Lesson Guide
Explanation
Here's the key that unlocks this whole lesson: the fraction bar is a division sign. Three quarters literally means 3 ÷ 4 — three pizzas shared fairly between four people. That means every fraction can be converted into a decimal, just by doing the division. The interesting question is what kind of decimal comes out — and it turns out there are only two possibilities.
Your Turn
Predict before moving on: what is 3 ÷ 4 as a decimal? What about 1 ÷ 3? Write your guesses down — one of them ends, and one never does.
Watch Out
If 3/4 and 3 ÷ 4 feel like unrelated objects to you, pause here until bar-means-divide clicks — everything else in this lesson is built on it.
Explanation
Let's actually do 3 ÷ 4 with short division. Four into three doesn't go, so move to the tenths: four into thirty goes 7, remainder 2. Carry the 2: four into twenty goes 5, remainder 0. That remainder of zero is the important moment — the division stops. 3/4 is exactly 0.75. When the remainder hits zero, we call the decimal terminating: the digits end.
Your Turn
Run 1 ÷ 8 yourself by short division. Check: 0.125. And ask yourself: what told you it was finished? The remainder reached 0.
Watch Out
Keep the language precise: it terminates because the REMAINDER reached zero. Hold onto that phrase — the next step is about what happens when it never does.
Explanation
Now try 2 ÷ 3. Three into twenty goes 6, remainder 2 — but wait: remainder 2 is exactly where we started. The same remainder must produce the same next digit, so the sixes repeat forever: 0.666… without end. That's a recurring decimal, and we write it with a dot over the repeating digit: 0.6̇. When a whole block repeats, like 5/11 = 0.454545…, the dots go on the first and last digits of the block: 0.4̇5̇.
Your Turn
Divide 5 by 11 until you catch the remainder cycle (5 → 6 → 5 → 6…). Practise reading the notation aloud: 0.4̇5̇ is "zero point four five recurring".
Watch Out
Here is the deep reason this works: a remainder can only be 0, 1, 2, …, one-less-than-the-divisor. So it MUST eventually repeat or hit zero — every fraction terminates or recurs. There is no third option for fractions.
Explanation
You can predict the outcome without dividing at all. Simplify the fraction, then factorise the denominator. If it's built only from 2s and 5s, the decimal terminates — because tens are made of 2s and 5s, so the denominator can be scaled up to a power of ten: 7/20 becomes 35/100, done. But if any other prime is hiding in there — a 3, a 7, an 11 — the decimal must recur.
Your Turn
Sort these without dividing, justifying each with prime factors: 3/8, 2/9, 11/40, 5/6, 13/50. Check: terminate, recur, terminate, recur, terminate.
Watch Out
Simplify FIRST. 3/6 looks like it should recur (6 contains a 3) but simplifies to 1/2, which terminates. Always judge the simplest form.
Explanation
Going the other way is pure place value. Say the decimal properly and it converts itself: 0.35 is 'thirty-five hundredths', so write 35/100, then simplify — divide top and bottom by 5 to get 7/20. The last decimal digit names the denominator: one place means tenths, two places hundredths, three places thousandths. Then simplifying is just tidying up.
Your Turn
Convert to fractions in simplest form: 0.8, 0.36, 0.125. Check: 4/5, 9/25, 1/8.
Watch Out
A marker will not accept 36/100 — simplify fully, every time. If getting from 36/100 to 9/25 felt slow, the skill to brush up is finding the HCF, not the conversion.
Explanation
Recurring decimals hide a beautiful trick. Call the number x: x = 0.777… Multiply by ten: 10x = 7.777… Now subtract the first line from the second — and the infinite tails line up perfectly and cancel. 9x equals exactly 7, so x = 7/9. If the repeating block is two digits long, multiply by 100 instead so the blocks line up: 0.4̇5̇ gives 99x = 45, so x = 45/99 = 5/11.
Your Turn
Work 0.4̇5̇ through line by line yourself: x = …, 100x = …, subtract, solve. Rule of thumb: multiply by 10 for every repeating digit, so the tails align before you subtract.
Watch Out
Doubting whether the tails REALLY cancel is normal — this is the stretch content. Write a few extra 7s in both lines and cross them out pair by pair: they match forever, so they cancel exactly.
Explanation
Step back and see the whole map. Terminating decimals are fractions. Recurring decimals are fractions too — you just proved it with the 10x trick. But there's a third kind of decimal: ones that go on forever without ever repeating a pattern — π, or √2. Those can never be written as fractions, and we call them irrational. So here's the big theorem of this lesson: a number is rational — writable as a fraction — exactly when its decimal terminates or recurs. That idea is the doorway to the next lesson.
Your Turn
Sort these: 0.272727…, π, 0.35, 3.1416. Check: rational (27/99 = 3/11), irrational, rational (7/20), rational — 3.1416 TERMINATES! It is an approximation of π, not π itself. Now do the quiz and exercises.
Watch Out
Any decimal you can finish writing is rational. Irrationality belongs to the exact number π, not to approximations like 3.1416 — a subtle distinction that examiners love.
📌 Key Points — this lesson in a nutshell
- A fraction IS a division: 3/8 means 3 ÷ 8.
- A simplified fraction terminates only when its denominator has no prime factors other than 2 and 5.
- Any other denominator makes the decimal recur — remainders must eventually repeat.
- Recurring → fraction: the 10x trick — shift by the cycle length and subtract.
- Rational = a fraction of integers; irrational decimals (π, √2) never terminate or repeat.
Step 1: Introduction
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