# Introductory Algebra - Second Term

#### Prerequisites

MTH 60 and placement into RD 90

## Course Description

Introduces algebraic concepts and processes with a focus on functions, linear systems, polynomials, and quadratic equations. Applications, graphs, functions, formulas, and proper mathematical notation are emphasized throughout the course. A scientific calculator is required. Prerequisites: MTH 60 and placement into RD 90. Audit available.

## Intended Outcomes

1. Recognize and differentiate between linear and quadratic patterns in ordered paired data, graphs, and equations.
2. Use variables to represent unknowns in linear or quadratic problems, create a linear system or quadratic equation that represents the situation, and find the solution to the problem using algebra.
3. Be successful in future coursework that requires the use of basic algebraic concepts and an understanding of functions.

## Outcome Assessment Strategies

Assessment shall include:

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

## Course Content (Themes, Concepts, Issues and Skills)

#### THEMES

1. Functions
2. Graphical understanding
3. Algebraic manipulation
4. Number sense
5. Problem solving
6. Applications, formulas, and modeling
7. Critical thinking
8. Effective communication

#### SKILLS

1.0 SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES
1.1 Solve and check systems of equations graphically and using the substitution and addition methods.
1.2 Create and solve real-world models involving systems of linear equations in two variables.
1.2.1 Properly define variables; include units in variable definitions.
1.2.2 Apply dimensional analysis while solving problems.
1.2.3 State contextual conclusions using complete sentences.
1.2.4 Use estimation to determine reasonableness of solution.

2.0 WORKING WITH ALGEBRAIC EXPRESSIONS
2.1 Apply the rules for integer exponents.
2.2 Work in scientific notation and demonstrate understanding of the magnitude of the quantities involved.
2.3 Add, subtract, multiply, and square polynomials.
2.4 Divide polynomials by a monomial.
2.5 Understand nonvariable square roots.
2.5.1 Simplify using the product rule of square roots.
2.5.2 Recognize like radical terms.
2.5.3 Rationalize denominators.
2.5.4 Estimate square roots.

3.0 FACTORING POLYNOMIALS
3.1 Factor the greatest common factor from a polynomial.
3.2 Factor a polynomial of four terms using the grouping method.
3.3 Factor trinomials that have leading coefficients of 1.
3.4 Factor trinomials that have leading coefficients other than 1.
3.5 Factor differences of squares.
3.6 Recognize and factor sums and differences of cubes.

4.0 QUADRATIC EQUATIONS IN ONE VARIABLE
4.1 Solve quadratic equations using the zero product principle (factoring).
4.2 Solve quadratic equations using the square root property (see Section 2.5).
4.3 Solve quadratic equations using the quadratic formula (see Section 2.5).
4.4 Make choices about the appropriate method to use when solving a quadratic equation.
4.5 Understand that the solutions satisfy the original equation by checking the solutions.
4.6 Distinguish between a linear and a quadratic equation and be able to solve both kinds of equations when mixed up in a problem set.
4.7 Create and solve real-world models involving quadratic equations.
4.7.1 Properly define variables; include units in variable definitions.
4.7.2 Apply dimensional analysis while solving problems.
4.7.3 State contextual conclusions using complete sentences.
4.7.4 Use estimation to determine reasonableness of solution.

5.0 QUADRATIC EQUATIONS IN TWO VARIABLES
5.1 Identify a quadratic equation in two variables.
5.2 Create a table of solutions for the equation of a quadratic function.
5.3 Emphasize that the graph of a parabola is a visual representation of the solution set to a quadratic equation.
5.4 Graph quadratic functions by finding the vertex and plotting additional points without using a graphing calculator.
5.5 Algebraically find the vertex, axis of symmetry, and vertical and horizontal intercepts and graph them by hand.
5.5.1 The vertex as well as the vertical and horizontal intercepts should be written as ordered pairs.
5.5.2 The axis of symmetry should be written as an equation.
5.6 Determine whether quadratic functions are concave up or concave down based on their equations.
5.7 Create, use, and interpret quadratic models of real-world situations algebraically and graphically.
5.7.1 Evaluate the function at a particular input value and interpret its meaning.
5.7.2 Given a functional value (output), find and interpret the input.
5.7.3 Interpret the vertex using proper units.
5.7.4 Interpret the vertical intercept using proper units.
5.7.5 Interpret the horizontal intercept(s) using proper units.

6.0 RELATIONS AND FUNCTIONS
6.1 Use the definition of a function to determine whether a given relation represents a function.
6.2 Determine the domain and range of a function given as a graph or as a table.
6.3 Apply function notation in graphical, algebraic, and tabular settings.
6.3.1 Understand the difference between the input and output.
6.3.2 Identify ordered pairs from function notation.
6.3.3 Given an input, find an output.
6.3.4 Given an output, find input(s).
6.4 Interpret function notation in real world applications.
6.4.1 Evaluate the function at a particular input value and interpret its meaning.
6.4.2 Given a functional value (output), find and interpret the input.

## Department Notes

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.

Proper usage of equal signs must be stressed at all times. Students need to be taught that equal signs are used to communicate multiple ideas and they need to be taught the manner in which equal signs are used to communicate these ideas.

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.

Instructors should demonstrate and emphasize the importance of performing operations in a vertical format. Equal signs must be used when changing the form of an expression. Examples of a vertical format are as follows:

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.