 Course Number:
 ALC 95B
 Course Title:
 Math 95 Lab  1 credit
 Credit Hours:
 1
 Lecture Hours:
 0
 Lecture/Lab Hours:
 0
 Lab Hours:
 30
 Special Fee:
 $12.00
Course Description
Provides an opportunity to practice and work towards a deeper understanding of individually chosen topics from Intermediate Algebra (MTH 95). Completion of this course does not meet prerequisite requirements for other courses. Audit available.Addendum to Course Description
This class is not intended to be a study hall for students to work on MTH assignments. The time needs to be spent working on material designated by your ALC instructor. If a student is coenrolled in an MTH class, then this may include targeted materials which are intended to support the concepts being taught in that MTH class.
Intended Outcomes for the course
Upon completion of the course students will be able to:

Perform appropriate intermediate algebraic computations in a variety of situations with and without a calculator.

Apply intermediate algebraic problem solving strategies in limited contexts.

Address intermediate algebraic problems with increased confidence.
 Demonstrate progression through mathematical learning objectives established between the student and instructor.
Course Activities and Design
Instructors may employ the use of worksheets, textbooks, online software, minilectures, and/or group work.
Outcome Assessment Strategies
Assessment shall include at least two of the following measures:
1. Active participation/effort
2. Personal program/portfolios
3. Individual student conference
4. Assignments
5. Pre/post evaluations
6. Tests/Quizzes
Course Content (Themes, Concepts, Issues and Skills)
Items from the course content may be chosen as appropriate for each student and some students may even work on content from other ALC courses as deemed appropriate by the instructor.
Intermediate Algebra (MTH 95)
Themes:
 Functions represented as graphs, tables, equations and in words, and connections among these representations
 Algebraic manipulation
 Graphical understanding
 Problem solving
 Effective communication
 Critical thinking
 Applications, formulas, and modeling
 Use of Technology
Skills:
 Calculator (integrated throughout the course)
 Use the home screen to carry out arithmetic operations
 Use the calculator's table feature to explore functions
 Graph functions
 Input the appropriate window settings to view the graph
 Use calculation tools
 Value
 Zero
 Maximum
 Minimum
 Intersect
 Understand that the calculator has limitations
 Functions
 Understand and apply the definition of function
 Determine whether one quantity is a function of another algebraically, graphically, numerically and within realworld contexts by applying the definition of a function
 Domain
 Understand the definition of domain (the set of all possible inputs)
 Determine the domain of functions represented graphically, algebraically, numerically and verbally
 Represent the domain in both interval and set notation, where appropriate
 Apply unions and intersections (AND and OR) when finding and stating the domain of functions
 Understand how the context of a function used as a model can limit the domain
 Range
 Understand the definition of range (the set of all possible outputs)
 Determine the range of functions represented graphically, numerically and verbally
 Represent the range in interval and set notation, where appropriate
 Function notation
 Evaluate functions with given inputs using function notation where functions are represented graphically, algebraically, numerically and verbally (e.g. evaluate \(f(7)\))
 Algebraically simplify and distinguish between different examples such as \(f(x + 2)\), \(f(x) + 2\), \(3f(x)\), and \(f(3x)\)
 Interpret \(f(a) = b\) in the appropriate context e.g. interpret \(f(3) = 5\) where \(f\) models a realworld function
 Solve function equations where functions are represented graphically, algebraically, numerically and verbally (i.e. solve \(f(x) = b\) for \(x\) and solve \(f(x) = g(x)\) for \(x\) where \(f\) and \(g\) should include but not be limited to linear functions, quadratic functions, and absolute value functions)
 Solve function inequalities algebraically (i.e. \(f(x) > b\), \(f(x) > g(x)\), and \(a < f(x) < b\) where \(f\) and \(g\) are linear functions and \(f(x) > b\) and \(f(x) < b\) where \(f\) is an absolute value function)
 Solve function inequalities graphically (i.e. \(f(x) > b\), \(f(x) > g(x)\), and \(a < f(x) < b\) where \(f\) and \(g\) should include but not be limited to linear functions, and \(f(x) > b\) for quadratic and absolute value functions)
 Graphs of functions
 Use the language of graphs and understand how to present answers to questions based on the graph (i.e. read the \(x\) value of an intersection to solve an equation and understand that \(f(2)\) is a number not a point)
 Determine function values, solve equations and inequalities, and find domain and range given a graph
 Apply function notation to prerequisite skill of finding linear equations given two ordered pairs
 Complex Numbers
 Perform operations using complex numbers (i.e. add, subtract, multiply, and divide)
 Rationalize complex denominators (e.g. \(\frac{2}{3+i}\), \(\frac{5}{i}\))
 Absolute Value
 Solve absolute value equations and inequalities
 Solve absolute value function equations where functions are represented graphically, algebraically, numerically and verbally (e.g. solve \(f(x) = b\) for \(x\) and solve \(f(x) = g(x)\))
 Solve absolute value function inequalities algebraically and graphically (e.g. \(f(x) > b\), \(f(x) < b\))
 Solve absolute value equations and inequalities
 Quadratics
 Recognize a quadratic equation given in standard form, vertex form and factored form
 Solve quadratic equations by completing the square
 Find complex solutions to quadratic equations by the quadratic formula or by completing the square
 Understand the graphical implications (e.g. when there is a complex number as a solution to a quadratic equation)
 Interpret the meaning in the context of an application
 Quadratic functions in vertex form
 Graph a parabola after obtaining the vertex form of the equation by completing the square
 Given a quadratic function in vertex form or as a graph, observe the vertical shift and horizontal shift of the graph \(y = x^2\)
 Connect graphing via vertex form with the prerequisite graphing methods (i.e. axis of symmetry, horizontal intercepts, vertex formula, vertical intercept, points found by symmetry)
 Determine the domain and range of quadratic functions algebraically and graphically
 Applications
 Understanding in context: given a quadratic function in algebraic or graphical form find and interpret, including units, the meaning of the:
 Vertex as a maximum or minimum
 Vertical intercept
 Zeros/horizontal intercepts/roots
 Inputs and outputs of functions (e.g. \(f(x) = 5\) and \(f(2)\))
 Clearly define variables including appropriate units
 State conclusions to applied problems in complete sentences including appropriate units
 Understanding in context: given a quadratic function in algebraic or graphical form find and interpret, including units, the meaning of the:
 Explore quadratic functions graphically using the graphing calculator. Convey results using function notation. Examine the following features:
 Vertex
 Vertical intercept
 Horizontal intercepts
 Radicals
 Understand \(n^{th}\) roots
 Determine the domain of radical functions with both even and odd roots algebraically and graphically
 Determine the range graphically
 Understand radicals as equivalent to expressions with rational exponents and vice versa
 Use rational exponents to simplify radical expressions (See addendum)
 Practice prerequisite skills of exponents rules in the context of rational exponents
 Rationalize denominators so students can recognize equivalent expressions (e.g. \(\frac{4}{2\sqrt{5}}\), \(\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}\)
 Solve radical equations algebraically and graphically
 Verify solutions algebraically
 Understand that extraneous solutions found algebraically do not appear as solutions on the graph
 Solve literal radical equations for a specified variable
 Calculator
 Approximate radicals as powers with rational exponents
 Find the domain and range of radical functions
 Solve radical equations graphically
 Use graphical solutions to check the validity of algebraic solutions
 Rational Functions
 Determine the domain of rational functions algebraically and graphically
 Simplify rational functions, understanding that domain conditions lost during simplification MUST be noted
 Rewrite rational expressions by
 Canceling factors common to the numerator and denominator
 Multiplying
 Dividing using both \(\frac{\frac{a}{b}}{\frac{c}{d}}\) and \(\frac{a}{b}\div \frac{c}{d}\)
 Simplify the following cases where \(a\), \(b\), \(c\), and \(d\) represent real numbers, linear polynomials or quadratic polynomials: \(\frac{a}{\frac{b}{c}}\), \(\frac{\frac{a}{b}}{c}\), and \(\frac{\frac{a}{b}}{\frac{c}{d}}\). (See addendum)
 Adding
 Subtracting
 Simplifying complex rational expressions
 The following forms of complex rational expressions shall be simplified: \(\frac{a}{\frac{b}{c}+\frac{d}{c}}\), \(\frac{\frac{b}{c}+\frac{d}{c}}{a}\), \(\frac{\frac{a}{b}}{\frac{c}{d}+\frac{c}{f}}\) where \(a\), \(b\), \(c\), \(d\), \(e\), and \(f\) represent real numbers, linear polynomials in one variable, or quadratic polynomials in one variable. (See addendum)
 Solve rational equations
 Check solutions algebraically
 Solve literal rational equations for a specified variable
 Introduce variables with subscripts
 Applications
 Solve distance, rate and time problems involving rational terms using well defined variables and stating conclusions in complete sentences including appropriate units
 Solve problems involving work rates using well defined variables and stating conclusions in complete sentences including appropriate units
ADDENDUM:
The mission of the Math ALC is to promote student success in MTH courses by tailoring the coursework to meet individual student needs.
Specifically, the Math ALC:

supports students concurrently enrolled in MTH courses;

prepares students to take a MTH course the following term;

allows students to work through the content of a MTH course over multiple terms;

provides an accelerated pathway allowing students to work through the content of multiple MTH courses in one term, allowing placement into the subsequent courses(s) upon demonstrated competency;

prepares students to take a mathplacement exam.
The intended goals from the MTH 95 CCOG follow:
Functions should be studied symbolically, graphically, numerically and verbally.
As much as possible, instructors should present functions that model realworld problems and relationships to address the content outlined on this CCOG.
Function notation is emphasized and should be used whenever it is appropriate in the course.
Students should be required to use proper mathematical language and notation. This includes using equal signs appropriately, labeling and scaling the axes of graphs appropriately, using correct units throughout the problem solving process, conveying answers in complete sentences when appropriate, and in general, using the required symbols correctly.
Students should understand the fundamental differences between expressions and equations including their definitions and proper notations.
All mathematical work should be organized so that it is clear and obvious what techniques the student employed to find his answer. Showing scratch work in the middle of a problem is not acceptable.
Since technology is used throughout the course, there is a required calculator packet for students that gives directions for several graphing calculators. The students should understand the limitations of calculator (i.e. when the calculator gives misleading information). Examples of the calculator limitations include the following: when nding horizontal intercepts, the calculator sometimes gives something like \(y=3\text{E}13\); the calculator rounds to 12 or fewer decimal places; some calculators appear to show vertical asymptotes on the graphs of rational functions; it appears that the graph of \(y = \frac{1}{x^2} touches the \(x\text{axis}\); the calculator does not show holes on rational function graphs; the calculator cannot handle very large numbers (e.g. \(10^{1000}\)).
For dividing rational expressions, focus on examples where the letters represent real numbers and linear polynomials. E.g. \(\frac{\frac{3}{x}}{5}\), \(\frac{x^2}{\frac{x}{5}}\), and \(\frac{\frac{3x}{4}}{\frac{x}{4x^4}}\).
Exploration of difficult rational exponents should be limited. Basic understanding is essential and a deep understanding takes more than one course to develop. Examples should be limited to one or two variables, keeping things as simple as possible while covering all possibilities. E.g. \(5x^{\frac{1}{2}}\), \(3x^{\frac{1}{3}}\), \(\frac{2x^{\frac{1}{3}}}{x^{\frac{1}{2}}}\), and \(4y^{\frac{1}{2}}y^{\frac{1}{3}}\).
When solving applications of quadratic equations, a complex solution should be interpreted as the graph never reaching a particular real world output value.
For simplifying complex rational expressions, a major emphasis shall be placed on cases where \(a\), \(b\), \(c\), \(d\), \(e\), and \(f\) (as above) represent real numbers, linear polynomials in one variable. For example, \(\frac{\frac{1}{x}+\frac{1}{x+2}}{5}\) and \(\frac{\frac{2}{3}}{\frac{1}{x}+\frac{5}{2}}\) would be good examples.
There is a required notation addendum and required problem set supplement for this course. Both can be found at spot.pcc.edu/math.