I draw great inspiration from the problem-based approach of distinguished mathematicians such as Gelfand in the Soviet mathematics education (see his trilogy: Algebra, Functions and Graphs, and Method of Coordinates) and Kodaira in the Japanese mathematics education (see his sequence: Mathematics 1, Mathematics 2, and Algebra and Geometry). 

Each class will consist of solving 5 to 10 interesting or/and challenging problems. The problems are chosen so that the students will:
-Develop their problem-solving and analytical skills
-Understand how algebra fits with the broader structure of mathematics
-Exploit fundamentals of arithmetics to extend their understanding of algebra (without memorizing algorithms)
-Discover how reasoning through a few fundamentals of algebra allows us to know far more
-Enjoy learning about algebra 
All of which will help to bridge the gap between high school and university math at an early stage. 

The ideal student is taking or has taken, a traditional Introduction to Algebra 1 course, usually with an emphasis on the presentation and execution of algorithms but now wants to be comfortable applying those fundamentals to solve novel problems. (Foerester’s book Algebra I: Expressions, Equations and Applications, provides a good foundation). 

I have arranged the problems that we will consider in each class to match the content of the first half of a typical honors Algebra 1 course. These topics include: Expressions and Equations, Solving One-Variable Linear Equations, Distributing: Axioms and Other properties, Harder Equations, Some Operations With Polynomials And Radicals, Quadratic Equations, Expressions And Equations Containing Two Variables.

How is the class structured? 
Students will be presented with a problem. When faced with an easier problem, the student will have a limited amount of time, after which we go over the solution and review relevant fundamentals. When faced with a particularly challenging problem, each student can work as a team or on their own. At such times, I will offer some suggestions and review the relevant fundamentals before allowing the class to complete the problem. At the end of class, we will discuss the bigger picture and establish deeper connections to Algebra.  

Students may join all of the following classes or may choose from the following topics according to their preference:

Lesson 1: Powers and Exponents
Laws of exponents for positive integer exponents
Powers of 10 and Scientific Notation
Word Problems and Graphs


Lesson 2: Dealing with Fractions
Definition of a fraction
Word Problems
Direct Proofs


Lesson 3: Translation of Verbal Information into Symbols
Translation
Introduction to Equations
Word Problems


Lesson 4: Solving One-Variable Linear Equations
Word Problems
Solving Equations


Lesson 5: Percent, ratio, and rate problems
Word Problems
Solving Equations


Lesson 6: Expressions And Equations Containing Two Variables
Word Problems
Solving Two Variables Linear Equations
Elimination Method
Substitution Method


Lesson 7: Some Operations With Polynomials And Radicals
Distributing: Axioms and Other properties
The square of a sum
The difference of squares
Multiplying Polynomials
Coefficient and values


Lesson 8: Some Operations With Polynomials And Radicals
Factoring Polynomials
The Greatest Common Factor
Factoring Polynomials That Have Common Factors
Common Binomial Factors


Lesson 9: Quadratic Equations
Solving Quadratic Equations by Factoring 
Introduction to the Quadratic Formula
Completing the Square
Complex numbers


Lesson 10: Expressions And Equations Containing Two Variables
The Cartesian Coordinate System
Graphs of Equations Containing Two Variables
Intercepts and Rapid Graphing
Slope and Rapid Graphing
Finding the Intersection of Two Graphs by Accurate Plotting