Spotlight On: Numbers

19 March 2019

This article first appeared in Queenwood News Weekly 15 March 2019.

This Thursday was pi day. Why was it pi day? Well pi equals approximately 3.14 (3.14159265359 or even to be precise) and we tend to follow America here where the date would have been written as 3/14. Give it four months and we will hit pi approximation day using our traditional approach to writing the date as 22/7.

Given any excuse I’ll happily tell you some fun maths facts:

  • 22/7 is the fraction with a single digit denominator that is closest to pi. It is accurate to three digits and has an error of 0.04%. The next best rational approximation with a small denominator is 355/113 which is accurate to 7 digits and has error 0.000008%.

  • Pi is a transcendental number, which is nice enough because it’s such a lovely word. What it means though is that there is no polynomial with solution pi. The square root of 2 isn’t transcendental, since x^2-2=0 has solution √2.

  • There actually aren’t that many numbers that are known to be transcendental. e is another well known one, as well as the sine, cosine and tangent ratios of non-zero algebraic numbers (which are essentially the opposite of transcendental numbers). Even nicer still, e^(π√(-1))+1=0, a beautiful little equation that combines all of the most famous maths constants. This is actually quite a popular mathematical tattoo, but I am not encouraging that.

  • Last time I checked, pi is known to around 10 decimal places. This puts it at about 10 to the power of 13 million miles in length when written down by hand. Queenwood girls on detention may have to test out this theory this year.

  • It turns out however that NASA only uses 15 digits of pi for calculations, this is enough to make their calculations accurate to the size of an atom.

  • If you divide the circumference of the sun by its diameter you’ll have a pi in the sky!

Pi is just a number but as you can see it is a reasonably interesting one. So, for that matter is the number 1, and the number 2 and if pushed I could probably find you something interesting about any number you might bring up. Even more interesting to the Mathematics Department this year is the connection between numbers and how the connections between them are used to perform calculations, both exact ones and approximations. Understanding the links between numbers and being able to use these connections to creatively and accurately perform mental calculations are important skills and have been linked to improvements in the achievements of students in Mathematics.

An area of focus for this year is numeracy development, particularly deepening girls’ number sense. While it is really satisfying to use calculus to find the equation of a tangent to a cubic polynomial in Year 12, you can’t do that if you can’t add, subtract, multiply and divide numbers. Fluent arithmetic is the most crucial building block in our mathematical development and a skill that seems to be diminishing among students. A strict regimen of regurgitating formal algorithmic procedures regardless of understanding is hardly desirable, but there are certain facts in mathematics, such as times tables, that must be known in order to progress to higher-order thinking.

We can point the finger of blame in many directions – the prevalence of technology, be it computers or the calculator, or a simple lack of interest among educators as new trends come in and out of fashion.

The fact remains, however, that without fast and accurate mental arithmetic, students’ mathematical careers can only go so far, as the cognitive gap beyond this foundational skill becomes too wide to bridge. Mathematics becomes technically dense very quickly and without a solid foundation of automated skills this technicality becomes almost impossible to master. If working memory is being used to calculate that 7 times 8 equals 56, the cognitive load of analytical thought required in the senior Mathematics courses becomes too heavy. These are facts that must be securely held in long-term memory and not recalculated on the fly every time that they are used.

If you would like to help your daughter to develop these foundations, a simple exercise of asking her to recite her times tables would be beneficial. Or, offers an interactive way for your daughter to get some practice in. Even worksheets, like the ones found here: can be helpful. Another small thing you could do is to pick two numbers and ask your daughter to sum them, find the difference, find the quotient and remainder on division and find any common factors. All of these activities will help your daughter to develop greater number fluency which will be beneficial in the classroom.

Throughout this exercise mistakes will no doubt be made, but the idea that mistakes aren’t a bad thing but rather an opportunity to learn, is one that I hope we can embrace at Queenwood.

Dr Paul Emanuel
Head of Mathematics