Portland Community College | Portland, Oregon

Upon completion of the course students should be able to:

• 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.
• Distinguish between linear and non-linear relationships represented by two variables when given a graph, table, verbal description or algebraic formula.
• Identify and interpret the rate of change in linear data, construct a model and use the model to make predictions.
• 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.

• Develop basic algebraic skills necessary to accurately simplify, evaluate, and/or factor polynomial and linear expressions and equations.
• Complete assignments, projects, test or quiz problems, and class activities which apply real-life situations to algebraic representations with both linear and quadratic expressions and equations.
• Show an understanding through the use of activities, exam problems, projects and/or discussions, multiple representations of linear systems and quadratics in a variety of forms.
• Participate in activities, assignments, and exams, which show an understanding of the connection between symbolic, graphical, numerical, and verbal representations of quadratics.
• 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
   • factoring quadratic binomials and trinomials
   • solving linear and quadratic equations
   • solving systems of linear equations
   • graphing linear and quadratic functions
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

Themes:

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

Skills:

1. RELATIONS AND FUNCTIONS
   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
2. LINEAR EQUATIONS AND FUNCTIONS
   1. Solve linear equations and inequalities in one variable
   2. Graph linear functions
      1. Graph a linear function by finding the intercepts
      2. Graph a linear function given its slope and the vertical intercept
      3. Create graphs where the axes are required to have different scales (e.g. Slope of 10 with scale of 1 on the \(x-\text{axis}\) and different scale on the \(y-\text{axis}\).)
      4. Graph a linear function given its slope and a point on the line
   3. Determine the slopes of lines from equations and graphs and interpret their significance as rates of change
   4. Determine linear functions
      1. Find an equation for a linear function given a graph of the function
      2. Find an equation for a linear function given two points
      3. Find an equation for a linear function given a verbal description of a linear relationship, first identifying the independent and dependent variables
   5. Solve systems of linear equations in two variables
      1. Solve systems by graphing
      2. Solve systems by the substitution method
      3. Solve systems by the addition (elimination) method
   6. Applications of linear equations of one variable, linear functions, and systems of linear equations
3. QUADRATIC EQUATIONS AND FUNCTIONS
   1. Review polynomial operations, laws of exponents, and factoring polynomial expressions
      1. Factor expressions having a greatest common factor
      2. Factor binomials
         1. Differences of squares
         2. Sums and differences of cubes
      3. Factor trinomials with leading coefficients of 1 as well as leading coefficients not 1
      4. Factor multivariable expressions
   2. Solve quadratic equations in one-variable
      1. Solve quadratic equations using the square root property
      2. Solve quadratic equations by factoring
      3. Solve quadratic equations using the Quadratic Formula including complex solutions
         1. Simplify and approximate non-variable square roots including complex solutions
         2. Use the Pythagorean Theorem to find missing sides of a right triangle.
   3. Graph quadratic functions
      1. Graph a quadratic function from an equation by creating a table and plotting points
      2. Graph a quadratic function by finding the axis of symmetry, the vertex and the intercepts
      3. 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, the vertical intercept, and any horizontal intercept(s) using proper units
4. LITERAL EQUATIONS AND FORMULAS
   1. Solve an equation for a specified variable in terms of other variables
   2. Input values into a formula and solve for the remaining variable

ADDENDUM:

The purpose of the MTH 70 class is to prepare students to be successful in MTH 95. Several topics in MTH 95 require that students have much more than an introductory understanding of MTH 60/65 material. For instance, students must be able to recognize and quickly factor quadratic expressions if they are to successfully understand rational expressions, the quadratic formula and completing the square in MTH 95. Algebraic concepts covered in MTH 70 will be used in MTH 95 with the expectation that students know and understand them.
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 70 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 introduced at the beginning of the course and continually revisited. Use linear and quadratic equations as examples of functions to reinforce the use of function notation, as well as the concepts of domain and range throughout the course.

Instructors should remind students that the topics discussed in MTH 70 will be revisited in "MTH 95 and beyond", but at a much faster pace while being integrated with new topics.