In this the third blog in the series of multiplying fractions we shall look at what is happening when two improper fractions are being multiplied. What we should not do at this stage is assume that what has been true for proper fraction multiplication and proper with improper multiplication will also be true for multiplication of two or more improper fractions. The concept in concrete or visual formats should enable the learner to see however, why it is possible to apply the same process. Starting by looking at the following multiplication problem:
As with all fraction operations it is essential to understand the nature of the ‘whole’ object from which the fractions are taken. It is possible again to make a model of the whole as an array using information from the denominators. As a reminder, pupils will have understood what a model of the ‘whole’ could look like by multiplying the denominators together to form an array.When a visual of the whole is then made we can start to understand what the fractions ‘of’ that whole, themselves look like. It would be prudent to explain also that the question is really asking, “What is 3/2 of 1 and then what is 3/2 of that result?”
As with the other types of fractions we should reference what is meant by the whole and as the problem involves halves it seems sensible to show the whole as a shape that is made with two very clear halves. We should also at this stage consider the alternative to the improper fraction; the mixed number; as this will possibly enable the learner to make more sense of what is being asked in the problem. As before the question we are looking at is really a short hand version of the following.
This Implies we are being asked to identify 1, then find 3/2 of it and finally continue the process to find 3/2 of that result. We can look at what this means in a number of ways.
Using mixed numbers
If we change the improper fractions to mixed numbers then there is a link to the distributive property of multiplication over addition that becomes useful.
The first part of the calculation should prove to be quite simple as when 1 is multiplied by any number n the result is n itself. The second part logically follows the same route and the opportunity to calculate what started out as essentially an improper fraction by improper fraction multiplication by now converting to an improper fraction multiplied by a mixed number problem. Here we can convert the multiplier to a mixed number. But what if we try to calculate keeping the fractions as improper?
Using improper fractions.
When calculating with fractions, each result is in effect a brand new whole entity. Take the following example of the multiplication of four proper fractions.
This can be broken down into single multiplication steps to illustrate how the whole is now being changed.
It is possible at this stage to solve the problem using language alone. “A half of two twenty-fourths is one twenty-fourth” but we will continue with the imagery for a moment.
The result of “two forty-eighths” implies the whole was initially made of 48 equal parts however in reality it is impossible to show or draw the object that it is, as it is a 4-dimensional object which we are not able to visualise as it would be an array of 4 x 2 x 3 x 2. By breaking the problem down into a 2-dimensional array at each stage as in the above diagrams it is possible to see where the result comes from. Importantly by doing this we are changing the nature of the whole at each step of the way. The same can be done with improper fractions. Look at the following example which is at the top of the page also.
As with all the multiplication of different fractions we have completed to this point we can make sense of the whole by making an array from the information given by the denominators.
In this example the whole is a 2 x 2 array made of four equal parts therefore. Making links to other mathematics is always to be encouraged when it is helpful in problem solving. The fraction 3/2 is exactly the same as the division 3÷2 but more importantly the understanding that the fraction is defined by (in this example) “3 whole objects divided by 2”. The image of 1 x 3/2 therefore can be understood from this in the following way.
This imagery can be reorganised visually in both of the following ways
We are now at liberty to see the first result as a temporary new ‘whole’ which can now be multiplied in the same way by 3/2.
In [a] we see the image of 1 x 3/2 (originally a 2 x 2 array) which we can temporarily regard as the ‘new whole’ so that in [b] when its is multiplied by 3/2, it is in effect showing three ‘new’ wholes divided by 2. The result of the division [c] shows 6 of the small squares (which we recall each represent 1/4 of the original model of the whole.) This can be reassembled [d] to show the resulting fraction in terms of the original whole. In the diagram it shows the result of 3/2 x 3/2 as 9/4 or the mixed number 2 1/4.
This reasoning shows that it is now safe to approach the problems of multiplication of two improper fractions in the same way as solving problems with proper fractions where multiplication of numerators followed by multiplication of denominators gets to a solution (but not always in its simplest form!)