## Course Content and Outcome Guide for ALC 95A

- Course Number:
- ALC 95A
- Course Title:
- Math 95 Lab - 0 credits
- Credit Hours:
- 0
- Lecture Hours:
- 0
- Lecture/Lab Hours:
- 0
- Lab Hours:
- 0
- Special Fee:

#### Course Description

Provides a review of individually chosen topics in Intermediate Algebra (Math 95). Completion of this course does not meet prerequisite requirements for other math courses.#### Intended Outcomes for the course

Upon successful completion of this course students will be able to:

- Creatively and confidently apply intermediate-level algebraic problem solving strategies.
- Be prepared for future course work.

#### Outcome Assessment Strategies

Assessment shall include at least two of the following measures:

1. Tests

2. Attendance

3. Portfolios

4. Individual student conference

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

**Intermediate Algebra (MTH 95)**

Themes:

Functions and function notation

Functions represented as graphs, tables, equations and in words

Connection between symbolic and graphical representations

Algebraic simplification of expressions and solving of equations

Problem solving and modeling, interpreting results in practical terms

Language of graphs

Skills:

Calculator (integrated throughout the course)

Use the home screen 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

ntersect

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 real life

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 (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 )

Algebraically simplify and distinguish between different examples such as , ,

and

nterpret in the appropriate context e.g. interpret where models a real-world function

Solve function equations where functions are represented graphically, algebraically, numerically and verbally (i.e.

solve for and solve for where and should include but not be limited to linear

functions, quadratic functions, and absolute value functions)

Solve function inequalities algebraically (i.e. , , and where and are

linear functions and and where is an absolute value function)

Solve function inequalities graphically (i.e. , , and where and should

include but not be limited to linear functions, and 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

value of an intersection to solve an equation and understand that 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

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 (i.e. when there is a complex number as a solution to a quadratic equation)

nterpret 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

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

Zeroes/horizontal intercepts/roots

nputs and outputs of functions (e.g. and )

Clearly define variables including appropriate units

State conclusions to applied problems in complete sentences including appropriate units

Explore quadratic functions graphically using the graphing calculator. Convey results using function notation. Examine

the following features:

. Vertex

. Vertical intercept

. Horizontal intercepts

Radical Functions

Understand nth roots

Determine the domain of radical functions with both even and odd roots algebraically and graphically

Determine the range graphically

Understand radicals as 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. )

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 and notation

Simplify the following cases where a, b, c, d represent real numbers, linear polynomials or quadratic polynomials:

, and . (See addendum)

. Adding

. Subtracting

. Simplifying complex rational expressions

The following forms of complex rational expressions shall be simplified: , , , and

where , , , , , and 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

Functions should be studied symbolically, graphically, numerically and verbally.

As much as possible, instructors should present functions that model real-world 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 calculatori.e. when

the calculator gives misleading information. Examples of the calculators limitations include the following: when

finding horizontal intercepts, the calculator sometimes gives something like y = 3E-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 touches the x axis; the calculator does not show holes on rational function

graphs; the calculator cannot handle very large numbers, e.g. etc.

For dividing rational expressions as in 5.3.3 and 5.3.3.1, focus on examples where the letters represent real

numbers and linear polynomials. E.g. , , and .

Exploration of difficult rational exponents, as in 4.5, 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. , , , .

As much as possible, instructors should present functions that model real-world problems and relationships to

address the content outlined on this CCOG.

In 3.3.1, when solving applications of quadratic equations, a complex solution should be interpreted as the graph

never reaching a particular real world y-value.

For simplifying complex rational expressions as in 5.3.6.1, a major emphasis shall be placed on cases where

, , , , , and (as above) represent real numbers, linear polynomials in one variable. For

example, or would be good examples.