Mathematics · Numbers and calculation · Grade 5

Fractions: from pizza to formula

Is 3/8 of a pizza more or less than 5/8? How do you add them?

The situation

You are in a pizzeria with friends. The pizza is cut into 8 equal slices: you took 3 slices and your friend took 2. Who ate more is obvious. But how do we compare 3/8 of one pizza with 1/2 of another, and how do we add them? We will cut slices, count them and discover it ourselves.

Topic

Fractions: equivalent fractions, comparing, adding and multiplying. Pizza and chocolate: a part of a whole, cutting pieces smaller, the common denominator and the area model

Key ideas

Fraction
Equivalent fractions
Common denominator

Formulas

\frac{a}{b} = \frac{a \cdot k}{b \cdot k}

The basic property of a fraction: if we multiply (or divide) the numerator and denominator by the same number, the fraction does not change.

\frac{a}{b} + \frac{c}{d} = \frac{a \cdot d + c \cdot b}{b \cdot d}

Adding: first a common denominator, then the numerators are added (the denominators are not added!).

\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}

Multiplying: numerator times numerator, denominator times denominator — in the area model, the number of squares.

\frac{n}{d} = q \frac{r}{d}, n = q \cdot d + r

Converting an improper fraction to a mixed number: $q$ is the whole pizzas, $r$ is the slices left over.

Theory

A fraction is a part of a whole

We cut the pizza into equal slices. The denominator of a fraction (the number below the line) shows how many slices there are in total, and the numerator (the number above the line) shows how many slices are taken: $\frac{3}{8}$ means three out of eight. If the slices are not equal, we cannot speak of a fraction. The larger the denominator, the smaller each slice: a $\frac{1}{8}$ slice is half the size of a $\frac{1}{4}$ slice.

Equivalent fractions: we keep cutting

Take half a pizza ( $\frac{1}{2}$ ) and cut each slice in two again: we get 2 slices out of 4 ( $\frac{2}{4}$ ), and the amount of pizza has not changed. So $\frac{1}{2} = \frac{2}{4} = \frac{4}{8}$ : on the number line they are all at the same point. This is the basic property of a fraction: multiplying or dividing the numerator and denominator by the same number does not change the fraction. Dividing is called cancelling (simplifying): $\frac{12}{20} = \frac{3}{5}$ .

Comparing: a common denominator

For fractions with the same denominator, the one with the larger numerator is larger. If the denominators differ, the slices are of different sizes, so first we make them the same: we take a number divisible by both denominators (the lowest common multiple) and rewrite each fraction with that denominator. For $\frac{3}{4}$ and $\frac{5}{8}$ the common denominator is 8: $\frac{3}{4} = \frac{6}{8} > \frac{5}{8}$ .

Adding and subtracting

Only identical slices can be counted and added: $\frac{1}{4} + \frac{2}{4} = \frac{3}{4}$ — the denominator stays the same and the numerators are added. If the denominators differ ( $\frac{1}{2} + \frac{1}{3}$ ), we first rewrite the slices with a common denominator: $\frac{3}{6} + \frac{2}{6} = \frac{5}{6}$ . The most common mistake is to add the denominators as well: $\frac{1}{2} + \frac{1}{2} \neq \frac{2}{4}$ , because two halves make a whole pizza.

Mixed numbers and multiplication

If the numerator is larger than the denominator, the fraction is improper: $\frac{7}{4}$ is one whole pizza and another $\frac{3}{4}$ , that is, $1 \frac{3}{4}$ . The whole part is the quotient of $7 : 4$ , and the remainder is the new numerator. Multiplying means “a part of a part”: $\frac{1}{2} \cdot \frac{3}{4}$ is half of three quarters. In the area model the width of a square is $\frac{3}{4}$ and its height is $\frac{1}{2}$ : there are $3 \cdot 1 = 3$ small squares out of $4 \cdot 2 = 8$ in total, which is $\frac{3}{8}$ .

About the model

This experiment deals only with positive fractions and up to 12 slices (more will not fit in the diagram); negative fractions and decimals come in later topics. The lowest common multiple is always used as the common denominator; any other common multiple also gives a correct answer, but there are more slices.

Did you know?

The word “kasr” (fraction) comes from the Arabic verb “kasara”, “to break”; the Latin “fractio” also means “breaking”. A fraction is a “broken” number, that is, a number split into pieces.
Jamshid al-Kashi, who worked at Ulugh Beg's observatory in Samarkand, used decimal fractions systematically in his book “Miftah al-hisab” (1427); in Europe this begins with Simon Stevin (1585).

Inside the experiment

The topic is explained step by step with animation, from simple to advanced: from a real-life picture to the formula and the proof. Then you drag the points yourself to check. Inside are missions, a quiz and a guided tour.

More experiments in this section

Prime numbers, HCF and LCM — Grade 7
The number line: from −7 to √2 — Grade 7
Powers: from folding to a formula — Grade 7
Roots: from squares to powers — Grade 8