Many people who have learned arithmetic successfully have some difficulty moving to algebra which is more abstract and symbolic.
In fact algebra is just arithmetic but using letters to be more general.
They say a picture is worth a thousand words, fortunately geometry can make the abstract ideas of algebra crystal clear and even obvious.
Multiplication can also be viewed as arranging things such as stones or pennies in a grid. Or if you have square paper just color the squares so
Perhaps geometry can give us a clue. How about if we cut up 6×8 square by drawing a line 2 squares from the left side and another 3 squares from the bottom side.
This tells us that
- one at a time take each number in the left hand brackets
- then also one at a time take each number in the right hand bracket
- and multiply each of these pairs
With any four numbers (a,b,c,d) we can draw a rectangle with sides of length (a+b) and (c+d) and divide up the rectangle as we just did
This is exactly the same rule for multiplying brackets that was given in words above, only this time its written more succinctly in letters.
One of the first things that is met in algebra is how to work out
- c with a
- d with b
But let’s give geometry a go first
- A square with sides of length a
- A square with sides of length b
- Two rectangles with sides of length a and b
In algebra it is normal to
- miss out multiplication signs as algebra uses single letters to represent different numbers when 2 letters are written next to each it is assumed that they are multiplied (i.e. ab means a x b).
- as the result of multiplying 2 numbers is the same whichever way they are written (i.e. a x b = b x a). When 2 or more letters are multiplied they are written alphabetic order (i.e. ba is reordered to ab).
- multiple occurrences of similar groups are collected together (i.e. ab + ab is written 2ab).
- letters that are multiplied by themselves are written as squares, cubes etc.
This result may not seem like much but it takes us about half the way to proving Pythagoras’ theorem. I am not going show how to do this in this post.
Instead I’m going to head back to Arithmetic and Multiplication and note that the rule
It is important to understand what is happening when multiplying numbers, this is illustrated by the large rectangle divided into 4 smaller rectangles and the algebra
- splitting the numbers into parts
- multiplying the parts
- adding the results
All that matters is that the method helps calculating the answer