Singapore Mental Math (6) – multiplying mixed numbers with identical fractions when the numerator is 1

[Part of the mental math series…]  This is the strategy for Week 27 in Grade 6.

Good: Pupils will be following an algorithm and are using vocabulary, while also doing some mental maths.

Bad: This is incredibly specific!  Is it really worth having an algorithm for this particular scenario?  The questions that follow all have numbers such that step 2 gives an integer.  This needn’t always be the case.

Missed opportunity:

Why does this work?  What is going on?  Is there a better way to do it?

I think the most interesting thing here is working out how to express the numbers in an algebraic form.  For some pupils it will be strange to realise that when we write numbers next to letters as algebra then it means we are multiplying, yet when we write a number next to a fraction it often means we are adding!

The scenario we have got can therefore be shown as:

The numerators are always 1 and the denominators are the same.

This is a quadratic!  If we expand we get this:

Now go back to the original algorithm and see how each part of this algebra is linked to the different steps.

Further thoughts:

How would we ordinarily multiply two mixed numbers?  We would turn them both into improper fractions and multiply.  Then if we wanted to we could turn the answer back into a mixed number.

The benefits of this are that it will work for all mixed numbers. 

Are there any drawbacks?  Well, in the example given above that gives us:

This isn’t straightforward to do mentally, and the calculations aren’t as easy as the calculations given above. 

If we want easy sums to do then the method given works very well.  The difficult part is remembering the method!