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Kim has a Ph.D. in Education and has taught math courses at four colleges, in addition to teaching math to K-12 students in a variety of settings.
An ordered pair such as (3,-1), is a shorthand way of writing two variables, such as x = 3 and y = -1. The order of the numbers in the pair is important: x always comes before y.
Let's take a look at an example. Pretend you want to study the relationship between the number of minutes you spend brushing your teeth each morning and the number of kisses your dog tries to give you. One day you brush your teeth for 2 minutes, and your dog gives you 5 kisses. The next day you brush for 1 minute, and your dog offers you 2 kisses. Then, on the third day, you totally forget to brush your teeth, and your dog offers you only 1 kiss. On the fourth day, you brush for 5 minutes, and your dog offers you 6 kisses.
Writing all that out verbally is a hassle. It also takes a lot of space. So, there are a few shortcuts. For example, you might put all your data (numbers) in a table. It would look like this:
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The table looks nice when you have a lot of data; however, if you just want to talk about one day's data, then a table is a bit cumbersome. For example, maybe you just want to refer to the second day. On that day, you brushed for 1 minute and got 2 dog kisses. Instead of using the cumbersome chart, you can call the minutes of brushing x and the dog kisses y. Then you can simply say, x = 1 and y = 2. That is a little better, but it still takes more space and time to write than necessary.
Mathematicians have found an even shorter way to write it, like this: (1,2). This format is called an ordered pair because the order of the pair of numbers matters: x always comes before y. With our teeth brushing and dog kissing example, the four days of data can be represented as four ordered pairs: (2,5), (1,2), (0,1), and (5,6). With this method, our order takes the place of labeling. You know the first number is always x, so, you don't have to write x = 2.
You might think of it as putting shoes on in the dark. If you can't see which is your left shoe and which is your right shoe, what do you do? Well, one choice is to feel your shoes. With some shoes that works better than others. A better option is to plan ahead and always put your shoes away with the left shoe on the left side and the right shoe on the right side. By being orderly with your shoes, you know which is which without labeling them, feeling them, or turning on the lights. It streamlines your life a bit. The same is true with an ordered pair.
Often, when you see an ordered pair, such as (-2,3), you might automatically think about graphing the two numbers on a two-dimensional graph - a Cartesian plane, as shown below. In that case, we might also call (-2,3) a point.
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Note that (-2,3) does not need to be graphed, for it to be called an ordered pair. A graph is simply a visual tool for seeing how various ordered pairs relate to each other.
If you graph the teeth brushing and dog kiss data, for example, you see a rough pattern: more teeth brushing (x) seems to relate to more dog kisses (y). The relationship isn't a perfectly straight line. Perfect lines rarely happen in the real world (unlike in algebra class).
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An ordered pair is a shorthand method for writing the values of x and y. Order matters in an ordered pair - the x always comes before the y. Graphing ordered pairs as points on the Cartesian plane allows you to more easily see the relationship between those points.
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