Last week, in Fundamental Understanding of Mathematics XIX, we took a another look at set theory, and why division isn't closed in the set of integers. This week we are going to take a look at the operation of division itself.
An example we used before was three friends sharing five cookies. In order to divide the cookies with no left overs, we first must cut the cookies into pieces, since three friends are sharing the cookies, we cut each cookie into three pieces, all exactly the same size.
Now, here is the wonderful, awesome thing about division.
If we write a division problem down correctly, we have solved it.
Follow me over the fold.
We are going to use fraction notation to write down our division problem. The problem is five cookies divided by three friends, so we write the five on top, and the three on the bottom. 5/3. That's the problem: five divided by three is written five over three.
That's also the answer: 5/3, five thirds.
No calculations needed. It's exactly right. Works every time. We're done.
I'll bet you had no idea division was so simple, right?
Let's try some more.
22 divided by 17? 22/17. Twenty-two seventeenths. Done.
77 divided by 125? 77/125. Seventy-seven one-hundred-and-twenty-fifths. Done.
345 divided by 7? 345/7. Three-hundred-and-forty-five sevenths. Done.
How about some unknown numbers?
a divided by b? a/b. a over b. Done.
Ok, that was easy, time to move on to some more fun stuff, like base 8 numbers, or square roots, or exponents...
But wait! What about that funny looking bracket thingy, with subtracting, and bringing down, and guessing which number to multiply by and all that confusing stuff? Isn't that division? It's was certainly called division when I was in school!
Here's what we are really doing with that bracket thingy: we are converting our fraction notation answer: 5/3, into a decimal notation answer: 1.66666... In decimal notation, we must be sure our columns line up, or we might be off by a factor of 10 or 100, we take some shortcuts, such as not writing in all the zeros, and we work from left to right, the opposite of adding or subtracting, since we want to deal with larger numbers first, and in a place value notation, the larger numbers are on the left.
It's not a bad method for converting a fractional number to a decimal number, although there are other ways (including a calculator,) but it's not the easiest to learn right out of the box. One big concern is that it hides the idea that division has something to do with separating into groups or same sized pieces, since the main thing students do is subtract and multiply to get an "answer."
Although the math is the same, there are two main ways of relating division to the real world. The first is dealing cards (separating into groups), the second is cutting string (getting same sized pieces).
Let's look at the first analogy: dealing cards.
You're the dealer, you have some number of cards, and some number of players at the table (groups). If you deal all the cards fairly, division will tell you how many cards each player has when you're done dealing (we're ignoring the leftover cards for now, dealing fairly means everyone gets the same number of cards.)
Let's take a look at the second analogy: cutting string.
You have a long piece of string and scissors. You want shorter pieces of string, all cut to the same size. If you cut the whole long string into pieces accurately, division will tell you how many short pieces of string you have when you're done cutting (again, ignoring the too short piece that might be left over.)
If we take a look at these two situations, we see why we want a fraction to decimal conversion process. When we deal 52 cards to 4 players, we want to know each player gets 13 cards, not 52/4ths cards, even though, mathematically, it's the same number. Likewise, if we have a yard of ribbon, and need four inches of ribbon to make a bow, it's nice to know that we can make 9 bows, not 36/4ths bows.
So, we start with 52 cards, and deal one round: one card to each player. We subtract the four cards we've dealt from the pack, and now we have 52 - 4 = 48 cards left to deal. We deal another round, we subtract four cards, so now the dealer has 44 cards, and each player has two cards. We can continue this process until we run out of cards (pausing to check at the end of each round to make sure we have enough to deal another complete round.) Then we count the number of times we subtracted four cards from the deck, it's 13, the number of cards in each player's hand, the number of rounds of dealing everyone one card. Just as multiplication is repeated addition, so division is repeated subtraction.
We have successfully converted 52/4 into 13. But it's a tedious process, and we can figure out some short cuts.
Rather than deal one card at a time, let's try dealing 10 at a time.
We start with 52 cards, each player gets 10 cards, we subtract 40 from the deck, and the dealer is left with 12 cards, and each player has 10 cards. We've dealt one round, but it's a round of ten, so we need to note that one round in the "ten's column." That's the first short cut: dealing with larger numbers first.
Now we have 12 cards to deal. We could, of course, deal the cards out one at a time, deal three rounds, and run out of cards. But instead, before we deal, we recall that 3 times 4 is 12, so we could get rid of all our 12 cards by dealing each player 3 cards each. So our second round is three cards, but they are single cards, so we put the digit "3" in the "one's column." That's our second short cut: guessing which number to deal so we run out of cards in only one round.
Now the dealer is out of cards, so we take a look at our notes: a "1" in the ten's column, and a "3" in the one's column. Looks like "13." Yes! Done it again, successfully converted 52/4ths into 13. If we organize our written notes around that bracket thingy, and make sure we keep the columns lined up, we've just walked through the "long division" algorithm.
I'm off to a professional development meeting today, so I won't be able to check in much, so play nice, you guys!
Have fun in the comments.