代数 1 の興味深い問題: パート B

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:


Week of January 21 - 27: Exponents
Review
Negative powers
Non-integer


Week of January 28 - February 3: Polynomials
Formulas for (a + b)...
Coefficients and values
Factoring
Rational expressions
Converting a rational expression into the quotient of two polynomials


Week of February 4 - 10: Polynomials
Rational fractions
Division of polynomials
Interpolation


Week of February 11 - 17: Quadratics Equations
Completing the Square and the Quadratic Formula 
Graphing


Week of February 18 - 24: Quadratics Equations
Optimizing Quadratics and AM-GM Inequality 


Week of February 25 - March 2: Functions


Week of March 3 - 9: Composition and Inverses 


Week of March 10 - 16: Graphing Functions


Week of March 17 - 23: Exponential Functions


Week of March 24 - 30: Special Functions
Logarithms
Radicals
Absolute Value 


Week of March 31 - April 6: Special Functions
Floor & Ceiling
Rational
Piecewise 


Week of April 7 - 13: Sequences & Series
Arithmetic and Finite Geometric 


Week of April 14 - 20: Sequences & Series
Infinite Geometric and Telescoping 


Week of April 21 - 27: More Interesting Problems