Calculate Like The Ancients
- By Aurel Nicolae
To an experienced mathematician or maths teacher, calculating the area of a rectangle comes naturally. It is simply the length of the rectangle multiplied by its width.
Given a rectangle with the length of 6 cm and the width of 4 cm, a teacher is usually quick to answer, as if on autopilot, “Well, the formula for area is length times width”. They then proceed to write out the formula A=l×w, substituting the given numbers i.e., A=6 cm ×4 cm, which obviously gives us the result of 24 〖cm〗^2.
Can you imagine the puzzled looks on the year 7 students’ faces? Put yourself in their shoes for a minute and think of all the new concepts that you have just bombarded them with. This is what a KS3 students might think or most likely vocalize out loud, followed by a few choice words from ‘brain rot’. Modern teachers have to keep up with the slang of Generation Alpha, as if teachers didn’t have enough to worry about.
Apologies, I digress. This is what a KS3 student would think or say: “I have seen multiplication before. That is ‘times-ing’ two numbers together, right? I kind of remember how to work out the area of the ‘square-thingy’ from KS2. ‘Formula’, what is that? We were told to use a 〖cm〗^2 in KS2, but what is it? And why is there a 2 above the cm? This is too much!”
Why do we use this formula? Why does it work? What are the centimetres squared? This is the story that I like to tell year 7 students when we cover basic geometry concepts such as area. Imagine that we could travel back in time to the ancient civilisations of Greece, Egypt, Mesopotamia, roughly where the present-day Iraq is, or China.
The ancients would use pebbles i.e., small stones, to count out lengths and distances. Likewise, they would tally animals or votes, do abacus arithmetic
i.e., adding and subtracting neatly ordered rows and columns of pebbles, and yes, they would count out the area of a rectangular piece of land, pyramid base and other shapes.
A pebble would stand for whatever quantity they wanted to measure, assuming the small stones were all roughly the same size. Given our first example, replacing centimetres with pebbles, the ancients would try to fill that shape with pebbles as close as possible. They would count the pebbles and what do you know, there are 24 pebbles that we can fit in the 6 pebbles by 4 pebbles rectangle. This is exactly the same result from our 6 times table with 4. Hence, the ‘formula’ for calculating area. Instead of formulas, the ancients had methods or ‘recipes’ for doing arithmetic.
Nowadays, instead of trying to fit small pebbles into a rectangle, we fit tiny squares having both the length and width equal to say, 1 centimetre, which we call a ‘square centimetre’. The notation of 〖cm〗^2 is shorthand for 1 cm ‘times’ 1 cm.
Interestingly, pebble in Greek is ‘χαλίκι’, pronounced ‘chaliki’, and in Latin it is called ‘calculus’. Since pebbles were used for counting out lengths, distances, areas, tallying possessions and doing arithmetic, the word ‘calculus’ came to mean ‘calculation’. Not surprisingly, multiplication has origins in the calculation of area for flat shapes.
In modern times i.e., since 1672, we have a whole branch of mathematics called ‘calculus’. Part of it is dedicated to calculating area under a curve. Even though calculus is beyond Key Stages 3 and 4, it is amazing to think how maths is built upon the mighty pebble.
By Aurel Nicolae
https://www.linkedin.com/in/aurelnicolae/
Images supplied by the author.
------
Keep up-to-date with the latest Maths Scholarships news:
Find us on Twitter, Instagram, LinkedIn, YouTube, and Facebook.
Join our mailing list or get in touch Here.