to work together on a shared project.  The finished products would make a great display!

Key questions

• How many separate pieces do you need to make a hat?
• What shape are they?
• How do you make sure the hat fits a person's head?
• Is there a 'best' hat?  What criteria might you use to judge hats?

Possible extension

Conical hats are more difficult to get right than cylindrical ones.  Once students are happy that a whole circle won't make a cone (why not?) they could investigate how changing the sector angle affects the kind of cone that results.  If students start from a semi-circle, what difference does it make to the finished cone as they increase or reduce the angle?  Students could investigate how the height of the finished hat depends on the angle of the sector - and whether the same hat can be achieved in more than one way.
This is an ideal opportunity to revise acute, obtuse and reflex angles.

Students could then investigate what hats they can make if there are constraints on the materials they can use, say a piece of A3 card or paper, or a particular length of fabric.  Drawing scale diagrams on squared paper might help them to investigate different ways in which the hats can be cut out of the card or fabric.  

Possible support

For students who struggle to understand how the 3-D hat is related to the 2-D net, it helps to have a collection of 'hats' which they can take apart and put back together again.  A simple cylindrical hat with just a top and no brim (like the fez) would be the most straightforward hat to make first.