Concurrent Enrollment Mathematics 1010 requires students to expand their knowledge of basic algebra concepts introduced in Beginning Algebra. The course is organized into several modules that will direct students as they attempt to understand and then apply mathematical concepts. These modules include: Linear Equations, Polynomial Equations, Rational Equations, Radical Equations, Exponential and Logarithmic Equations, Systems of Equations, Functions, and Real World Application of mathematics. The course necessitates that students reason abstractly and quantitatively to understand each of these modules. Further, students will use appropriate tools, including technology, to model their mathematical thinking, and use structure and regularity to describe mathematical situations and solve problems.

Mathematics 1010 consists of eight modules: Linear Equations and Inequalities, Functions, Systems of Linear Equations, Polynomial Expressions and Equations, Rational Expressions and Equations, Radical Expressions and Equations, Quadratic Equations, and Exponential and Logarithmic Equations. The modules are presented in the syllabus below in a recommended order; however, it is up to the Institution of Higher Education (IHE) with which a secondary institution has a contractual agreement to decide the order of the modules or whether to let the instructor decide for him or herself the order.

At the end of each module, it is recommended that students take the Utah System of Higher Education (USHE) module assessment as a formative indicator of student progress. At the end of the course, there is also a required USHE final assessment. Based on this final assessment, students will receive one of three designations: “exceeded expectations,” “met expectations,” or “did not meet expectations.” Individual high schools and IHE’s determine high school and college credit and grades for Math 1010.

Each module identifies the “Mathematical Foundation” and “Application of Mathematics” for module topics. The Mathematical Foundation identifies the content that is to drive the understanding, communication, and decision contexts identified by the Mathematical Foundation list. The Mathematical Foundation topics only identify the broad mathematical concepts within the course. It is understood and expected that the course involve rich mathematics and real application. The ultimate goal (as explicit in the Expected Learning Outcomes section below) is that students leave the course with a fundamental ability to apply and use the mathematical foundation to a variety of contexts they are likely to experience in future math courses.

Text: Although the materials being prepared for this course are meant to suffice for instruction, we recommend, where feasible, adopting the text used in many Utah higher education institutions:

Intermediate Algebra, by M. Sullivan and K. Struve, Second Edition, Pearson Publishing

Standard I: Students will use linear expressions, equations, and inequalities

Mathematical Foundation:

Linear Expressions
Linear Equations
Linear Inequalities
Problem solving
Compound Inequalities
Absolute Values

Applications of Mathematics:

Objective 1: Understand how to evaluate an expression

1. Understand order of operations
2. Evaluate an expression with one variable
3. Evaluate an expression with more than one variable

Objective 2: Understand how to solve linear equations

1. Understand the addition and multiplication properties of equality
2. Solve an equation in one variable
3. Solve for a variable in a formula
4. Graph linear equations in one variable
5. Graph linear equations in two variables utilizing point-plotting, slope, and intercepts
6. Understand how two lines can be parallel or perpendicular and find the equations of lines parallel or perpendicular to other lines

Objective 3: Understand how to solve linear inequalities

1. Understand the addition and multiplication properties of inequality
2. Solve an inequality in one variable, expressing the answer in inequality notation, set builder notation, and interval notation
3. Graph an inequality in one variable
4. Solve an inequality in two variables by graphing

Objective 4: Use linear equations to solve problems

1. Translate English sentences into mathematical statements
2. Model and solve direct translation problems
3. Model and solve mixture problems
4. Model and solve uniform motion problems
5. Model and solve geometry problems

Objective 5: Understand how to solve compound inequalities

1. Determine the intersection or union of two sets
2. Solve compound inequalities involving “and”
3. Solve compound inequalities involving “or”
4. Solve problems using compound inequalities

Objective 6: Understand how to use absolute values

1. Solve absolute value equations
2. Solve absolute value inequalities

Standard II: Students will understand how relations and functions work

Mathematical Foundation:

Relations: Inputs and outputs, domain and range, ways to express
Functions: Definitions, domain and range, ways to express
Linear Functions: Graphing, Solving, Zeros

Applications of Mathematics:

Objective 1: Use Relations to express mathematical models

1. Express a relation as a map
2. Express a relation as a set
3. Express a relation as a graph
4. Express a relation as an equation
5. Define and find domain and range of a relation

Objective 2: Use Functions to express mathematical models

1. Determine if a relation is a function as a map, a set, a graph, or an equation
2. Find the domain and range of a function with and without graphing
3. Evaluate a function for a particular input
4. Graph a function

Objective 3: Model Linear Functions

1. Understand the relation between linear equations in two variables and functions
2. Evaluate linear functions
3. Find the zeros of a linear function
4. Graph linear functions
5. Build linear models from verbal descriptions and data

Standard III: Students will use systems of equations to solve mathematical models

Mathematical Foundation:

Systems of linear equations: Two variables systems, Substitution, Elimination, Graphing
Problem solving with systems of equations
Systems of linear equations: Three variables systems, Elimination, Gaussian elimination
Systems of linear inequalities: Graphing, Boundaries, Boundary Points

Applications of Mathematics:

Objective 1: Understand how to solve a linear system of equations in two variables

1. Determine if a given ordered pair is a solution to a system of linear equations
2. Solve a system of equations in two variables by graphing
3. Solve a system of equations in two variables by substitution
4. Solve a system of equations in two variables by using elimination
5. Identify inconsistent and dependent systems of equations

Objective 2: Use system of linear systems in two variables to solve problems

1. Model and solve direct translation problems involving two linear equations containing two unknowns
2. Model and solve geometry problems involving two linear equations containing two unknowns
3. Model and solve mixture problems involving two linear equations containing two unknowns
4. Model and solve uniform motion problems involving two linear equations containing two unknowns

Objective 3: Systems of Linear Equations in Three Variables

1. Solve systems of three linear equations containing three variables
2. Identify inconsistent and depending systems
3. Model and solve problems involving three linear equations containing three unknowns

Objective 4: Systems of Linear Inequalities in Two Variables

1. Determine whether an ordered pair is a solution to a system of linear inequalities
2. Graph a system of linear inequalities
3. Solve problems involving systems of linear inequalities
   

Standard IV: Students will be able to recognize and use polynomial expressions

Mathematical Foundation:

Laws of Exponents: Product Rule, Quotient Rule, Zero-Exponent Rule, Negative Exponents, Power Rule, Product to a Power Rule, Quotient to a Power Rule
Arithmetic of Polynomials: Add, subtract, multiply, and divide polynomials; long and synthetic division
Factor Polynomials: Greatest Common Factors, Factoring Trinomials, Special Products
Polynomial Equations
Polynomial Functions

Applications of Mathematics:

Objective 1: Use the Laws of Exponents to simplify expressions

1. Understand the Product Rule
2. Understand the Quotient Rule
3. Understand the Zero-Exponent Rule
4. Understand the Negative Exponent Rule
5. Understand the Product to a Power Rule
6. Understand the Quotient to the Power Rule
7. Simplify complex expressions using the Laws of Exponents

Objective 2: Simplify polynomial expressions through arithmetic operations

1. Define a polynomial by degree and type (monomial, binomial, trinomial, polynomial)
2. Add and subtract polynomials by combining like terms
3. Multiply polynomials together, including monomials by monomials, monomials by polynomials, binomials by binomials, and polynomials by polynomials as well as special cases including perfect square trinomials and difference of squares
4. Divide polynomials involving division by a monomial, division by long division, and division using synthetic division

Objective 3: Factor polynomials into prime factors

1. Find the greatest common factor of a polynomial and factor it out
2. Factor a large polynomial by Grouping
3. Factor a trinomial of the form ax^2+bx+c
4. Factor a trinomial of the form ax^2+bx+c through Trial and Error or by Grouping
5. Factor special cases of polynomials such as perfect square trinomials, difference of squares, and sum or difference of cubes
6. Factor polynomials through substitution

Objective 4: Use polynomial equations to solve mathematical models

1. Solve polynomial equations using the Zero-Product Property
2. Solve equations involving polynomials
3. Model and solve problems involving polynomials

Objective 5: Use polynomial functions to solve mathematical models

1. Relate polynomials to functions
2. Use the Remainder Theorem to find the remainder of Polynomial functions
3. Use the Factor Theorem to decide if a polynomial function is factorable for specific factors
4. Evaluate polynomial functions
5. Add, subtract, multiply, and divide polynomial functions

Standard V: Students will solve Rational equations

Mathematical Foundation:

Rational expressions: simplify through addition, subtraction, multiplication, and division; simplify complex rational expressions
Rational equations
Rational inequalities
Rational functions

Applications of Mathematics:

Objective 1: Simplify rational expressions

1. Find the domain of a rational expression
2. Simplify a rational expression through factoring
3. Multiply a rational expression
4. Divide a rational expression
5. Add or subtract rational expressions with a common denominator
6. Find the lowest common denominator of a rational expression
7. Add or subtract rational expressions with an uncommon denominator
8. Simplify a complex rational expression by either simplifying the numerator and denominator separately before dividing or by simplifying each piece of the rational expression by its denominator

Objective 2: Solve rational equations

1. Solve equations containing rational expressions
2. Solve for a variable in a rational equation
3. Model and solve ratio and proportion problems
4. Model and solve work problems
5. Model and solve uniform motion problems

Objective 3: Solve rational inequalities

1. Understand and use sign tables
2. Solve a rational inequality by simplifying the rational expressions

Objective 4: Use Rational functions to solve mathematical models

1. Relate rational equations to functions
2. Find the domain of a rational function
3. Solve equations involving rational functions
4. Model and solve problems using rational functions

Standard VI: Students will simplify, solve, and graph radicals

Mathematical Foundation:

Radical expressions: nth roots and rational exponents, simplifying, adding, subtracting, multiplying, and rationalizing
Radical equations
Radical functions
Complex numbers

Applications of Mathematics:

Objective 1: Simplify radical expressions

1. Define and evaluate nth roots, simplify expressions in radical form and in rational form
2. Use the Laws of Exponents to simplify expressions of rational exponents
3. Use the Laws of Exponents to simplify radical expressions
4. Factor expressions containing rational exponents
5. Use the product property to multiply radical expressions
6. Use the product property to simplify radical expressions
7. Use the quotient property to simplify radical expressions
8. Multiply radicals with unlike indices
9. Add, subtract, and multiply radicals
10. Rationalize radicals with a one-term denominator or with a two-term denominator

Objective 2: Solve radical equations

1. Find the domain of a radical expression
2. Solve radical equations containing one radical
3. Solve radical equations containing two radicals
4. Solve for a variable in a radical equation

Objective 3: Use radical functions to model mathematical problems

1. Relate radical functions to radical equations
2. Find the domain of a radical function
3. Evaluate a functions whose rule is a radical expression
4. Graph functions involving square roots
5. Graph functions involving cube roots

Objective 4: Understand and use complex numbers

1. Evaluate the square root of negative real numbers
2. Add or subtract complex numbers
3. Multiply complex numbers
4. Divide complex numbers
5. Evaluate the powers of i

Standard VII: Students will solve and graph quadratic equations

Mathematical Foundation:

Solve quadratic equations: use the square root property, completing the square, and the quadratic formula; solve equations that are quadratic in form
Graph quadratic functions: through transformations and through properties
Solve quadratic inequalities

Applications of Mathematics:

Objective 1: Solve quadratic equations

1. Use the square root property to solve equations
2. Use completing the square to solve equations
3. Use the Pythagorean theorem
4. Use the quadratic formula to solve equations
5. Use the discriminant to determine the nature of solutions in a quadratic equation
6. Model and solve problems involving quadratic equations
7. Solve equations that are quadratic in form using substitution

Objective 2: Graph a quadratic function

1. Use transformations to graph a quadratic function through shifting horizontally and vertically and by stretching or shrinking a basic quadratic graph
2. Use the properties of a quadratic function to graph the function
3. Find the minimum or maximum of a quadratic function and solve problems involving quadratic functions

Objective 3: Solve quadratic inequalities

1. Use a sign table
2. Solve a quadratic inequality through graphing
3. Solve a quadratic inequality algebraically

Standard VIII: Students will solve and graph exponential and logarithmic equations

Mathematical Foundation:

Functions: One-to-one functions, composition, and functional inverses
Exponential functions: Definition, domain and range, evaluating, graphing, and solving
Logarithmic functions: Definition, converting to exponentials, common and natural logs, domain and range, evaluating, graphing, and solving

Applications of Mathematics:

Objective 1: Find a composite function and determine if a function is one-to-one or not

1. Form the composite function
2. Determine whether a function is one-to-one or not
3. Determine the inverse of a function defined by a map or a set of ordered pairs
4. Obtain the graph of the inverse function from the graph of a function
5. Find the inverse of a function defined by an equation

Objective 2: Use exponential functions to solve mathematical models

1. Evaluate exponential expressions
2. Graph exponential functions
3. Define the number e
4. Solve exponential equations
5. Use exponential models

Objective 3: Use logarithmic functions to solve mathematical models

1. Change exponential expressions into logarithmic expressions and logarithmic expressions into exponential expressions
2. Evaluate logarithmic functions
3. Graph logarithmic functions
4. Work with natural and common logarithms
5. Use logarithmic models
6. Use the properties of logarithms
7. Write a logarithmic expression as a sum or difference of logarithms and write a logarithmic expression as a single logarithm
8. Evaluate logarithms whose base is neither 10 nor e
9. Solve logarithmic equations using the properties of logarithms
10. Solve equations using exponential models

Standard IX: Students will apply formulas related to circles and will be able to graph circles

Mathematical Foundation:

Distance formula
Midpoint formula
Circles: graphing, converting from standard to general form, converting from general form to standard form

Applications of Mathematics:

Objective 1: Use formulas to find out information about line segments

1. Use the distance formula to find the length of a line
2. Use the midpoint formula to find the midpoint of a line

Objective 2: Graph circles

1. Find the radius and center point of a circle written in standard form
2. Graph a circle written in standard form
3. Convert from the general form of a circle to standard form