It’s possible that what I am about to explain is already completely obvious to many a maths teacher. But after 5 years of teaching, it’s only just dawned on me (a recurrent theme of this blog by the way!) so I thought I’d share.

I like teaching the concept of Bounds. It highlights a fundamental difference between concepts that exist in Maths and the real world. We measure things using numbers as continuous variables. When we measure the length of a table, say, as being 120cm, we are rounding to the nearest centimetre. We could be more accurate and say it is 120.4cm or even 120.4187234cm but this is still an approximation. It reminds me of my engineering days, looking at milling machines that were capable of cutting to the nearest 0.1mm. I used to wonder about how those milling machines were made and the tolerances required in the dimensions of their components. What machines were used to make the components? And what about the machines that were used to make those machines? Where does it stop?!

But one of the stumbling blocks with bounds is always with the Upper bound.

“But you told us that 5 rounds up!” is the usual complaint. And then we get into a heated arguement about whether:

Which of course it is, for the same reason:

Earlier this week, I was marking this GCSE question, which most of my students, including those sitting Foundation were answering no problem.

I’m pretty sure that I have previously suggested that students state the “Lower Bound” and the “Upper Bound” almost as though they are two separate answers. But expressing the **range** of possible values in this way makes more sense. It is also entirely consistent with the well-understood rule that “5 rounds up”. In the question above, if the actual length was 13.5, it **would** be rounded up to 14, but a value below 13.5 would not. Whereas a measured value of 12.5 would be recorded as 13.

All makes perfect sense now!