real world introduction
One of the most important types of mathematical objects in algebra is an object called a "variable name" (or variable for short). As it turns out, we regularly use variable names in our every day lives. We just don't know we are, because doing so comes quite naturally to us.
To see what I mean, we're going to consider two examples- one involving a bag of candy, and one involving fishing.
Bag of Candy Example
imagine that you have a bag filled with two types of candy- let's say the first
kind of candy is "jelly beans" and the second is "gummy
bears". Now suppose you close your eyes, and pick a candy out of the bag.
Imagine doing this. Now, while your eyes are still closed, without looking at
the object you've grabbed, can you still talk about what you've grabbed from
Question: How would you talk about the object you are holding before you've opened your hand? Try saying something about it. What sort of temporary name or label might you give it? (Once you’ve tried to work it out yourself, see Answer 1, bottom of page)
Question: Without looking at it, what are some facts you can tell about "The Candy in my Hand"? (Once you’ve tried to work it out yourself, see Answer 2, bottom of page)
Question: Without bothering to looking at it, what are some things you could do with "The Candy in my Hand"? (Once you’ve tried to work it out yourself, see Answer 3, bottom of page)
One of the things you could do with "The Candy in my Hand" is tell someone else about it, even if neither you nor they have looked at the particular candy that is in your hand. For example, you could hold out your hand and say "Here, is The Candy in my Hand. It's sweet and tasty. Would you like to have "The Candy in my Hand"?
This sort of situation happens more often than you might think.
For example, now suppose you are fishing, and a fish bites your rod, but then gets off the hook. In that case, you would talk to your friends about "the Fish that Got Away". You could say something like "Man- I bet the Fish That Got Away was huge!"
In this case,
you would be talking about a particular fish, and you would know some facts
about it (it has gills, it swims in the water, it was pretty big!) even though
you don't actually know where that fish is, or exactly which type of fish it
In mathematics, the names that you give to these unknown objects are called variable names, or variables for short. "The Candy in My Hand" is a variable name (or variable), and so is "The Fish that Got Away".
Answer 1: Some
possible answers for what you could call the candy in your hand: "My
Candy" "The Candy I picked out of the bad" "The Candy"
In this case, let's suppose I gave it the name "The Candy in my Hand".
Answer 2: Some possible answers for what you could say about the candy without looking at it: it is small, it is light, it is sweet, it is colourful, it is edible, either it is a gummy bear or it is a jelly bean.
Answer 3: Some possible answers for what you could do with the candy: eat it, give it to someone else, put it back in the bag, throw it out.