Course Title:
Algebra I for Students of the Classical Liberal Arts Academy
Course Description:
This course, Algebra I for Students of the Classical Liberal Arts Academy, offers a complete and rigorous introduction to the science of algebra, presented within the framework of a classical Catholic education. Rooted in the tradition of the liberal arts, this course does not treat algebra merely as a set of mechanical procedures or a modern academic requirement, but as a rational science that disciplines the intellect and prepares the mind for higher studies in mathematics, philosophy, and the natural sciences.
Students will study algebra through structured, text-based lessons written in formal, paragraph style—emphasizing the mastery of all terms, definitions, rules, and methods by which the science operates. Rather than rushing into symbolic manipulation, the course teaches students to understand what they are doing and why, forming concepts in clear language before expressing them in algebraic notation. Mathematical symbols are introduced only after the student has understood their meaning through reasoned explanation.
The course begins with a philosophical and historical introduction to algebra as the science of quantity in general, distinguishing it from arithmetic and geometry. It then progresses through all major topics required for high school and college preparation: the basic operations of algebra, expressions and equations, ratios and proportions, exponents and roots, polynomials and factoring, rational expressions, coordinate graphing, and an introduction to quadratic equations.
This course does not follow a fixed 36-week calendar but offers a comprehensive and flexible structure that supports deep, deliberate study. It includes regular opportunities for review, oral memory work, and classical applications, helping students to see not only the utility of algebra, but its beauty and order within the created world.
Students who complete this course will not only be prepared for Algebra II and future mathematical studies but will have developed habits of logical thinking, careful expression, and intellectual discipline essential to the life of the classical scholar.
Part I – Foundations of Algebraic Science
Objective: Establish the purpose, nature, and language of algebra.
Definition and scope of algebra as the science of quantity in general.
Comparison with arithmetic and geometry.
Historical origins and significance in mathematical development.
The Nature of Mathematical Terms and Propositions
Definition of terms, propositions, axioms, and postulates.
Importance of definitions and logical reasoning.
Mathematical language as a precise form of philosophical discourse.
Symbols and Notation in Algebra
Introduction to algebraic symbols as a shorthand language.
Use of letters to represent general numbers (variables and constants).
The equals sign, plus and minus signs, and the concept of identity.
Part II – The Basic Operations
Objective: Understand the foundational operations of algebra and their rules.
Addition and Subtraction of Like Terms
Definition and examples of like and unlike terms.
The principle of combining like terms.
The meaning of subtraction as addition of opposites.
Multiplication and the Law of Signs
Multiplication as repeated addition.
Distributive property.
Law of signs: positive × positive, etc., with verbal reasoning.
Division and Rational Expressions
Division as the inverse of multiplication.
Understanding fractional expressions.
Simplifying algebraic fractions.
Properties of Zero and the Identity Element
Zero in addition, multiplication, and division.
The additive and multiplicative identities and inverses.
Part III – Expressions and Equations
Objective: Introduce the structure of algebraic expressions and the solving of equations.
Constructing and Simplifying Algebraic Expressions
Combining terms, applying operations.
Writing expressions from verbal statements.
Parentheses and order of operations.
Linear Equations in One Variable
Definition and nature of equality.
Solving by inverse operations.
Checking solutions and the concept of identity vs. contradiction.
Word Problems and Translation of Language into Algebra
Verbal reasoning and formulation of equations.
Classical emphasis on understanding the question asked before solving.
Equations with Parentheses and Fractions
Use of distributive property.
Clearing fractions by multiplication.
Part IV – Proportions, Ratios, and Percents
Objective: Understand proportional reasoning in the context of algebra.
The Nature of Ratios and Proportions
Ratio as a comparison of quantities.
Proportion as an equation of two ratios.
Means and extremes, cross-multiplication.
Applications of Proportions in Problem Solving
Classical problems involving parts and wholes.
Proportional reasoning in geometric and real-world settings.
Percentage as a Ratio to 100
Definition of percent.
Problems involving percent increase, decrease, and part-whole relationships.
Part V – Powers, Roots, and Exponents
Objective: Introduce the concept of powers and their properties.
Exponents and Powers
Definition of a power.
Laws of exponents explained with examples.
Negative and Zero Exponents
Inverse operations and the nature of zero exponents.
Use of reciprocals.
Roots and Radicals
The square root and cube root.
Properties of radicals.
Simplifying radicals and combining like radicals.
Part VI – Polynomials and Factoring
Objective: Develop mastery in working with polynomial expressions.
What Is a Polynomial?
Terms, degrees, and classification.
Addition and subtraction of polynomials.
Multiplying Polynomials
Distributive law extended.
FOIL method (introduced as a memory aid after conceptual understanding).
Special Products
Square of a binomial, difference of squares.
Verbal reasoning behind common patterns.
Factoring Techniques
Common factor extraction.
Factoring trinomials.
Factoring special products.
Solving Quadratic Equations by Factoring
Zero-product property.
Applications in word problems.
Part VII – Rational and Irrational Expressions
Objective: Master operations with algebraic fractions.
Operations with Rational Expressions
Adding, subtracting, multiplying, and dividing.
Finding common denominators and simplifying.
Complex Rational Expressions
Simplifying nested fractions.
Interpreting results.
Part VIII – Linear Equations and Inequalities in Two Variables
Objective: Introduce the concept of functions and coordinate representation.
The Cartesian Plane and Ordered Pairs
Definition of a plane, axes, and origin.
Plotting and interpreting points.
Graphing Linear Equations
Table of values method.
Slope and intercept defined in words and visualized.
Slope-Intercept and Standard Forms
Deriving equations from given data.
Translating between forms.
Systems of Linear Equations
Solving by graphing, substitution, and elimination.
Word problems involving multiple conditions.
Inequalities and Their Graphs
Definition and notation.
Graphing one-variable and two-variable inequalities.
Part IX – Introduction to Quadratic Functions
Objective: Prepare students for Algebra II with introductory study of nonlinear equations.
The Nature of Quadratic Equations
Definition and standard form.
Solving by completing the square.
The Quadratic Formula
Derivation through reasoning.
Application and interpretation.
Graphs of Quadratic Functions
Parabolas: vertex, axis of symmetry.
Relation to solutions.
Part X – Classical Applications of Algebra
Objective: Connect the study of algebra with practical and philosophical applications.
Proportionality in Classical Geometry
Application of algebra in the study of classical constructions.
Examples from Euclid and Archimedes.
Problems of Motion and Mixture
Classic verbal problems analyzed algebraically.
Problems involving speed, distance, and concentration.
Number Theory and Algebra
Perfect numbers, prime numbers, and Diophantine equations.
A Review of the Science of Algebra
Summary of all principles, with emphasis on their rational foundations.
Preparation for continued study in Algebra II, Geometry, and Logic.
