Fraction Pictures: Pictorial Representations of Fractions

A fraction like \( \frac{3}{4} \) is really just two numbers stacked together, but a picture makes it easy to see what those numbers mean. In a pictorial representation, a whole shape or a whole group of objects is split into equal parts. The bottom number of the fraction, the denominator, tells you how many equal parts the whole was split into. The top number, the numerator, tells you how many of those parts are being shaded, counted, or chosen.

Before drawing fraction pictures, it helps to already know what a fraction is describing. If you need a refresher on that idea first, take a look at Introduction to Fractions.

The most common fraction picture is an area model. You start with one whole shape (often a rectangle, a bar, or a circle), divide it into equal-sized pieces, and shade some of them. The shape below is split into 4 equal parts, and 3 of them are shaded, so it shows \( \frac{3}{4} \).

The same fraction can be drawn with a completely different shape. Here is a circle split into 4 equal sectors, again with 3 sectors shaded. Even though the shape changed, the fraction shown is still \( \frac{3}{4} \), because the whole was still divided into 4 equal-sized pieces.

Two rules must always be followed when you draw an area model: the whole must be split into parts that are equal in size, and the number of equal parts drawn must match the denominator exactly. If the pieces are not equal, the picture does not correctly represent the fraction. For more area, set, and number line pictures side by side, see Fractions - Region, Sets & Linear Models.

A fraction can also describe part of a group instead of part of a single shape. This is called a set model. Instead of splitting one shape into equal regions, you split a group into a certain number of items, and then count how many of those items have a certain property (shaded, a certain color, chosen, and so on).

Notice the pattern is the same as before: count the total number of objects in the set for the denominator, then count the objects with the property you care about for the numerator. Here, 6 circles total gives the denominator, and 4 shaded circles give the numerator, so the picture shows \( \frac{4}{6} \).

Whenever you're given a picture and asked to name the fraction it shows, use the same two-step check every time.

Step 1: Count the total number of equal parts (or total objects in the set). This becomes the denominator.

Step 2: Count how many of those parts are shaded, marked, or picked out. This becomes the numerator.

Example. A rectangle is split into 5 equal columns, and 2 of the columns are shaded. What fraction is shown?

Total equal parts: 5, so the denominator is 5. Shaded parts: 2, so the numerator is 2. The picture represents \( \frac{2}{5} \).

Area models and set models are the two most common fraction pictures, but they aren't the only ones. A fraction can also be shown as a labeled point sitting between two whole numbers on a line, which is a very useful picture for comparing fractions or lining them up in order. That method has its own detailed lesson at Fractions - Number Lines.

Once you're comfortable turning a picture into a fraction and a fraction into a picture, pictures also become a powerful tool for spotting when two differently drawn fractions are actually worth the same amount, or for solving problems written in words rather than numbers.

A circle is divided into 8 equal sectors. Five sectors are shaded blue. What fraction of the circle is shaded?

Total equal sectors: 8 (denominator). Shaded sectors: 5 (numerator). The shaded part of the circle is \( \frac{5}{8} \).

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