Learning Addition and Multiplication with Algebra
Date: 3/16/96 at 20:5:52 From: Anonymous Subject: The (a) in algebra Dear Doctor Math, I am 8 years old. How do you know what the (a) is in algebra? I saw some algebra on a picture. I like math, so send me a good letter so that an 8-year-old can understand. My name is Jeremiah. Jeremiah
Date: 3/19/96 at 14:23:39 From: Doctor Aaron Subject: Re: The (a) in algebra Hi Jeremiah, I like math too. I'm taking a guess here, but the picture you saw probably looked something like: 5 + a = 12 or 6 x a = 18. Just think of the (a) as a (?). Then if the equation is: 5 + a = 12 or 5 + ? = 12, we have to figure out what number a or ? could be to make the equation true. Maybe you can just look at the equation and see the answer, but if you can't, there are tricks you can use to solve the problem anyway. Since we know that adding 5 to (a) gets us 12, and that adding 5 is just counting upward from a by 5, we can count backward from 12 by 5 to get back to (a). It might help to think of a real world example. Let's say that you have 12 apples (if you're bored of fruit in math use 12 hamsters). Now you're going to close your eyes and someone else is going to put some of the apples into a black box so you can't see them. You can count the apples that aren't inside of the box. You count 5 apples. Well, you know that you started with 12 apples. If you take away one apple and completely forget about it, that brings your total number of apples down to 11, the apples outside of box down to 4, and you haven't touched the apples inside of the box. If you take another apple away, your total number is down to 10, you have 3 apples that you can see, and you still haven't touched the box. We could keep doing this until you get down to zero apples that you can see and all of them are in the box. Then your total number has to be the same as the number of apples in the box, so we know what a is. This could take a while if I asked you to solve 25 + a = 27. That's why it's a good thing that somebody thought of subtraction. When we keep taking apples away, we're really just subtracting the apples that we can see from the total number. So if 5 + a = 12, then 5 - 5 + a = 12 - 5, but 5 - 5 = 0, so a = 12 - 5 = 7. If you understood all of that there are some more key concepts that I don't want to leave out. (a) can be in more that one equation, and stand for different numbers. This is really important. If I ask you to find (a) when 4 + a = 7, and then I ask you to find (a) when 4 + a = 9, you are going to get two different answers for (a). This doesn't mean that you did one of them wrong, or that (a) is both 3 and 5, it just means that (a) only makes sense with the problem that you're working on at the time. If we think about the black boxes again, it would be silly to think that every time you closed your eyes and someone put some apples into a black box, it would always be the same number of apples. Well, since the black box was just our real-world way of thinking about (a), (a) can also be different depending on the problem that we're working on. The last thing I'm going to talk about is multiplication - because its a little trickier than addition. Let's say that 6 x a = 18. This is like saying that we have 18 apples. When we close our eyes, someone comes and puts the apples into 6 separate black boxes each with the same number of apples. Another way of saying this is that 18 apples are divided equally among 6 boxes. Good thing we have division to tell us how many apples are in each box. Then 18 / 6 = 3 is the number of apples in each box. I'm going to stop here. If you want me to give a better explanation of algebra with addition and multiplication, or if you understand it and want me to talk about more complicated algebra, please write back. -Doctor Aaron, The Math Forum
Date: 3/29/96 at 15:50:19 From: Anonymous Subject: Thanks for the letter Dear Doctor Math, My name is Jeremiah. You sent me some info on "the (a) in algebra." What I said was not like this: 5 + a = 12. This is what I saw: a(1/a) = 1/a(a) = 1. I don't understand that. Can you send me some good info? I'm 8 years old. I thought it was funny when you said that if I'm getting tired of using fruit in math, I can use hamsters. I like chess a lot. Is chess linked with math? Jeremiah
Date: 5/7/96 at 13:7:28 From: Doctor Ken Subject: Re: Thanks for the letter Hello! That "a" is essentially the same kind of thing that Aaron was talking about. It's what algebra is all about: using letters to stand for numbers when we don't know exactly what the numbers are. The reason we do that is that lots of times when we write down some equations (like a + 5 = 7 or 3/a = 6) we can actually figure out what a is. Incidentally, we don't HAVE to use "a". We can use x, or y, or a little balloon shape, or whatever we want to. Let me help you translate the thing that you saw. It said that a times 1/a equals 1, and 1/a times a equals 1 also (it becomes customary in algebra to leave out the x symbol for times in most cases - do you think that's good?). So in order to understand that, you have to know what the expression 1/a means. Do you know what fractions like 1/2 and 1/8 are? Well, 1/a is exactly the same thing, but now we just don't know what number a is. We do know that whenever we see an a, we'll replace it with some number (a represents the SAME number everywhere, not 6 in one place and 3 in another place). We just don't know what that number is. So the equation a(1/a) = 1 means that if you take a number, and you multiply it by 1 over itself, you'll get 1, guaranteed. Unless that number is 0. But that's another matter. Just remember this: DON'T EVER, EVER DIVIDE ANYTHING BY ZERO! It'll get you into a lot of trouble. Now, about your other question. There is some connection between chess and math. Most of it has to do with the way people think. Lots of people say that people who like to play chess also like to do math, because they like to figure out puzzles and solve things. There are also some more concrete math things going on in chess. For instance, number the columns of a chess board 1 through 8, and number the rows 1 through 8. Now, put a piece on a black square, and add the row number to the column number. What do you get? What do you get if you're on a white square? Can you find something general to say about what kind of numbers you get on black squares and what kind of numbers you get on white squares? -Doctor Ken, The Math Forum
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