Lesson 6

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

The lines in a double number line diagram are similar to the number lines students have seen in earlier grades in that:

• Numbers correspond to distances on the line (so that the distance between, say, 0 and 12 is three times the distance between 0 and 4);
• We can choose what scale to use (i.e., whether each interval represents 1 unit, 2 units, 5 units, etc.);
• 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 choose a scale on each line so equivalent ratios line up vertically.

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

Because they represent quantities with length on a number line rather than with counts of objects, double number lines are both more abstract and more general than discrete diagrams. Later in this unit, students will learn an even more abstract representation of equivalent ratios—the table of values. Connecting the concrete to the abstract helps students connect quantitative reasoning to abstract reasoning (MP2). Though some activities are designed to hone students’ facility with particular representations, students should continue to have autonomy in choosing representations to solve problems (MP5), as long as they can explain their meaning (MP3).