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TEACHER'S GUIDE
This activity is aimed at enriching the algebra strand by bringing to the classroom a medical application of systems of linear equations. These problems go beyond the traditional interpretation of systems of equations as intersections of lines and planes and furnish a nice mix of algebraic and geometric concepts. The activity is designed for small-group work.
Grade levels: 10-12
Materials: Calculators or software with the capability to solve matrices or linear systems
Prerequisites: Students must be familiar with at least one method for solving systems of equations involving three or four variables. The traditional high school techniques--linear combinations or substitution--are usually too cumbersome for larger systems, but in this activity, since the coefficients are always 1 and each equation includes only two variables, those techniques will work.
The activity on sheet 3 is recommended as an introduction to, or an application of, more general and powerful techniques that are based on the use of technology. Many graphing calculators and software packages include the ability to solve systems of linear equations, either by a direct entry of the equation or through the manipulation of matrices. Taking advantage of this feature allows the teacher to spend more instructional time on interesting mathematical questions and less time on error-prone routine manipulations.
Directions: Sheets 1 and 2 introduce and apply the basic concepts and techniques. If time or technology is not available for sheet 3, sheets 1 and 2 can stand alone. Completing sheet 3 is recommended, however, since it extends the activity into rich mathematical territory.
Sheet 1: The purpose of this sheet is to introduce the idea behind CAT scanners and to explain how they are used in medical research and diagnosis. Color photographs that create interest in the topic are available in such publications as the National Geographic, whose January 1987 cover story, "Medicine's New Vision," featured this topic (Sochurek and Miller 1987).
This sheet is not difficult, but students should not rush through it. Instead, it should be used as an opportunity to present the following information, which creates a context for sheets 2 and 3. The figures on sheet 1 focus the presentation and encourage discussion. Displaying an overhead transparency of the sheet facilitates whole-class discussion.
Medical scanners use X-rays measured...