no matter how thin, has a thickness to it that makes it a solid 3-D shape rather than only 2-D. Measurements of actual physical objects are never exact, either. Even so, valid reasoning about ideal geometric shapes and ideal theoretical measurements can be aided with approximate physical shapes and measurements.

Measurement

In its most basic form, measurement is the process of determining the size of an object. But the size of an object can be described in different ways, depending on the attribute one chooses. For example, the size of a tower made of cube-shaped blocks might be described by the height of the tower (a length) or in terms of the number of blocks in the tower (a volume). The size of the floor of a room that is covered in square tiles can be described in terms of the number of tiles on the floor (an area). The most important measurable attributes in mathematics are length, area, and volume.

To measure a quantity (with respect to a given measurable attribute, such as length, area, or volume), a unit must be chosen. Once a unit is chosen, the size of an object (with respect to the given measurable attribute) is the number of those units it takes to make (the chosen attribute of) the object.

For length, a stick, for example, 1 foot long, could be chosen to be a unit. With respect to that unit of length, the length of a toy train is the number of those sticks (all identical) needed to lay end to end alongside the train from the front to the end.

For area, a square tile, such as a tile that is 1 inch by 1 inch, could be chosen to be a unit. With respect to that unit of area, the area of a rectangular tray is the number of those tiles (all identical) it takes to cover the tray without gaps or overlaps. Although squares need not be used for units of area, they make especially useful units because they line up in neat rows and columns and fill rectangular regions completely without gaps or overlaps.

For volume, a cube-shaped block, such as a block that is 1 inch by 1 inch by 1 inch, could be chosen to be a unit. With respect to that unit of volume, the volume of a box is the number of those cubes (all identical) it takes to fill the box without any gaps. Although cubes need not be used for units of volume, they make especially useful units because they line up in neat rows and columns and stack in neat layers to fill box shapes completely without gaps.

Once a unit has been chosen, a measurement is a number of those units (e.g., 3 inches, 6 square inches, 12 cubic inches). So measurement is a generalization of cardinality, which describes how many things are in a collection. For young children, measurements will generally be restricted to whole numbers, but measurement is a natural context in which fractions



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