Models for Fractions
Models for teaching about fractions can be classified as shown in the table below. Using a linear model is very important for several reasons:
- It helps in building the concept of fraction as a number.
- A linear model, where the size of a fraction is modelled by the length of a line, is the beginning of the number line. Students will need to use both horizontal and vertical number lines when they progress in mathematics to learning about Cartesian graphs.
- A linear model is the best model for highlighting the concept of 'number density', that is, between any two numbers, there are many other numbers. If we are using only whole numbers, there is no number between 3 and 4, but once we allow fractions and decimals, then there are an infinite number of possibilities.
- Many types of numbers can be represented with a linear model or shown on the number line: whole numbers, fractions and decimals. This means they are all examples of real numbers.
| Models | Fraction size represented by: | Examples | Comments |
|
3 dimensional (volume model) |
relative volume |
objects (eg apples cut into pieces, drink containers fully or partially filled) |
Advantage is that the 'whole' is obvious, and it is very compact. This is also a disadvantage as amount of material inside is hidden from view. Not all young students have a good idea of volume and so may be observing something else (e.g. surface area). |
|
2 dimensional (area model) |
relative area |
Circles, rectangles, squares, 'pizzas'. |
Whole can be made obvious. Medium compactness. Often hard to compare amounts. Not all young students have a good idea of area and so may be observing something else (eg perimeter). |
|
1 dimensional (linear model) |
relative length |
fraction walls, fraction strips, number lines |
A linear model gives a good feel for relative size of the numbers. Least compact, so larger numbers can be very long on the page. Need to be clear about what length is the 'whole'. |
|
discrete (subset/set model) |
relative numbers
|
counters, children |
Need to be clear about what set of objects is to be regarded as the 'whole' and then identify fractions of this whole. This can reinforce the idea that fractions are two numbers rather than one (2 of the 5 counters are black). |
| number line | position of a point on a line | number line |
A linear model (e.g. a fraction strip) represents numbers by length. A number line is made by representing each number by a point at that distance (length) from the origin. The number line is a more sophisticated concept than a linear model, which is an important pre-requisite. |