This lesson introduces the double number line diagram, a useful and efficient tool for reasoning about equivalent ratios.

The number lines that make up this new representation are similar to those seen in earlier grades in that:

• Each number corresponds to a point located that many units away from 0 on the number line.
• Because numbers correspond to distances, the distance between, for example, 0 and 12 is three times the distance between 0 and 4.
• We can choose the scale to use—whether each interval represents 1 unit, 2 units, 5 units, and so on. 
• The lines can be extended as needed.

In a double number line diagram, we use two parallel number lines—one line for each quantity in the ratio—and line up the 0s vertically. We choose a scale on each line so that equivalent ratios also line up vertically.

For example, if the ratio of the number of eggs to the cups of milk in a recipe is 4 to 1, we can draw one number line to represent the number of eggs and another for the cups of milk. On the two number lines, the positions of 4 eggs and 1 cup of milk would line up vertically, as would 8 eggs and 2 cups of milk, and so on.

Double number lines are more abstract and general than are discrete diagrams because quantities are represented by locations on a number line rather than with counts of objects. Relating the concrete to the abstract allows students to practice reasoning quantitatively and abstractly (MP2).