This post is inspired by a blog post from Ed Southall (@solvemymaths) where he writes about the typical unit conversion problem.

I have of course the same experience of students struggling with unit conversion problems and I have also tried different approaches of „making them understand“.

For older students (16+) I have tried a more formal approach which at least for some works quite well. It requires a closer look and deeper understanding of how we write physical quantities and what we actually mean by saying the area of a rectangle is 24 cm^{2}.

## What is the meaning of 24cm^{2 }?

Let’s look at how we calculate the area of a rectangle:

$$ Area = length \times width $$

*Length* and *width* are both physical quantities. Physical quantities are not just numbers, they are a number plus a unit of length which is somehow *attached to the number*.

So what does „attached“ mean?

$$ length = 6 cm $$

Attached means that what we actually mean is this:

$$ length = 6 \cdot 1 cm$$

We have unit of length (think of it as a ruler or yardstick) and we measure by applying it 6 times the the side of the rectangle.

Let’s go back to calculating the area of the rectangle (and let’s say *width* = 6cm):

$$ Area = (6 \dot 1 cm) \cdot (4 \cdot 1 cm) $$

Rearranging gives us

$$ Area = (6 \cdot 4) \times (1 cm) \cdot (1 cm) $$

$$ Area =24 \cdot (1 cm)^2 $$

$$ Area =24 cm^2 $$

## Two important insights

- We can separate the amount (the number) from the unit of measure:

$$ 6 cm = 6 \times 1 cm $$ - By using the longhand notation we can work with units the same way we work with regular variables:

$$ 3 cm \cdot 2 cm = (3 \cdot 2)\times (1 cm) \cdot (1 cm) = 6 \times (1cm)^2 $$

## Let’s convert some units

How do we use these two insights for the task of converting units? Let’s say we want to convert the area of a rectangle from m^{2} to mm^{2}:

$$ A = 17m^2 = … mm^2$$

First we separate the amount from the unit

$$ A = 17 \cdot 1 m^2 $$

Next we rewrite the shorthand notation m^{2} to the explicit form:

$$ A = 17 \cdot (1 m \times 1 m) $$

Now we start the stepwise conversion process by substituting for the unit of 1m the equivalent of 100cm:

$$ A = 17 \cdot (1 m \times 1 m) = 17 \cdot (100 cm \times 100 cm) $$

100cm can again be written as \( 100 \cdot 1cm \):

$$ A = 17 \cdot (100 \cdot 1 cm) \times (100 \cdot 1 cm) $$

Now we rearrange the terms and multiply the numbers:

$$ A = 17 \cdot 100 \cdot 100 \times ( 1cm \cdot 1 cm) = 170000 cm^2$$

Since we want to convert to mm^{2} we need to do one more step and start the same procedure again – this time substituting 1 cm with 10mm:

$$ A = 170000 cm^2 = 170000 \cdot (1 cm \cdot 1 cm)$$

$$ A = 170000 \cdot (10 mm \cdot 10 mm) $$

$$ A = 170000 \cdot 100 \cdot (1 mm \cdot 1mm)$$

$$ A = 17000000 mm^2$$

## It’s a bit lengthy and formal, but…

Yes, the process is a bit lengthy but the upside is:** It works for any kind of unit conversion! **

Suppose we want to convert yards per minute into km/h.

$$ v = 13 yd / min = … km / h$$

$$ v = 13 \times \frac{1 yd }{1 min} $$

1 yd equals 0.9144 m which we can substitute:

$$ v = 13 \cdot \frac{0.9144 m }{1 min} $$

1 m is 1/1000 km and 1 min is 1/60 of 1h:

$$ v = 13 \cdot \frac{0.9144 \cdot 1/1000 (1 \cdot km) }{1/60 (1 \cdot h)} $$

Multiply all the numbers:

$$ v = 13 \cdot \frac{ \frac{0.9144}{1000} } { \frac{1}{ 60} } \cdot (1 km) / (1 h) $$

$$ v = 0,713232 km/h$$