Geometry Can Help Make Clear What Algebra and Multiplication Mean

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

looks like

How about

this must also be 48 as

eqn2Add4And3Add5Is there a way for us to do the sum by doing the multiplication first rather than the addition?  The answer must still be 48.

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.

We end up with 4 rectangles whose areas add up to 48 –  the area of the whole rectangle.

This tells us that

eqnExpand2Add4Times3Add5And the rule is

  • 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

The area of the whole rectangle will be the sum of the areas of the individual rectangles so


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

Now this is similar to eqnaAddbTimescAdddand it would be possible to just rewrite this replacing

  • c with a
  • d with b

But let’s give geometry a go first

The square with sides of length a+b has been split up into 2 squares and 2 rectangles

  • A square with sides of length a
  • A square with sides of length b
  • Two rectangles with sides of length a and b

So An explanation of how we get from second to third line

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

eqnExpandaAddbTimescAdddcan be used to multiply numbers such as 15 x 23 if we take the numbers in units and tens columns separately.

This is just what is now taught in primary schools as the ‘grid method’ though laid out differently.  Something like this would be more usual

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

eqnExpandaAddbTimescAdddThe grid method, or any other method, is just a tool for performing the calculation. Once the idea of multiplying larger numbers by

  • splitting the numbers into parts
  • multiplying the parts
  • adding the results

All that matters is that the method helps calculating the answer

  • simply
  • accurately
  • reliably

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