Portland Community College | Portland, Oregon

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 co-enrolled in an MTH class, then this may include targeted materials which are intended to support the concepts being taught in that MTH class.

Upon completion of the course students will be able to:

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

• Apply algebraic problem solving strategies in limited contexts.

• Address algebraic problems with increased confidence.

• Demonstrate progression through mathematical learning objectives established between the student and instructor.

Instructors may employ the use of worksheets, textbooks, online software, mini-lectures, and/or group work.

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

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.

Introductory Algebra II (MTH 65)

THEMES:

Introductory algebraic and geometric manipulations useful in STEM courses

   ​
SKILLS:

1. Polynomial Expressions and Exponents
   1. Develop 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. E.g. \(\sqrt[3]{x^7}=\cdots=x^2\sqrt[3]{x}\), \(\sqrt{x}\cdot\sqrt[3]{x}=\cdots=\sqrt[6]{x^5}\).
   6. Rationalize denominators with square roots in them. E.g. \(\frac{5}{\sqrt{2}}\), \(\frac{5}{1+\sqrt{2}}\).
   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. Verify solutions algebraically and graphically, noting when extraneous solutions may result.
   5. Solve a formula for a specific variable.
   6. Solve linear, quadratic, and radical equations when mixed up in a problem set.
   7. 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.
6. Solving Equations and Inequalities Graphically
   1. Given an equation, solve using a graphing utility by finding points of intersection.
   2. Given an inequality, solve using a graphing utility and express the solution in interval notation.

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 math-placement exam.

The intended goals from the MTH 65 CCOG follow:

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 them throughout the course to get better at performing operations with fractions and negative numbers, as it will make a difference in this and future math courses.

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.

There is a required notation addendum and required problem set supplement for this course. Both can be found at: https://www.pcc.edu/programs/math/course-downloads.html