Wednesday, March 2, 2011

Further exploring the properties of one

To find an equivalent fraction of a decimals, one way to explain it is to take the decimal part of the original number and place it over the lowest place value. Leave any whole numbers in front. (This only works for non-recurring decimals)

eg 0.123

The lowest place value is thousandths, the decimal part is 123.

therefore:

0.123 = 123/1000

An alternative way to explain it is using properties of one. The idea is that
a) numerators of fractions should be whole numbers and;
b) the fraction should be equivalent to the decimal.

We can ensure the fraction is equivalent if we only multiply or divide by 1 or more importantly a fraction that is equivalent to 1.

To satisfy part a)
To make 0.123 a whole number we have to multiply it by a power of 10 - 1000 (10^3). This was a concept we had investigated earlier.

..but if we multiply by 1000 we will change the original number from 0.123 to 123 - it will no longer be equivalent.

So to satisfy part b)
We multiply by 1000/1000 (or 1!)

Thus:
.0123 = .123/1 x 1000/1000
= 123 / 1000

I like this because it continues to explore how fractions are constructed, the connection between decimals and fractions and why decimal conversion works. I wouldn't try it in classes with low ability due to the possibility for high levels of confusion if understandings of multiplication and commutative properties are not properly understood.

An earlier article exploring one and fractions can be found here.

Viola.