The purpose of this multi-level task is to engage students in an investigation of the area of circles.

The background knowledge presumed for this task is outlined in the diagram below:

The task can be presented with graded expectations to provide appropriate challenge for individual learning needs. The word establish has been used so that students can receive much guidance to verify the given rule, or can deduce a rule, using the radius and circumference of the circle. The extension of the algebraically able student, in this task, is intended to develop the thinking needed to understand calculus. The early ideas of decreasing the size of the sectors, leading to greater accuracy and approaching a true model is a necessary concept for students to understand integration (Level 8 mathematics) though the Level 5 content of areas of a circle.

Task: Find the area of one of these circles and use this to establish a general rule for any circle of radius r.

### The arithmetic approach (show more)

- The student carries out directed calculations that will lead them to verify the formula for the area of a circle.

### The procedural algebraic approach (show more)

- The student carries out an algebraic investigation that would allow them to verify and to show, by example, the derivation of the area of a circle formula.

### The conceptual algebraic approach (show more)

- The student develops a rule, using familiar processes and a chain of reasoning.