In this lesson, students explicitly connect and contrast double number lines and tables. Students consider the ways in which the two representations are alike (for instance, labels are needed and numbers that form equivalent ratios line up) and how they are different. Students examine the relationship between values in different rows in a table. Two features of tables make them more flexible than double number lines:

• On a double number line, differences between values are represented by lengths. This feature can help support reasoning about relative sizes, but it can also be a limitation when large or small numbers are involved. A table removes this limitation because relative sizes of values are no longer represented by the geometry of a number line.
• On a number line, values are listed in the order of their size. In a table, pairs of values can be written in any order, allowing for more nimble problem solving. Suppose 5 pounds of coffee cost \$40. How much does 8.5 pounds cost? The table here shows that we can scale up and down as needed to find the cost of 8.5 pounds.

At this point in the unit, students should have a strong sense of what it means for two ratios to be equivalent, so they can fill in a table of equivalent ratios with understanding instead of just by following a procedure (such as dividing and multiplying the values in the table in a particular order). Students can also use other representations to support or check their reasoning, if needed.

As students analyze different representations and identify and explain correspondences between them, they practice making sense of problems (MP1).