FMCC Closing Ideas

Section 18.4 Closing Ideas

On the last worksheet, you derived the general formula for adding two fractions. There's a similar one for subtraction.

Theorem 18.4.1. The Sum and Difference of Fractions.

The sum of the fractions \(\frac{a}{b}\) and \(\frac{c}{d}\) is given by the following formula:

\begin{equation*} \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}. \end{equation*}

The difference of the fractions \(\frac{a}{b}\) and \(\frac{c}{d}\) is given by the following formula:

\begin{equation*} \frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}. \end{equation*}

On the very last problem of the worksheet, you also saw the downside to having such a formula. If you had no concept of common denominators, you would have ended up making the problem unnecessarily difficult for yourself because the numbers in your calculation would end up being quite large.

Some students simply learn the "rule" for adding and subtracting fractions with different denominators. These are also the students that tend to forget over time, and they will often start to guess wildly at solutions. Here are some non-examples of adding fractions together:

\begin{equation*} \frac{a}{b} + \frac{c}{d} \overset{\times}{=} \frac{a + c}{b + d} \qquad \frac{a}{b} + \frac{c}{d} \overset{\times}{=} \frac{a + c}{bd} \end{equation*}

The major trapping is for students to fall into the habit of simply manipulating the symbols without understanding what's happening. It's very easy to think you understand something because you can execute the proper procedure. But mathematical reasoning goes beyond execution.

Take another look at the two errors. Suppose that a friend showed you that work and asked you for help. Would you just tell them that they did it wrong and show them the right way to do it? Would be able to explain the right ideas to them? In the mathematical world, the goal is to both be able to execute the calculation and explain the reasoning behind it. This is a theme that we will keep returning to throughout the book.