Fractions greater than one (mixed numerals)
Amounts bigger than one whole can be written in two ways: as an improper fraction such as \(\frac{8}{5}\) or as the mixed numeral \(1\frac{3}{5}\). In Tasmanian Year 4 maths this subtopic reads, writes and converts these amounts using shapes and number lines (Australian Curriculum AC9M4N04).
Students count the full wholes and the leftover parts, change mixed numerals back into improper fractions, such as \(3\frac{1}{3} = \frac{10}{3}\), find how many thirds or quarters make 2 or 3 wholes, and compare amounts greater than one. Pizza slices, recipe scoops and jugs of juice show why it matters. It is part of Year 4 Fractions.
Theory
A fraction greater than one can be written as an improper fraction like \(\dfrac{4}{3}\) or as a mixed numeral like \(1\dfrac{1}{3}\). This Year 4 maths page shows how to read them from shapes and number lines, how to change an improper fraction into a mixed numeral and back again, and how to compare amounts greater than one.
An improper fraction has a top number (the numerator) that is the same as or larger than its bottom number (the denominator). It is worth one whole or more, like \(\dfrac{4}{3}\).
A mixed numeral names the same amount as a whole number and a fraction together, like \(1\dfrac{1}{3}\). The whole number counts the full groups and the fraction counts the leftover parts.
The two forms are equal: \(\dfrac{4}{3}=1\dfrac{1}{3}\). Remember that one whole is a fraction with the same top and bottom, so \(\dfrac{4}{4}=1\), \(\dfrac{3}{3}=1\) and \(\dfrac{5}{5}=1\).
These facts help you swap between improper fractions and mixed numerals.
How to change an improper fraction to a mixed numeral
- Find one whole. The denominator tells how many parts make one whole, so \(4\) quarters make a whole.
- Take out the wholes. Count how many full groups of that many parts you have.
- Write the leftover. Put the whole number and the leftover fraction side by side.
To go back the other way, change each whole into parts and add them on: \(2\dfrac{1}{5}=\dfrac{10}{5}+\dfrac{1}{5}=\dfrac{11}{5}\).
How to compare fractions greater than one
- Same bottom number? Compare the top numbers: more parts is the larger amount.
- Use the wholes. Find how many parts make each whole number. In sixths, \(1=\dfrac{6}{6}\), \(2=\dfrac{12}{6}\), \(3=\dfrac{18}{6}\). A fraction is more than that whole number when its top number is bigger.
- Parts still needed. To reach the next whole number, take the parts you have away from the parts in that whole.
| \(\text{one whole}\) | \(=\) | \(\dfrac{5}{5}\) |
| \(\dfrac{5}{5}+\dfrac{3}{5}\) | \(=\) | \(\dfrac{8}{5}\) |
The total is \(\dfrac{8}{5}\).
| \(\dfrac{11}{3}\) | \(=\) | \(\dfrac{9}{3}+\dfrac{2}{3}\) |
| \(\dfrac{9}{3}\) | \(=\) | \(3\) |
| \(\dfrac{11}{3}\) | \(=\) | \(3\dfrac{2}{3}\) |
Three wholes and two thirds: \(3\dfrac{2}{3}\).
| \(2\) | \(=\) | \(\dfrac{10}{5}\) |
| \(2\dfrac{1}{5}\) | \(=\) | \(\dfrac{10}{5}+\dfrac{1}{5}\) |
| \(2\dfrac{1}{5}\) | \(=\) | \(\dfrac{11}{5}\) |
The improper form is \(\dfrac{11}{5}\).
| \(7\text{ halves}\) | \(=\) | \(\dfrac{7}{2}\) |
| \(\dfrac{7}{2}\) | \(=\) | \(\dfrac{6}{2}+\dfrac{1}{2}\) |
| \(\dfrac{7}{2}\) | \(=\) | \(3\dfrac{1}{2}\) |
She has \(3\dfrac{1}{2}\) cups of jelly.
| \(3\text{ wholes}\) | \(=\) | \(3 \times 5 = 15\text{ fifths}\) |
| \(14 < 15,\ \ 15 = 15\) | \(,\) | \(17 > 15\) |
Only \(\dfrac{17}{5}\) is more than 3 wholes. \(\dfrac{15}{5}\) is exactly 3 and \(\dfrac{14}{5}\) is less.
| \(2\dfrac{3}{5}\) | \(=\) | \(\dfrac{10}{5}+\dfrac{3}{5}=\dfrac{13}{5}\) |
| \(3\) | \(=\) | \(\dfrac{15}{5}\) |
| \(15 - 13\) | \(=\) | \(2\) |
He needs 2 more fifths.
Common pitfalls
Frequently asked questions
What is an improper fraction?
An improper fraction has a top number the same as or bigger than the bottom number, so it is worth one whole or more. For example, \(\dfrac{13}{4}\) is thirteen quarters, which is more than three wholes.
What is a mixed numeral?
A mixed numeral is a whole number and a fraction written together, like \(3\dfrac{1}{4}\). It shows the same amount as an improper fraction.
How do you change an improper fraction to a mixed numeral?
Find how many whole groups you can make, then write the leftover as a fraction. For \(\dfrac{13}{4}\), twelve quarters make three wholes and one quarter is left over, so \(\dfrac{13}{4}=3\dfrac{1}{4}\).
How do you change a mixed numeral to an improper fraction?
Change each whole into parts and add on the extra parts. For \(2\dfrac{1}{5}\), two wholes are \(\dfrac{10}{5}\), plus \(\dfrac{1}{5}\) is \(\dfrac{11}{5}\).
Is 13/4 bigger than 3 1/4?
No. They are the same amount written two different ways. \(\dfrac{13}{4}=3\dfrac{1}{4}\).
How many sixths make 2 wholes?
Six sixths make one whole, so two wholes need twelve sixths. That is \(\dfrac{12}{6}\).