1 Why use algebra?
You may well have come across formulas before, such as:
This example allows you to calculate the speed for any distance travelled and associated time taken to travel that distance. Similarly, if you knew the speed and distance travelled by rearranging the equation you could calculate the time taken. This is a relatively simple example of how algebra can be used, and has been used for centuries.
Algebra is a powerful mathematical tool that enables generalisation – something may seem to be true for many sets of specific data values, but is it true for all such sets?
Before discussing equations in more detail, here’s one example of how algebra is used to generalise.
Generalisation using algebra
Find the sum of three consecutive whole numbers or integers. Divide your answer by 3. What do you notice? (You may need to try this for several sets until you find a pattern.) You should notice that the result appears to be divisible exactly by 3, and be the middle of the three consecutive numbers.
The pattern can be explained by working out a formula for this calculation. Suppose you call the first number n, that means as the three numbers are consecutive the second number can be written as n + 1, and the third as n + 2.
So, the sum of any three consecutive numbers is:
n + (n + 1) + (n + 2)
This can be simplified by carrying out the sum.
n + (n + 1) + (n + 2) = n + n + n + 3
Since there are 3 lots of n, this can be shown as 3n (3n = 3 × n).
Thus n + (n + 1) + (n + 2) = 3n + 3
The next instruction is to divide the sum by 3.
Therefore, the product of three consecutive numbers is always divisible by exactly by three, and n + 1 – the middle of the three numbers.
This example uses simplification to obtain the generalisation. You will cover this technique in the next section.