Re: Multiplying Fractions.

It helps if we think of multiplication as finding the area in square units of a rectangle - i.e. multiplying the number of squares in one row (the length) by the number of rows (width).

Then we need to define a unit square to be a square with length and width each = 1 unit in length.

Say we wanted to evaluate $\displaystyle \displaystyle \begin{align*} \frac{1}{2} \times \frac{1}{2} \end{align*}$, we need to evaluate the area of a square that has a length $\displaystyle \displaystyle \begin{align*} \frac{1}{2} \end{align*}$ unit in length, and width $\displaystyle \displaystyle \begin{align*} \frac{1}{2} \end{align*}$ unit in length.

http://i22.photobucket.com/albums/b3.../halfxhalf.jpg

How much of the unit square $\displaystyle \displaystyle \begin{align*} \frac{1}{2} \times \frac{1}{2} \end{align*}$? Why, $\displaystyle \displaystyle \begin{align*} \frac{1}{4} \end{align*}$ of it. Therefore $\displaystyle \displaystyle \begin{align*} \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \end{align*}$.

Now what if we tried something like $\displaystyle \displaystyle \begin{align*} \frac{3}{4} \times \frac{2}{3} \end{align*}$? We would need to evaluate the area of a rectangle that has a length of $\displaystyle \displaystyle \begin{align*} \frac{3}{4} \end{align*}$ of a unit, and a width of $\displaystyle \displaystyle \begin{align*} \frac{2}{3} \end{align*}$ of a unit.

http://i22.photobucket.com/albums/b3...xtwothirds.jpg

How much of the unit square is covered by this $\displaystyle \displaystyle \begin{align*} \frac{3}{4} \times \frac{2}{3} \end{align*}$ rectangle? Why, $\displaystyle \displaystyle \begin{align*} \frac{6}{12} \end{align*}$. Therefore $\displaystyle \displaystyle \begin{align*} \frac{3}{4} \times \frac{2}{3} = \frac{6}{12} \end{align*}$.

I'm sure you're seeing that it looks like you can multiply tops and multiply bottoms. Why is this?

It's because in order to multiply the fractions, you need to split the "length" of your unit square into as many pieces as defined by the denominator of the first fraction, and you need to split the "width" of your unit square into as many pieces as defined by the denominator of the second fraction. That means the unit square is divided into as many pieces as the product of the denominators, and so this becomes your new denominator.

Then you need to count as many of these pieces along the "length" of your unit square as in the numerator of the first fraction, and you need to count as many of these pieces along the "width" of your unit square as in the numerator of the second fraction. So the number of pieces you will be counting in total is the same as the product of the numerators, and so this becomes your new numerator.

Therefore, to multiply fractions, you can multiply the numerators and multiply the denominators.

Re: Multiplying Fractions.

Another way to think about this- if you have a rectangle with one side of length "3 meters" and another of length "5 meters", then the area is 15 **square** meters. That is, we multiply the lengths and we "multiply" the units.

In a very real sense, the denominator is a "unit". The fraction "3/5" says that we have 3 things, each the size of a "1/5".

Re: Multiplying Fractions.

Awesome, that is perfect and makes sense. THanks so much, you guys rock as always.