Model unit fractions (halves, quarters, eighths)
Theory
Modelling unit fractions is a Year 3 maths skill: cutting a whole into equal parts and naming one part as a unit fraction such as one half \(\left(\dfrac{1}{2}\right)\), one quarter \(\left(\dfrac{1}{4}\right)\) or one eighth \(\left(\dfrac{1}{8}\right)\). This page explains that the bottom number names the number of equal parts, how to show fractions with bar and circle models, how halving makes quarters and eighths, how to name parts that are shaded or not shaded, how to compare unit fractions, and how to find a fraction of a group. It also shows how to join eighths, such as one eighth and six eighths. Curriculum: VC2M3N03 / AC9M3N02.
A fraction shows equal parts of one whole. To make a fraction you cut the whole into equal parts that are all the same size. The bottom number (the denominator) tells how many equal parts there are, and the top number tells how many parts you take.
A unit fraction is just one equal part, so it has 1 on top. Cut a whole into 2 equal parts and take one to get one half, \(\dfrac{1}{2}\). Cut it into 4 equal parts to get one quarter, \(\dfrac{1}{4}\). Cut it into 8 equal parts to get one eighth, \(\dfrac{1}{8}\).
It takes 2 halves, 4 quarters or 8 eighths to rebuild one whole. The more equal parts a whole is cut into, the smaller each part is, so \(\dfrac{1}{2} > \dfrac{1}{4} > \dfrac{1}{8}\).
Halving makes the smaller parts. Cut a half into 2 equal pieces and each piece is a quarter. Cut a quarter into 2 equal pieces and each piece is an eighth.
When more than one part is shaded, the top number counts the shaded parts. If 6 of 8 equal parts are shaded, the shaded fraction is \(\dfrac{6}{8}\) and the part not shaded is \(\dfrac{2}{8}\). Some fractions name the same amount: 2 quarters cover the same part of a strip as 1 half.
Unit fractions do not need formulas, but these facts help you build and compare halves, quarters and eighths. Each one says how many equal parts rebuild one whole.
How to name a unit fraction from a picture
- Check the parts are equal. Look at the whole and make sure every part is the same size. If the parts are not equal, you cannot name a fraction yet.
- Count the equal parts. The number of equal parts is the bottom number: 2 for halves, 4 for quarters, 8 for eighths.
- Count the shaded parts. For a unit fraction just one part is shaded, so the top number is 1. If more parts are shaded, the top number is how many are shaded.
- Find the part not shaded. Take the shaded parts away from all the parts. That number goes on top, with the same bottom number.
How to find a fraction of a group
- Share into equal groups. For one half make 2 equal groups, for one quarter make 4, for one eighth make 8.
- Count one group. The number in one group is the unit fraction of the collection.
- Working back to the whole. If you know one part, multiply by the number of parts. If one quarter is 5, the whole is \(4 \times 5 = 20\).
| \(\text{equal parts}\) | \(=\) | \(8\) |
| \(\text{shaded}\) | \(=\) | \(\dfrac{7}{8}\) |
| \(\text{not shaded}\) | \(=\) | \(8 - 7 = 1\text{ part, so }\dfrac{1}{8}\) |
Seven eighths are shaded and one eighth is not. Together they make \(\dfrac{8}{8}\), one whole.
(i) How many equal parts are there?
(ii) What is each part called?
| \(\text{1st fold}\) | \(\to\) | \(2\text{ parts (halves)}\) |
| \(\text{2nd fold}\) | \(\to\) | \(2 \times 2 = 4\text{ parts (quarters)}\) |
| \(\text{3rd fold}\) | \(\to\) | \(4 \times 2 = 8\text{ parts (eighths)}\) |
(i) There are 8 equal parts. (ii) Each part is one eighth, \(\dfrac{1}{8}\). Every fold cuts each part in half.
(ii) One eighth of a bag of lollies is 2 lollies. How many lollies are in the whole bag?
| \(\text{(i)}\;\; 16 \div 4\) | \(=\) | \(4\text{ in each group}\) |
| \(\dfrac{1}{4}\text{ of }16\) | \(=\) | \(4\) |
| \(\text{(ii)}\;\; 8\text{ eighths} \times 2\) | \(=\) | \(16\) |
(i) One quarter of the tray is 4 muffins. (ii) 8 eighths make the whole bag, so the bag holds 16 lollies.
| \(\dfrac{1}{4}\) | \(:\) | \(4\text{ equal parts}\) |
| \(\dfrac{1}{8}\) | \(:\) | \(8\text{ equal parts}\) |
| \(\dfrac{1}{4}\) | \(>\) | \(\dfrac{1}{8}\) |
Fewer equal parts make a bigger part, so one quarter is larger.
| \(1\text{ eighth and }6\text{ eighths}\) | \(=\) | \(7\text{ eighths}\) |
| \(7\text{ eighths}\) | \(=\) | \(\dfrac{7}{8}\) |
The pieces are all eighths, so count them: they ate \(\dfrac{7}{8}\) of the bar. The bottom number stays 8. One more eighth would complete the whole bar, because \(8\) eighths make one whole.
Common pitfalls
Frequently asked questions
What is a unit fraction?
A unit fraction is one equal part of a whole. It has 1 on top, like \(\dfrac{1}{2}\), \(\dfrac{1}{4}\) or \(\dfrac{1}{8}\). The bottom number tells how many equal parts the whole is cut into.
What does the bottom number of a fraction mean?
The bottom number, called the denominator, tells how many equal parts the whole is cut into. For eighths the whole is cut into 8 equal parts, so the bottom number is 8.
Why do the parts have to be equal?
A fraction names equal shares of a whole. If a whole is cut into 4 parts of different sizes, no part is a quarter. A quarter is 1 of 4 equal parts.
Which is bigger, one quarter or one eighth?
When the wholes are the same size, one quarter is bigger. More equal parts make smaller parts, so \(\dfrac{1}{2} > \dfrac{1}{4} > \dfrac{1}{8}\).
How many halves make one whole?
It takes 2 halves to make one whole, because a whole cut in half has 2 equal parts. In the same way, the number of equal parts always rebuilds one whole: quarters need 4 and eighths need 8.
How do you find one quarter of a group of objects?
Share the objects into 4 equal groups. One quarter is one of those groups. For 20 objects, \(20 \div 4 = 5\), so one quarter of 20 is 5.
Can two different fractions show the same amount?
Yes. If a strip is cut into 8 equal parts and 6 are shaded, the shaded part is \(\dfrac{6}{8}\). Mark the same strip in quarters and the shading covers 3 quarters, so \(\dfrac{6}{8}\) and \(\dfrac{3}{4}\) are the same amount.