Students roll circles of different sizes out flat, divide the circumference by the diameter each time, and discover the constant π ≈ 3.14, with its approximations and a quick estimate C ≈ 3d (AC9M7M03, node 2).
Load these into Waypoints Prepare: choose Load URL and paste the copied address, or download the file and import it. In Prepare, the worksheet's Teacher edition shows the answers; the student copy hides them.
Predict and Run 10 min · Investigate 10 min · Modify 10 min · Practise and Make 10 min
Do the Run board yourself first, then try it with real objects: string round tins and plates, measured against their diameters. The board is the clean version; the hands-on measurement shows why measured values wobble around 3.14.
The one idea: C ÷ d is the same for every circle, and that number is π. The common mistake is thinking π is exactly 3.14 or 22/7 (Kai's mistake).
Believing the ratio is the same for every circle before seeing it. Ask students to predict C ÷ d for a much bigger circle before they roll it, and let the table decide.
| What you will see | A question to ask | The fix |
|---|---|---|
| A student thinks π is exactly 3.14 (Kai's mistake). | Is any ruler measurement exact? | π = 3.14159… never ends; 3.14 is rounded. |
| A student divides d by C. | How many diameters fit round: is the answer more or less than 1? | C ÷ d, about 3. |
| A student thinks bigger circles have bigger π. | If the diameter doubles, what happens to the circumference? | Both double, so the ratio stays the same. |
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 out how Archimedes trapped π between 3 10/71 and 3 1/7 using polygons inside and outside a circle.
Support: Use string and tins; keep the table of C, d and C ÷ d on 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 7 student in Australia. Help me understand ONE idea only, from the lesson "Discovering π". The idea: Discovering π as a constant ratio: I can Students investigate the ratio C/d for multiple circles and recognise that it is always approximately 3.14 — this constant is π. What understanding looks like: - Measures and calculates C/d for at least two circles and observes the result is approximately 3.14. - States that π ≈ 3.14 and explains what it represents. - Does not treat π as just a button on a calculator without understanding its meaning. Mistakes students often make here: - A student thinks π is exactly 3.14 (Kai's mistake). - A student divides d by C. - A student thinks bigger circles have bigger π. 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. - If I seem to be missing the idea that comes before this one (Features of circles and their relationships), check it quickly, then come back. - 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 "Discovering π".
What is π? Estimate the circumference of a circle with diameter 20 cm.
π is the circumference divided by the diameter, the same for every circle, about 3.14. C ≈ 3.14 × 20 = 62.8 cm.
Next: Circumference (node 3): C = πd and C = 2πr, and finding d from C.
Where it leads: the area of a circle (Year 8), and π as an irrational number (Year 9).