We often use rules of thumb to give rough answers but sometimes they can short circuit the thinking process. There are two rules of thumb associated with using a ruler.

• The uncertainty is half the smallest division.
• There is an uncertainty at both ends.

So if the smallest division is 1 mm then each measurement has an uncertainty of ± 0.5 mm, the ruler has two ends so the total uncertainty is ± 1 mm. But when we use a ruler we don't measure two positions we line zero with one end and measure the other.  I think we can line up more accurately than we can measure. I'd say the top measurement is 3.20 ± 0.05 cm and the bottom one is 3.2 ± 0.1 cm.

The half the smallest division rule is also dubious, Take a look at the image below.

What do you think the length is?

0.5 ± 0.5 cm?

I think you can do much better than that, it's a bit less than 0.5 cm, probably around 0.4, I think I can estimate within 0.1 cm but I wouldn't bet my life on it, maybe 0.4 ± 0.2 cm to be on the safe side. The half a division rule is fine for small divisions but doesn't really apply in this example. How small is small? Maybe it's got something to do with the size of the line you are measuring and the width of the scale markers?

I think this one is a bit more uncertain.

At the end of the day we want our students to have an appreciation of the uncertainties. It is better that they think about the scale and the way it is used rather than just apply a rule of thumb, unless they are using the right hand grip rule which is a rule where they have to use their thumb.