Mixed numbers and improper fractions: what they mean and how to convert between them, with pictures
A mixed number is a whole number and a proper fraction written together: 2 1/3 means 2 whole units plus 1/3, and the curriculum asks students to understand a mixed number as the sum of a whole number and a proper fraction. An improper fraction has a numerator no smaller than its denominator: 7/3 means 7 thirds. They are two ways of writing the same number. To turn an improper fraction into a mixed number, divide the numerator by the denominator: the quotient is the whole number and the remainder is the new numerator. To turn a mixed number into an improper fraction, multiply the whole number by the denominator and add the numerator; the denominator stays the same. This is part of the Primary 4 unit "Fractions (II)", and in Primary 5 mixed numbers must be changed into improper fractions before multiplying or dividing. Answers may be given as a mixed number or an improper fraction in its simplest form.
- A mixed number and an improper fraction are two ways of writing the same number; a mixed number is the sum of a whole number and a proper fraction.
- Improper to mixed: divide the numerator by the denominator; the quotient is the whole number and the remainder is the numerator.
- Mixed to improper: multiply the whole number by the denominator and add the numerator; keep the denominator.
- For adding and subtracting, the whole and fraction parts can be worked separately; for multiplying and dividing, change mixed numbers into improper fractions first.
- Answers may be mixed numbers or improper fractions, but always in simplest form.
- Turning the answer back into the other form is a quick check against careless mistakes.
- Primary 4Year for converting mixed numbers and improper fractions (unit 4N6)Source: 小學數學科學習內容（數學教育學習領域課程指引補充文件，2017）
- 9 hoursSuggested teaching time for "Fractions (II)"Source: 小學數學科學習內容（數學教育學習領域課程指引補充文件，2017）
- 1Most mixed numbers allowed when multiplying or dividing three fractionsSource: 小學數學科學習內容（數學教育學習領域課程指引補充文件，2017）
Where this sits in the primary curriculum
Mixed numbers and improper fractions are taught in Primary 4. In the Education Bureau primary maths curriculum, learning unit 4N6 "Fractions (II)" has a suggested teaching time of 9 hours, and its first two learning objectives are recognising the concepts of proper fractions, improper fractions and mixed numbers, and converting between improper fractions and mixed numbers; the conversions include whole numbers such as 2 and 3. The same unit also covers expanding and reducing fractions and adding and subtracting fractions with the same denominator.
In Primary 3, "Fractions (I)" first treats a fraction as part of one whole or of a set of objects; in Primary 5, "Fractions (III)" to "Fractions (V)" go on to unlike denominators, multiplication and division. The curriculum notes in each of these units that results may be given as mixed numbers or improper fractions in their simplest form. So converting is not a one-lesson topic: every later fraction calculation depends on it.
| Year | Learning unit | What it covers about mixed numbers |
|---|---|---|
| Primary 3 | 3N5 Fractions (I) | Fractions as parts of a whole, equivalent fractions, sums and differences not greater than 1 |
| Primary 4 | 4N6 Fractions (II) | Proper, improper and mixed fractions and converting between them, simplest form |
| Primary 5 | 5N2 Fractions (III) | Adding and subtracting unlike fractions; results as mixed numbers or improper fractions |
| Primary 5 | 5N3 Fractions (IV) | Multiplying fractions; at most one mixed number when three fractions are multiplied |
| Primary 5 | 5N5 Fractions (V) | Dividing fractions and mixed operations; at most one mixed number among three fractions |
What each one means: "how many wholes" versus "how many parts"
A fraction can be read as "split one whole into equal parts and take some of them". A fraction whose numerator is smaller than its denominator, such as 3/4, is a proper fraction and is less than 1. Once the parts taken reach or pass one whole, there are two ways to write the number.
A mixed number writes the complete wholes and the leftover part separately: 2 3/4 means 2 whole units plus 3 of 4 equal parts. The curriculum asks students to understand that a mixed number can be seen as the sum of a whole number and a proper fraction, so 2 3/4 = 2 + 3/4.
An improper fraction puts every part into the numerator at once: 11/4 means each whole is cut into 4 parts and 11 parts are taken. A numerator no smaller than the denominator shows the number is at least 1; when the numerator equals the denominator, as in 4/4, the number is exactly 1.
Parents can remind children that a mixed number and an improper fraction are not two different numbers but two ways of writing the same number, just as $1.50 and 15 ten-cent coins are worth the same.
Seeing it in pictures: pie charts and a number line
Drawing is the best way in. For 2 1/3, draw three circles of the same size: shade the first two completely, split the third into 3 equal parts and shade 1 of them. Children can see at a glance that it is "a little more than two".
The improper fraction 7/3 uses the same picture: split each of the three circles into 3 parts, 9 parts in all, and shade 7 of them, which is two whole circles and 1 more part. That shows 7/3 equals 2 1/3.
A number line works just as well. Mark 0, 1, 2 and 3, and split each gap into 3 equal steps; 7/3 is 7 steps from 0, landing on the 1st step after 2.
The point of a picture is not neatness. Shading parts and counting steps helps children absorb the idea that the denominator says how many parts each whole is split into and the numerator says how many parts are taken.
Improper fraction to mixed number: divide, read, write
Turning an improper fraction into a mixed number answers the question "how many whole units are inside, and how much is left?" First, divide the numerator by the denominator; second, the quotient is the whole-number part; third, the remainder becomes the numerator and the denominator stays the same.
Example: 17/5. 17 divided by 5 is 3 remainder 2, so 17/5 = 3 2/5. Check: 3 wholes of 5 parts each make 15 parts, and the 2 parts left over make 17 parts exactly.
Another example: 8/4. 8 divided by 4 is 2 remainder 0, so the answer is 2. When the remainder is 0 the result is a whole number and needs no fraction part. When the numerator equals the denominator, as in 6/6, the quotient is 1 remainder 0, so the answer is 1. These "divides exactly" cases deserve their own practice, because children often add an extra fraction part.
Whole numbers can also be written as improper fractions. The conversions in the curriculum include whole numbers such as 2 and 3: for example 2 = 6/3 and 3 = 12/4. This is handy when adding or subtracting fractions with the same denominator.
Mixed number to improper fraction: convert, then add
Going the other way, the key is to turn the whole-number part into parts first. Take 4 3/5: each whole is 5 parts, so 4 wholes are 4 × 5 = 20 parts; add the 3 parts already there to get 23 parts, so the answer is 23/5.
This sums up as one rule: multiply the whole number by the denominator, add the numerator, and keep the denominator. Behind the rule is the "convert, then add" process above; once children see where it comes from, they rarely get it wrong.
One more: 1 7/8. 1 × 8 + 7 = 15, so the answer is 15/8. A mixed number close to 2 gives a numerator close to twice the denominator, which is a handy self-check.
At first, ask children to check each answer with a picture or an estimate: the result should be no smaller than the whole-number part and smaller than the whole-number part plus 1.
When do you need to convert in a calculation?
When adding or subtracting, mixed numbers need not be converted: work out the whole numbers and the fractions separately, then put them together. For 1 1/3 + 2 1/2, the whole numbers give 1 + 2 = 3; for the fractions, use a common denominator: 2/6 + 3/6 = 5/6, so the answer is 3 5/6. If the fraction parts add up to more than 1, carry the extra whole.
When multiplying or dividing, change mixed numbers into improper fractions first. For 1 1/2 × 2/3, change 1 1/2 into 3/2, then 3/2 × 2/3 = 6/6, which is 1. A common mistake is to multiply the whole-number part and the fraction part separately, which leaves out the cross terms and gives a wrong answer.
Division works the same way: for 2 1/4 ÷ 3/4, change 2 1/4 into 9/4, then multiply by the reciprocal of the divisor: 9/4 × 4/3 = 36/12, which is 3. The curriculum also keeps questions manageable: multiplying three fractions should not involve more than one mixed number, and the same holds for division and mixed multiplication and division.
After calculating, the answer may stay as an improper fraction or go back to a mixed number, but either way it must be in its simplest form. If the question or teacher asks for one form, use that form.
- PrimaryMixed numbers and improper fractions213 = 2 × 3 + 13 = 73Whole number times denominator, add the numerator, keep the denominator
- PrimaryAdding unlike fractions12 + 13 = 36 + 26 = 56Make the denominators the same, then add the numerators
- PrimaryMultiplying fractions23 × 34 = 612 = 12Numerator times numerator, denominator times denominator, then simplify
- PrimaryDividing fractions34 ÷ 25 = 34 × 52 = 158Dividing by a fraction is multiplying by its reciprocal
- Primary 5 maths: Fractions (III), unlike denominators
- Primary 5 maths: Fractions (IV), multiplying fractions
- Primary 5 maths: Fractions (V), dividing fractions
Common mistakes and how to check
The most common mistake when turning a mixed number into an improper fraction is to multiply without adding, for example writing 2 1/3 as 6/3. The second half of the rule, "add the numerator", is the reminder; or ask the child to count again using a picture.
The second kind of mistake is swapping the quotient and the remainder, for example writing 17/5 as 2 3/5. Compare sizes to check: 17/5 is greater than 3, but 2 3/5 is less than 3, so it cannot be right.
The third kind is not simplifying. The curriculum asks students to understand the concept of the simplest form of a fraction, so after writing an answer, check whether the numerator and denominator share a common factor: 14/6 should be reduced to 7/3, which is 2 1/3. Simplifying and converting are separate steps and are best practised separately.
Parents can help children build the habit of checking back: turn the answer into the other form and see whether it returns to the original number. It takes a few seconds and catches most careless mistakes.
Practice: three short, effective methods
First, picture to number: a parent draws shaded pie charts and the child writes both the mixed number and the improper fraction. Writing two forms for the same picture best builds the idea of one number, two ways of writing it.
Second, set questions for each other: the child writes an improper fraction and the parent turns it into a mixed number, then swap. Setting a question takes more understanding than answering one, and children check their own steps to get it right.
Third, quick estimates: looking only at the question, say which two whole numbers the answer lies between. For 20/7, think: 14/7 is 2 and 21/7 is 3, so the answer is between 2 and 3; then work it out: 2 6/7.
Short, frequent practice works best, checking answers straight away and giving reasons, rather than doing many questions in one go. Fractions are the base for decimals, percentages and algebraic fractions later, so it is worth getting this step right.
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When extra support helps
Once children understand the pictures and the rule, most can convert reliably with textbook practice. Extra support is worth considering only if a child stays confused about fractions, for example about the roles of the numerator and the denominator.
Common kinds of support in Hong Kong include tutoring centres, private tutors, home tutors and online tutoring. Each has trade-offs; choose by the child's personality, weak areas and the family's timetable.
Whichever you choose, judge it by whether the child can explain why the steps work, not by how many pages are finished. Drilling without understanding means a new number brings the same mistake back.
Frequently asked questions
Should answers be mixed numbers or improper fractions?
The curriculum notes that results may be given as mixed numbers or improper fractions in their simplest form, so both are correct. If the question or teacher asks for one form, use that. One form is enough for an answer: write 7/3 or 2 1/3, not both.
Why multiply and then add when turning a mixed number into an improper fraction?
Because the whole-number part has to become parts first. In 3 1/4, each whole is 4 parts, so 3 wholes are 12 parts; adding the 1 part already there gives 13 parts, so it is 13/4. Multiplying converts the wholes; adding brings in the leftover part.
Which year are mixed numbers taught in?
Primary 4. Learning unit 4N6 "Fractions (II)" in the Education Bureau curriculum covers proper fractions, improper fractions and mixed numbers and converting between improper fractions and mixed numbers, with a suggested teaching time of 9 hours. Multiplying and dividing fractions in Primary 5 uses the same skill.
My child calculates correctly but often forgets to simplify. What can I do?
Make simplifying the fixed last step of every answer, and in practice ask the child to say "is there a common factor?" out loud after each answer. Once the habit forms, simplifying becomes part of the routine without reminders.