Portland Community College | Portland, Oregon

A scientific calculator and access to a graphing utility may be required.

Students are no longer required to have physical graphing calculators in MTH 60, 65, 70, 95, 111, and 112. Where physically possible instructors will demonstrate using Desmos, GeoGebra, or other online programs in class. Assessments requiring the use of a graphing utility may be done outside of proctored exams.

There is a required notation addendum which may be found at pcc.edu/programs/math/course-downloads.html.

Upon completion of the course students should be able to:

1. Recognize and apply the operations necessary to simplify expressions and solve equations.
2. Perform polynomial addition, subtraction, and multiplication and perform polynomial division by a monomial.
3. Use exponent and radical properties to simplify expressions and solve radical and quadratic equations.
4. Distinguish among perimeter, area, and volume and apply the formulas and appropriate units in contextual situations.
5. Perform unit conversions.
6. Distinguish between quadratic and linear relationships in symbolic, graphical, and verbal forms.
7. Create quadratic models, make predictions, and interpret the meaning of intercepts, vertices, and maximum or minimum values.

Assessment Requirements

1. The following must be assessed in a proctored, closed-book, no-note, and no-calculator setting:
   1. Simplifying polynomial and radical expressions
   2. Solving equations using the quadratic formula
   3. Solving radical equations
   4. Graphing quadratic equations in two variables
2. At least two proctored, closed-book, no-note examinations (one of which is the comprehensive final) must be given. 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 conclusions using proper mathematical procedures and notation. Application problems must be answered in complete sentences.
4. At least two of the following additional measures must also be used.
   1. Graded homework
   2. Quizzes
   3. Projects
   4. In-class activities
   5. Portfolios

Introductory algebraic and geometric manipulations useful in STEM courses

1. Polynomial Expressions and Exponents
   1. Å·ÖÞ±­¾öÈü¾º²Âapp_Å·ÖÞ±­×ãÇòÍø-Ͷע|¹ÙÍø exponent rules including for negative exponents and apply them when helpful in algebraic manipulations.
   2. Add, subtract, multiply and square polynomials.
   3. Divide polynomials by a monomial.
   4. Convert between scientific notation and standard form to demonstrate an understanding of magnitude.
   5. Perform multiplication and division operations in scientific notation in context.
2. Radical Expressions
   1. Evaluate \(n\)th roots numerically with and without technology.
   2. Recognize that an even root of a negative number is not real.
   3. Convert radical expressions to expressions with rational exponents and vice versa.
   4. Simplify, add, subtract, multiply and divide radical expressions.
   5. Use rational exponents to simplify radical expressions. 
   6. Rationalize denominators with square roots in them. 
   7. Use a calculator to approximate radicals using rational exponents.
3. Solving Equations in One Variable
   1. Solve quadratic equations using the square root property.
   2. Solve quadratic equations using the quadratic formula including complex solutions.
   3. Solve radical equations that have a single radical term.
   4. Solve an equation using a graphing utility by finding points of intersection.
   5. Verify solutions algebraically and graphically, noting when extraneous solutions may result.
   6. Solve a formula for a specific variable.
   7. Solve linear, quadratic, and radical equations when mixed up in a problem set.
   8. Solve real-world models involving quadratic and radical equations.
4. Quadratic Equations in Two Variables
   1. Algebraically find the vertex (using the formula \(h=-\frac{b}{2a}\)), the axis of symmetry, and the vertical and horizontal intercepts.
      1. The vertex and intercept(s) should be written as ordered pairs.
      2. The axis of symmetry should be written as an equation.
   2. Graph by hand a quadratic equation by finding the vertex, plotting at least two additional points on one side and using symmetry to complete the graph.
   3. Create, use, and interpret quadratic models of real-world situations algebraically and graphically.
      1. Interpret the vertex as a maximum or minimum in context with units.
      2. Interpret the intercept(s) in context with units.
   4. In a mixed problem set, distinguish between linear and quadratic equations and graph them.
5. Geometry Applications and Unit Analysis
   1. Know and apply appropriate units for various situations; e.g. perimeter units, area units, volume units, rate units, etc.
   2. Explore, understand, and apply the formulas for perimeter; area formulas for rectangles, circles, and triangles; and volume formulas for a rectangular solid and a right circular cylinder.
   3. Use similar triangles to find missing sides in a triangle.
   4. Use the Pythagorean Theorem to find a missing side in a right triangle.
   5. Use estimation to determine reasonableness of solution.
   6. Use unit fractions to convert time, length, area, volume, mass, density, and speed to other units, including metric/non-metric conversions.

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 students throughout the course to perform arithmetic operations with fractions and negative numbers without calculators.

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.

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.

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.

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. \(5x^{1/2}\), \(3x^{1/3}\), \(\frac{2x^{1/3}}{x^{1/2}}\), and \(4x^{1/2}x^{1/3}\).

Rationalizing the denominator should be limited to the following types of problems: \(\frac{2}{\sqrt{5}}\), \(\frac{6}{\sqrt{7}-2}\).

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.

Unit Conversions Reference Sheet: The following reference sheet may be provided to students during all exams and all other assessments. Items may be removed at an instructor’s discretion; however, nothing may be added.

https://spot.pcc.edu/math/download/65/mth65-reference.pdf