Geometric Mean - beyond right triangles
Description
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