At a glance. One lesson of about 60 minutes. Students explain why C ÷ d is the same for every circle, use C = πd and C = 2πr with 3.14 or the key, and work backwards from a circumference.
Calculator: scientific calculator. A scientific calculator takes the arithmetic. The lesson fixes π at 3.14 so everyone gets the same answer, and asks for an estimate of three diameters first so that a radius slip is caught.
What students will learn
Know
- C = πd = 2πr.
- π is about 3.14, and C ÷ d is the same for every circle.
- d = 2r.
Understand
- Every circle is the same shape, so its edge is a fixed number of diameters.
- A wheel rolls its circumference in one turn.
- An answer in terms of π is exact; 3.14 gives a close decimal.
Do
- Find a circumference from a diameter or a radius.
- Find a diameter or radius from a circumference.
- Find the length of part of a circle's edge and combine it with straight edges.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: C = πd = 2πr, radius and diameter (questions 1 to 9).
- Criterion B. Investigating patterns: measure the circumference and diameter of five round objects with string, and look for the pattern in C ÷ d (the practical below, and the first figure).
- Criterion C. Communicating: show the diameter and the radius on a sketch, and write the formula before substituting (questions 4 and 6).
- Criterion D. Applying mathematics in real-life contexts: wheels, tables and window frames, where the quantity asked for is a distance (questions 2, 5 and 6).
The one thing to do with the figure
Do the string first. Before showing the first figure, have students wrap string round a tin and lay it along the tin's width. They find a little over three. Then show the figure and ask whether a bigger circle would change that.
In the second figure, hide the readouts and ask for the distance after 3 turns for a 40 cm wheel before sliding. Then ask what changes if the wheel is twice as wide.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | String and tins, then the bicycle and the prediction. First figure. |
| 10 min | Key idea. Collect class measurements of C and d in a table and divide. |
| 10 min | Second figure, worked examples, then the faded one in pairs. |
| 5 min | Question 1, the check. If many choose B, return to the common-mistake card. |
| 15 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9, on paper or on screen. |
The answers
| Warm-up | 38 cm; 3.14; x = 18. |
|---|---|
| Worked examples | 157 cm; 14π and 43.96 cm; diameter 20 cm and radius 10 cm. |
| Faded example | 94.2 m. |
| 1. Circumference from the radius | A. B uses the radius as the diameter, C is the area, D adds. If many choose B, return to the common-mistake card and compare with “a little over three diameters”. |
| 2. Circumference from the diameter | 282.6 cm. 3.14 × 90 = 282.6. |
| 3. In terms of pi | 16. C = 16π cm. |
| 4. Missing diameter | 8 cm. 25.12 ÷ 3.14 = 8. |
| 5. Turns of a wheel | 30 turns. 56.52 m = 5652 cm, and 5652 ÷ 188.4 = 30. |
| 6. Part of a circle | 445.6 cm. 320 + 125.6 = 445.6. |
| 7. The extra length of a ring | 12.56 m. 138.16 − 125.6 = 12.56, or 2 × 3.14 × 2. |
| 8. Doubling the diameter | 21.98 cm. 43.96 − 21.98 = 21.98. |
| 9. Same shape, same ratio | A. B adds the change in diameter, C squares the factor, D thinks π depends on the size. If many choose D, go back to the first figure and change the diameter. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B in question 1 | Uses the radius as the diameter. | Ask for an estimate of three diameters, and compare. |
| 141.3 or 565.2 in question 2 | Halves, or doubles a diameter. | Ask which length is given, the radius or the diameter. |
| 8 or 64 in question 3 | Leaves the radius alone, or squares it. | Write C = 2 × π × r and substitute. |
| 78.8768 or 4 in question 4 | Multiplies instead of dividing, or gives the radius. | Write C = 3.14 × d and ask what undoes a multiplication. |
| 0.3 or 60 in question 5 | Mixes metres with centimetres, or uses the radius. | Convert the distance to centimetres first. Then find one turn. |
| 571.2 or 525.6 in question 6 | Uses a whole circle, or includes the diameter. | Trace the frame with a finger. Does it go across the top? |
| 138.16 or 6.28 in question 7 | Reports one circumference, or drops the 2. | Ask what the question compares. |
| 43.96 or 7 in question 8 | Reports one circumference, or differences the diameters. | Write both circumferences, then subtract. |
| B, C or D in question 9 | Adds the change in diameter, squares the factor, or thinks π grows with the circle. | Put 50.24 and 150.72 side by side and divide each by its diameter. |
Questions worth asking
- Why is C ÷ d the same for a coin and a bicycle wheel? They are the same shape, one is just bigger. Listen for “an enlargement” or “the same shape”.
- Is 3.14 equal to π? No, it is a rounded value; π has decimals that never end. Listen for “close but not equal”.
- Which is easier to measure on a real circle, the diameter or the radius? Usually the diameter, across the widest part. Listen for the centre being hard to find.
- When is an answer in terms of π better than a decimal? When you need it exact, or you are going to use it again. Listen for “no rounding”.
Differentiation
A way in for anyone stuck
- Give string, tins and a ruler so students can find the number for themselves.
- Ask for the estimate three diameters before every calculation and write it next to the answer.
- Mark the diameter in colour on every diagram before multiplying.
Stretch
- A circle's circumference is 31.4 cm. What is its diameter, its radius, and what number of π is the circumference? (10 cm, 5 cm, 10π.)
- Show that if you add 1 m to a belt round any circle, the radius grows by about 16 cm, whatever the size of the circle. (100 ÷ 6.28 = 15.9 cm.)
Students learning in English as an additional language
“Circumference” is a long word with a plain meaning: the distance all the way round a circle. “Diameter” goes across through the centre; “radius” goes from the centre to the edge. Say them together, in a chart, with a drawing and the word in the home language beside each one.
| Word | Plain meaning | Example |
|---|---|---|
| circumference | the distance all the way round a circle | The length of string round a tin |
| diameter | a straight line across a circle through the centre | The width of a wheel |
| radius | a straight line from the centre to the edge, half of the diameter | A spoke of a wheel |
| π | the number you get when you divide a circle's circumference by its diameter, about 3.14 | 3.14 is a rounded value of π |
| in terms of π | an answer written with the symbol π in it, not as a decimal | Radius 3 cm gives a circumference of 6π cm |
| semicircle | half of a circle | The top of an arched window |
Sentence starters for explaining aloud or in writing:
- The diameter is … so the circumference is …
- The circumference must be a little more than …, because …
- I used the radius / diameter because …
Two things not to say
“π is 22 over 7.” 22/7 is close, but it is a fraction, and π is not a fraction. Say “about 3.14”.
“Radius times 3.14 times 2, or diameter times 3.14, whichever you like.” Both are right, but the choice should come from what is given, and students should say why.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- FluencyA circle has a diameter of 10 cm. Find its circumference. Use 3.14.
31.4 cm. - FluencyA circle has a radius of 5 cm. Find its circumference in terms of π.
10π cm.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA cyclist in Dhaka has wheels 70 cm across. How far does the bike go in 10 turns of a wheel? Use 3.14.
2198 cm, which is 21.98 m. - SpacingFrom lesson 2: a regular hexagon has sides of 6 cm. What is its perimeter?
36 cm.
3Challenge
Reasoning, generalizing and working backwards.
- MissingA circular track has an inner edge of 314 m. A runner in the outer lane runs along an edge 4 m further out from the centre. How much longer is the outer lap? Use 3.14.
25.12 m: the extra radius is 4 m and 2 × 3.14 × 4 = 25.12. - ReasoningTwo circles have diameters of 8 cm and 24 cm. How many times as long is the circumference of the second as the first?
3 times: 75.36 ÷ 25.12 = 3, because the diameter is 3 times as large.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Every figure works with the keyboard (tab to a slider, then the arrow keys) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Squared paper, scissors, string and a few sheets of card do the same job: cut, turn, wrap and fold.
String, a ruler and five round objects (a tin, a cup, a coin, a plate, a bowl) are enough for the investigation. Students wrap string round each, measure it, and measure the width.