Understanding Fraction Division #1

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Fractions.  

Most students – and people in general – cringe at the sound of this word.  It seems to draw up the worst memories they had with mathematics.  I can probably spend a long time exploring the reasons behind this hate and disgust (and I probably will in the future), but I will do something else here instead.

I have always pushed and emphasized understanding of mathematics instead of memorization.  It is a much more enriching experience for the students, and much more interesting for me as well.  So I am going to spend a few blog posts exploring the different ways of understanding and exploring fraction.

The concept of “fractions” is a rich one.  There is so much to talk about.  E.g. what is a fraction? What are the different representations of fractions?  I will probably touch on the meaning of fractions in general in the future.  

I’ve recently attended the ministry funded CAMPPP (Collective Activities for MathGAINS Personalization, Precision,Professional learning).

At the conference, the theme was on fractions “with an emphasis on representing, ordering.”  While it was an excellent – one that I would come back to next year – it left me a bit unsatisfied in terms of exploring ways of explaining dividing fractions to students.  Namely: why does it make sense to “invert” the second fraction when we divide two fractions.

Enough context, let’s get to the first way I am going to attempt to explain the division of fractions.

Understanding Dividing Fractions #1

What does it mean to have “something” divide by 2?  It means we’re taking the “something” and dividing it into 2 equal groups.

So if the “something” was a number like 6, then the 2 groups would each contain 3 members.

In this approach, we are not only trying to divide the 6 into two equal groups, each containing 3.  We are specifically asking what is in one of those equal groups. In this example, it would be 3. Let’s set up this as our way of understanding division in general.

Therefore we can say: 

(Ok now I am going to take a bit of a leap, because the following is not immediately clear to me why this would make sense in an intuitive way)

But note that we can effectively write this as:

We can see that the “1” as a denominator means we are taking one of the groups from the initial set.

If you accept the above, then we can extend this idea to dividing fractions in general: If you have any set, dividing by a fraction x/y, you are essentially dividing it into x number of equal groups, and then keeping y number of those groups as what you end up with.

Ok let’s take a different example:

Applying the same logic as before, we have a set of 18 things (in this case, 18 blue circles on your computer screen).  First we divide it into 3 equal groups.

Note that each group has 6 members.  But now we look at the denominator, and it says 10.  Extending the same pattern and way of think, this means we need 10 groups of what we had before!  In which case, we are getting more than what we started out with (and if you rearranged the equation it should make sense that there are more).

And so we actually end up with 60.  There are some additional algebraic ways of looking at this as well.  We have effectively changed our operations to

Which is hopefully a less scary representation than the fractions in the beginning.

I suspect there are additional ways of playing with this idea that makes this clearer.  There are some lingering questions about intuitive approaches to this method… but I will leave that to another time.

Read on to Understanding Fraction Division #2