Consider the following that I took from a student’s book. Is this something you have seen before? What do you think the student’s reasoning is?
What’s the Problem?
Dividing would not be many students favourite mathematical topic. It seems that we are far happier counting up than sharing out. So, if dividing by lovely whole numbers bamboozles youngsters, you can imagine the nightmare that occurs when the divisor is a decimal. For the example above, I have seen answers of 1 (“anything divided by 1 is 1”), 0.5 (“1 divided by anything equals itself”) and even 17 (“dunno sir, I just guessed”).
What’s the Solution?
Often in mathematics, the solution to common mistakes and misconceptions can be found by relating the problem to real life. However, that is all well and good when talking about sharing 24 cakes equally between 6 people, but things are not so conceptually straight forward when one cake is somehow to be split between half a person (perhaps the other half is on a diet?).
However, there is one scenario that may just make sense to students. Imagine you have a big six litre jug full of Vimto. How many two litre bottles can you fill? Easy, it’s 3, so 6 ÷ 2 = 3. Okay, so now how about if you only have a one litre jug, how many two litre bottles can you fill? Hmmm, well you can probably only fill up half of one, so 1 ÷2 = 0.5. And now its crunch time: if we still had a one litre jug, but this time we have 0.5 litre bottles, how many could we fill? The only answer that makes sense is 2, so 1 ÷ 0.5 = 2. And if there is still any doubt, off down to the supermarket you go!
Once students have accepted that it is possible to divide something and end up with an answer bigger than they started with, their minds are then open to cracking this particularly tricky topic, and you can begin to introduce more efficient techniques.
Craig Barton is an AST from Thornleigh Salesian College, Bolton. He is also the creator of www.mrbartonmaths.com and TES Secondary Maths Adviser. He can be found on Twitter using @TESMaths
Resources to help
Check out these resources and collections on TES which may help tackle to problem:
MEP – Division of Decimals: http://www.tes.co.uk/teaching-resource/Division-of-Decimals-MEP-Year-7-Unit-8-6051087
Collection: Tarsia – Number Resources: http://www.tes.co.uk/article.aspx?storyCode=6107408
Decimal Division Activity: http://www.tes.co.uk/teaching-resource/Division-of-Decimals-MEP-Year-7-Unit-8-6051087
Dividing Decimals Lesson: http://www.tes.co.uk/teaching-resource/Dividing-decimals-lesson-6173431/
N2 – Evaluating Statements about Number Operations: http://www.tes.co.uk/teaching-resource/N2-Evaluating-statements-about-number-operations-6086832