Geometric Mean - beyond right triangles

Geometric mean is a topic in geometry that is only used to calculate a pointless length in right triangles. It is a throwaway topic, right? If your students are tired of having their time wasted memorizing calculations they will never use again, they will be demanding to know why they should learn about the geometric mean. Included are three mini-lessons -- one each for geometry honors, precalculus honors, BC calculus. You are saying, "But I only teach, geometry. Why would I bother paying for the other lessons?" Because you know someone who teaches precalculus or calculus and they would like to present a lesson at least once that does not involve trig identities and you are a good friend. Also, you are intrigued. It is simple enough to be understood by gradeschool students, but unless you have a graduate degree in Pure Math, you have never seen this before.

Pre-planning:

3 mini-lessons, each two pages: geometry honors, precalculus honors, BC calculus

40-90 minutes depending on class level and focus

A scientific (geometry) or graphing (precalc, calc) calculator

Put students into pairs or groups

Prerequisite Knowledge:

(Geometry)

average

arithmetic and geometric sequences and series

(Precalculus)

logarithm laws

sigma notation

(Calculus)

integration by parts

definition of definite integral

definition of derivative

exponential models

factorial notation

complex numbers in polar form

Learning Objectives:

learn the definition of geometric mean and practice calculating it

review arithmetic ideas and generalize them by analogy

deepen understanding of geometric sequences and series including a proof of the Root Test

use mathematical analysis to provide evidence for historical analysis

appreciate the grand scope of mathematics that is not contained in the textbook