Some people consider the idea of proportional relationship as the culmination of the elementary school mathematics and the cornerstone of the middle school mathematics. This standards is one example of how proportional relationships play a role in different parts of the middle school mathematics.
Let's think about a situation of converting linear measurements between inches and feet.
You can easily see that as the numbers for inches become 2, 3, 4,... times as much, the numbers for inches also become 2, 3, 4, ... times as much. Therefore, these numbers are in a proportional relationship. Thus, we can use all the tools we have discussed previously to convert from one unit to another.
Suppose you want to find out how many inches 34 feet may be, you can set up the double number line representations in this way.
This representation shows that we know the per-one (or per-unit) quantity and you want to know the number corresponding to 34 units. So, you can use multiplication to find the missing number: ? = 12 x 34.
Going the other direction, for example, converting 104 inches to feet, can be represented in the same way.
Again, we know the per-unit quantity, and you want to know how many units would correspond to 104. Thus, this is a quotitive (measurement) division situation. So, you can find the missing number by division: ? = 104 ÷ 12.
To solve all these unit conversion problems, students do need to know (or be able to look up) one relationship between the two units - and it doesn't have to be 1 to something else. If you know that 2 feet = 24 inches, that's good enough to set up a double number line representation. You can solve it like you do with other proportion problems.
In principle, the situation remains the same whether you are converting within or across different measurement systems. If you know that 1 inch is approximately 2.5 cm, that is enough information for students to convert between inches and centimeters - approximately, but all measurements are approximation, anyway. Although students in earlier grades should be able to convert measurements from one unit to another in some simple cases, once students learn about proportional relationships, they no longer have to think of it in isolation. The idea of proportional relationships, thus, unifies many of the ideas students have learned previously. And, helping students to revisit some of those ideas and look at them from a new perspective is something we need to emphasize, not just the procedure of solving proportional problems.
