arise. To fill a bucket with sand, a child might pour in 4 full cups of sand and another cup that is only half full of sand, so that the volume of the bucket is approximately 4½ cups.

An important but subtle idea about units, which children learn gradually, is that when measuring a given object, the larger the unit used to measure, the smaller the total number of units. For example, suppose there are two sizes of sticks to use as units of length: short sticks and longer sticks. More short sticks than long ones are needed to measure the same length. In other words, there is an inverse relation between the size of a measuring unit and the number of units needed to measure some characteristic.

Young children may also not grasp the importance of using standard units, which allow one to compare objects that are widely separated in space or time (see Chapter 3 for further discussion).

Shapes found in nature, such as flowers, leaves, tree trunks, and rocks, are complex, intricate, and 3-D rather than 2-D. In contrast, the familiar 2-D shapes studied in geometry, such as triangles, rectangles, and circles, are relatively simple. Compared with most shapes in the natural world, these shapes are relatively easy to draw or create and also to describe and analyze. Many manufactured objects, such as tabletops and appliances, have parts that are approximate triangles, rectangles, or circles. Many shapes in the natural world are approximate combinations of parts of these simpler geometric shapes. For example, a birch leaf might look like a triangle joined to a half-circle.

Although geometric shapes can be described and discussed informally and children can simply be told the names of some prototypical examples of these shapes (for ease of reference and discussion), these shapes also have mathematical definitions, which teachers should know.

Geometric shapes have parts and features that can be observed and analyzed. The shapes all have an “inside region” and an “outer boundary.” Distinguishing the inside region of a 2-D shape from its outer boundary is an especially important foundation for understanding the distinction between the perimeter and area of a shape in later grades. Except for circles, the outer boundary of the common 2-D geometric shapes consists of straight sides, and the nature of these sides and their relationships to each other are important characteristics of a shape. One can attend to the number of sides and the relative length of the sides: Are all the sides of the same length, or are some longer than others? Where two sides meet, there