Fractions are Hard!

1.3 Yet another story of fractions

Eventually we are told – either explicitly or implicitly – that fractions just are numbers. Specifically, that they are answers to division problems.


\(\dfrac{7}{3}\) is the answer to \(7 \div 3\). This is two with one more still to be divided by three: \(\dfrac{7}{3} = 2 \frac{1}{3}\).

\(\dfrac{40}{11}\) is the answer to \(40 \div 11\). This is three with seven more still to be divided by eleven: \(\dfrac{40}{11} = 3 \frac{7}{11}\).


These numbers have places on the number line.


We learn that there are many ways to express the same fraction. By sneaking back to our parts-of-a-whole thinking, we might draw pictures to see that two-thirds is the same as four-sixths, which is the same as six-ninths, and so on.


or we might be asked to look back at the number line instead to see this.


We are told that adding fractions of “like denominator” is just as easy adding apples:


\(\dfrac{2}{9} + \dfrac{5}{9}\), two ninths plus five ninths, is \(\dfrac{7}{9}\), just as two apples plus five apples is seven apples.


And for fractions without a like denominators we are told to rewrite them with common denominators.


Example: Compute \(\dfrac{1}{2}+\dfrac{2}{3}\).

Answer: Think of \(\dfrac{1}{2}\) as \(\dfrac{3}{6}\) and  \(\dfrac{2}{3}\) as \(\dfrac{4}{6}\). Then \(\dfrac{1}{2}+\dfrac{2}{3} = \dfrac{3}{6}+\dfrac{4}{6} = \dfrac{7}{6}\).


UPSHOT: The mechanics of fraction arithmetic becomes the focus and we are weaned away from the intuitive picture of parts of a whole. (After all, one cannot add kittens and stars.)    


Next something very strange typically occurs …



After being weaned off of the parts-of-a-whole thinking we are asked to consider parts of a part of a whole!


Example: Draw a picture of \(\dfrac{2}{3}\) of \(\dfrac{4}{5}\).

Answer: First, here is \(\dfrac{4}{5}\) of a pie.


Now think of the shaded region as a whole. Two-thirds of this four-fifths of a pie is two copies of one-third of the shaded part. Here’s one-third of the four-fifths.


And here is two copies of one-third of four-fifths.


If we draw in extra lines we see that this new shaded region matches 8 parts out of total of 15 parts.


So \(\dfrac{2}{3}\) of \(\dfrac{4}{5}\) looks like \(\dfrac{8}{15}\) in our parts-of-a-whole thinking.

MOREOVER … our picture looks like an area.


We like to associate the operation of multiplication with areas. So let’s just say that “of means multiply” and write

\(\dfrac{2}{3} \times \dfrac{4}{5} = \dfrac{8}{15}\).

We practice lots of examples like these and realize, in the end, that the mechanics of this work has us multiplying numerators and multiplying denominators. That is, we see, in general

\(\dfrac{a}{b} \times \dfrac{c}{d} = \dfrac{ac}{bd}\).

UPSHOT: Defining the multiplication of fractions feels random. (We bring in parts-of-a-whole thinking whenever it just seems convenient, but only for a mixture of line segments and areas. Can we apply this thinking to kittens and stars?)


The following rule is drilled into us:

To divide two fractions, multiply by the reciprocal.

For example:

\(\dfrac{4}{5} \div \dfrac{2}{3} = \dfrac{4}{5} \times \dfrac{3}{2} = \dfrac{12}{10}\).

\(\dfrac{10}{17} \div \dfrac{1001}{97} = \dfrac{10}{17} \times \dfrac{97}{1001} = \dfrac{970}{17017}\).

I am yet to meet someone off the street who can explain why this rule works.

UPSHOT: We are told a rule about dividing fractions. If we were given an explanation as to why that rule is appropriate it doesn’t seem to have been satisfactory enough to “stick” and stay in long-term memory. 

There is more to say about division in the next lesson.

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