Primary Mathematics/Working with fractions

Working with fractions
Mixed numbers and top-heavy fractions:

A mixed number is an integer and a fraction, eg two and a half, a top heavy fraction is a fraction with a larger numerator than denominator, eg 3/2. For multiplication and division you must not use mixed numbers. They can be used in addition and subtraction (do the integers and fractions separately and combine the result) if this is not too confusing.

Simplest terms: Multiplying (or dividing) the numerator and denominator by the same number will not change the fraction. A fraction is put into its simplest terms by dividing both by the largest integer that goes into both a whole number of times, this can be done in stages using small integers such as 2 and 3, 5 and 10 repeatedly.

Multiplication:

$$\frac{a}{c}*\frac{b}{d} = \frac{ab}{cd}$$

Where ab=a*b.

Division:

To divide fractions; remember that a fraction is a division. Fractions can be stacked like so:

$$\frac{\frac{a}{b}}{\frac{c}{d}}$$

The rule to remember to simplify is that anything under an odd number of lines goes on the bottom and anything under an even number of lines goes on the top (0 is even for this purpose) thus:

$$\frac{ad}{bc}$$

It is easier (perhaps) to remember that the second fraction in the division gets turned upside down.

Addition and subtraction:

For addition the denominators of the two fractions must be the same - if this is so then simply add (or subtract) the numerators:

$$\frac{a}{b}+\frac{c}{b} = \frac{a+c}{b}$$

If the denominators are not the same then a common denominator must be found, the simplest way to do this is take the product of the denominators although if one is a multiple of the other simply multiplying it by the applicable integer works as well. Remember to multiply the numerators by the same number. Generally:

$$\frac{a}{b}+\frac{c}{d} = \frac{ad+cb}{bd}$$

But for example

$$\frac{a}{5}+\frac{b}{10} = \frac{2a+b}{10}$$

Is possible.

Remember that young children will not understand the algebraic formulation here. The art of primary teaching is finding a way to make this make sense to young children. However, teachers need to know these general rules.