**ERIC Number:**EJ853816

**Record Type:**Journal

**Publication Date:**2009

**Pages:**7

**Abstractor:**ERIC

**Reference Count:**2

**ISBN:**N/A

**ISSN:**ISSN-0819-4564

Putting a Classroom Spin on Invariance in Circles

Staples, Ed

Australian Senior Mathematics Journal, v23 n1 p51-57 2009

An old chestnut goes something like this. The surface area of a pond in the form of an annulus is required, but the only measurement possible is the length of the chord across the outer circumference and tangent to the inner circumference. It is a beautiful example of invariance. Invariance in mathematics usually refers to a quantity that remains unchanged despite changes to other attributes related to that quantity. In this example, one could start drawing right angled triangles from the common centre, and use a little algebra to find the surface area--but there is an easier way. Those familiar with this problem will know that for a fixed chord length, the inner and outer radii can vary without changing the area of the annulus. This means that one can reduce the inner radius to zero, so that the chord becomes a diameter of the outer circle. Therefore, the area of the annulus is the area of the circle with that diameter. This article puts a classroom spin on invariance in circles. (Contains 8 figures.)

Descriptors: Mathematics Instruction, Mathematical Concepts, Geometric Concepts, Measurement Techniques, Algebra, Validity, Mathematical Logic, Equations (Mathematics), Problem Solving, Secondary School Mathematics

Australian Association of Mathematics Teachers (AAMT). GPO Box 1729, Adelaide 5001, South Australia. Tel: +61-8-8363-0288; Fax: +61-8-8362-9288; e-mail: office@aamt.edu.au; Web site: http://www.aamt.edu.au

**Publication Type:**Journal Articles; Reports - Descriptive

**Education Level:**Secondary Education

**Audience:**N/A

**Language:**English

**Sponsor:**N/A

**Authoring Institution:**N/A