Students divide fractions out on a long-division board, watch the remainders, and see why decimals either stop or repeat, writing recurring decimals with bars and dots (AC9M8N03, node 1).
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Predict and Run 8 min · Investigate 8 min · Modify 10 min · Practise and Make 9 min
Do the Run board yourself first. Students need fractions as decimals (Number Foundations).
The one idea: the remainders are always less than the denominator, so either one is 0 (the decimal stops) or one comes back (the digits repeat). The common mistake is calling a recurring decimal irrational (Zoe's mistake): a recurring decimal is a fraction.
The board shows the first 12 steps; longer blocks (1/17, 1/19, 1/23, 1/29) are written out in full on the decimal line.
Wider: why 0.999… = 1; the longest blocks (1/7 has 6, 1/17 has 16: the block is at most one less than the denominator).
Trusting the remainders over the calculator. Ask students to predict the next remainder before the board shows it.
| What you will see | A question to ask | The fix |
|---|---|---|
| A student says a long calculator decimal never stops, so it is irrational. | Do the digits repeat? | A repeating decimal is a fraction: rational. |
| A student puts the bar over the wrong digits. | Which digits come round again? | The bar goes over the whole repeating block, and only that. |
| A student thinks 0.33 is exactly 1/3. | What is 3 × 0.33? | 0.99 is not 1: 0.33 is an approximation. |
One possible solution for each task. Other correct programs are fine.
Open task: any program that does what the step asks.
Open task: any program that does what the step asks.
Open task: any program that does what the step asks.
Open task: any program that does what the step asks.
Open task: any program that does what the step asks.
Stretch: Find the length of the repeating block of 1/17, 1/19 and 1/23. What is the longest a block can be for denominator b, and why?
Support: Do 1 ÷ 3 and 1 ÷ 4 on paper side by side before the board.
For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.
You are a patient maths coach for a Year 8 student in Australia. Help me understand ONE idea only, from the lesson "Recurring decimals". The idea: Identifying terminating and recurring decimals: I can Students distinguish terminating from recurring decimals and use correct notation for recurring decimals. What understanding looks like: - Correctly classifies a fraction's decimal expansion as terminating or recurring. - Writes a recurring decimal using correct notation. - Does not confuse recurring with irrational (non-terminating, non-recurring). Mistakes students often make here: - A student says a long calculator decimal never stops, so it is irrational. - A student puts the bar over the wrong digits. - A student thinks 0.33 is exactly 1/3. How to coach me (rules from the Waypoints Kit coach): - Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question. - Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit). - If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go. - Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back. - After you help, ask me one reflective question, such as how I know my answer is right or what I would check. - Praise my effort and method, not my ability. - You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so. - Use simple words. - Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Recurring decimals".
Does 5/6 terminate or recur? Write it with a dot.
It recurs: 5/6 = 0.8333… = 0.83 with a dot over the 3.
This starts AC9M8N03.
Where it leads: predicting from the denominator (next lesson), irrational numbers, and converting recurring decimals to fractions (Year 9 and 10).