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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.
A description of how to make the five Platonic solids out of paper.
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.