I have often found that students really struggle to understand what is going on when we convert units of area and volume. It’s pretty straight forward to demonstrate conversion of units of distance: 1 metre = 100 cm etc just by using simple instruments like a ruler or metre stick, or measuring tape. Whilst some may initially struggle with this, really it’s just a question of scale / ratio etc.

When dealing with area however, it gets a lot more confusing. 1 m^{2} is frequently misunderstood as being equivalent to 100cm^{2}. It’s understandable why: it feels intuitive, and maps nicely against existing knowledge that 1m = 100cm. Unpicking this misconception can be tricky. In what I’d consider to be the least appealing scenario, students are left just memorising that they multiply/divide by the conversion ratio ‘squared’ so 1m2 would be equivalent to (1 x (100)^{2}) cm2 and so forth. This, as stand-alone information, feels a little empty – and of course I’m keen to expand on why this works. The most common explanation I’ve seen is using a diagram of, say, a square (below is taken from BBC Bitesize):

I have no issue with this explanation. It makes sense and it helps students overcome the somewhat counter-intuitive result upon first inspection, that 50000cm^{2} = 5m^{2}

However, what I’ve often found is that even with this explanation, students resort back to making the same misconception/error regardless of demonstrations to try and make it feel more intuitive. Perhaps then, the issue needs to be tackled further by adjusting the way in which we approach these problems.

Take the following example:

“Convert 250cm^{2} to mm^{2}”

Students who get it right will (again, in my experience) typically mull it over, recall something about areas being a bit different to distances, and then decide upon a multiplier. They would literally write down 250 x 10^{2}, or 250 x 100, then arrive at the answer and move on.

I prefer a pictorial approach, which I’ve found tends to bring out more right answers from those other students, and maintains the intuitiveness of the answer being right.

Forget the conversion element of the question for a moment. We are dealing not just with a number (250), but an area of 250cm^{2}. We can represent that as follows:

This gives us our required area.

Now convert the units for each side into our new units, millimetres:

Our rectangle is the same size, it has the same area, and the lengths are unchanged. We’re just using equivalent amounts in different units of measure.

So now the area, which is unchanged, is represented as 250 x 100 = 25000mm^{2}

Students would literally draw both rectangles, one below the other like so:

Now, all is well and good so far, however, what if we get a question like this:

“Convert 17m^{2} to mm^{2}”

17 is prime, and less easy to put into a nice rectangle. Or is it? In fact, this gives us the perfect adaptation to the rectangle representations. Rather than continuously adapting the heights depending on the numbers we’re given in the question, just make them all of height ‘1’, forever, for every question:

“Convert 25mm^{2} to cm^{2}”

The idea can be further adapted for volume conversions:

“Convert 3m^{3} to cm^{3}”

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