Model thirds & fifths
Theory
Modelling thirds and fifths is a Year 3 maths skill: splitting one whole into 3 equal parts to make thirds or 5 equal parts to make fifths, then naming the fraction shaded. This page explains how the denominator counts the equal parts, how to find a fraction of a group, why one third is bigger than one fifth, and how cutting every part in half turns thirds into sixths and fifths into tenths. It also covers ordering unit fractions and joining thirds or fifths. Curriculum: VC2M3N03 / AC9M3N02.
A fraction shows equal parts of one whole. To make a fraction you split the whole into parts that are all the same size, then count how many parts you have.
Thirds are made by splitting a whole into 3 equal parts. One part is one third, written \(\dfrac{1}{3}\). Fifths are made by splitting a whole into 5 equal parts. One part is one fifth, written \(\dfrac{1}{5}\).
Every fraction has two numbers. The bottom number, the denominator, names how many equal parts the whole is cut into. The top number, the numerator, tells how many of those parts you have. In \(\dfrac{2}{3}\) the 3 means 3 equal parts and the 2 means 2 of them are counted.
If you cut every part in half, there are twice as many parts and each one is smaller. Thirds become sixths (6 equal parts) and fifths become tenths (10 equal parts). Each third is now 2 sixths, and each fifth is now 2 tenths.
Thirds and fifths do not need formulas, but these facts help you build and check them.
How to name a fraction of a shape
- Count the equal parts. Check that the whole is split into parts that are all the same size, then count them. This number is the denominator.
- Count the shaded parts. Count how many of those equal parts are shaded or chosen. This number is the numerator.
- Write the fraction. Put the shaded count on top and the total equal parts on the bottom, like \(\dfrac{2}{5}\).
- Not shaded? Take the shaded parts away from all the parts. In fifths, if 1 part is shaded then \(5 - 1 = 4\) parts are not, so \(\dfrac{4}{5}\) is not shaded.
How to find a fraction of a group
- Share into equal groups. Make as many equal groups as the bottom number: 3 groups for thirds, 5 groups for fifths.
- Count one group. That is the unit fraction, such as \(\dfrac{1}{3}\) of the collection.
- Take the top number of groups. For \(\dfrac{2}{3}\) take 2 groups; for \(\dfrac{4}{5}\) take 4 groups.
How to cut every part in half
- Double the number of parts. 3 parts become 6 (sixths) and 5 parts become 10 (tenths).
- Double the shaded parts too. Each shaded part becomes 2 smaller shaded parts, so the shaded amount stays the same.
| \(\text{equal parts}\) | \(=\) | \(5\text{ (fifths)}\) |
| \(\text{shaded parts}\) | \(=\) | \(4\) |
| \(\text{fraction}\) | \(=\) | \(\dfrac{4}{5}\) |
Four of the five equal parts are shaded, so \(\dfrac{4}{5}\).
| \(6 \div 3\) | \(=\) | \(2\) |
| \(\dfrac{1}{3}\text{ of }6\) | \(=\) | \(2\) |
| \(\dfrac{2}{3}\text{ of }6\) | \(=\) | \(2 \times 2 = 4\) |
Two of the three equal groups hold 4 counters, so \(\dfrac{2}{3}\) of 6 is 4.
| \(\text{top strip}\) | \(=\) | \(2\text{ of }3\text{ parts}\) |
| \(\text{bottom strip}\) | \(=\) | \(2\text{ of }5\text{ parts}\) |
| \(\dfrac{2}{3}\) | \(>\) | \(\dfrac{2}{5}\) |
Both strips have 2 parts shaded, but a third is bigger than a fifth, so two thirds is bigger.
| \(\text{parts in all}\) | \(=\) | \(3\) |
| \(\text{not shaded}\) | \(=\) | \(3 - 1 = 2\) |
| \(\text{fraction}\) | \(=\) | \(\dfrac{2}{3}\) |
Two of the three equal parts are not shaded, so \(\dfrac{2}{3}\).
(i) How many tenths are shaded now?
(ii) Write the shaded part as a fraction in fifths and in tenths.
| \(\text{parts}\) | \(=\) | \(5 \times 2 = 10\text{ (tenths)}\) |
| \(\text{shaded}\) | \(=\) | \(2 \times 2 = 4\text{ tenths}\) |
| \(\dfrac{2}{5}\) | \(=\) | \(\dfrac{4}{10}\) |
(i) 4 tenths are shaded, because each shaded fifth became 2 tenths. (ii) \(\dfrac{2}{5}=\dfrac{4}{10}\): the shaded length has not changed.
| \(\text{more parts}\) | \(\to\) | \(\text{smaller parts}\) |
| \(5 > 3 > 2\text{ parts}\) | \(\to\) | \(\dfrac{1}{5} < \dfrac{1}{3} < \dfrac{1}{2}\) |
From smallest to largest: \(\dfrac{1}{5}\), \(\dfrac{1}{3}\), \(\dfrac{1}{2}\).
| \(2\text{ fifths and }2\text{ fifths}\) | \(=\) | \(4\text{ fifths}\) |
| \(4\text{ fifths}\) | \(=\) | \(\dfrac{4}{5}\) |
The parts are all fifths, so count them. She has painted \(\dfrac{4}{5}\) of the fence. One more fifth would finish it, because \(5\) fifths make one whole.
Common pitfalls
Frequently asked questions
What is a third?
A third is one part of a whole that is split into 3 equal parts. It is written \(\dfrac{1}{3}\). Three thirds make one whole.
What is a fifth?
A fifth is one part of a whole that is split into 5 equal parts. It is written \(\dfrac{1}{5}\). Five fifths make one whole.
What does the bottom number of a fraction tell you?
The bottom number, called the denominator, tells you how many equal parts the whole is cut into. In \(\dfrac{1}{5}\) the 5 means 5 equal parts.
Which is bigger, two thirds or two fifths?
Two thirds is bigger. Both have 2 parts, but when the same whole is cut into fewer parts each part is bigger, and a third is bigger than a fifth. So \(\dfrac{2}{3} > \dfrac{2}{5}\).
How do you find a fifth of a group of objects?
Share the group into 5 equal parts. One of those equal parts is one fifth. For example, \(\dfrac{1}{5}\) of 25 counters is 5, because 25 shared into 5 equal groups gives 5 in each group.
Does cutting every part in half change the shaded amount?
No. Each shaded part becomes 2 smaller shaded parts, so there are twice as many parts and twice as many shaded parts, but the shaded length stays the same. For example, \(\dfrac{4}{5}\) is the same as \(\dfrac{8}{10}\).
Do the parts of a fraction have to be equal?
Yes. Thirds and fifths only work when all the parts are the same size. Three parts of different sizes do not show thirds.