Equivalent fractions (as lengths)
Theory
Equivalent fractions as lengths is a Year 4 maths skill: seeing that fractions like \(\dfrac{1}{2}\), \(\dfrac{2}{4}\) and \(\dfrac{4}{8}\) name the same length. This page uses fraction strips, a fraction wall and number lines to line up equal lengths, and shows how to rename fractions, write them in simplest form, and compare and order them. Curriculum: VC2M4N03 / AC9M4N03.
Equivalent fractions are fractions that name the same amount. When you draw them on strips of the same length, their shaded parts reach the same place. So \(\dfrac{1}{2}\), \(\dfrac{2}{4}\) and \(\dfrac{4}{8}\) are all the same length.
The top number, the numerator, counts the shaded parts. The bottom number, the denominator, counts all the equal parts. To make an equivalent fraction, split every part into the same number of smaller parts. Splitting each part in two turns \(\dfrac{1}{2}\) into \(\dfrac{2}{4}\), then \(\dfrac{4}{8}\).
A bigger denominator does not mean a bigger fraction. \(\dfrac{4}{8}\) has more pieces than \(\dfrac{1}{2}\), but each piece is smaller, so the total length is the same. Renaming fractions this way lets you compare and order them.
You can also go the other way and join small parts into bigger ones. A fraction is in its simplest form when it uses the fewest, biggest parts possible. For example \(\dfrac{2}{8}\) in simplest form is \(\dfrac{1}{4}\). When the top and bottom numbers are the same, the fraction is one whole, like \(\dfrac{4}{4} = 1\).
Fractions only compare fairly when they come from the same-sized whole. Half of a long ribbon is longer than half of a short ribbon.
These equivalent-fraction facts come straight from lining up equal lengths. Learn them and you can rename fractions quickly.
How to find an equivalent fraction as a length
- Draw equal strips. Make both strips the same length so you can compare the shaded parts fairly.
- Split the parts. Cut each equal part into the same number of smaller parts. This changes the denominator.
- Count the shaded parts. The shaded length does not move, so count the new shaded parts to read the new numerator.
- Check they line up. If both shaded lengths reach the same place, the fractions are equivalent.
How to write a fraction in simplest form
- Look for equal groups. Find a number that shares both the top and the bottom equally, such as 2 or 5.
- Divide both. Divide the top and bottom by that number. The length does not change.
- Repeat if you can. Stop when no number bigger than 1 shares both equally.
How to compare using lengths
- Rename to the same parts. Change the fractions so they have the same bottom number, then compare the top numbers.
- Or line up equal strips. The shaded part that reaches further is the bigger fraction.
| \(\dfrac{1}{4}\) | \(=\) | \(\dfrac{1 \times 2}{4 \times 2}\) |
| \(\dfrac{1}{4}\) | \(=\) | \(\dfrac{2}{8}\) |
Two eighths reach the same place, so the answer is \(2\).
| \(2\text{ eighths}\) | \(=\) | \(1\text{ quarter}\) |
| \(\dfrac{2}{8}\) | \(=\) | \(\dfrac{2 \div 2}{8 \div 2} = \dfrac{1}{4}\) |
Two eighths reach the first quarter mark, so the point is also named by \(\dfrac{1}{4}\).
| \(\dfrac{2}{4}\) | \(=\) | \(\dfrac{2 \times 2}{4 \times 2}\) |
| \(\dfrac{2}{4}\) | \(=\) | \(\dfrac{4}{8}\) |
| \(\dfrac{4}{8}\) | \(\neq\) | \(\dfrac{3}{8}\) |
Splitting each quarter in two gives \(\dfrac{4}{8}\), not \(\dfrac{3}{8}\). So no: \(\dfrac{3}{8}\) is one eighth shorter.
| \(\dfrac{1}{4}\) | \(=\) | \(\dfrac{2}{8}\) |
| \(\dfrac{1}{2}\) | \(=\) | \(\dfrac{4}{8}\) |
| \(\text{eighths}\) | \(:\) | \(2 < 3 < 4\) |
In order: \(\dfrac{1}{4},\ \dfrac{3}{8},\ \dfrac{1}{2}\).
| \(\text{shaded}\) | \(=\) | \(\dfrac{5}{10}\) |
| \(\dfrac{5}{10}\) | \(=\) | \(\dfrac{5 \div 5}{10 \div 5}\) |
| \(\dfrac{5}{10}\) | \(=\) | \(\dfrac{1}{2}\) |
Joining the tenths into groups of 5 makes 2 equal parts with 1 shaded, so the simplest form is \(\dfrac{1}{2}\).
Common pitfalls
Frequently asked questions
What are equivalent fractions?
Equivalent fractions are fractions that name the same amount. On equal-length strips their shaded parts reach the same place, so \(\dfrac{1}{4}\) and \(\dfrac{2}{8}\) are the same length.
How do you show that 1/4 equals 2/8?
Line up two strips of the same length. Split one into quarters and shade one part. Split the other into eighths and shade two parts. The two shaded lengths reach the same place, so \(\dfrac{1}{4} = \dfrac{2}{8}\).
How do you make an equivalent fraction?
Split every equal part into the same number of smaller parts. This multiplies the top and bottom by the same amount, so \(\dfrac{1}{4}\) becomes \(\dfrac{2}{8}\), and the length stays the same.
Is 5/10 bigger than 1/2?
No. \(\dfrac{5}{10}\) has more pieces than \(\dfrac{1}{2}\), but each piece is smaller, so the total length is the same. \(\dfrac{5}{10} = \dfrac{1}{2}\).
What does simplest form mean?
A fraction is in simplest form when it uses the fewest, biggest equal parts. To get there, divide the top and bottom by the same number. For example \(\dfrac{2}{8} = \dfrac{1}{4}\).
How do you compare fractions with different denominators?
Rename them so they share the same denominator, then compare the top numbers. For example \(\dfrac{1}{4} = \dfrac{2}{8}\) and \(\dfrac{1}{2} = \dfrac{4}{8}\), so \(\dfrac{1}{4}\) is smaller than \(\dfrac{1}{2}\).
Do equivalent fractions point to the same place on a number line?
Yes. \(\dfrac{1}{2}\) and \(\dfrac{5}{10}\) point to the same spot, halfway between \(0\) and \(1\), because they name the same length.