Division vs. Ratio

I’ve noticed that the teachers of fractions tend to make a strong distinction between division and ratios, but I haven’t entirely understood why. In my mind, ratios and division are intimately related, even inextricably so.

However, my reflections on the abstract unit has brought me to a realization that there is one significant difference between ratios and division, or rather, that ratios are a special kind of division with regards to unit.

A canonical example of division involves separating a group of objects among a group of people. Yum, pizza! We have three pizzas, each of which has eight slices. We have six diners. If the diners all eat the same amount of pizza, how much do they each have?

The answer, of course, is four pieces, which represents half a pizza per person. Using explicit units, where P = pizzas and D = diners, we might write: \[3P \div 6D = 3 \div 6 PpD = \frac{1}{2} PpD\]

In division with implied units, then, the solution involves “items per group” in some fashion. That’s how we tend to teach division, and reinforce the teaching.

When we speak of ratios or scaling, though, we’re generally comparing two items of the same unit. If we have a recipe for two people and we want to make it for six people, we use a scale factor of three. If we have a map where each centimeter represents a hundred meters in the world, we use a scale factor of 10,000:1. If we’re mixing cocktails, we’re using 4 parts of gin for every 2 parts of vermouth. These are all ratios.

In this perspective, scale is indifferent to the actual measure used. If a map has a scale factor of 10,000:1, it doesn’t matter whether I measure distances on the map in centimeters or inches. A measurement of 2 inches is equal to 20,000 inches (about 3/10ths of a mile) in the real world

The thing is, though, that the mathematics of the numbers themselves is the same. If your company’s salary is $10,000 per month for every manager, then your total monthly salary for managers will be 10,000 times the number of managers. If you’re using a 10,000:1 map, then your “real world” distance will be 10,000 times the measurement on the map.

The units of scale factors are perhaps more abstract than the units of division, and the purpose is different. When two objects are in a ratio-based relationship, then every measurement-based aspect of the objects are in a ratio-based relationship (albeit one tied to the dimensions involved). Division, by its name, suggests creating equally sized subgroups of a larger group. Conceptually, then, the function of ratios is different than the function of division.

However, what is the goal of teaching fractions? This is a serious question: I’m not sure what the clear answer is, and I’m even less convinced that most math teachers have reflected on it. If the goal is to teach a mathematical process, independent of the units we generally so heartily disregard, than ratios and division can be taught in parallel, if not as the same thing. If the goal is to teach the conceptual difference between the process of dividing and the process of scaling, then why isn’t that concept more explicitly separated from the mathematical process?

Regardless, I do not think that students should be moving into high school with the impression that division and ratio fractions are inherently such different things that they’re unconnected in student minds.

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