Circles in Action

Overview

Students investigate how the radius, diameter and circumference of a circle are related, then use this relationship to calculate circumference. The lesson builds on prior understanding of measuring length, multiplying decimals and substituting values into formulas.

Learning intentions

• WALT identify and measure the radius and diameter of a circle.
• WALT understand that π is the circumference divided by the diameter.
• WALT use (C=\pi d) and (C=2\pi r) to calculate circumference.
• WALT give exact answers and appropriate decimal approximations.
• WALT explain our reasoning and check whether an answer is reasonable.

Success criteria

• I can correctly identify the radius, diameter and circumference.
• I can choose and use a suitable circumference formula.
• I can leave an answer in terms of π or round it appropriately.
• I can explain my method and check my answer using an estimate or a second method.

Prior knowledge and vocabulary

Students should know how to measure length in millimetres and centimetres, distinguish between a radius and a diameter, multiply by decimals, and substitute values into a formula.

Key vocabulary: circle, centre, radius, diameter, circumference, π (pi), exact form, approximation, decimal places, measure, estimate, reasonable.

Curriculum links

• Phase 4 Measurement and Geometry: identify and use the radius, diameter and circumference of circles.
• Phase 4 Measurement and Geometry: understand π as the constant ratio of circumference to diameter.
• Phase 4 Measurement and Geometry: use (C=\pi d) and (C=2\pi r) to solve measurement problems, including appropriate rounding.
• Te Mātaiaho Mathematics and Statistics: Mathsteasers — develop higher-order thinking, reasoning and challenge through mathematical investigation.
• Mathematical practices: represent, calculate, communicate, justify and check mathematical thinking collaboratively.

Lesson structure (60 minutes)

1. 0–5 min · Hook and retrieval. Open with the circle comparison hook showing a bicycle wheel and a round sports-field marking, and ask: “Which has the greater distance around it, and how could we find out?” Students complete a quick think-pair-share, label a circle with centre, radius, diameter and circumference, and recall the relationship between radius and diameter.

2. 5–15 min · Explicit modelling. Use the worked modelling sequence to draw a circle and highlight that the diameter is twice the radius. Wrap a strip of string around a circular lid, mark its circumference, straighten it and compare it with the diameter. Model that (\pi=\frac{C}{d}), so (C=\pi d), then derive (C=2\pi r). Work through:

• (d=8\text{ cm}): (C=8\pi\text{ cm}) exactly, approximately (25.1\text{ cm}) to 1 decimal place.
• (r=5\text{ cm}): (C=2\pi(5)=10\pi\text{ cm}) exactly, approximately (31.4\text{ cm}).

Emphasise writing units, selecting the known measurement, and using the calculator’s π key rather than 3.14 unless instructed. Students answer mini-checks: “If the radius is 7 cm, what is the diameter?” and “Which formula would you choose if diameter is given?”

3. 15–28 min · Practical investigation. Organise students into groups of three, with roles of measurer, recorder and checker. Distribute the circle measurement investigation and provide three to four safe circular objects per group, such as lids, plates, tape rolls or containers. Students measure each object’s diameter and circumference using a ruler, string and tape measure, record units, calculate (\frac{C}{d}), and compare their results with π. Pause midway for a measurement check: students show the teacher one measured diameter and explain how they ensured it passed through the centre.

4. 28–43 min · Differentiated collaborative practice. Students complete the appropriately marked section of the circumference problem set in pairs, explaining each answer to a partner. Core questions include finding circumference from radius or diameter, choosing between the two formulas, and rounding to the nearest centimetre or tenth. Support students use a formula box, labelled diagrams and partially completed substitutions. Students ready for greater challenge solve contextual problems involving a circular garden edge, bicycle wheel travel and a circular table, including one question requiring an exact answer and one requiring a justified approximation. Circulate and ask, “What measurement do you know?”, “Why is your formula suitable?” and “How can you check this?”

5. 43–53 min · Reasoning and misconception check. Display the statements from the discussion and misconception slides:

• “The diameter is the same as the radius.”
• “A circle with diameter 12 cm has circumference about 12 cm.”
• “(C=2\pi d) is the correct formula.”
• “(6\pi) cm and (18.85) cm can represent the same circumference.”

Groups decide whether each statement is true or false, correct false statements and justify their thinking. Address common misconceptions: confusing radius and diameter, measuring across a circle without going through the centre, using area formulas, omitting units, rounding too early, and assuming an answer close to the diameter is reasonable. Reinforce that circumference is a little more than three times the diameter.

6. 53–60 min · Plenary and exit ticket. Return to the hook and ask students to explain how they could compare the distances around the two circles without measuring the entire circumference. Students complete the exit ticket on the final section of the exit ticket questions:
7. Find the circumference of a circle with (d=9\text{ cm}), giving an exact answer and an approximation to 1 decimal place.
8. A circle has circumference approximately (37.7\text{ cm}). Estimate its diameter.
9. State one way to check that your answer is reasonable.

Collect responses as students leave and use them to plan the next lesson.

Resources

• the circumference investigation slide deck
• the circle measurement and problem-solving worksheet
• Rulers and metre rulers
• String or flexible tape measures
• Calculators with a π key
• Safe circular objects: lids, plates, tape rolls or containers
• Blu-tack or masking tape
• Board and markers
• Coloured pencils

Assessment

• Listen to student explanations during modelling and practical measurement; check whether they distinguish radius from diameter and use units consistently.
• Check group investigation tables for sensible measurements and values of (\frac{C}{d}) close to π.
• Use the exit ticket to identify students who can calculate, round, explain and check; revisit formula choice or radius–diameter relationships where needed.

Differentiation

• Support: provide a labelled circle, a formula box, a radius-to-diameter reminder and sentence starters such as “I used ___ because ___ was given.”
• Pair students strategically and assign practical roles so all students contribute; allow a calculator, ruler with clear markings and additional time for recording or reading.
• For EAL learners, use the visual vocabulary on the slides, gesture and demonstrate “around” versus “across”; accept oral explanation before written recording.
• Extension: students investigate why all circles give approximately the same value for (\frac{C}{d}), compare calculator rounding with 3.14, or create a real-world circle problem with an exact and rounded solution.