Really understanding fractions: beyond the pizza model

Fractions are where many students’ relationship with maths starts to go wrong. A child who is comfortable with whole numbers suddenly struggles to understand why $\frac{1}{8}$ is smaller than $\frac{1}{6}$, why $\frac{1}{2} + \frac{1}{3}$ doesn’t make $\frac{2}{5}$, or why dividing a number by $\frac{1}{2}$ makes the answer bigger. In school, fractions are usually introduced with slices of pizza or cake. That is a good start, but on its own it is not enough. In this article we look at where the pizza model helps, where it gets stuck, and how to really grasp fractions as numbers. It is written for students, and just as much for the parents and teachers who want to help them.

How many different things is a fraction?

What we mean when we write $\frac{3}{4}$ can change with the context. Researchers in maths education have long stressed that the idea of a fraction has several different, related meanings. The most important are:

1. Part–whole: A whole has been divided into 4 equal parts and 3 parts have been taken. This is exactly the pizza model.
2. Measure (number): $\frac{3}{4}$ is a particular point on the number line between 0 and 1. It is a number, just like 2 or 5.
3. Quotient: $\frac{3}{4}$ is also the result of $3 \div 4$. If you share 3 pizzas equally among 4 people, each gets $\frac{3}{4}$ of a pizza.
4. Ratio: If a class has 4 boys for every 3 girls, the ratio of girls to boys is $3:4$.
5. Operator: When we say “$\frac{3}{4}$ of 120”, the fraction acts like an operation that transforms a number: $120 \cdot \frac{3}{4} = 90$.

The pizza model represents only the first meaning strongly. The problem is when a student starts to see a fraction only as “slices of something” and can’t think of it as a real number.

Strengths and weaknesses of the pizza model

The pizza model’s greatest strength is that it makes the idea of equal parts concrete. If you cut a pizza into four slices, one very big and the others small, the big slice is not $\frac{1}{4}$. Fractions are built on “fair sharing”, and slices show this nicely.

But the model gets stuck at several points:

• Fractions greater than one: What is $\frac{7}{4}$? A pizza can’t have 7 quarter slices, so students usually have to draw two pizzas, and the idea of “the whole” gets blurred.
• Comparing different denominators: Comparing $\frac{5}{8}$ and $\frac{3}{5}$ by drawing slices is almost impossible; the drawings look about the same.
• Division: How do you explain “$3 \div \frac{1}{2}$” with pizzas? For most students the model stops working here, and all that is left is the memorised “flip and multiply” rule.
• The idea that fractions are numbers: A pizza does not show that a fraction has a place on the number line.

The number line: turning a fraction into a number

That is why many researchers and teaching guides recommend representing fractions on the number line as well. When you divide the stretch from 0 to 1 into 4 equal parts, the third mark is $\frac{3}{4}$. If you go past 1 and keep counting the same way, you reach $\frac{5}{4}$, $\frac{6}{4}$ and $\frac{7}{4}$. Fractions greater than one are no longer strange at all; they are just numbers to the right of 1.

The number line also makes a powerful idea visible: between any two whole numbers there are infinitely many fractions. Between $\frac{1}{2}$ and $\frac{3}{4}$ there is $\frac{5}{8}$; between $\frac{5}{8}$ and $\frac{3}{4}$ there is $\frac{11}{16}$, and so on for ever. Whole numbers have the idea of “the next number”; fractions don’t. This is an important shift in a student’s thinking.

The three most common mistakes and where they come from

Most mistakes with fractions are not random. Most come from applying rules that are true for whole numbers to fractions. In the research literature this tendency is called whole number bias.

Mistake 1: “8 is bigger than 6, so 1/8 is bigger than 1/6”

As the denominator gets bigger, the pieces get smaller. If you share a pizza among 8 people, you get a smaller slice than if you share it among 6. Thinking it through once with a sharing story lasts much longer than memorising the rule.

Mistake 2: “I add the numerators and the denominators”

Instead of repeating the rule to a student who writes $\frac{1}{2} + \frac{1}{3} = \frac{2}{5}$, ask: “What is $\frac{1}{2} + \frac{1}{2}$?” The same method gives $\frac{2}{4}$, which is $\frac{1}{2}$ again. Adding half a pizza to half a pizza and getting half a pizza makes no sense. This small contradiction gets the student to question their own method.

The logic of the correct method is this: we can only add the same units. We add 3 apples and 2 apples; but “halves” and “thirds” are different units. We need to convert them into a common unit, namely sixths:

Mistake 3: “Multiplying makes bigger, dividing makes smaller”

With whole numbers (numbers greater than 1), multiplying makes the result bigger and dividing makes it smaller. But multiplying by $\frac{1}{2}$ means “taking half” and makes the result smaller, while dividing by $\frac{1}{2}$ makes it bigger. We will make sense of this in the next section.

Let’s think it through with examples

Worked example 1: Which is bigger, 5/8 or 3/5?

The first way is to use a common denominator. A common multiple of 8 and 5 is 40:

So $\frac{5}{8}$ is a little bigger than $\frac{3}{5}$.

The second way is to use a benchmark such as distance from 1. $\frac{5}{8}$ needs $\frac{3}{8}$ more to make 1; $\frac{3}{5}$ needs $\frac{2}{5}$ more. Since $\frac{3}{8} = 0.375$ and $\frac{2}{5} = 0.4$, $\frac{5}{8}$ is closer to 1, so it is bigger.

This benchmark idea sometimes gives an answer very quickly. Let’s compare $\frac{7}{8}$ and $\frac{8}{9}$: the missing piece is $\frac{1}{8}$ in the first and $\frac{1}{9}$ in the second. Since $\frac{1}{9}$ is a smaller gap, $\frac{8}{9}$ is bigger. No calculation needed.

Worked example 2: How many halves are there in 3?

Read $3 \div \frac{1}{2}$ as a question: “How many $\frac{1}{2}$s are there in 3?” Every whole has 2 halves, so 3 wholes have 6 halves:

You can show this on the number line too: how many steps do you take from 0 to 3 in half steps? 6 steps. When division is thought of as “how many fit?”, it becomes obvious why dividing by a small piece gives a big result.

Let’s go one step further: what is $\frac{3}{4} \div \frac{1}{8}$? “How many eighths fit into $\frac{3}{4}$?” Since $\frac{3}{4} = \frac{6}{8}$, the answer is 6. The “flip and multiply” rule gives the same result: $\frac{3}{4} \cdot \frac{8}{1} = \frac{24}{4} = 6$. But now we know why the rule works.

Worked example 3: The area model for multiplying fractions

To picture $\frac{2}{3} \cdot \frac{3}{4}$, draw a rectangle. Divide it into 4 equal columns with vertical lines and shade 3 columns; that is $\frac{3}{4}$. Then divide it into 3 equal rows with horizontal lines and mark 2 rows of the shaded region a second time. There are $3 \cdot 4 = 12$ small boxes in total, and the number of boxes marked twice is $2 \cdot 3 = 6$:

This model shows directly why the rule “multiply the numerators, multiply the denominators” is true: the product of the denominators is the total number of boxes, and the product of the numerators is the number of double-shaded boxes.

Equivalent fractions: different names for the same number

$\frac{1}{2}$, $\frac{2}{4}$, $\frac{3}{6}$ and $\frac{50}{100}$ are the same number. On the number line they all land on exactly the same point. This idea is at the centre of understanding that fractions are numbers. Multiplying the numerator and denominator of a fraction by the same number really means cutting the parts into smaller equal parts:

Taking half a pizza and cutting each slice in two doesn’t change how much pizza you have; it only increases the number of slices. Here the pizza model is useful again; the point is to use it not as the only tool, but as one of many.

Decimals and percentages: other faces of the same idea

For a student who has started to see fractions as numbers, decimals and percentages are not new topics but different ways of writing the same idea. If we read the fraction $\frac{3}{4}$ as a quotient, we get $3 \div 4 = 0.75$. A percentage means “how many parts per hundred”; since $\frac{3}{4} = \frac{75}{100}$, $\frac{3}{4}$ equals 75 per cent. All three point to the same place on the number line.

When this connection isn’t made, students fall into whole number bias with decimals too. For example, many students think $0.25$ is bigger than $0.3$, because “25 is bigger than 3”. But $0.3 = \frac{30}{100}$ and $0.25 = \frac{25}{100}$, so $0.3$ is bigger. Reading decimals as fractions makes this mistake disappear on its own.

Some fractions can’t be written as terminating decimals. $\frac{1}{3} = 0.333\ldots$ repeats for ever. Students often find this surprising, and it starts a good discussion: fractions whose denominators have only 2 and 5 as prime factors give terminating decimals; the others recur. When students work out an observation like this for themselves from a few examples, the link between fractions and decimals stops being a memorised rule and becomes a discovery.

A few tips for parents and teachers

• Bring fractions into everyday language: Recipe measurements, how much of a journey is done, or how much of a match has been played are natural chances to think of fractions as numbers.
• Use the number line early and often: Ask students to place fractions on the line; ask “Is this number bigger or smaller than 1?”
• Meaning before rules: Introduce rules like “flip and multiply” after students have seen what the operation means in a few examples.
• Turn the mistake into a question: Instead of correcting a mistake straight away, have students try their own method on a simple example.
• Start with an estimate: Questions like “Is $\frac{5}{8} + \frac{4}{9}$ bigger than 1?” build number sense before any calculation. (Yes: $\frac{5}{8}$ is $\frac{1}{8}$ more than a half, while $\frac{4}{9}$ is only $\frac{1}{18}$ less than a half. The surplus is bigger than the shortfall, so the total goes past 1.)

A student who really understands fractions stands on much firmer ground later on with ratios, percentages, probability and algebra. For more examples, take a look at our maths articles and the maths topics. In Kuazara’s interactive fraction lessons you can split a whole into parts yourself and place fractions on the number line, so you understand the rules without memorising them.