My emphasis for the last week has been on establishing an idea of "one" with my year 9 academic class. We examined how our idea of one influences how we deal with fractions and algebra.

Firstly we looked at common denominator problems and examined in more detail the method for adding fractions with different denominators.

A common idea is to find common multiples or factors of the denominator and then multiply both the numerator and denominator of the fractions until common denominators are found.

eg. 1/2 + 1/3 -> common denominator of 6 (LCM of 2 and 3)

We then need to find equivalent fractions with denominators of six.

eg 1/2 x 3/3 = 3/6

1/3 x 2/2 = 2/6

Now we have common denominators we can add the fractions..

eg 2/6 + 3/6 = 5/6

But.. why does multiplying by 2/2 and 3/3 work??? Understanding "One" is the answer!!!

1/2 x 1 = 1/2

3/3 = 1

Therefore by substitution 1/2 x 3/3 is just multiplying 1/2 by one. Any number multiplied by one is equal to the original value thus any resulting fraction must be equal to 1/2!

This illustrates two different ideas related to one.. "Multiplying by One" and "Dividing a number by itself".

We also looked at cancelling and why it works..

2m / 3m, we commonly use the skill cancel the m's and 2/3 is what is left.

By re-examining how multiplication works with fractions we find that we can rewrite

2m/3m

as

2/3 x m/m

..but we know that anything divided by itself is 1 (other than zero of course!)

Therefore we can simplify to

2/3 x 1

and we know that anything multiplied by one is equal to the original value.... thus we can see why cancelling works..

Quite a fun little lesson.

Russ.

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