Lesson 15 Circle functions

# Circle functions

Overview Resources
Pupils explore the area of the circle in relation to the radius (or diameter) and then compare circumference and area as linear and quadratic functions of the radius.

Pupils work first on their intuitive understanding of the circle relationship, then verbalise and clarify their constructions, before exploring features of the graphs of these functions. The lesson follows on from TM14: Tents.
Worksheet 1, Worksheets 2 and 3 photocopies side by side on A3 paper. If possible 4 x 4 grid on board or OHT Circle drawn on 4 x 4 grid
Linking concrete experiences with linear and quadratic functions and their graphical representation.
• Area and circumference of circle as a function of radius.
• Pi
• Continuity.
• EPISODE 1
Intuitively relating area of circle to square on radius
Pupils look at a way for estimating areas of circles to complete the square on the radius, and see how much of the circle it covers. They try this with different size circles to establish that the area of a circle is slightly more than three times the area of the square on the radius. They consider the use of the approximate zvalues, which they met in TM14: Tents.

The approach in this lesson is in contrast to the algorithmic form of A = ar. This quadratic formula is even more onerous to most pupils than the linear one for the circumference. It reinforces pupilsâ€™ view of maths as a subject with fixed rules and formulae on the page with the vaguest correspondence to reality. Pupils need time to ponder this relationship, since it is more complex than the circumference to diameter comparison in TM14.
EPISODE 2
Comparing area and circumference as functions
Pupils consider what happens to the area when the radius changes, and how that compares to what happens to the circumference. They plot values which they have produced for both functions on a graph for comparisons. Pupils attempt to describe the differences between the two graphs, and to estimate mentally areas and circumferences of some circles with easy measurements.
EPISODE 3 (EXTENSION)
Does twice the circumference mean twice the area?
Pupils consider a real-life situation of daffodil bunches sold in two different sizes. One size is half the circumference of the other and half its price. Pupils recognise that they must compare areas and use newly acquired knowledge to argue their ideas.