# Errors and Inequalities Recently I was revising y = mx + c with my Year 10s.  I was using an app that demonstrated the gradient calculation for different lines.  The line started with a vertical change of 2 units and a horizontal of 1 giving a gradient of 2 ÷ 1 = 2.  I could then drag the ends of the line to demonstrate different gradient calculations such as 8 ÷ 4 = 2.  What was interesting was that before I reached 8 and 4 the app was showing 8.1 ÷ 4.1 = 2.

Obviously 8.1 ÷ 4.1 is not equal to 2, but this gave me a chance to look at errors in measurement in a ‘real life’ context.  The app was obviously giving its measurements correct to one decimal place so we looked at what the ‘real’ length of the lines might be. It could be anything from 4.05 through to 4.14999999999…

Ask your students to suggest what the two exact lengths may have been that gave exactly 2 when divided? It could have been 8.12 and 4.06 or maybe 8.122 and 4.061. How many different possible such answers could there be?

Another context where this can crop up is when using geometry software to measure angles. For example, when studying circle theorems you may find that opposite angles of a cyclic quadrilateral add up to 179.8° or that the angle at the centre is apparently not exactly twice the angle at the circumference.

The new GCSE requires that students can express rounded values using inequality notation and these errors in measurement provide a good stimulus for discussion.

All the best,

Steve Cavill

Ste ve Cavill BSc(Hons) PGCE FCIEA has taught maths in both state and independent schools. He spent a few years as an Associate Lecturer for the OU and has written a number of GCSE maths books, workbooks and revision guides as well as being a senior examiner and moderator for GCSE and IGCSE.