Math 0995 Course Objectives
Students will:
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Be familiar with the basic definition of a set and the notation used to define a set.
Be able to give examples and non‐examples of natural numbers.
Be able to give examples and non‐examples of whole numbers.
Be able to give examples and non‐examples of integers.
Be able to give examples and non‐examples of rational numbers.
Be able to give examples and non‐examples of irrational numbers.
Be able to identify problems that can be addressed with each subset of the real numbers.
Be familiar with the basic definition of the absolute value of a real number.
Be able to determine the absolute value of a given real number.
Be able to determine the additive inverse of a given number.
Be able to simplify absolute value expressions.
Be able to add and subtract positive and negative integers.
Be able to multiply and divide positive and negative integers.
Be able to determine all of the factors of a given natural number.
Be able to determine the prime factorization for a given natural number.
Be able to determine the least common multiple for a given set of natural numbers.
Understand why no number can be written as a fraction with a denominator of zero.
Be able to write any fraction as a reduced fraction where the numerator and denominator have no common
prime factors.
Be able to multiply and divide fractions and write the result as a reduced fraction.
Be able to create an equivalent fraction with a given denominator.
Be able to add and subtract fractions and write the result as a reduced fraction.
Be able to write a fraction as a decimal.
Be able to use the order of operations to evaluate and simplify an expression.
Be able to evaluate an algebraic expression for specified values of the variables.
Be able to simplify variable expressions using the algebraic properties of addition and
multiplication.
Be able to simplify variable expressions using the distributive property.
Understand and be able to utilize the rules for multiplying and dividing exponential expressions.
Understand and be able to utilize the rule for simplifying the power of an exponential expression.
Understand and be able to utilize the rule for simplifying the powers of products and quotients.
Be able to interpret and simplify an exponential with zero as an exponent.
Be able to interpret and simplify an exponential with a negative number exponent.
Be able to simplify monomial expressions by using properties of exponents.
Be able to distinguish between polynomial and non‐polynomial expressions.
Be able to determine the degree of a polynomial, and for a polynomial of a single variable, the leading term, leading
coefficient, and constant term. Students will also be able to recognize and distinguish between monomials,
binomials, and trinomials.
Understand that there are many forms that a polynomial can be expressed in and there are advantages
to different forms of a polynomial in different contexts.
Learn that the distributive property allows us to change the form of an expression. It is an expression of a
relation and should not be understood as a mandate.
Learn to change the form of an expression by using the distributive property to expand terms in an expression.
Learn how to multiply polynomials and combine like terms to simplify the product.
Be able to explain how dividing a polynomial by a polynomial is associated with the process of dividing
numbers and writing an improper fraction as a mixed number.
Learn to divide a polynomial by a monomial.
Be able to use their understanding of addition of fractions to justify the method for dividing a polynomial by a
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monomial.
Learn the algorithm for long division.
Learn what a factor is and what it means to factor an expression. Building on their experiences with
factoring integers, be able to determine the factors for a given monomial expression.
Learn to change the form of an expression by using the distributive property to factor terms in an expression.
Be able to identify the greatest common factor for all terms of an expression and be able to factor out the
greatest common factor to create a factored form of the expression.
Be able to identify examples and non‐examples of expressions that are written in a factored form and those
that are not. Students will be able to identify the individual factors of an expression in a factored form.
Learn the method of factoring by grouping.
Learn the method of factoring a trinomial where the leading coefficient is 1.
Learn the method of factoring a trinomial where the leading coefficient is not 1.
Learn to recognize and factor difference of squares binomials.
Learn to recognize and factor perfect square trinomials.
Learn the importance of factoring expressions as a prerequisite for reducing a fraction.
First review the idea of reducing non‐variable fractions by determining factors and then progress to reducing
rational expressions.
Learn why it is inappropriate to cancel like terms that exist in the numerator and denominator of a
fraction.
Learn that the domain of a rational expression often changes when a rational expression is reduced.
Learn that their work with rational expressions is not considered completely simplified unless all of the
factors of all of the numerators and denominators are identified and that no numerators and denominators
have a common factor.
Progress to multiplying rational expressions building on their experiences with multiplying rational
numbers.
Learn the benefits of writing the numerators and denominators of rational expressions in factored form when
multiplying or dividing.
Be exposed to different products that can exist in rational expressions and be able to combine the factors of a
given product and reduce the resulting fractions.
Review the idea that division is the same as multiplication by the reciprocal.
Progress to adding rational expressions building on their experiences with adding rational numbers.
Learn the benefits of writing the numerators and denominators of rational expressions in factored form when
adding or subtracting.
Learn that an exponential expression with a rational exponent is equivalent to a radical expression.
Learn that positive real numbers have two real square roots and that negative real numbers do not have
real square roots.
Learn that most principle nth roots of real numbers are irrational and that any decimal representation of an
irrational number is an approximation.
Learn and be able to justify the properties associated with multiplying and dividing radicals.
Learn to reduce a radical.
Learn to simplify expressions by combining radical terms that are alike.
Learn to rationalize the denominator of an expression.
Learn why it is often beneficial to rationalize the denominator of an expression that contains a radical.
Learn the relationship between logarithms and exponents.
Learn to simplify logarithms that are equal to rational numbers.
Learn set-builder notation as a way to describe a set of numbers, both finite and infinite, that satisfy a given
condition.
Learn interval notation as a way to describe continuous sets.
Be able to graph a set of numbers on the real number line.
Learn that equations are expressions of a relation and they indicate two forms of the same thing.
Learn examples of conditional equations, identities, and contradictions.
Determine the solutions of equations of many different forms that are sufficiently simplified so that the
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solutions can be determined without the need to manipulate an equation. These equations should include
linear, polynomial, rational, and radical equations.
Be able to determine if a specified value is or is not a solution to a given equation.
Consider application problems in mathematics where the solution to the problem is found by solving an
equation.
By manipulating equations, learn and be able to justify the following properties of equations:
addition/subtraction property, multiplication/division property, zero‐factor property, nth-roots property, powers
property, and the absolute value property.
Be able to solve linear equations that require distributing, combining like terms, arithmetic with fractions.
Be able to identify linear equations that are identities and contradictions.
Understand that a linear equation can be solved without carrying the simplification to the point where the
unknown is isolated. Students will understand that solving an equation is not the process of getting 𝑥 by itself; it
is the process of doing whatever is necessary to determine the value(s) of 𝑥.
Be able to solve equations that can be simplified to a form that includes a single absolute value term that
is equal to a real number.
Be able to identify absolute value equations that are contradictions.
Be able to justify when and why we can rewrite an absolute value equation as two equations associated with a
positive and negative value for the argument of the absolute value.
Be able to use the zero factor property to solve polynomial equations that can be factored.
Be able to solve polynomial equations that require factored terms to be expanded and like terms combined, in
order to use the zero‐factor property.
Be exposed to equations that have imaginary solutions. Imaginary numbers will be mentioned as a topic of
future study. Students will learn that such polynomials have no real number solutions, and depending on the
context we then must determine if it is prudent to determine the imaginary solutions.
Learn the importance of recognizing the domain of a rational equation as they solve rational equations.
Be able to determine the least common multiple of the denominators of a given rational equation.
Use the multiplication principle to rewrite a rational equation into an equivalent form with no fractions. Be
able to identify the assumptions that are made about the domain of the equivalent non‐fractional form.
Be able to simplify and solve rational equations that can be simplified into linear and polynomial equations.
Be able to solve radical equations that can be written in a form where a single radical term is equal to a real
number or a linear expression.
Be able to justify the need to check for extraneous solutions after an equation is transformed by raising
both sides of the equal sign to a power.
Solve the general quadratic equation 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0.
Be able to use the quadratic formula to solve quadratic equations that are originally presented in many
different forms.
Be able to solve inequalities using the method of test intervals and test points.
Be able to solve linear inequalities that require distributing, combining like terms, arithmetic with fractions.
Be able to justify the need to change the direction of the inequality when multiplying or dividing by a negative
number.
Justify and rewrite inequalities that can be written in the form |𝑓(𝑥)| < 𝐴 and |𝑓(𝑥)| > 𝐴 where 𝐴 is a positive
number.
Identify absolute value equations with no solutions and infinite solutions.
Describe some of the solutions to two‐variable equations using only ordered pairs.
Be able to determine if a given ordered pair is or is not a solution to a given equation.
Review the Cartesian plane.
Learn to plot the solutions of two variable equations as points in the Cartesian plane.
Be able to determine if a given point is or is not a point on the graph of a given equation.
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Understand that the graph of an equation is a continuous set of points, each of which is a solution to the
equation.
Be able to find the midpoint of a line segment between a given pair of points.
Be able to find the distance between a given pair of points.
Be able to determine the domain and range of a given relation.
Be able to determine if a given relation is or is not a function.
Be able to evaluate a function at a given input.
Be able to define and interpret a function using function notation.
Be able to plot points that are on the graph of a function.
Learn that the solutions of linear equations all belong to a given line.
Be able to determine examples and non‐examples of linear equations.
Learn to summarize all of the solutions to a linear equation by identifying two solutions.
Be able to determine the x‐intercept and y‐intercept of the graph of a given linear equation.
Learn that the solutions of an equation of the form 𝑦 = 𝑎 is the set of points on a horizontal line that all
have the 𝑦‐coordinate of 𝑎.
Learn that the solutions of an equation of the form 𝑥 = 𝑏 is the set of points on a vertical line that all
have the 𝑥‐coordinate of 𝑏.
Be able to determine the slope of a line given two points on the line.
Be able to determine the slope of a line given the equation of the line.
Be able to solve application problems associated with average rates of change.
Be able to graph a line given a point and the slope of the line.
Be able to determine the slope of a line by converting the equation to slope‐intercept form.
Find the equation of a line given enough information to determine the slope and a point on the line.
Learn the relationship between parallel/perpendicular lines and their slopes.
Draw rough approximations of graphs on non-linear functions by plotting a subset of ordered pairs of the
function.
Be able to recognize basic graphs of the following function types: quadratic, absolute value, and principle square
root.
Be able to recognize when the graph of an equation can be obtained through a vertical or horizontal shift.
Be able to recognize when the graph of an equation can be obtained through a vertical reflection.
Learn to graph equations on their calculators.
Learn to use tables or the “value” feature to evaluate equations at different values on their calculator.
Learn to use trace and zoom on their calculators.
Be able to find x‐intercepts on their calculators.
Learn to find intersections of two graphs on their calculators.
Be able to solve equations using a graphing calculator.
Be able to solve inequalities using a graphing calculator.
Be able to determine if a given ordered pair is or is not a solution to a 2x2 system of linear equations.
Be able to use the method of substitution and elimination to solve a 2x2 system of linear equations.
Be able to use the graphs of equations to solve a 2x2 system of linear equations.
Be able to graphically describe the equations and the solutions to a dependent or inconsistent 2x2 system
of linear equations.