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Problem Solving in Algebra

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Eligibility: CTY-level or Advanced CTY-level math score required

Prerequisites: Completion of at least one semester of Algebra I

Course Format: Individually Paced

Course Length: Typically 3 months

Course Code: PS3

Course Description


CTY's Problem Solving courses sharpen investigative skills, broaden mathematical understanding of concepts, and enhance reasoning skills. Designed around performance objectives that reflect national and state mathematical standards and drawing on video lectures provided by Thinkwell, these courses demonstrate how mathematical issues arise out of real-life situations. Concepts are assessed through challenging quizzes and chapter tests. With the guidance of their instructors, students typically select enrichment or supplementary math courses of the appropriate level in between their accelerated courses to ensure a solid foundation and mastery of math concepts at each level. 

In Problem Solving in Algebra, students focus on the symbols and rules of algebra and how they are used to represent relationships. Students master skills and concepts through practice problems, interactions within every lesson, and workout questions. Students apply what they have learned to solve sets of questions at varying levels of difficulty. This course provides an excellent foundation for students who will be advancing into Honors Algebra II.

Each student is assigned to a CTY instructor to help them during their course. Students can contact their instructors via email with any questions or concerns at any time. Live one-on-one online review sessions can be scheduled as well to prepare for the graded assessments, which include homework, chapter exams, and a cumulative midterm and final.

Instructors use virtual classroom software allowing video, voice, text, screen sharing, and whiteboard interaction. Students are strongly encouraged to work on the course at least 1 hour a day, 5 days a week (for a 6-month enrollment), and email their instructors at least once per week.

Upon successful completion of this course, a student will be able to understand concepts typically covered in first-semester algebra, such as:

  • operations on polynomials
  • solving absolute value equations 
  • working with linear equations
  • writing functional relationships
  • examining perpendicular and parallel lines
  • solving equations using multiple operations
  • applications of linear equations and systems
  • solving systems of linear equations and inequalities
  • working with quadratic equations and expressions

For a detailed list of topics, click the List of Topics tab.


Problem Solving in Algebra 1 banner


Materials Needed

There are no required materials for this course.

List of Topics

Chapter 1: The Language of Algebra

Introduction to Polynomials

  • Determining Components and Degree
  • Adding and Subtracting Polynomials
  • Multiplying Polynomials
  • The FOIL Method
  • Multiplying Big Products
  • Using Special Products
  • Factoring Perfect Square Trinomials
  • Factoring the Difference of Two Squares
  • Factoring Sums and Differences of Cubes
  • Factoring by Any Method

Absolute Value

  • Properties of Absolute Value
  • Evaluating Absolute Value Expressions
  • Matching Number Lines with Absolute Value
  • Solving Absolute Value Equations
  • Solving Equations with Two Absolute Value Expressions

Chapter 2: Linear Equations and Functions

Graphing and Writing Linear Equations

  • An Introduction to Slope
  • Finding the Slope Given Two Points
  • Graphing Linear Functions
  • Graphing Important Functions

Introduction to Functions

  • Introducing Relations and Functions
  • Function Notation and Values
  • Functions and the Vertical Line Test
  • Using Intercepts to Graph Lines
  • Working with Specific Lines
  • Writing an Equation in Slope-Intercept Form
  • Writing an Equation Given Two Points
  • Writing an Equation in Point-Slope Form
  • Matching a Slope-Intercept Equation with its Graph
  • Slope for Parallel and Perpendicular Lines
  • Solving Word Problems Involving Functions

Chapter 3: Systems of Linear Equations

Graphic Solutions of Linear Systems

  • An Introduction to Linear Systems
  • Solving a System by Graphing

Algebraic Solutions of Linear Systems

  • Solving a System by Substitution
  • Solving a System by Elimination
  • Investments
  • Solving a Business Problem

Chapter 4: Linear Inequalities

Inequalities in One Variable

  • An Introduction to Solving Inequalities
  • Solving Inequalities
  • Solving Absolute Value Inequalities
  • Solving Word Problems with Inequalities
  • Sets, Intersections, and Unions
  • Solving Compound Inequalities

Inequalities in Two Variables

  • An Introduction to Graphing Inequalities in Two Variables
  • Graphing Linear and Nonlinear Inequalities
  • Graphing the Solution Set of a System of Inequalities

Chapter 5: Quadratic Functions 

The Basics of Quadratics

  •  An Introduction to Quadratics
  •  Solving Quadratics by Factoring

Graphics of Quadratics

  • Finding x- and y-Intercepts
  • Nice-Looking Parabolas
  • Graphing Parabolas

Solving by Completing the Square

  • Finding the Vertex by Completing the Square

Writing Quadratic Equations

  • Using the Vertex to Write the Equation
  • Building a Polynomial from Its Solutions

Solving with the Quadratic Formula

  • Proving the Quadratic Formula
  • Using the Quadratic Formula
  • Predicting Types of Solutions from the Discriminant
  • Using the Discriminant to Graph Parabolas

Equations Quadratic in Form

  • Solving for a Squared Variable
  • Finding Real Number Restrictions

Formulas and Applications

  • Solving a Quadratic Geometry Problem
  • Solving with the Pythagorean Theorem
  • Solving a Motion Problem
  • Solving a Projectile Problem
  • Solving Other Problems


Problem Solving Help Room

Help Room

Each week, all students are invited to a Problem Solving Help Room run by a rotating staff of instructors. Students are encouraged to come with questions or just to meet other online students. Topics reviewed vary each week. 

The Problem Solving Help Room meets each Tuesday from 7 – 8 p.m. ET. 


Sample Video Lecture

Sample Video

Problem Solving in Algebra video

Technical Requirements

Technical Requirements

This course requires a properly maintained computer with high-speed internet access and an up-to-date web browser (such as Chrome or Firefox). The student must be able to communicate with the instructor via email. Visit the Technical Requirements and Support page for more details.

Zoom online virtual classroom
This course uses an online virtual classroom for discussions with the instructor. The classroom works on standard computers with the Zoom desktop client and also tablets or handhelds that support the Zoom Mobile app. Students who are unable to attend live sessions will need a computer with the Zoom desktop client installed to watch recorded meetings. The Zoom desktop client and Zoom Mobile App are both available for free download.