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.
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.
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"):
π=dC≈3.14
Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Diameter 10
Picture walking round a circle compared with walking across it.
TryTry it on the board and compare the result with your prediction.
Roll three circles
Change the radius, then press Roll it out. Each size adds a row to the table.
Try it yourselfYour edits are saved in this browser.
Roll out at least three circles of different sizes. Look at C ÷ d in the table. Was your prediction right?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
The same number every time
Here is what π is, and what it is not.
Worked exampleBefore each step, say what you think comes next.
π=C÷dπ is how many diameters fit round the circumference.
π=3.14159265…Its digits never end and never settle into a repeating pattern.
C≈3×dA quick estimate: a bit more than 3 diameters.
Now: what does the table show?
Read Kai's reasoning. Is a measurement ever exact?
Kai measures a plate to find π. Click the line where his reasoning goes wrong.
A wheel has diameter 60 cm. Estimate its circumference in cm, using π ≈ 3.14.
ModifyChange the problem on purpose and see what happens.
One change at a time
Each task changes one thing.
Need a hint?
Diameter 1 means radius 0.5.
Try it yourselfYour edits are saved in this browser.
Roll out a circle with diameter 1 cm. How long is its circumference?
Use the board. This ticks when you get there.
A bin lid measures circumference 47.1 cm and diameter 15 cm. What is C ÷ d?
PractiseA few on your own, to check you have it.
Π
A circle has circumference 31.4 cm. Using π ≈ 3.14, what is its diameter, in cm?
Review: a circle has diameter 14 cm. What is its radius, in cm?
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
A circumference between 12 and 13
Predict the diameter before you roll.
Try it yourselfYour edits are saved in this browser.
Need a hint?
12 ÷ 3.14 is about 4. What radius gives diameter 4?
Another hint
Predict first, then roll to check.
Find and roll out a circle whose circumference is between 12 cm and 13 cm.
Use the board. This ticks when you get there.
For teachers
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.
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.
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).
Running the lesson
Predict: take estimates; most say 20 or 30.
Run: ask students to read down the C ÷ d column aloud.
Investigate: collect real measurements from tins and plates; their ratios cluster near 3.1 or 3.2.
Modify: diameter 1 gives π itself: link to the definition.
Make: insist on a prediction before rolling.
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.
Stretch and support
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.
Where it 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).
What each step is for
diameter 10: Committing to an estimate for the circumference of a 10 cm circle shows whether they expect a fixed multiple of the diameter, and roughly what it is.
roll three circles: Rolling circles of three sizes and dividing C by d each time shows the ratio is the same, about 3.14, whatever the size.
the same number every time: C ÷ d is the same for every circle; Kai's measurement gives 3.14, but π is not exactly 3.14. A quick estimate uses C ≈ 3d.
one change at a time: 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.
π: Estimating and working out circumferences and the meaning of π, mixed; π as exact is the planted slip; the review keeps radius and diameter fresh.
a circumference between 12 and 13: Searching for a circle with a circumference in a given range uses C ≈ 3.14d backwards as a prediction, checked by rolling.
Now you can
Explain what π means: how many diameters fit round a circle.
Estimate a circumference from a diameter.
Say why 3.14 and 22/7 are close to π but not exact.