Circles and Pi

Year Level

Year 7 (Australian Curriculum v9)

Duration

45 minutes

Class Size

12 students

---

Learning Objectives

By the end of this lesson, students will be able to:

• Describe the relationship between π and the features of circles including circumference, radius, and diameter.
• Recognise and label parts of a circle (centre, radius, diameter, circumference).
• Understand and use formulas connecting circumference with radius and diameter.
• Investigate how π acts as a constant in the ratio of circumference to diameter.
• Explore cultural applications of circles, including Indigenous Australian perspectives.

Australian Curriculum Alignment:

• AC9M7M03: Describe the relationship between π and the features of circles including the circumference, radius, and diameter.

---

Curriculum Elaborations to Address

• Labelling and recognising circle features; diameter twice the radius; radius equals circumference divided by 2π.
• Using measuring tools (compass, string) to compare circumference and diameter and empirically establish formulas.
• Investigating π as a constant and its historical approximations.
• Applications of circle concepts in daily life and Indigenous Australian contexts.

---

Lesson Outline

1. Introduction & Engagement (5 minutes)

• Begin with a brief discussion: "What shapes do you see every day that are circles?"
• Show real-life circular objects (e.g., CDs, lids).
• Ask: "What do you notice about the sizes of these circles? How might we describe their sizes mathematically?"

2. Teaching and Demonstration (10 minutes)

• Draw a large circle on the board.
• Label the features: centre, radius (r), diameter (d), circumference (C).
• Explain:
  • Diameter = 2 × radius.
  • Circumference is the distance around the circle.
• Introduce π (pi) as the special constant ratio where Circumference = π × Diameter.
• Derive the key formula: ( C = 2πr ).
• Relate the formulas clearly linking circumference, diameter, and radius.

3. Hands-on Investigation Activity (15 minutes)

Materials: Compasses, rulers, strings, circular objects (tin lids, paper plates), calculators.

• Students draw three different sized circles using compasses.
• Using string, measure each circumference (wrap string around and then measure string length).
• Measure diameter of each circle with ruler.
• Calculate the ratio circumference / diameter for each (using calculators).
• Record results to observe π ≈ 3.14.
• Discuss findings; reinforce π as a constant ratio regardless of circle size.

4. Cultural Connection and Reflection (5 minutes)

• Briefly share how circles and circular designs are significant in some First Nations Australian art, basketry, and architecture.
• Discuss how understanding circle properties can help in practical life and cultural crafts.
• Invite students to share or imagine other uses of circles in daily life or other cultures.

5. Independent Practice & Formative Assessment (7 minutes)

• Worksheet with problems:
  • Label circle parts.
  • Given radius or diameter, calculate circumference using π.
  • Fill in missing lengths.
  • One question asking to explain the relationship between π, diameter, radius, and circumference in their own words.
• Circulate and provide feedback as students work.

6. Conclusion and Summary (3 minutes)

• Recap key points:
  • π links circumference and diameter.
  • Diameter is twice the radius.
  • Working out circumference uses π and radius/diameter.
• Invite questions.
• Highlight the importance of π in mathematics and everyday life.

---

Resources and Materials

• Whiteboard and markers.
• Compasses, rulers, string pieces.
• Variety of circular objects.
• Calculators.
• Student worksheet (prepared before class).

---

Differentiation

• Provide measuring aids and guided steps for students who need support.
• Extension: Challenge some students to research historical approximations of π or create their own circle art applying learnt concepts.
• Encourage group collaboration for shared investigation and peer teaching.

---

Assessment

Formative:

• Observation during measuring activity.
• Correct use of formulas in worksheet.
• Students' verbal explanation of π relationship during discussion.

---

Reflection for Teachers

• Did students grasp that π is a constant ratio?
• Were students able to measure and calculate accurately?
• Engage with students’ cultural perspectives to enrich understanding.
• Consider follow-up lessons on circle area or coordinate plotting of circle features.

---

This lesson integrates inquiry-based learning and cultural context relevant to Australian students while closely aligning with Year 7 content description AC9M7M03 of the Australian Curriculum (v9).