## Monday, 31 August 2015

### Explaining Significant Figures - why are they significant?

I have always been somewhat dissatisfied with how myself and other teachers explain significant figures to pupils, in particular why certain figures are significant and others aren't, and why certain bits of logic only apply in certain situations. The only really acceptable link I have found up to now is to that of standard form, in that when represented in standard form the digits multiplying the power of 10 are only the significant one. However I am still not very happy with this, as we often teach significant figures after standard form - I feel like a proper understanding of significant figures should inform a pupil's understanding form, not the other way around. Thankfully I think I have found a way of describing significant figures to pupils, which should allow them to begin to understand...

"In any number, figures are significant if they are needed to describe a number using separate columns without skipping any columns."

Now this seems a little bit of overkill I must admit, but given that the next most satisfactory definition I could find was to have four separate rules to cover situations without and without decimal points.

Lets examine some numbers to see how this can be applied:

456 - this would be described as 4 hundreds, 5 tens and 6 units, so all 3 figures are significant.

4057 - this would be described as 4 thousands, 0 hundreds, 5 tens and 7 units, so all 4 figures are significant (note we need 0 hundreds as we cannot skip the hundreds column)

300000 - this would be 3 hundred-thousands and so would have a single significant figure.

000456 - this would still be 4 hundreds, 5 tens and 6 units, so again has 3 significant figures.

0.00042 - this would be 4 ten-thousandths and 2 hundred-thousandths, so would have 2 significant figures

0.0506 - this would be 5 hundredths, 0 thousandths and 6 ten-thousandths, so would have 3 significant figures (again, the thousandths is needed so as to not skip columns).

For me the only place where this definition is a stretch is with decimal numbers that have trailing zeroes, as it would be tempting not to include the trailing 0s, such as in the number 300000 above. To counter this one would need to ensure that pupils had a proper understanding of place-holding. The zeroes in 300000 are place holders, they are there only to ensure the 3 lands in the correct column. The zeroes to the right in 0.4600 are not place holders (or shouldn't be) as they do nothing to ensure numbers find their way into correct columns. These digits must have a separate purpose (and of course when it comes to measurement they do - they tell us the accuracy applied, which is a natural point to eventually take significant figure understanding to) and it is this purpose which makes them significant. This allows us to show that the number 0.4600 will have 4 significant figures, and indeed that while the number 300000 only has one significant figure, the number 300000.0 will have 7 significant figures.

So if you are struggling to justify to pupils why certain digits are significant, and others are not, try first of all giving them a proper understanding of place-holding zeroes, and then applying the logic above - it may not be perfect, but I haven't seen anything better in 10 years of teaching maths.