The way we define area is in terms of little squares. (We could use little equilateral triangles or little right-angled isosceles triangles or some other unit i

The way we define area is in terms of little squares. (We could use little equilateral triangles or little right-angled isosceles triangles or some other unit instead, but we don't, conventionally preferring squares.) So the way to learn about areas in general is to start with the area of a unit square = 1, and then clarify the properties of area in general, notably that we can cut and paste shapes without changing the total area, as long as we don't overlap them, which means we can count up the number of unit squares and that tells us the area of a figure. Most of the rest can be figured out by students if guided by a well organized set of problems which build on each-other. We can proceed to first finding areas of shapes made using axis-aligned sides of natural number lengths on a square grid. For these the area can be found by directly counting squares, and e.g. by natural number multiplication in the case of rectangles. Then we can look at the areas of other shapes with vertices on a square grid. Then we can look at the areas of rectangles with rational side lengths (no longer nicely on the original unit grid; instead we need to make a finer grid to count, resulting in rational-number areas, since we must relate the grid units). Then we can next proceed to parallelograms – in a square-area-unit conventional world these are arguably more fundamental than triangles, but we could deal with right-angled triangles at about the same time. Triangles where we can find base and height can come next; these can be split into a pair of right-angled triangles or duplicated and glued into a parallelogram. After that finding areas of other shapes (e.g. with irrational side lengths) takes more machinery from Euclidean geometry (starting from the Pythagorean theorem), and can be delayed a while. While talking about parallelograms it would be nice to take some squares and other rectangles outside and look at the shadows they cast on the ground. This is a nice chance to introduce the concept of affine transformations, which preserve parallel lines and area ratios, as we can see by observing the shadow cast by a square grid (e.g. made of wire or drawn on a transparency sheet), especially if we have some translucent shapes lying on the grid. Basic transformation geometry (especially discussions of reflection/rotation/translation [isometries] and snowflakes, wallpaper tilings, etc., and affine transformations) is accessible at quite a young age and some hugely valuable topics to introduce early so that kids will be prepared to extend them later. More general projective transformations can wait a while.