Fraction division

Everyone knows how to divide fractions, right? You just flip-and-multiply.

So

\frac{2}{3} \div \frac{2}{9} = \frac{2}{3} \times \frac{9}{2} = 3

Simple.

Unfortunately, although this will enable you to solve problems all day long, it is an example of a procedure performed ‘without understanding’ and so that is bad.

If you know that \frac{2}{9} means the same as “two divided by nine” then I can demonstrate how it works. Of course, if you don’t know this then I will have to prove it to you first so that you have ‘understanding’. No, scratch that, I will have to create some rich task that enables you to come to that understanding for yourself or something. But let’s assume you get it. We can say

Let \frac{2}{3} \div \frac{2}{9} = \frac{a}{b}

If so, it must also be true that \frac{2}{3} = \frac{a}{b} \times \frac{2}{9}

However, given that \frac{2}{9} = 2 \div 9 we can write this as

\frac{2}{3} = \frac{a}{b} \times 2 \div 9

If so, it must also be true that \frac{2}{3} \times 9 = \frac{a}{b} \times 2

Continuing, we can see that \frac{2}{3} \times 9 \div 2= \frac{a}{b}

Given that 9 \div 2 = \frac{9}{2} we can rewrite as

\frac{2}{3} \times \frac{9}{2} = \frac{a}{b}

If we wanted to, we could extend this logic to any pair of fractions by replacing the numbers with algebraic terms.

So, are you happy now? Do you understand? Are we all OK to go away and flip-and-multiply?

Perhaps not. After all, do you really understand? I mean, like really? Can you actually picture what is going on? No? OK then, here’s a picture:

By completing our fraction division this way, we can avoid flip-and-multiply and instead spend our time drawing lots of diagrams and colouring them in. Is this better? What if you tried to do \frac{4}{5} \div \frac{2}{7} this way?

Note a few important points. Firstly, flip-and-multiply is a very simple rule to apply. Once you have it in long-term memory alongside procedures for simplifying fractions (e.g. \frac{2}{2} = 1 or \frac{9}{3} = 3) you can tackle problems involving fraction division with relative ease.

In contrast, visually solving fraction division problems involves a large number of interacting elements and will therefore be high in cognitive load. Similarly, following the explanation of why flip-and-multiply works requires tracking through many steps and following the logic at each step. This imposes a high cognitive burden (which ironically will be reduced if you already have a lot of rules in your long-term memory – see the discussion here).

Imagine a school makes the decision that ‘understanding must come first’. In this school, flip-and-multiply is forbidden and students have to solve all such problems using diagrams like the one above. This will have two effects. Firstly, it will limit the number of problems that students may be asked to solve to those that can be feasibly represented visually. Secondly, it will impose a far higher cognitive burden on students. Many students who may have successfully deployed flip-and-multiply will struggle with the diagram approach.

This seems counterproductive and doctrinaire. Why not teach students the rule and allow them to use it? What harm would come of that? There is nothing to stop you explaining it and you could use both a diagram approach and the kind of rough proof I have given above in order to do this. Some students may follow it and some may not. Some may learn the rule and later come to appreciate how it works. In fact, that’s quite likely because the more they have in long-term memory, the easier it is to follow the proof. However, pretty much all students should be able to apply the rule in solving problems, provided they have the prerequisite skills.

If you insist that understanding must always come first then you are placing limits on students. By raising the cognitive burden, such an approach creates confusion and, for many students, will result in the exact opposite of the intended outcome.

28 thoughts on “Fraction division

  1. I teach the procedure, and include an explanation. It is an explanation that goes in one ear and out the other. “Just let us do the problems”. Of course, someone will ask “Do I flip the first or second fraction” to which “understanding uber alles” types laugh and point and say “There, you see? That’s what happens when you teach w/o understanding.

    I teach the derivation of the rule also in my algebra classes, using the tools of algebra. Even the brightest students cannot reproduce this method with ease. It might be a few years later that it all comes together when after much practice with the procedure they can absorb it more easily. Or some may never understand it.

    I once posted a blog in which I admitted that I only learned the reason about ten years ago–and I majored in math. A twitter conversation ensued, in which a rather popular blogger said he didn’t like the “I never learned it and I turned out just fine” message that he thought I was promulgating.

    See also: https://www.educationnews.org/k-12-schools/poster-children-of-math-education/

  2. I’m not a maths teacher, but whenever I’ve been relieving a class or just talking about maths with my form class, I’ve got a long way with things like flip and multiply. The kids love it. Another one is percentages. To express one number as a percentage of another, divide the small number by the big number and times by 100.

    1. To be gentle consider changing the smaller number by larger number bit as that could cause problems as I might want to compare them the other way round. (Try if I want this number A as a percentage of that number B I do A/B x 100). . This is why being specialists in our fields is so important and why my pottery teaching is atrocious.

  3. If we know that 2/9=2 divided by 9 and
    2/9= 2*(1/9)
    Then we can easily see that dividing by a number is the same as multiplying by its reciprocal.
    That should be an easier explanation to remember.

    1. Not sure it is. I think it would have confused me as a child. Also it is not obvious that 2/9 is 2 divided by 9 to a lot of people, and children are likely to have been told that it is two parts out of nine, with a fraction cake, just to be more confusing.

    1. Sure, but for use with fractions, don’t they need to know the reciprocal of a fraction i.e. that 1/(2/9) = 9/2. In order to understand why this is the case, they would need to already know the flip and multiply rule, right?

  4. 2/3 is the same as 6/9 (simple fraction rule)
    6/9 divided by 2/9 is the same as 6/1 divided by 2/1, or 6/2
    which is a/b, the answer
    (equivalent fractions and nothing else)

    1. Yes, I am familiar with the common denominator approach and I nearly mentioned it. In my experience, it makes the calculation harder than flip and multiply because it adds additional steps. I am also not convinced that it aids understanding so I’m not sure it is worth this cost.

      1. I’m surprised you discarded this so quickly and deemed that it doesn’t aid understanding. I think that once you show this visually it really does aid understanding. The irony is that your visualisation did actually show 6/9 rather than 2/3.
        I teach disengaged, low attainment learners. If I use a common denominator approach then that will work for 3 of their 4 operations.
        Even though multiplication doesn’t work that way, it does use the same picture (one denominator cutting vertically and the other horizontally).
        So that’s a total of one picture; one method for three operations; another method (the simplest) for the other.

  5. All justifications, explanations or formal proofs of this involve how associativity, commutivity work with sequences of multiplication and division along with some other property such as multiplicative identity. If these are new then the explanation involves a lot of new moving parts.
    But that is just an error in teaching too much in one step. If a,b and c are fractions there are lots of problems that can be simplified using a x b x c = (a x c) x b. And lots of the form a / b x c = a x c / b.
    If students are really comfortable handling these because they have done a lot of them then a proof for the case given is not that far off.

  6. I can only speak for myself here, but the problems I see in my students is applying the flip-and-multiply procedure beyond fraction division.

    There’s no question that flip-and-multiply is the most efficient procedure (that I know of) for dividing fractions. The question is how we can teach it so that the simple procedure is connected to the appropriate problem domain. Otherwise kids start flipping and multiplying while adding fractions — I see this all the time.

    (See also: cross multiplication.)

    That’s where I’d make the case for some non-algorithmic work that comes along with the procedure (I don’t care much about whether it’s before or after, though before is fine). I think some work with images related to problems in context can be useful (it’s also just good math).

    I think mental math can be a good environment for gearing up towards something like flip-and-multiply. There are useful algorithms that can lay the groundwork for the full procedure, things like “if the numerators are equal then you can just look at the denominators.” Or evaluating more complex fractions, like 1/8 over 1/2.

    My thinking here is that these other mini-algorithms and mini-skills help students understand the problem domain and see more readily the necessary connection between the flip-and-multiply procedure and fraction division.

    1. Most misconceptions in mathematics are over-generalisations. I doubt whether experience with non-algorithmic ‘conceptual’ work will prevent students from over-generalising flip-and-multiply, but I would be interested to see the evidence on that. Instead, this is where Engelmann’s DISTAR programmes may have something to tell us with their use of non-examples.

      We need to instruct students when to use certain procedures and, crucially, when not to. This then defines the set of problems for which this step is appropriate. If we leave it for students to infer this boundary then that is a form of discovery learning. How do we know which non-examples to teach? From compiling common errors and misconceptions.

      Of course, simply teaching a student a non-example is not enough, in the same way that teaching a true example is not enough. We have to cycle back, practise retrieval, distribute practice etc.

      1. I’m imagining having them learn the rule and practice a little bit, followed by interleaved practice with other operations they already know. Hopefully, this would prevent overgeneralization.

  7. I think the rule can be easily justified based on:
    * Division is the inverse operation to multiplication
    * The rule for multiplying fractions

    So take (a/b) / (c/d). This means to find x such that

    (c/d) * x = (a/b).

    You can check that x = (a/b) * (d/c) = ad/bc works:

    (c/d) * x
    = (c/d) * ad/bc
    = cad/dbc [rule for multiplying fractions]
    = a/b * cd/dc [rule for multiplying fractions]
    = a/b

    (The commutative and associative laws for multiplication are assumed for integers throughout.)

    Giving a visual explanation based on bar models may be useful, but I suspect it is hard to make this cover all the cases: quotitive division (“how many halves in 6?”), partitive division (“what is 6 one half of?”), whether the divisor is a unit fraction, and whether the quotient is an integer or fractional.

    Your “proof” has some holes in it:

    2/3 = (a/b) * (2/9)

    You make this

    2/3 = (a/b) * 2 / 9

    but the order of operations is not clear: is it (a/b) * [2 / 9] or [(a/b) * 2] / 9? You need to justify why the second expression is equivalent before you can multiply through by 9.

    Likewise, when you divide through by 2, you need to justify that

    [(2/3) * 9] / 2 = (2/3) * (9/2).

    1. Here is my version of this

      2/3 / 2/9
      A number multiplied by 1 is unchanged
      2/3 / 2/9 = (2/3 x 1) / 2/9$
      A fraction multiplied by its reciprocal is 1.
      2/3 x 1 / 2/9 = 2/3 x 9/2 x 2/9 / 2/9
      Multiplication is associative and division is left associative
      2/3 x 9/2 * 2/9 / 2/9 = 2/3 x 9/2 x (2/9 / 2/9)
      A non-zero number divided by itself is 1 and a number multiplied by 1 is unchanged.
      2/3 x 9/2 x (2/9 / 2/9) = 2/3 x 9/2 x 1 = 2/3 x 9/2

      Skipping the trivial steps we have just two:

      2/3 x 1 / 2/9 = 2/3 x (9/2 x 2/9) / 2/9
      2/3 x 1 / 2/9 = 2/3 x 9/2

      The first inserts a multiplication by 1. The second uses associativity to do the division of a number by itself and discards the resulting multiply by 1 term.
      The properties used are all either items that work for integers or a result from multiplication of fractions.
      Anyone could learn this as a two step procedure that can be shortened to a single step. Like anything you would have to practice recalling it to remember it. No one should be expected to remember it after one explanation.

  8. I’d like to think that teaching for understanding first equips the student to be able to become better problem solvers in the long run. Particularly for younger students (KS3) I think we should try to encourage understanding to create curiosity and allow students more than one way to solve a problem. Ratio problem solving comes to mind where I’ve recently changed my teaching to incorporate bar modelling to help students solve all kinds of ratio problems that are diagrammatic and dispose of a simple rule that just let’s them solve a ‘sharing into ratio’ problem as an example.

  9. I agree with those saying understanding looks a lot like knowing more facts. What I don’t get here is why people think it is important to know you can do the flip trick but not know any justification for it.
    The justification (in more detail in my other comment) is just
    2/3 / 2/9 = 2/3 x 1 / 2/9 = 2/3 x 9/2 x 2/9 / 2/9 = 2/3 x 9/2 x 1 = 2/3 x 9/2

    The facts you need to know are t
    – you can multiply by 1 without changing a number
    – a non zero fraction multiplied by its reciprocal is 1
    – a non zero fraction divided by its self is 1
    – you can reorder the operations here to evaluate the fraction divided by itself first.

    These are all facts kids should know. Knowing you can use them here is also not difficult to remember if it is taught with repetition by the students.

    It is far more useful to know that you can do these sorts of manipulations to get an answer than it is to know the flip trick. People just don’t have to divide fractions that often. Students who want to get through high school or further math do need to know what manipulations work.

    This is a charitable view of the point made by those saying understanding is more important than knowing some trick. They are correct. Just teaching the flip trick without ensuring students can remember why it works is pointless.
    Again ask anyone when they last divided two fractions. Then ask anyone doing well in high school calculus if they need to know these sorts of manipulations.

  10. In my classes I introduce the flip and multiply with a few examples. I know that students will soon mix up which fraction to flip. After a short whole class discussion, I’ll be able to draw out the question from at least one student “why do we flip only the second fraction?” Many are wondering this and I’m determined to demonstrate that there isn’t any magic behind what we’re doing.

    I then give three simple examples using grids like in your blog, each one used to further convince the students that our new rule works. The class will practice the flip and multiply skill over the next few days/weeks, focusing on the skill and not on the graphic, they don’t need the extraneous load. Some will still flip the wrong fraction but at least for many kids, they’ll believe or understand why we flip the 2nd fraction. I don’t expect the kids to reproduce the graphic. I do think my students will display a range of comfort in talking through what the grids show and I’m ok with that – it’s not a primary learning objective.

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