Algebra Simplification

Algebra can seem abstract and confusing to pupils when they first meet it.

The fact that you are expecting quite young children to imagine letters representing numbers can be tough enough. Then making it harder by confusing the matter with methods that frankly do not make much sense.

Initially pupils are expected to write expressions

e.g. Consider the number y

Write an expression for:-

“twice the number”  2y
“four more than the number”  y + 4

“y less than 20”  20 - y

The pupils get the hang of these and hopefully see the logic by relating the y to the number 6 for example.

The next challenge involves simplifying expressions.

Gathering Like terms

Consider 2a+3b+4a+6b = 6a+9b

Some textbooks talk about “adding 2 apples and 4 apples to get 6 apples then 3 bananas and 6 bananas to get 9 bananas”. And stressing that you can’t add apples to bananas.

The children have just about got used to a letter representing a number when suddenly it becomes an apple, banana, egg or car!

Attempting to simplify 7ab -3a - 5b - 6a - 2b + 5 + ab using apples and bananas makes no sense. (negative 5 bananas subtract 2 bananas ?? 7 apple/bananas ??)

A Better Way:

Using the listing method
Starting off with 2x + 6x think of x as a pile of sand, x metres above ground level. Adding 2 piles high of this sand with 6 piles high will give a pile 8 high.

Negative amounts of x are then thought of as holes in the ground. 

-3x is thought of as a hole three lots of x metres down.

Looking at the examples below.

1) a – a is thought of as a pile of sand a metres high filling in a hole of giving level ground.

i.e. 0 metres.

2) 2a – 3a is thought of as a pile of sand 2a metres high filling in a hole deep leaving us with a hole of a metres deep 

i.e. -a

This is then used to explain the Listing Method:

A negative will cancel out a positive.  (filling in holes) 

Two positives will combine to make a bigger positive.   (a higher pile)

Two negatives will combine to make a bigger negative.  (a deeper hole)

Consider 1. 

List 3a - 4b + 5a - 2b

 

= 8a - 6b

Consider 2. 

List -3y - 2z + 4y + z

 

= y–z (or -z+y )

This method is especially useful when simplifying expressions with powers of letters. 

Consider 3.

This method has been used successfully for the last few years. The pupils are more accurate and are more likely to understand the logic of gathering like terms.

It can also be used to understand basic negative number calculations.

R. Farnan 18/11/19