Math 407, Fall 2006, Algebra Module

Goals

In the algebra module we aimed to look at the logical and structural underpinnings of algebra. We studied two examples: solving equations as a process of logical proof, and at the structure of algebraic expressions. My goal in each case was for you to think about how to get students in high school to look behind symbolic procedures and make explicit their hidden meaning.

Syllabus

Week 1: Extended analysis

Extended analysis of a high school mathematics problem (see sidebar). Generalization of the problem to a line through an arbitrary point (p,q) in the first quadrant, and to oblique axes. Consideration of algebraic and geometric solutions, and different forms of algebraic expressions of the area.

Week 2: Equations

The relation between solving equations and logical deduction.

Week 3: The quadratic equations and the quadratic formula

The reasoning behind the factoring approach to solving quadratic equations. Relationship with the quadratic formula. Fine structure of the quadratic formula. Assignment 1 was due Tuesday September 19.

Week 4: Algebraic expressions

Looking at the structure of algebraic expressions. Equivalent forms reveal different aspects of the same calculation. Writing problems that make students think about the structure of algebraic expressions.

Of all the lines through the point (5,2), find the line that cuts off the triangle of smallest area in the first quadrant.

Parameterize the equation of the line with one of the parameters m (slope), b (vertical intercept), a (horizontal intercept), c (base of lower small triangle), d (height of vertical triangle), ϑ (angle of line with horizontal axis).
Express the area of the triangle in terms of the parameter you chose.
Find the value of the parameter and the equation of the line that minimises the area.

From Rethinking Secondary Mathematics, part of the TEXTEAMS materials of The Charles A. Dana Center, The University of Texas at Austin.