University of Arizona

Math 407, Fall 2007, Algebra Module

Goals

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

Syllabus

Aug 20, 22, 27: Equations

The relation between solving equations and logical deduction. The reasoning behind the factoring approach to solving quadratic equations. Relationship with the quadratic formula. Fine structure of the quadratic formula. Assignment 1 is due Wednesday, September 5.

What was on the smartboard


September 3: Labor Day


Aug 29, Sep 5, 10, 12: 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. Reasoning behind and purpose of completing the square. Logical structure of the exponent rules.


What is this?

Quadratic Formula

Click on the picture for a larger view. Send your answers to the listserv.


Properties of numbers

Send an email to the listserv by Wednesday, August 22 with your proposed addition for this list, or with a comment on or rewording of somebody else's addition.

  • You can add and multiply any two numbers and get another number.
  • You can do addition in any order and any grouping (this combines the commutative and associative laws of addition).
  • You can do multiplication in any order and any grouping (ditto for multiplication).
  • If A=B and B = C then A=C (and same property for > and <).
  • If a + b = a + c then b = c.
  • For any real number x, there exists a number y such that xy = 1.
  • For any real number R, there exists a number S such that R+S = 0.
  • For any real numbers a, b, and c, we have (a+b)c = ac + bc.
  • There exists an additive identity, 0, such that x+0=x and 0+x=x for any x.
  • There exists a multiplicative identity, 1, such that x×1=x and 1×x=x for any x.
  • For any number A, we have A × 0 = 0.