A second strategy that children use to determine which figure has a larger area is to decompose the region covered by one figure and to try to fit the parts into the other. If the parts fit exactly, the two figures cover the same area. This strategy also works well for familiar shapes like the two rectangles displayed below, but it is often difficult to decompose a shape like a hand into units that can be easily perceived. Through experience with strategies like superposition or decomposition, children develop ideas about measuring areas. Measuring area allows children to say which hand has the larger area, and also to know how much larger it is.
Children often start systematically measuring with a direct modeling strategy.
They can then apply these rules with understanding. Teaching children these rules before they have experiences leads to rote measurement with little understanding of why the rule works. For example, fifth grade children sometimes tell us that the area of a 3 by 3 square is 9 square inches. Here children are simply applying rote formulas without understanding that the area of the square cannot be less than the area of the triangle. As children build their ideas about measurement of area, they often entertain a variety of ideas that capture some of the key ideas of measurement but violate others.