--- title: "Discovering π" subtitle: "Area, volume, circles and Pythagoras · Circles · Lesson 7" --- # Discovering π **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** 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. :::teacher-note **In one sentence:** 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). **Students should be able to say:** 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. **Width:** the unmarked parts are the floor every student does (predict, run, investigate and make). Typical adds the modify practice; wider adds the real-world contexts. They add contexts, not difficulty. Students work in the online tutorial alongside this sheet. ::: ## Predict Picture walking round a circle compared with walking across it. :::keep-together **A circle has diameter 10 cm. Roughly how long is its circumference?** ☐ About 20 cm ☐ About 31 cm ☐ About 100 cm ☐ About 10 cm My prediction, and why: {{lines-3}} ::: :::teacher-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. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## The same number every time :::keep-together ### What π means Here is what π is, and what it is not. 1. $\pi = C \div d$ — π is how many diameters fit round the circumference. 2. $\pi = 3.14159265\ldots$ — Its digits never end and never settle into a repeating pattern. 3. $\pi \approx 3.14 \quad \text{or} \quad \pi \approx \tfrac{22}{7}$ — Two handy approximations (22/7 = 3.142…). 4. $C \approx 3 \times d$ — A quick estimate: a bit more than 3 diameters. Now: what does the table show? **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 ::: :::teacher-note **Answer:** It is about 3.14 for every circle C ÷ d is the same number for every circle: π ≈ 3.14. ::: :::keep-together ### 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. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 3 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. ::: :::keep-together ### Estimate **A wheel has diameter 60 cm. Estimate its circumference in cm, using π ≈ 3.14.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 188.4 C ≈ 3.14 × 60 = 188.4 cm (quick check: 3 × 60 = 180, a bit more). ::: :::typical ## One change at a time **Diameter 1.** Each task changes one thing. **π from a measurement.** **A quick estimate.** **Bigger circles.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Π :::keep-together ### 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 ::: :::teacher-note **Answer:** 22 cm 22/7 × 7 = 22 cm. ::: :::keep-together ### Backwards: the diameter **A circle has circumference 31.4 cm. Using π ≈ 3.14, what is its diameter, in cm?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 10 C ÷ π = d: 31.4 ÷ 3.14 = 10 cm. ::: :::keep-together ### 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 ::: :::teacher-note **Answer:** π is a little more than 3.14 π = 3.14159…: a little more than 3.14, a little less than 22/7. ::: :::keep-together ### Review: radius **Review: a circle has diameter 14 cm. What is its radius, in cm?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 7 14 ÷ 2 = 7 cm. ::: ## Make: a circumference between 12 and 13 Predict the diameter before you roll. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **What is π? Estimate the circumference of a circle with diameter 20 cm.** {{lines-3}} ::: :::teacher-note **Answer:** π is the circumference divided by the diameter, the same for every circle, about 3.14. C ≈ 3.14 × 20 = 62.8 cm. :::