Unit Conversions Part 1: Language and Questioning

Hi folks! This is the second part in my series on units and conversions. Last week’s instalment on base prefixes can be found here.

This time round, we’re going to have a look at the wording behind conversions, as a prelude to a deep dive into some possible methods of converting.

Converting units is often perilous, whether in primary or secondary, because it is one of the first times that our pupils experience an inverse relationship. The unit gets larger, but the number gets…smaller? What on earth? For this reason, I really think concrete experiments are going to be useful here, particularly if you have a primary class or a lower set in secondary.

Get both a mm/cm ruler and a metre rule. Measure a load of things in both units (you’d be amazed even at secondary how many will benefit practicing accurate measurements in different units. Let them record these in tables, and get them used to seeing a greater number of mm than cm, and much greater than m.

You can even make the scale bigger, and measure the UK. 1 United Kingdom is made of 4 countries. These countries consist of 92 counties (or 99, yay ceremonial vs historic). The counties house lots of town and cities. If you want to really delve into this with your class, the House of Commons Library has a great resource here.

All that experimenting is to show, that when units are smaller, you can fit a lot more of them into a given space, whereas the larger units, because they are so much bigger, can only fit a few in.

From this, we can derive 4 questions to help us convert a unit quickly:

· Is the unit larger or smaller?

· So can we fit more of them or fewer of them?

· Does that mean we multiply or divide?

· Which number do we multiply/divide by (using our knowledge of base prefixes)?

This will hopefully stop students from taking a wild stab at metric conversions, and lets them use logic to determine, rather than the coin flip guesses that usually occur when not fluent in measures.

But how about metric to imperial? Here we have horrible scale factors, and sometimes less than zero which throws our logic out the window. Unlike metric conversions, these ones are actually given by a key in the question, and we can exploit this.

The most common to find here is a 1:n relationship. The most frequently used of these, at secondary at least, is that 1in = 2.54cm. When they look like this, I again ask questions, and refer to n as ‘the weird number’:

· Where is the weird number, in or cm?

· Am I becoming cm, or moving away from cm?

· So do I multiply or divide by 2.54?

Say I’m asked to convert 5 in to cm…

· Where is the weird number in the key, in or cm? Answer: cm

· Am I becoming cm, or moving away from cm? Answer: becoming cm

· So do I multiply or divide by 2.54? Answer: multiply

So 5 x 2.54 = 12.7 cm

The next question goes the other way, converting 508cm to in…

· Where is the weird number in the key, in or cm? Answer: cm

· Am I becoming cm, or moving away from cm? Answer: moving away from cm

· So do I multiply or divide by 2.54? Answer: divide

So 508 ÷ 2.54 = 200in

Have a currency exchange? Same thing. Given £1 = $1.25, convert £4.50 to dollars

· Where is the weird number in the key, £ or $? Answer: $

· Am I becoming $, or moving away from $? Answer: becoming $

· So do I multiply or divide by 1.25? Answer: multiply

1.50 x 1.25 = $5.63 (to nearest cent)

Have a go, if you like. 1 mile = 1.6km, convert 5 miles to km, then convert 32km to miles. Answers on a postcard.

How about non-unitised relationships? Glad you asked.

We know the classic, 5 miles = 8km. This one I keep in equation format and look for the scale factor.

Let’s say we’re converting 15 miles to km…

Start by writing out the original equation: 5 miles = 8 km

We now write the question into a parallel equation underneath:

5 miles = 8km

15 miles = km

Now, we draw arrows to find the scale factor:

Having found the scale factor of 3, we find that the 8km becomes 24km. Try it now, if you like, by converting 56 km back into miles.

Now you may be saying ‘um…this looks like a ratio table.’ And it does! So, next week, we look at, now they are more confident in the logic behind conversions, how can we get there quickly. And one answer, at least, is the ratio table!

If you liked this, please consider sharing the series, and any feedback is greatly received as a comment here, an email to jwrightmaths@gmail.com, or a message on Bluesky to jwrightmaths.

See you next week!