Lesson 12 — Circles and the Number π

Summary

Learners measure round objects and discover that circumference is always about 3.14 times the diameter — the number π (pi). They use C = πd (and C = 2πr) to find circumference, diameter, and radius in real problems, and meet π’s approximations across cultures.

Objectives

• Use the relationship between circumference and diameter (π) to solve real problems with circles. (D05.S3.07.02)

Connection

A wheel rolls one full turn, and it moves forward a distance equal to its circumference — the distance around. If you measure a wheel’s distance across (its diameter) and then wrap a string around its edge, the string is always about 3.14 times the diameter — no matter how big or small the wheel. That constant is π. A potter making a lid, a tailor measuring a round hem, a builder setting a round column, a driver reading an odometer — all use the same circle fact. And people in many lands found it: the Egyptians used 256/81 (about 3.16), Archimedes bounded it between 3 10/71 and 3 1/7, and the Chinese mathematician Zu Chongzhi found 355/113, accurate to six decimal places, in the 5th century. π belongs to everyone.

Materials

• Round objects and string
• Circumference recording page
• Rulers or measuring tape

Preparation

• Gather round objects (cans, lids, wheels) and string per pair.
• Have worked examples ready: a circle with diameter 10 cm → C = π × 10 ≈ 31.4 cm.
• Recall from Lesson 3: proportional relationships (C and d are proportional).

Facilitator note

This lesson is written to the learner (“you”). The ideas to land: (1) circumference is the distance around a circle; diameter is the distance across through the centre (twice the radius); (2) for every circle, C ÷ d is the same constant, π ≈ 3.14; and (3) the formulas C = πd and C = 2πr turn that fact into a problem-solving tool. Because circumference is a foundational skill, use explicit instruction, worked examples, and guided practice before independent work (Kirschner, Sweller & Clark, 2006). Watch for the slips of confusing radius and diameter (d = 2r), and of using the wrong formula. Retrieval: ask learners to recall the constant of proportionality (Lessons 8–9) — π is the constant of proportionality between circumference and diameter: C = πd is a y = kx relationship. The global lens: π has been approximated across cultures — Egypt (256/81), Archimedes (bounded by 22/7), Zu Chongzhi in China (355/113, c. 5th century) — a shared human achievement, not one culture’s. The egalitarian lens (docs/philosophy.md §4): one string and one ruler are enough for anyone to find π — the discovery is open to all. The critical-thinking lens (docs/philosophy.md §7): measuring several circles and testing whether the ratio really is constant is doing science honestly.

Procedure

1. Gather (5 min). You have learned that some relationships are proportional. Today you find a famous one hiding in every circle.
2. Measure and discover (15 min). For each round object: wrap the string around the edge, mark and measure it — that is the circumference C. Measure straight across the middle — the diameter d. Record both. Then divide: C ÷ d. What do you notice? It is always about 3.14.
3. Meet π (10 min). That constant is π (pi), about 3.14. So C = π × d, written C = πd. Because the radius r is half the diameter (d = 2r), you can also write C = 2πr.
4. Worked example (10 min). A round table has a diameter of 10 cm (say, a model). Its circumference is C = π × 10 ≈ 3.14 × 10 = 31.4 cm. Given a circumference of 62.8 cm, find the diameter: d = C ÷ π = 62.8 ÷ 3.14 = 20 cm.
5. Practice with a partner (5 min). Take turns: one names a circle’s diameter or radius, the other finds the circumference (and back again). Check each other’s formulas and units.
6. Close (5 min). Every circle keeps C = πd. Measure around and across to find π yourself, then use C = πd and C = 2πr to solve real circle problems.

Differentiation

• Support: Use only the formula C = πd with a measuring string; describe each step aloud so learners with low vision can follow by listening.
• Extension: Solve for radius or diameter given circumference, and estimate a tree’s girth or a wheel’s distance travelled per revolution.

Assessment

• Formative (observation/performance): Can the learner measure C and d, find their ratio near π, and use C = πd (or C = 2πr) to solve for an unknown in a real circle problem?
• Self-check: The learner asks, “Did I measure around for C and across for d? Is my ratio near 3.14? Did I use the right formula (C = πd, or 2πr), and are my units correct?”

Home connection

At home, find a round object, measure its diameter and its circumference with string, and check that the circumference is about 3.14 times the diameter — then use it to predict another circle’s circumference.

Resources

• Circumference and π are standard in middle-grades geometry; see John A. Van de Walle, Elementary and Middle School Mathematics (10th ed., 2019). The notes that π was approximated by the Egyptians (256/81), by Archimedes (bounded by 22/7), and by Zu Chongzhi in China (355/113, c. 5th century, accurate to six decimals) are documented history (MacTutor, “A history of Pi,” S-291); treating all traditions as worthy is a value (docs/philosophy.md §4). The claim that π’s discovery should be open to all is a value commitment.