length and on number lines (in around second or third grade). A number line is much like an infinitely long ruler, so number lines can be viewed as unifying measurement and number in a one-dimensional space. A number on a number line can be thought of as representing the length from 0 to the number (see Figure 2-11).

Because of the close connection between number lines and length, number lines are difficult for children below about second grade. In contrast, the number paths on most number board games used for preschoolers are a count model, not a number line. There is a path of squares, circles, or rocks, each has a number on it, and players move along this path by counting the squares or other objects or saying the number on them as they move. These are appropriate for younger children because they can support their knowledge of counting, cardinality, comparing, and number symbols.

In measurement, there is an important relationship between the size of a unit and the number of units it takes to make a given, fixed quantity. For example, if the triangle in Figure 2-5 is designated to have 1 unit of area, then the hexagon has an area of 6 units. But if one picks a new unit of area, such as designating the area of the rhombus in Figure 2-5 to be 1 unit, which is twice the size of the triangle, then the hexagon has an area of only 3 units.

Later in elementary school (in around second grade), children see this inverse relationship between the size of a unit of measurement and the number of units it takes to make a given quantity reflected in the inverse relationship between the ordering of the counting numbers and the ordering of the unit fractions (see Figure 2-12).

To use data to answer (or address) a question, one must analyze the data, which often involves classifying the data into different categories,