Why is “per” so important?

When we teachers talk about proportional situations, we often use special words such as per. Even native speakers get confused by terms like these. Be alert, and if you need to, take a moment in class to be explicit about how you use these words.

Per

Per means, “for each” or “for every.”

In one usage of the word, we mean that we're going to share whatever comes before the word per equally among the things that come after. So if we have three people and twelve cookies, we ask, how many cookies per person?

In that situation, most students will see that to get the numerical answer, we divide. We compute 12 ÷ 3 = 4. Four cookies per person. Notice how the computation is easy but has no context. When we use the “units” (cookies, people) and the word “per,” the context and meaning come back.

When a “naked computation” appears in a context-rich problem, ask students what it means. In this case, “the four means four cookies per person” is an excellent answer.

For other uses of per, the sharing metaphor can get a little strained—but we still divide. Divide whatever before per by whatever comes after:

• Cookies per person? Divide the number of cookies by the number of people.
• Miles per gallon? Divide the number of miles by the number of gallons.
• Feet per second? Divide the number of feet by the number of seconds.

After the division, the result is a unit rate.

Two more things worth noticing:

First, you can turn any of these around. Suppose we wanted people per cookie. Then we would divide people (3) by cookies (12) and get 0.25 or 1/4. If the cookies were sharing the people, that’s how many people each cookie would get. It still makes sense, although we don’t usually think of this situation that way. The numerical insight is that if we turn the “per” statement around, the number we get is the reciprocal.

Second, the word per is in the word percent—and cent means 100. So what is 74 percent? Simply 74 divided by 100, or 0.74.

A look ahead to high school:

If you’re plotting a line from some situation, the slope of that line is in units of the y-axis per the x-axis. For example, if you put distance (miles) on y and time (hours) on x, the slope is the speed—e.g., miles per hour. If you put the weight of a bunch of beans on y and the number of beans on x, the slope is grams per bean—the weight of a single bean.

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