Last week, in Fundamental Understanding of Mathematics XL, we took a second look at the standard long division algorithm, showing how it was used to divide by multi-digit numbers. There is a further extension, getting rid of the remainders by using digits beyond the decimal point, but we are going to leave that for a later diary. I'd like to revisit fractions. When we took a first look at fractions, I assumed readers knew what they were, and talked about how to manipulate them, multiply or divide them.
This week I'm going to go back a bit, and talk about what fractions are.
A fraction is a non-whole number. If we divide 3 cookies by five friends, nobody gets a whole cookie. We break the cookie into pieces in order to distribute the cookies fairly (equally.) In order to describe how much cookie each friend gets, we need two related numbers: the first number tells us how large the pieces are, the second number tells us how many pieces that size each friend gets. We call these numbers "rational" because relating two numbers is called a "ratio."
Saying the first number tells how large the pieces are is a bit of a problem, because the actual size varies with the size of the original cookies. Let's change the example to coins, to illustrate this.
Suppose you have 3 coins and you want to divide the money equally among five friends. Now, we aren't going to actually take a hammer and chisel and physically break the coins into pieces, we are going to use the value of the coins and make change. Since we have five friends, and they get the same amount from each coin, we must "break" each coin into five equal "pieces," that is to say, we must get five smaller equal value coins in change.
Now, if our coins are quarters, the smaller pieces will be nickels, and each friend gets three nickels. A nickel is 1/5 th of a quarter.
On the other hand, if our coins are nickels, the smaller pieces will be pennies, and each friend gets three pennies. A penny is 1/5th of a nickel.
If our coins happen to be half dollars, the smaller pieces are dimes, and each friend gets three dimes, or thirty cents.
In all three cases, a single smaller piece is called "1/5th", but the actual cash value of 1/5th of a coin depends on the value of the original coin. When the "whole" is different, the actual value of the fraction will be different. When we deal with fractions, we must be very clear in identifying the "whole."
When we use a number line, the "whole" is defined as our unit distance, the distance from 0 to 1, (or between any two consecutive whole numbers) on the line.
In this situation, we divide all the unit distances into segments of equal length. If our cookies are divided among five friends, each unit distance would represent one cookie, and it is divided into five equal segments so we can distribute that cookie fairly among the five friends. So our "unit" is "cookie." In other situations, our "unit" could be quarters, half dollars, nickels, or any other measurable or countable thing.
Each small segment is 1/5 th the unit distance, and we can count and lable the number line to show the distance of each mark from zero.
Now each friend takes one piece from each cookie,
and we find out how much that friend has by lining the pieces up, end to end, starting at zero.
Each friend ends up with three fifths of a cookie. We could also have noticed that the total number of pieces in the three cookies was equal to 15, a multiple of five, so we could divide 15 by 5 to get 3 with no remainder, giving us five segments 3/5ths long, one for each friend.
This also illustrates how to think about adding fractions. Just as we add whole numbers on a number line by placing lengths end to end, so we add fractions.
3/5 + 4/5 would look like this:
We take a segment 3/5ths long, and another segment 4/5ths long, place them end to end beginning at zero,
and there's the sum: 7/5ths. This also helps drive home the idea that only the number on top (the numerator) but not the number on the bottom (the denominator) is added when adding fractions, 3/5 + 4/5 is not 7/10, a common mistake.
Subtracting fractions with the same denominator (the bottom number) works the same as subtracting whole numbers. Notice that this only works for adding or subtracting fractions with like denominators. Next week we will take a look at equivalent fractions, which will let us add and subtract fractions with different denominators.
Have fun in the comments.