It was a great pleasure to host Rob Eastaway at the London branch ATM/MA meeting this morning. His theme was Arithmetic, and how some techniques are almost becoming a lost art. There was so much energy in this session, the room was positively buzzing with pencils and pens scratching away! We covered so much ground in two and a half hours, I’m not going to attempt to write about everything but I am going to pick the thing that resonated most with me.
A number of techniques we explored were about getting exact answers, but this section was as rough as you like! Rob introduced us to his idea of “Zequals“. When teaching rounding, I always enjoy introducing my students to the “approximately equals” sign, ≈. I hadn’t really considered how this symbol, on its own, doesn’t give the complete story. All of these statements are true…
7.3 ≈ 7 7.3 ≈ 10 and even 7.3 ≈ 7.4
but they don’t give an explanation of what you have actually done to the 7.3 and in the last example here, it really would require quite a bit of explanation!
So Rob proposes “Zequals” which has a precise method. It looks like this:
and it involves rounding the numbers you calculate with AND the result to 1 significant figure.
The Numberphile video explains in more detail here. An interesting question to ask might be, what calculation would give the biggest discrepancy between the accurate calculation and the Zequals calculation? And what would the % error be? The blog post explains this beautiful graphical representation of that % error, which turns out to be a fractal.
Now, to be honest, I would be hesitant to “teach” non-standard notation and methods as part of the regular timetable of maths. There is already so much to learn and time is so precious, why would I take a lesson explaining something they are unlikely to ever encounter again in this form? But dismissing it on that basis, misses the point, I feel.
Estimation as a topic features in a fairly minor way at GCSE but is a critically important skill in many jobs and life in general. There was some discussion amongst the attendees this morning that as students progress through KS3 to KS4 and A-level they become more and more reliant on their calculator. With the demise of the C1 paper, there is no longer a requirement for a non-calc paper at A level which is inevitably going to mean that our students will get weaker at this skill rather than stronger. This seems like a real mismatch between our education system and the requirements of employees and our broader society.
An idea which might help is to explicitly teach estimation as a technique to be employed when doing calculations with large numbers or decimals. Typically these types of calculations would involve some sort of “ignoring” the decimal point or the zeros, doing the calculation, and then “putting it back”
3.23 × 3 323 × 3 = 949, so 3.23×3 = 9.49 (counting 2 d.p.) 23.1 × 0.31 231 × 31 = 7161, so 23.1 × 0.31 = 7.161 (counting 3 d.p.) 3200 × 40 32 × 4 = 128, so 3200 × 40 = 128000 (counting 3 zeros)
Maybe instead of, or as well as, “counting the decimal places” when doing these calculations we could do some rounding / estimation. So
3.23 × 3 ≈ 3 × 3 = 9 323 × 3 = 949, so 3.23×3 = 9.49 (same order of magnitude) 23.1 × 0.31 ≈ 20 × 0.3 = 6 231 × 31 = 7161, so 0.23 × 0.031 = 7.161 3200 × 40 ≈ 3000 × 40 = 120000 32 × 4 = 128, so 3200 × 40 = 128000
Now I am not claiming that this is a more efficient or reliable method. It does depend to a certain extent on the examples chosen and “counting the decimal places” is a method that will always work. But I feel that the approximation step helps with number sense: the idea that 20 × 0.3 is a bit less than half of 20 so must be 6 is really valuable for life beyond exams. It provides an opportunity to practise these skills, practice which I believe our students currently have precious little of.