1 Jan 2010
Why invert and multiply when working on fraction divisions
Ho Math and Chess Learning Centre
Canada, BC certified math teacher
Vancouver, BC, Canada
One of the biggest problems many math teachers facing when working fraction division is how to explain to students the reason of why change “division” sign to “multiplication” and then invert the second fraction?
For example, ½ divided by ¼ (How many 1/4s in ½?).
With this example being simple, so it is easy to show that ½ has 2 of ¼ so the answer is 2.
One can also demonstrate it by using a number line. It is easy to show that in ½ there are 2 of ¼.
How about showing ¾ divided by 2/3 graphically?
It is much easier if we can convert both fractions to the same LCD when using number line. It shows that ¾ divided by 2/3 = 9/12 divided by 8/12 and in this case the denominator has nothing to do with solution. It shows that the result has quotient 1 and the remainder is 1/8 (not 1/12).
We can also show it by using 2rectangles. ¾ has 9 cells in 4 by 3 rectangle and 2/3 has 8 cells in 4 by 3 rectangle so the division result of ¾ divided by 2/3 is 9/8.
But the above conceptually explanations do not exactly demonstrate why the computing algorithm of changing 3/4 divided by 2/3 to 3/4 times 3/2. Many teachers also attempt to use different ways of conceptually showing fraction divisions to link “invert and multiply”. None of them really make sense to directly explain why invert and multiply.
So why we invert and multiply when working on fractions division? The reason is it simply is much easier for us to find the solution.
When working on fraction division, we do not really teach children any division concept in actual computation, we simply convert fraction division to fraction multiplication so trying to explain fraction division by using invert and multiply will only hit a roadblock. This is one of the classic examples the computing algorithms do not really match the meaning of concepts. It is simply a shortcut to do fraction division. Many times, the reason that students do not do well is they cannot perform these math shortcuts.
There are other examples that we use shortcuts to do computing. Are we really doing whole number division when working on 6 divided by 2? We can teach children it takes 3 times to subtract 2 from 6 so the answer is 3 but in reality, what we teach children in computing is to ask what times 2 is equal to 6, so it is a reverse of multiplication children are doing. This is another example that the concept teaching does not match computing algorithm.
Why we change the divisor of decimal division to whole number when working on division? The answer is again it is easier to do
In contrast, there are also cases that the computing algorithm matches concept teaching, for example 23 times 21 can be thought as 23 times (20+1) so in the vertical format computing, it demonstrates this expanded from of computing.
We must show children, in some cases, the computing procedures may not match their meaning of real concepts for the reason of performing shortcuts and “invert and multiply” fraction division is one of these cases.
(2 divided 3) divided (3 divided 4) can be calculated much easier using 1’s property to convert division to multiplication, so the purpose of invert is simply to convert division to multiplication so we have a chance to simplify fraction in case there are common factors.
(3 / 4)/ (2/3) = (3/4) x (1 / (2/3)) (here we uses 1 to find the reciprocal of (2/3))
= 3/4 times 3/2
So we achieve the purpose of (3 / 4)/ (2/3) = 3/4 times the reciprocal of (2/3).
The idea of using 1 has beautifully changing a division problem to a multiplication problem and this way we allow many options of simplifying and also get the answer easily by simply using “across multiplication”.
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