Education Technology

Arithmetic and Geometric means

Published on 08/18/2007

Activity Overview

This activity relates the concepts of the arithmetic and geometric means of two numbers. Students, with the aid of their TI calculators and TI-Navigator system, compute the arithmetic and geometric means for four different pairs of numbers. They send their results to the teacher's computer where they are plotted. The teacher then sends the graph of all of the data back to the students' calculators. The students try to write an equation of a line that borders the region.

Before the Activity

Teacher notes: Required technology: * TI-Navigator 3.0 * Computer projector * Each student will need a TI-83 Plus (operating system 1.19 or higher) or TI-84 Plus (operating system 2.40 or higher) loaded with the applications LearnChk, NavNet, θAlgACT, and θnavstk. Step 1: Start a new class session with the TI-Navigator 3.0 and have the students log in. Step 2: Load the activity settings and press Start Activity to send the forms to the students Step 3: Once the students have completed sending in their data, press Stop Activity. In the upper left-hand corner, change Contribute from Forms to Equations to enable students to submit an equation of a line that borders the graph. Press Start Activity and encourage students to feel comfortable revising their equation based on its graph. Introduction This activity relates the concepts of the arithmetic and geometric means of two numbers. Students, with the aid of their TI calculators and TI-Navigator system, compute the arithmetic and geometric means for four different pairs of numbers. They send their results to the teacher's computer where they are plotted. The teacher then sends the graph of all of the data back to the students' calculators. The students try to write an equation of a line that borders the region.

During the Activity

Student worksheet: Iheoma is 16 years old. Her grandmother is 64. Iheoma has two uncles: James who is 40 and Enobong who is 32. James says that his age is half way between Iheoma's and her grandmother's because 40 is one half of 16 plus 64. Enobong argues that his age is half way between Iheoma's and her grandmother's because he is twice Iheoma's age and that her grandmother is twice his age. Who is correct? The answer is: Both are correct. James's reasoning is based on the average or arithmetic mean of two numbers. Enobong's reasoning is based on the geometric mean. Both means are used extensively in mathematics and in everyday life. The average or arithmetic mean of two numbers is found by calculating the sum of the two numbers and dividing by two. The geometric mean of two numbers is found by calculating the product of the two numbers and then taking the square root of the product. This activity explores the relationship between the arithmetic and geometric means of two numbers

After the Activity

Discussion: The students should conclude that the arithmetic mean (the x-coordinate) is bigger than or equal to the geometric mean (the y-coordinate) of the two numbers. Ask them to determine when the two means are the same. Ask them to determine which birthdays would yield the biggest difference between the two means. Extension: 1) Students can do this activity with either three numbers (using cube roots) or four numbers (using fourth roots) at a time. 2) Students can compare the arithmetic and geometric means with the harmonic mean. The harmonic mean of two numbers, a and b, is . They can then find the relationship between these three means: . An example of the harmonic mean: If you drive 120 miles at an average of 60 miles/hour and then turn around and return at an average of 40 miles/hour then you have traveled a total of 240 miles in a total of 5 hours. Thus your average speed is 48 miles per hour. 48 is the harmonic mean of 60 and 40 since