Mathematical goals

This lesson unit is intended to help students to apply their knowledge of the relationship between the radius and the area of a circle. In particular, students will:

• Sketch and construct loci involving arcs of circles.
• Calculate the area of a circle, given its radius.
• Calculate the radius of a circle, given its area.

Introduction

The lesson unit is structured in the following way:

• Before the lesson, students work individually on an assessment task designed to reveal their current understanding and difficulties. You then review their solutions and create questions for students to help them to improve their work. They receive this feedback in the follow-up lesson.
• After a whole-class introduction students work in small groups on a collaborative task, finding the rope length necessary to enable a goat to eat a specified area of grass.
• In the same small groups, students evaluate and comment on sample responses, identifying strengths and weaknesses and comparing them with their own work. After a whole-class discussion, they then try to improve their collaborative work.
• In an optional final whole-class discussion, students discuss the methods they have seen and used.
• Finally, students work alone on a new assessment task, or try to improve their original responses.

Materials required

• Each individual student will need a copy of the assessment tasks The Goat and the Grass and The Goat and the Grass (revisited). Calculators, rulers and compasses should be made available.
• Each small group will need plain paper and copies of the Field and Sample Responses to Discuss.

Time needed

15 minutes before the lesson, a 100-minute lesson (or two 60-minute lessons), and 15 minutes in a follow-up lesson. Timings are approximate and will depend on the needs of the class.