Interesting Problems in Algebra 1: Part A

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

Lesson 2: Solving One-Variable Linear Equations
Solving Equations
Word Problems

Lesson 3: Percent, Ratio, and Rate Problems
Solving Equations
Word Problems

Lesson 4: Multi-Variable Linear Equations
Word Problems
Solving Two Variables Linear Equations
Elimination Method
Substitution Method

Lesson 5: Powers and Exponents
Laws of exponents for positive integer exponents
Powers of 10 and Scientific Notation
Word Problems and Graphs

Lesson 6: Some Operations With Polynomials And Radicals
Distributing: Axioms and Other properties 
Multiplying polynomials

Lesson 7: Some Operations With Polynomials And Radicals
Special Factorization
Difference of squares
Sum and difference of cubes
Simon’s favorite factoring trick

Lesson 8: Quadratic Equations - Part 1
Solving Quadratic Equations by Factoring 
Word Problems
Vieta

Lesson 9: Quadratic Equations - Part 2
Solving Quadratic Equations by Factoring 
Completing the Square
Introduction to the Quadratic Formula

Lesson 10: Quadratic Equations - Part 3
Complex Numbers
The Quadratic Formula

Lesson 11: Expressions And Equations Containing Two Variables
The Cartesian Coordinate System
Graphs of Equations Containing Two Variables
Intercepts and Rapid Graphing
Slope and Rapid Graphing

Lesson 12: Introduction to Inequalities
Linear inequalities 
Direct Proofs
Graphing inequalities 
Optimization