The Forty-Yard-Line is Opposite the Forty-Yard-Line?

The Forty-Yard-Line is Opposite the Forty-Yard-Line?

One of the Common Core Standards for Mathematical Practice includes a focus on students knowing and using correct mathematical language and using clear definitions in discussions with others. There are times when everyday words are used in special ways in school mathematics, and it is important that students come to understand the precise mathematical meaning of these words.

The word “opposite” is one such word. A dictionary I consulted says that it means set over against something that is at the other end or side of an intervening line or space. Interestingly, the two examples the dictionary gives are from mathematics—opposite interior angles and opposite ends of a diameter. Clearly the dictionary definition fits with the mathematical meaning.

A second definition given was “diametrically different as in opposite meanings.” This definition fits well our use of opposites in connection with integers. We say that 2 and -2 are opposites of each other. When we view them on the number line the first definition fits well—the two numbers are “on the other side” of the zero point from each other. However, 2 and -2 are not only on opposite sides of the zero point, these numbers also have diametrically different meanings. For example, 2 might tell you how much money you have and -2 how much money you owe. Or 2 might tell you how far you are above sea level, while -2 tells you how far you are below sea level. So the word opposite tells us something about location as well something about different meanings.

There are many situations in our real world where these opposites come up and where we don’t designate them as positive or negative, but we could. For example, streets running north and south often have a dividing line so that the same house number may appear on either side of that dividing line. Instead of putting a plus or minus sign to differentiate between the two numbers, we designate one a south and the other as north. We might have two addresses numbered 2415 on Chestnut Avenue, where one is designated as 2415 North Chestnut and the other 2415 South Chestnut.

In football, the 50-yard line is the dividing line. Instead of plus or minus, we use language like the Oklahoma 40-yard line, which is on the opposite side of the 50-yard line as the Texas 40-yard line.

The comic that I’ve attached is one we did several years ago for the purpose of looking at these kinds of situations. Notice that in every situation there was a dividing line or point. Our hope for the comic was that it would reinforce the use of the word opposite as it is used within the context of the integers.

Opposites Comic page1 Opposites Comic page2 Opposites Comic page3

As a final comment, I want to point out that whenever we use the word opposite with a pair of integers, we are saying something about how the two numbers are related. We are saying that they are on different sides of the zero point, and are the same distance from zero. We can say that “is the opposite of” is a relation defined on the integers, much as are “is less than” or “is equal to.” Just as for any two integers a and b we can ask, “is ‘a’ less than ‘b’?” So too for any two integers a and b we can ask “is ‘a’ the opposite of ‘b’?”

The opposite of every positive number is a negative number and the opposite of every negative number is a positive number. Zero is the dividing point; it is the only integer that does not have an opposite.

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