Portland Community College | Portland, Oregon

Course Number:
MTH 65
Course Title:
Introductory Algebra - Second Term
Credit Hours:
Lecture Hours:
Lecture/Lab Hours:
Lab Hours:
Special Fee:

Course Description

Introduces algebraic concepts and processes with a focus on linear systems, polynomials, quadratic equations, and functions. Emphasizes applications, graphs, formulas, and proper mathematical notation throughout the course. A scientific calculator may be required. The TI-30X II is recommended. Prerequisites: (MTH 60 or MTH 62) and (RD 80 or ESOL 250). Recommended that MTH 60 or MTH 62 be taken within the past 4 terms. The PCC math department recommends that students take MTH courses in consecutive terms. Audit available.

Intended Outcomes for the course

Upon completion of the course students will be able to:

  • Convert between different polynomial forms using operations and factoring.
  • Represent quadratic functions in various forms: symbolic, graphical, numerical, and verbal.
  • Use variables to represent unknown quantities and create models using linear equations, quadratic equations, and systems of equations.
  • Make predictions and interpret the results for models using linear equations, quadratic equations, and systems of equations.
  • Recognize and differentiate between linear and quadratic patterns in ordered paired data, graphs, and equations.
  • Interpret properties such as ordered pairs, and maximum and minimum values in a quadratic relationship.

Outcome Assessment Strategies

  • Develop skills needed to simplify and factor polynomials.
  • Complete activities, assignments, and exams, which show an understanding of the connection between symbolic, graphical, numerical, and verbal representations of quadratics.
  • Complete assignments, projects, test or quiz problems, and class activities which apply real-life situations to linear systems and quadratics.
  • Show an understanding of linear systems and quadratics in a variety of forms through the use of activities, exam problems, projects and/or discussions,
    multiple representations.
  • Participate in, and contribute to, class discussions and activities.
  • Take all scheduled examinations.

Assessment Requirements

  1. The following must be assessed in a proctored, closed-book, no-note, and no-calculator setting:
    • simplifying expressions
    • binomial and trinomial factoring
    • extracting roots
    • solving quadratic equations by factoring
    • solving equations using the quadratic formula
    • graphing quadratic functions
    • solving systems of linear equations
  2. At least two proctored, closed-book, no-note examinations (one of which is the comprehensive final) must be given. These exams must consist primarily of free response questions although a limited number of multiple choice and/or ll in the blank questions may be used where appropriate.
  3. Assessment must include evaluation of the students ability to arrive at correct conclusions using proper mathematical procedures and notation. Additionally, each student must be assessed on their ability to use appropriate organizational strategies and write appropriate conclusions. Application problems must be answered in complete sentences.
  4. At least two of the following additional measures must also be used
    1. Take-home examinations
    2. Graded homework
    3. Quizzes
    4. Projects
    5. In-class activities
    6. Portfolios

Course Content (Themes, Concepts, Issues and Skills)


  • Algebraic manipulation
  • Graphical understanding
  • Problem solving
  • Effective communication
  • Critical thinking
  • Applications, formulas, and modeling


    1. Solve and check systems of equations graphically and using the substitution and addition methods
    2. Create and solve real-world models involving systems of linear equations in two variables
      1. Properly define variables; include units in variable definitions
      2. Apply dimensional analysis while solving problems
      3. State contextual conclusions using complete sentences
      4. Use estimation to determine reasonableness of solution
    1. Apply the rules for integer exponents
    2. Work in scientific notation and demonstrate understanding of the magnitude of the quantities involved
    3. Add, subtract, multiply, and square polynomials
    4. Divide polynomials by a monomial
    5. Understand nonvariable square roots
      1. Simplify using the product rule of square roots including complex numbers (i.e. \(\sqrt{-72}=6i\sqrt{2}\))
      2. Recognize like radical terms
      3. Rationalize denominators (i.e. \(\frac{1}{\sqrt{2}}\) but not \(\frac{1}{3+\sqrt{5}}\))
      4. Estimate square roots
    1. Factor the greatest common factor from a polynomial
    2. Factor a polynomial of four terms using the grouping method
    3. Factor trinomials that have leading coefficients of 1
    4. Factor trinomials that have leading coefficients other than 1
    5. Factor differences of squares
    6. Recognize and factor sums and differences of cubes
    1. Solve quadratic equations using the zero product principle (factoring)
    2. Solve quadratic equations using the square root property
    3. Solve quadratic equations using the quadratic formula including complex solutions
    4. Make choices about the appropriate method to use when solving a quadratic equation
    5. Understand that the solutions satisfy the original equation by checking the solutions
    6. Distinguish between a linear and a quadratic equation and be able to solve both kinds of equations when mixed up in a problem set
    7. Create and solve real-world models involving quadratic equations
      1. Properly define variables; include units in variable definitions
      2. Apply dimensional analysis while solving problems
      3. Use the Pythagorean Theorem to find missing sides of a right triangle, then use the lengths to write the sine, cosine, and tangent of an angle within the triangle
      4. State contextual conclusions using complete sentences
      5. Use estimation to determine reasonableness of solution
    1. Identify a quadratic equation in two variables
    2. Create a table of solutions for the equation of a quadratic function
    3. Emphasize that the graph of a parabola is a visual representation of the solution set to a quadratic equation
    4. Graph quadratic functions by finding the vertex and plotting additional points without using a graphing calculator
    5. Algebraically find the vertex, axis of symmetry, and vertical and horizontal intercepts and graph them by hand
      1. The vertex as well as the vertical and horizontal intercepts should be written as ordered pairs
      2. The axis of symmetry should be written as an equation
    6. Determine whether quadratic functions are concave up or concave down based on their equations
    7. Create, use, and interpret quadratic models of real-world situations algebraically and graphically
      1. Evaluate the function at a particular input value and interpret its meaning
      2. Given a functional value (output), find and interpret the input
      3. Interpret the vertex, vertical intercept, and any horizontal intercept(s) using proper units
    1. Use the definition of a function to determine whether a given relation represents a function
    2. Determine the domain and range of functions given as a graph, given as a set of ordered pairs, and given as a table
    3. Apply function notation in graphical, algebraic, and tabular settings
      1. Understand the difference between the input and output
      2. Identify ordered pairs from function notation
      3. Given an input, find an output
      4. Given an output, find input(s)
    4. Interpret function notation in real world applications
      1. Evaluate the function at a particular input value and interpret its meaning
      2. Given a functional value (output), find and interpret the input


MTH 65 is the second term of a two term sequence in beginning algebra. One major problem experienced by beginning algebra students is difficulty conducting operations with fractions and negative numbers. It would be beneficial to incorporate these topics throughout the course, whenever possible, so that students have ample exposure. Encourage them throughout the course to get better at performing operations with fractions and negative numbers, as it will make a difference in this and future math courses.

Vocabulary is an important part of algebra. Instructors should make a point of using proper vocabulary throughout the course. Some of this vocabulary should include, but not be limited to, inverses, identities, the commutative property, the associative property, the distributive property, equations, expressions and equivalent equations.

The difference between expressions and equations needs to be emphasized throughout the course. A focus must be placed on helping students understand that evaluating an expression, simplifying an expression, and solving an equation are distinct mathematical processes and that each has its own set of rules, procedures, and outcomes.

Equivalence of expressions is always communicated using equal signs. Students need to be taught that when they simplify or evaluate an expression they are not solving an equation despite the presence of equal signs. Instructors should also stress that it is not acceptable to write equal signs between nonequivalent expressions.

Instructors should demonstrate that both sides of an equation need to be written on each line when solving an equation. An emphasis should be placed on the fact that two equations are not equal to one another but they can be equivalent to one another.

The distinction between an equal sign and an approximately equal sign should be noted and students should be taught when it is appropriate to use one sign or the other.

The manner in which one presents the steps to a problem is very important. We want all of our students to recognize this fact; thus the instructor needs to emphasize the importance of writing mathematics properly and students need to be held accountable to the standard. When presenting their work, all students in a MTH 65 course should consistently show appropriate steps using correct mathematical notation and appropriate forms of organization. All axes on graphs should include scales and labels. A portion of the grade for any free response problem should be based on mathematical syntax.

The concept of functions should be covered before quadratic equations, and continually revisited. Use quadratic equations as an example of a function to reinforce the use of function notation, and the concepts of domain and range throughout the course.

Instructors should remind students that the topics discussed in MTH 60 and MTH 65 will be revisited in MTH 95 and beyond, but at a much faster pace while being integrated with new topics.

There is a required notation addendum and required problem set supplement for this course. Both can be found at spot.pcc.edu/math.