Here’s a page from a forthcoming book I’m working on about the core geometric truths (first principles) that underlie–and led to the development of– all our layout tools, including the sector. If you have kids around the house, you might find it interesting (maybe even enlightening) to put the sector into their hands to show them how to physically (i.e. by hand and eye) manipulate a tool to illustrate and quickly grasp the intuitive concept of the fraction.
As you’ll see when the book comes out, you can use every one of your layout tools to illustrate–that is, to perform–the reality behind the abstract geometry that your kids are being injected with at school.
If you don’t have a sector, you can download (for free) a template to make one on the “shop” page of this site. You’ll find a manual for using this three-scale sector on the page as well…for a small fee to help George and I keep the lights on and the electrons flowing.