How this unit works
It's organised by year group, so you can start where your child or class actually is rather than where they're supposed to be. Each section has teaching, worked examples and practice — the answers, common mistakes and teaching notes are at the end.
| Year | What's covered |
|---|---|
| Year 3 | Naming fractions, unit fractions, tenths, equivalent fractions |
| Year 4 | Hundredths, decimal equivalents, dividing by 10 and 100 |
| Year 5 | Improper fractions and mixed numbers, adding and subtracting, percentages introduced |
| Year 6 | All four operations, FDP equivalence, percentages of amounts |
Year 3 What a fraction is
A fraction is a way of describing part of a whole. When we write ¼, the two numbers each have a job: the top number (1) is the numerator — how many parts we have; the bottom number (4) is the denominator — how many equal parts the whole was split into.
If you cut a carrot into four pieces and one piece is enormous, those aren't quarters. Quarters must all be the same size.
Fraction names
| Fraction | Name | The whole is split into… |
|---|---|---|
| ½ | one half | 2 equal parts |
| ⅓ | one third | 3 equal parts |
| ¼ | one quarter | 4 equal parts |
| ⅕ | one fifth | 5 equal parts |
| ⅒ | one tenth | 10 equal parts |
The bigger the denominator, the smaller each piece. Share one apple between 2 people — big pieces. Share one apple between 10 people — small pieces. So ⅒ is smaller than ½, even though 10 is bigger than 2. Children find this genuinely surprising, and it's worth spending time on.
Finding a fraction of an amount
To find ¼ of something, share it into 4 equal groups. There are 12 guinea pigs and ¼ are brown: 12 ÷ 4 = 3 brown. To find ¾, find ¼ first, then take three of them: ¼ of 12 = 3, so ¾ = 3 × 3 = 9.
Divide by the bottom, times by the top. ⅗ of 20 → 20 ÷ 5 = 4 → 4 × 3 = 12.
Equivalent fractions
Some fractions look different but are worth the same: ½ = 2/4 = 4/8 = 5/10 — all mean "half." You can see it with a bar:
Same amount shaded, different names. To make an equivalent fraction, multiply the top and bottom by the same number: ⅓ × 2/2 = 2/6; ⅓ × 3/3 = 3/9; ⅓ × 4/4 = 4/12.
✅ Year 3 practice
- What is ½ of 16?
- What is ⅓ of 18?
- What is ¼ of 20?
- What is ⅖ of 15?
- What is ¾ of 8?
- Which is bigger: ¼ or ⅕?
- Write two fractions equivalent to ½
- There are 24 animals. ⅙ are birds. How many birds?
Year 4 Tenths, hundredths & decimals
A tenth is one whole split into 10 equal parts: ⅒ = 0.1. A hundredth is one whole split into 100 equal parts: 1/100 = 0.01.
| Fraction | Decimal |
|---|---|
| ⅒ | 0.1 |
| 3/10 | 0.3 |
| 7/10 | 0.7 |
| 1/100 | 0.01 |
| 25/100 | 0.25 |
| 50/100 | 0.5 |
The first place after the point is tenths; the second is hundredths. 0.4 = four tenths; 0.04 = four hundredths — ten times smaller.
The ones to know by heart
These come up constantly and should be instant: ½ = 0.5 · ¼ = 0.25 · ¾ = 0.75 · ⅒ = 0.1 · ⅕ = 0.2.
Dividing by 10 and 100
| Start | ÷ 10 | ÷ 100 |
|---|---|---|
| 6 | 0.6 | 0.06 |
| 45 | 4.5 | 0.45 |
| 380 | 38 | 3.8 |
It's more accurate — and much more useful later — to say the digits move one place, and the point stays still. The point marks where the whole numbers end. It doesn't wander about.
Counting in fractions
Counting in quarters: ¼, 2/4, ¾, 1, 1¼, 1½, 1¾, 2… Notice that 4/4 = 1 — when the numerator matches the denominator, you have a whole one.
✅ Year 4 practice
- Write ⅒ as a decimal
- Write 0.7 as a fraction
- Write ¾ as a decimal
- What is 60 ÷ 10?
- What is 45 ÷ 100?
- Which is bigger: 0.3 or 0.03?
- Count on in quarters from ¾ for four more numbers
- A hedgehog weighs 0.4 kg. Write that as a fraction of a kilogram.
Year 5 Mixed numbers, adding & percentages
Improper fractions and mixed numbers
An improper fraction has a numerator bigger than the denominator (7/4). A mixed number has a whole number and a fraction (1¾). They mean the same thing.
Improper → mixed: divide the top by the bottom. 11/4 → 11 ÷ 4 = 2 remainder 3 → 2¾. Mixed → improper: multiply the whole by the denominator, add the numerator. 3⅖ → 3 × 5 = 15, + 2 = 17 → 17/5.
If a pizza is cut into quarters, 7/4 means seven quarter-slices. Four make one whole pizza; three are left over. So 7/4 = 1¾ — same slices, described differently.
Adding & subtracting with the same denominator
Add the numerators. Leave the denominator alone. 2/7 + 3/7 = 5/7. 5/8 − 2/8 = 3/8.
The denominator says what size the pieces are. You aren't changing the size, only how many you have. Two eighths plus three eighths is five eighths — still eighths. 2/8 + 3/8 = 5/16 is wrong, and it's the most common fraction error in KS2.
Different denominators — find a common one
½ + ¼: change ½ into quarters (½ = 2/4), then 2/4 + ¼ = ¾. ⅓ + ⅙: change ⅓ into sixths (⅓ = 2/6), then 2/6 + ⅙ = 3/6 = ½.
Percentages — a new name for hundredths
Per cent means "out of 100." 50% means 50 out of 100.
| Percentage | Fraction | Decimal |
|---|---|---|
| 10% | 10/100 = ⅒ | 0.1 |
| 25% | 25/100 = ¼ | 0.25 |
| 50% | 50/100 = ½ | 0.5 |
| 75% | 75/100 = ¾ | 0.75 |
| 100% | 100/100 = 1 | 1.0 |
Finding a percentage of an amount: 50% of 40 = half of 40 = 20; 25% of 40 = quarter of 40 = 10; 10% of 40 = 40 ÷ 10 = 4.
✅ Year 5 practice
- Change 9/4 into a mixed number
- Change 2⅓ into an improper fraction
- 3/8 + 4/8 =
- 7/10 − 3/10 =
- ½ + ¼ =
- What is 50% of 60?
- What is 25% of 80?
- What is 10% of 350?
- Write 30% as a fraction in its simplest form
- There are 40 animals at a centre. 25% are reptiles. How many reptiles?
Year 6 All four operations & FDP equivalence
The full equivalence table — learn these
They appear constantly at KS2, in SATs, and all the way through GCSE.
| Fraction | Decimal | Percentage |
|---|---|---|
| ½ | 0.5 | 50% |
| ¼ | 0.25 | 25% |
| ¾ | 0.75 | 75% |
| ⅕ | 0.2 | 20% |
| ⅖ | 0.4 | 40% |
| ⅒ | 0.1 | 10% |
| 3/10 | 0.3 | 30% |
| ⅓ | 0.333… | 33.3% |
| ⅛ | 0.125 | 12.5% |
Converting between them
Fraction → decimal: divide the top by the bottom. ⅗ = 3 ÷ 5 = 0.6. Decimal → percentage: multiply by 100. 0.6 = 60%. Percentage → fraction: put it over 100 and simplify. 40% = 40/100 = ⅖.
Multiplying fractions
Multiply the tops. Multiply the bottoms. ⅔ × ¼ = (2×1)/(3×4) = 2/12 = ⅙. Fraction × whole number: ⅗ × 4 = (3×4)/5 = 12/5 = 2⅖.
Children expect multiplying to make numbers bigger; with fractions it usually doesn't. ½ of ½ is ¼ — half of a half, smaller than either. The word "of" is the clue: ⅔ of ¼ means a part of a part.
Dividing fractions by whole numbers
Multiply the denominator by the whole number. ⅓ ÷ 2 = ⅙. Think of it as one third, shared between two people, giving each a sixth.
Percentages of amounts — build from the ones you know
50% = ÷2 · 25% = ÷4 · 10% = ÷10 · 5% = half of 10% · 1% = ÷100. Find 35% of 240: 10% = 24, 30% = 72, 5% = 12, so 35% = 72 + 12 = 84.
Simplifying fractions
Divide the top and bottom by the same number until you can't any more. 12/18 → both ÷ 6 → ⅔. 25/100 → both ÷ 25 → ¼. 8/12 → both ÷ 4 → ⅔.
✅ Year 6 practice
- Write ⅘ as a decimal and a percentage
- Write 0.35 as a fraction in its simplest form
- Write 60% as a fraction in its simplest form
- ⅔ × ⅗ =
- ¾ × 8 =
- ¼ ÷ 3 =
- Simplify 18/24
- Find 20% of 350
- Find 15% of 60
- Which is bigger: ⅗ or 0.55? Show how you know.
Reasoning problems
These are the SATs-style questions. They use the same maths but need thinking about first. Worked answers are in the answer key below.
Problem 1 · Year 4/5
An animal centre has 30 animals. ⅓ are reptiles, ½ are mammals, and the rest are birds. How many birds are there? Show your working.
Problem 2 · Year 5
A bag of animal food weighs ¾ kg. (a) How much do 4 bags weigh? (b) A keeper uses ¼ kg a day. How many days will one bag last?
Problem 3 · Year 5/6
A hedgehog weighs 420 g in March. By June it weighs 25% more. How much does it weigh in June?
Problem 4 · Year 6
Three children are counting animals. Amir says ⅖ are reptiles. Beth says 40% are reptiles. Cal says 0.4 are reptiles. Who is correct? Explain your answer.
Problem 5 · Year 6
A tortoise eats 2¼ kg of greens each week. (a) How much does he eat in 4 weeks? (b) A sack holds 15 kg. For how many complete weeks will one sack last?
Problem 6 · Year 6, multi-step
There are 60 animals at a centre. 25% are reptiles. ⅓ of the rest are birds. The remaining animals are mammals. How many mammals are there?
Answers
Year 3
1. 8 2. 6 3. 5 4. 15 ÷ 5 = 3, 3 × 2 = 6 5. 8 ÷ 4 = 2, 2 × 3 = 6 6. ¼ — the smaller the denominator, the bigger the piece 7. any two of 2/4, 3/6, 4/8, 5/10, 6/12 8. 24 ÷ 6 = 4 birds
Year 4
1. 0.1 2. 7/10 3. 0.75 4. 6 5. 0.45 6. 0.3 — three tenths is bigger than three hundredths 7. 1, 1¼, 1½, 1¾ 8. 4/10 (accept ⅖)
Year 5
1. 9 ÷ 4 = 2 r 1 → 2¼ 2. 2 × 3 = 6, + 1 = 7/3 3. 7/8 4. 4/10 (accept ⅖) 5. ½ = 2/4, so 2/4 + ¼ = ¾ 6. 30 7. 20 8. 35 9. 30/100 = 3/10 10. 40 ÷ 4 = 10 reptiles
Year 6
1. 4 ÷ 5 = 0.8 = 80% 2. 35/100 = 7/20 3. 60/100 = ⅗ 4. (2×3)/(3×5) = 6/15 = ⅖ 5. (3×8)/4 = 24/4 = 6 6. 1/12 7. both ÷ 6 = ¾ 8. 10% = 35, so 20% = 70 9. 10% = 6, 5% = 3, so 15% = 9 10. ⅗ = 0.6, bigger than 0.55 → ⅗ is bigger
Reasoning problems
Problem 1. ⅓ of 30 = 10 reptiles; ½ of 30 = 15 mammals; 10 + 15 = 25; 30 − 25 = 5 birds.
Problem 2. (a) ¾ × 4 = 12/4 = 3 kg. (b) ¾ ÷ ¼ = 3 days.
Problem 3. 25% of 420 = 105; 420 + 105 = 525 g. (Or 420 × 1.25 = 525 g.)
Problem 4. All three are correct. ⅖ = 0.4 = 40% — three ways of writing the same amount. This tests equivalence; the expected wrong answer is choosing just one. Full marks require naming all three and explaining why.
Problem 5. (a) 2¼ × 4 = 9/4 × 4 = 36/4 = 9 kg. (b) 15 ÷ 2.25 = 6.67, so 6 complete weeks (round down — you can't have two-thirds of a week's food and call it a week).
Problem 6. 25% of 60 = 15 reptiles; 60 − 15 = 45; ⅓ of 45 = 15 birds; 45 − 15 = 30 mammals.
The mistakes to watch for
Adding the denominators. 2/8 + 3/8 = 5/16. The most common fraction error in KS2. Keep returning to what size are the pieces?
Thinking a bigger denominator means a bigger fraction. ⅒ looks bigger than ½ because 10 > 2. Use a bar or a chocolate bar until it clicks.
"Adding a zero" when multiplying by 10. Works for whole numbers, breaks immediately with decimals — 4.5 × 10 is not 4.50. Teach digits moving place, not zeros appearing.
Expecting multiplication to make things bigger. ½ × ½ = ¼. Use the word "of" — half of a half.
Forgetting to simplify. Many mark schemes want the simplest form. Make it a habit: can both numbers be divided by the same thing?
Not reading the question in reasoning problems. Problem 6 asks for mammals; children who work out the reptiles and birds correctly often stop there.
Teaching notes
Fractions need to be physical for longer than most schemes allow. Fraction walls, paper folding, cutting real things. A child who has folded a strip into eighths and seen that two of them cover a quarter has understood something no worksheet delivers.
Teach ⅒ is smaller than ½ explicitly, and expect resistance. This single misconception blocks progress in comparison, ordering and addition. It's worth a whole lesson.
The equivalence table in Year 6 should be memorised. ½ = 0.5 = 50% ought to be as automatic as a times table. It saves time in every subsequent topic and all the way through GCSE.
Reasoning problems should be talked through before they're written. Ask what the question is actually asking for. Problem 6 in particular — most children can do all three steps and still answer the wrong question.
For mixed-age settings, the year headings are a guide, not a rule. A Year 6 child insecure on equivalent fractions should be working in the Year 3 section, and there's no shame in it. Fractions are cumulative — the later work simply does not function without the earlier.
Animal contexts help, but keep the numbers honest. A hedgehog that weighs 420 g and gains 25% is a realistic hedgehog. Made-up numbers that produce neat answers teach children that maths problems aren't about anything real.
This unit feeds directly into GCSE. The Year 6 percentage methods — 10%, halve for 5%, add — are exactly the methods used at GCSE. A child taught them properly at 11 doesn't have to learn anything new at 15.