How I wish I'd always taught... Fractions.

Fractions can be a tricky concept for children to learn and adults to teach effectively.

Fractions are a part of a whole but before we move on, this concept of a 'part' and a 'whole' can be the first problem when introducing fractions. For example, the whole in any given context may change and so might the part. If the world is the whole, then the continent of Europe is part of that whole. If Europe is the whole, then the United Kingdom is part of the whole. If the United Kingdom is the whole, then England is part of the whole. The concept of different wholes and parts can be tricky and requires a lot of attention and conceptual variation to demonstrate it. Even with the same whole, half may look very different.

Parts and whole.

Concept of one half.

Same whole, different part (half)

Types of fractions

Just as natural numbers have names, so do fractions. Even the parts of fractions are referred to by particular names and later categorised. Fractions are inextricably linked to division. Even the symbols are linked.

Numerator: This referrs to how many equal parts there are of a whole.

Denimonator: This referrs to how many equal parts a whole (or set) is divided into.

Division bar: The line which divides the numerator by the denominator.

The different types of fractions are below:

Proper Fraction: a fraction where the numerator is less than the denominator.	Common fraction: Has a numerator and denominator which are both integers.
Improper fraction: a fraction where the numerator is greater than the denominator (the fraction is greater than a whole).	Complex fraction: Has a fraction as either or both the numerator and denominator.
Mixed number: A whole number alongside a fraction.	Equivalent fraction: Fractions which are equal.
Reduced fraction: a fraction written in its lowest terms; the numerator and denominator are divided by their highest common factor.	Unit fraction: Fractions with a numerator of 1.

Fractions as operators (addition and subtraction)

When we start to teach adding and subtracting fractions, it seems intuitive that 1/4 + 1/4 = 2/4. We may represent this type of question with a bar. It is easy to see how this progresses to addition of proper fractions (non-unit fractions) such as 1/5 + 2/5 = 3/5.

The next phase of adding (and subtracting) fractions requires an understanding of equivalents and common factors. For example: 1/3 + 1/6 = 3/6. This may then be simplified to one half.

When adding (and subtracting) with different denominators. In this example, 1/2 + 1/5, we must find a common multiple of 2 and 5 to become our denominator and then make each fraction equivalent to the common multiple of 2 and 5. Therefore, we must add tenths. Strong conceptual understanding of equivalents is needed to do this.

Multiplication and fractions.

When multiplying fractions, we may read the above example as 'one-half of one-quarter equals one-eighth.' Here, we can see one-half of one-quarter.

Using the area model can be useful to teach the concept of multiplying fractions. Here. we shade in 1/4, then 1/2 and the answer is the parts shaded in both colours out of the whole. Again, if you look closely, you'll see that you have shaded half of a quarter of the shape (1/8).

Fractions and division.

Dividing a fraction by a whole number works in much the same way as multiplication/ For example, 1/4 ÷ 2 = 1/8. As 1/2 is the same as dividing by 2, we can see how 1/2 x 1/4 and 1/4 ÷ 2 both equal 1/8. Here, language matters. On the left, we can see the algorithm which shows '1/2 of 1/4 = 1/8' and on the right we say '1/4 divided by 2 = 1/8'.

Fractions, decimals and percentages.

We may divide a numerator by a denominator to calculate its decimal value.

We may illustrate this using a number line. In more recent times, children may be more familiar with percentages as an amount out of 100 due to looking at phone and tablet batteries. We may teach children that percent means 'per 100'.

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