Witch's Hat
What shapes should Elly cut out to make a witch's hat? How can she make a taller hat?
Age
11 to 16
Challenge level
Problem
Image
Choose one of the hats shown here, or design your own.
What shapes would you need to cut out to make it?
If you're not sure how to make a witch's hat, you could start with a top hat or a fez, and then see how you need to adapt them to make a witch's hat.
You could start by drawing the shapes you think she might need on squared paper, then cut them out to see if they fold into the right sort of hat.
Sellotape will do at this stage to stick your hat together.
When you're happy with your design, try making it with paper, card or fabric.
If you use A4 or A3 paper, try different ways of laying out the pieces you want on the paper.
What is the tallest witch's hat you can make from a sheet of A3 paper?
Here are some questions you might like to consider:
If you wanted to make the tallest hat possible from your A3 paper, would you choose to make a witch's hat, a wizard's hat, a top hat or a fez?
How much difference does having a brim make?
The fez could be modelled as a cylinder or as the frustum of a cone. How does this change the maximum possible height?
Images in the public domain, and taken from Clker.com and Wikimedia Commons (the fez).
Getting Started
Start with cylindrical hats.
How could you make a cylinder? Try cutting and unfolding a cylindrical container to see how it's made.
Now explore conical hats.
Roll up a piece of paper and cut it to make a cone, then unroll it. What do you notice?
How could you make a cylinder? Try cutting and unfolding a cylindrical container to see how it's made.
Now explore conical hats.
Roll up a piece of paper and cut it to make a cone, then unroll it. What do you notice?
Student Solutions
Thanks to Cubes, Aisling and Brandon for their ideas on how to make a wizard's hat or a witch's hat. Cubes suggested cutting out two triangles and sticking them together. Aisling and Brandon both suggested rolling a piece of paper into a cone.
Either of these methods would work to make a pointy hat.
But there is a better way to make a conical hat.
Alice worked through this problem and found a very surprising answer. She sent in some work which you can find here.
She only found the tallest wizard's hat, can you do something similar to find the witch's hat, fez or top hat?
Teachers' Resources
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
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.