# Mathematical debates, and Bounds

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:

(these images from Don Steward can be useful for this one).
Sometimes that argument can be fun, but I think that as teachers we need to beware of going down rabbit holes. The more inquisitive students might find this philosophical debate stimulating but it can be a turn-off for others. And even the ones that do actively engage in that discussion may not be convinced at the end of it.  It’s just another one of those decisions that we make in the moment as we read the mood in the class.  In making that decision we should ensure that we are taking into account the feelings and needs of all students, not just the more vociferous ones.
So, to the point of this post, how to move this debate on.

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!