Any comments, suggestions, or stories of how you used this lesson would be greatly appreciated. I can be contacted through the MSTE Webmaster

### Level and Materials:

Level:

This lesson is intended for students 9-12 who are exploring the statistical concepts of variance and covariance in whatever class they may be currently enrolled. The students should have the following prerequisite knowledge.

• Ability to calculate sums and means.
• Ability to interpret distances.
• Ability to calculate areas of squares and rectangles
• Experience with the slope of a line.
• Some experience with Microsoft Excel and Geometer's Sketchpad.

Materials:

• A Macintosh Computer

### Objectives of lesson:

• To understand the concept of variation as the mean of the sum of squares.
• To calculate means, deviations and variances for data sets.
• To understand the concept of covariation as the mean of the sum of products of deviations.
• To calculate covariances for various data sets.
• To interpret data and make basic inferences based on data and results.
• To become more acquainted with powerful technology such as The Geometer's Sketchpad, Excel and the Internet.
The variance and covariance lessons are independent of each other and can be done separately. However, I suggest that they be taught together since there are many similarities between the two concepts.

Problem SolvingCommunicationReasoningConnections
Algebra . ..

Geometry

Statistics

Probability...

Functions...

### Problem Solving

In the lesson, students are confronted with a problems involving baseball players salaries (variance) and temperature changes with respect to longitude and latitude (covariance). The students learn problem solving techniques using geometric and statistical approaches to infer conclusions about the problems above.

### Communication

Many questions asked of the students involve writing conjectures and conclusions in their own words. Also, students are asked to address the statistical concepts of variance and covariance in writing. Also, students are asked to interpret data and express this in writing.

### Reasoning

Students formulate their own conjectures about variance and covariation and are allowed to test them in both the geometry and statistic settings. Also, their are guided in relating how the procedures in one method are related to the other. Students also are given data in tabular and graphical form and will be able to relate the two.

### Connections

There are numerous connections between various math areas and this lesson. For example, the idea of functions, in regards, to equations of lines is addressed. Also, students will be taught the connection between the graphical and statistical interpretations of variance and covariance. Students will also explore how using algebra is essential to doing statistical problems.