How do you write a computer program that creates the illusion of stretching elastic bands between pegs of a Geoboard? The answer contains some surprising mathematics.
This article for students gives some instructions about how to make some different braids.
This article for pupils gives an introduction to Celtic knotwork
patterns and a feel for how you can draw them.
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.