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# Discovering π

Area, volume, circles and Pythagoras · Circles

<!-- Speaker note: 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). About 40 minutes (Predict and Run 10 min · Investigate 10 min · Modify 10 min · Practise and Make 10 min). -->

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## Today

**We are learning to** roll any circle out flat and its circumference is just over 3 of its diameters. C ÷ d is the same number for every circle: π (pi), about 3.14. π's digits never end: 3.14 and 22/7 are close, not exact.

- I can explain what π means: how many diameters fit round a circle.
- I can estimate a circumference from a diameter.
- I can say why 3.14 and 22/7 are close to π but not exact.

<!-- Speaker note: Before class: 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). -->

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## Why it matters

> A bike wheel, a coin and the Earth's equator: wildly different sizes, but for each one the distance round is the same number of times the distance across. People have known this for four thousand years.

<!-- Speaker note: The **circumference** of a circle is hard to measure with a ruler, because it is curved. But you can roll the circle along a line for one full turn: the track it leaves is exactly its circumference, laid flat. Compare that track with the **diameter**, and something remarkable happens: for every circle, big or small, the circumference is the same number of diameters. That number is called **π** (the Greek letter pi, said "pie"): $$\pi = \frac{C}{d} \approx 3.14$$ -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Predict

**A circle has diameter 10 cm. Roughly how long is its circumference?**

- About 20 cm
- About 31 cm
- About 100 cm
- About 10 cm

<!-- Speaker note: What happens: About 31 cm: just over 3 diameters (31.4 cm). 20 cm would be 2 diameters, which is less than the way round. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: take estimates; most say 20 or 30. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Fmeasurement%2Fdiscovering-pi%2Fstep-2&code=widget%3A+circle%0Aslider+r%3A+0.5%2C+3%2C+1%2C+0.5%0Aroll%3A+on%0Ashow%3A+none%0Aunits%3A+cm%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/measurement/discovering-pi/?full=step-2#step-2)

<!-- Speaker note: Change the radius, then press **Roll it out**. Each size adds a row to the table. -->

<!-- Speaker note: Teacher: ask students to read down the C ÷ d column aloud. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## What happened?

* About 31 cm: just over 3 diameters (31.4 cm). 20 cm would be 2 diameters, which is less than the way round.

<!-- Speaker note: Reveal after running. Ask who predicted it, and what made them think so. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## What π means

Here is what π is, and what it is not.

Now: what does the table show?

$$\pi = C \div d$$

1) $\pi = 3.14159265\ldots$ — Its digits never end and never settle into a repeating pattern.
2) $\pi \approx 3.14 \quad \text{or} \quad \pi \approx \tfrac{22}{7}$ — Two handy approximations (22/7 = 3.142…).
3) $C \approx 3 \times d$ — A quick estimate: a bit more than 3 diameters.

<!-- Speaker note: Model this live on the board before revealing anything: write each line and say why, pausing to ask the class what comes next. Then click to reveal the lines and compare. Script. Start: write = C ÷ d and say "π is how many diameters fit round the circumference." Line 1: write = 3.14159265 and say "Its digits never end and never settle into a repeating pattern." Line 2: write 3.14 or 22 ÷ 7 and say "Two handy approximations (22/7 = 3.142…)." Line 3: write C 3 × d and say "A quick estimate: a bit more than 3 diameters." -->

<!-- Speaker note: Teacher: collect real measurements from tins and plates; their ratios cluster near 3.1 or 3.2. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## What π means

**What does the table show about C ÷ d?**

- It is about 3.14 for every circle
- It gets bigger for bigger circles
- It changes every time
- It is 2, because the diameter is two radii

* Answer: It is about 3.14 for every circle

<!-- Speaker note: C ÷ d is the same number for every circle: π ≈ 3.14. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Find the mistake

**Kai measures a plate to find π. Click the line where his reasoning goes wrong.**

```
 1  I measured a plate: circumference 78.5 cm, diameter 25 cm.
 2  78.5 ÷ 25 = 3.14
 3  So π is exactly 3.14.
```

* Answer: line 3

<!-- Speaker note: Line 3: a measurement gives π only to a few places. π = 3.14159… has digits that never end or settle into a pattern; 3.14 is an approximation. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Estimate

**A wheel has diameter 60 cm. Estimate its circumference in cm, using π ≈ 3.14.**

* Answer: 188.4

<!-- Speaker note: C ≈ 3.14 × 60 = 188.4 cm (quick check: 3 × 60 = 180, a bit more). -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## One change at a time



- **Diameter 1:** Each task changes one thing.
- **π from a measurement:** 
- **A quick estimate:** 
- **Bigger circles:** 

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/measurement/discovering-pi/?full=step-4#step-4)

<!-- Speaker note: What this step is for: One change at a time: a circle of diameter 1 (so C is π itself), π from a measurement, a quick estimate, and the edge case of a bigger circle having a bigger π. Hints, in order: Diameter 1 means radius 0.5. For students who need more support: Use string and tins; keep the table of C, d and C ÷ d on the board. For early finishers: Find out how Archimedes trapped π between 3 10/71 and 3 1/7 using polygons inside and outside a circle. The hard part, on purpose: 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. When students get stuck: A student thinks π is exactly 3.14 (Kai's mistake). Ask: Is any ruler measurement exact? Fix: π = 3.14159… never ends; 3.14 is rounded. / A student divides d by C. Ask: How many diameters fit round: is the answer more or less than 1? Fix: C ÷ d, about 3. / A student thinks bigger circles have bigger π. Ask: If the diameter doubles, what happens to the circumference? Fix: Both double, so the ratio stays the same. -->

<!-- Speaker note: Teacher: diameter 1 gives π itself: link to the definition. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## 22/7

**Using π ≈ 22/7, about how long is the circumference of a circle with diameter 7 cm?**

- 7 cm
- 14 cm
- 22 cm
- 49 cm

* Answer: 22 cm

<!-- Speaker note: 22/7 × 7 = 22 cm. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Backwards: the diameter

**A circle has circumference 31.4 cm. Using π ≈ 3.14, what is its diameter, in cm?**

* Answer: 10

<!-- Speaker note: C ÷ π = d: 31.4 ÷ 3.14 = 10 cm. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## What π is

**Which is true?**

- π is exactly 22/7
- π is exactly 3.14
- π is a little more than 3.14
- π is 3

* Answer: π is a little more than 3.14

<!-- Speaker note: π = 3.14159…: a little more than 3.14, a little less than 22/7. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Review: radius

**Review: a circle has diameter 14 cm. What is its radius, in cm?**

* Answer: 7

<!-- Speaker note: 14 ÷ 2 = 7 cm. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Make: a circumference between 12 and 13

Predict the diameter before you roll.

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/measurement/discovering-pi/?full=step-6#step-6)

<!-- Speaker note: What this step is for: Searching for a circle with a circumference in a given range uses C ≈ 3.14d backwards as a prediction, checked by rolling. Hints, in order: 12 ÷ 3.14 is about 4. What radius gives diameter 4? / Predict first, then roll to check. -->

<!-- Speaker note: Teacher: insist on a prediction before rolling. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Exit ticket

**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.

<!-- Speaker note: Students answer on paper or a mini whiteboard before the reveal. Anyone who cannot answer needs the main idea again next lesson. Where this leads: 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). -->