Estimation – an undervalued skill.

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:Screen Shot 2017-10-07 at 17.10.11

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.




Rational and Irrational numbers

I only recently properly considered why all fractions are either terminating or recurring decimals.  Fundamentally, this is because there are a finite number of options for the remainder, which is a maximum of 1 less than the divisor.  This is most easily seen when dividing by 7.  All six potential remainders are used and the 7th division goes back to the beginning of the sequence as shown on this task from Don Steward.


Maybe this is an important thing to establish before talking about irrational numbers. With irrational numbers, we are effectively trying to convince students that there is a separate class of numbers on the number line that can’t be expressed as a division of two integers.   

There is a proof for why ∏ is irrational but it’s not pretty. I’m taking Peter Mitchell’s word for it on that who presented on this topic at the recent MEI conference.  He has a proof here, but in his own words “it’s really, really tedious!”  So maybe surds are a better place to look for an example of a proof that irrational numbers exist. Although this is an A level topic, I think with the right class this could be used at KS4.

Proof that √2 is irrational

This is a proof by contradiction, which in itself is a bit strange.  But the logic is sound: if I assume something to be true and then work through it to show that there is something inherent within it that is false, then I have proved that thing cannot be true therefore it must be false.

In this case, we are going assume that √2 is a rational number, prove that that is false, thereby proving that √2 must be irrational.

If √2 is a rational number, then we can write it √2  = a/b where ab are whole numbers, b not zero.

We additionally assume that this a/b is simplified to lowest terms, since that can obviously be done with any fraction. Notice that in order for a/b to be in simplest terms, both of a and b cannot be even. One or both must be odd. Otherwise, we could simplify a/b further.

Going back to our first statement:

√2  = a/b

we can square both sides to get:

2 = a2/b2


a2 = 2b2.

So the square of a must be an even number since it is two times something.   If is even then a itself must also be even. Any odd number time an odd number creates an odd number (some more of these here).

Okay, if a itself is an even number, then a is 2 times some other whole number. In symbols, a = 2k where k is this other number.

If we substitute a = 2k into the original equation 2 = a2/b2:

2 = (2k)2/b2

2 = 4k2/b2

2b2 = 4k2

b2 = 2k2


Again, because b is 2 times something, b must be an even number.

We have shown that a and b are both even numbers, but we started saying that a/b was a fraction in its simplest form.

I might want students to explore what happens with √4 in this same proof, i.e. prove why √4 is not irrational.  From there we could go on to look at √3.  It’s a bit harder, but only really requires that all odd numbers can be written in the form 2n+1.  Here is a spoiler if you are stuck.





Generalising Surface area

What is the surface area of a cube of side length 1?
If we then cut this cube in half, and throw one of the halves away, what is surface area of the remaining cuboid?
Repeat the process, cut the shape in half along the same plane. What pattern can you see?

Screen Shot 2017-07-09 at 16.59.12

What is the general formula for the surface of a cuboid of width 1, depth 1 and height h?

What is the general formula for a cuboid of width 1, depth d and height h?

What is the general formula for a cuboid of width w, depth d and height h?

What other 3D shapes can you find the general formula for the surface area? Try:

  • A tetrahedron, side length a
  • A square based pyramid, base length a, height, h
  • A cylinder radius r, length l


Student generated data

I know many teachers find teaching Statistics at KS3 & 4 a bit dry.  One way to make it a bit more interesting is to make the data somehow relevant to the students.  I’m not talking about football scores or download charts here. I’ve gone to great lengths to source that kind of data and create beautiful resources with it only to find that, while it engages some students, it has the opposite effect on others if it is something they are definitely not into.

So, having given up trying to get down with the kids, here is another approach which involves them generating data so they feel they have some ownership of it.  It’s quick, they can do it in their seats, they get a bit competitive and it’s interesting.


How many dots?

Display this for about 2 seconds.  Tell them what they are going to see and make really clear that they are not to discuss it but write down their estimate.  Then go round the class capturing those estimates on a spreadsheet.  I might use Geogebra for this as it is great a creating box plots, but Excel would be equally good.

You must plead with your students not to cheat and change their estimates from the one they wrote down.  Tell them they will get a second chance.  Do it all again to get a second second set of data.  You now have two sets of randomly generated data that you can use to compare using averages, box plots, standard deviation, etc. It should be a great example of regression to the mean.  Also “The wisdom of the crowd” – always interesting to see how wise your crowd actually is!

Oh, and how many dots? 46. The bare bones of a Powerpoint is here.

Trigonometry, another way

Trigonometry falls firmly into the camp of one of those areas which I don’t feel I have cracked yet.  If “cracked” means finding a bomb-proof way to introduce it to any class I encounter, then maybe I never will!  Either way, I am always interested in different approaches to teach this topic which many students seem to struggle with at first.

This approach builds on the presentation that Mike Ollerton gave at the recent Mixed Attainment Maths conference in Sheffield (keep an eye on the site for details of the next conference in November!)

I didn’t actually attend Mike’s presentation – I was too busy giving my own – but Mike has kindly shared his ideas and I have been thinking about how I might use Geogebra as a tool to aid learning.

As with most of my use of Geogebra, I am using it as an exposition tool to structure whole-class questioning and discussion around. In an ideal world, I might get students to do this themselves, but that is not practical in my classroom. I feel that starting Geogebra from a blank sheet can be nearly as powerful as them doing it themselves and is likely to be a much more efficient use of lesson time.

The basic principle used is that of rotating a fixed line segment, a “spinner” if you like, around a point.  We are aiming to explore the co-ordinates of the point at the end of the line as the angle increases from 0º to 90º (and beyond) in a table.

I must say that from this point onwards this post is not Mike’s recommended approach (which is here) – but my interpretation of it using Geogebra.

So, first step is to form the spinner by by plotting points at (0,0) and (1,0), zooming in and connecting the points with a line segment. There is something in observing what happens to the scale on the axes as we zoom in.

Jul-02-2017 17-44-33

Next, we create the angle by using the Angle with Given Size tool. As the prompt says when you hover, the tool: select leg point, then vertex.  Rather than fix the angle, I want to make a slider so I can easily change it. I set the angle “a” making sure to leave the degree symbol in place (otherwise you get radians).  The slider then needs tweaking by double-clicking to set the max, min and increment.

Jul-02-2017 17-45-35

Next, I need to make the line segment a bit bolder by right clicking on it…

Jul-02-2017 17-45-45

..and change the properties of the point so it shows the coordinates to 2 decimal places, also using right click.

Jul-02-2017 17-45-54

We now have a tool that can tell us the co-ordinates.  Before using this, however, I think I would want pupils to do some work on paper, using Mike’s handout to get a feel for the numbers and get their own results.  To fill in this table:

Screen Shot 2017-07-04 at 19.15.45

I feel that it’s useful that they have the opportunity to correct any measurement inaccuracies before the next step and this is where the computer helps.

As per the worksheet, a series of questions can be posed before using the calculator’s Sin and Cos functions to complete the following:

Screen Shot 2017-07-04 at 19.16.44.png

From there we can easily start exploring what happens when the side length is not 1 and use the ideas of scaling and similar triangles.

Jul-02-2017 17-46-04

And then, of course change the angle again.

Jul-02-2017 17-46-14

Once this is all set up, it’s easy to display / hide the coordinates and maybe so some miniwhiteboard work to assess how well the class has grasped the use of the Sin and Cos functions. And then keep the coordinates, but hide the angle to demonstrate inverse Sin and Cos.

With some practise and familiarity with Geogebra you are spending less than 3 minutes on the computer.  If you like the idea of “Geogebra from a blank sheet”, click on the Geogebra category link at the top to see posts on using the same idea for other topics.

Ideas for better maths teaching