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# MTH 60/65/70/95 CCOGs (2019 revisions)

The subcommittee for revising 60/65/70/95 CCOGs will post drafts here of what content has been approved by the subcommittee. SAC members may comment.

# MTH 60

## Course Description [SAC approved, slightly modified by Curriculum Committee and then approved]

Introduces algebraic concepts and processes with a focus on linear equations, linear inequalities, and systems of linear equations. Emphasizes number-sense, applications, graphs, formulas, and proper mathematical notation. Recommended: MTH 20 be taken within the past 4 terms. The PCC math department recommends that students take MTH courses in consecutive terms. Audit available.

### Addendum to the Course Description

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.

## Intended Outcomes [SAC approved, Curriculum Committee approved]

Upon completion of the course students should be able to:

1. Identify the differences between an expression and an equation.
2. Simplify and evaluate algebraic expressions.
3. Solve linear equations and inequalities in one variable, and linear systems in two variables.
4. Identify and interpret the slope as a rate of change in linear relationships.
5. Create linear equations, inequalities, and systems that model contextual situations and use the model to make predictions.
6. Represent linear relationships between two variables using a graph, table, verbal description, or algebraic formula.

## Course Content (Themes, Concepts, Issues and Skills) [Subcommittee approved]

### Themes

• Number sense
• Algebraic Manipulation
• Graphical understanding
• Problem solving
• Effective communication
• Critical thinking
• Applications, formulas, and modeling

### Skills

1. Algebraic Expressions and Equations

1. Simplify algebraic expressions using the distributive, commutative, and associative properties
2. Evaluate algebraic expressions
3. Translate phrases and sentences into algebraic expressions and equations, and vice versa
4. Distinguish between factors and terms
5. Distinguish between evaluating expressions, simplifying expressions, and solving equations
2. Linear Equations and Inequalities in One Variable
1. identify linear equations and inequalities in one variable.
2. Use the definition of a solution to an equation or inequality to check if a given value is a solution.
3. Solve linear equations and non-compound linear inequalities symbolically.
4. Express inequality solution sets graphically and with interval notation.
5. Create and solve linear equations and inequalities in one variable that model real life situations
1. Properly define variables; include units in variable definitions
2. State contextual conclusions using complete sentences
3. Use estimation to determine reasonableness of solutions
6. Solve an equation for a specified variable in terms of other variables
7. Solve applications in which two values are unknown but their total is known; for example, a 10-foot board cut into two pieces of where one piece is 2.5 feet longer than the other piece
3. Introduction to Tables and Graphs
1. Plot points on the Cartesian coordinate system, including pairs of values from a table.
2. Determine coordinates of points by reading a Cartesian graph.
3. Create a table of values from an equation or application. Make a plot from the table. When appropriate, correctly identify the independent variable with the horizontal axis and the dependent variable with the vertical axis.
4. Classify points by quadrant or as points on an axis; identify the origin
5. Label and scale axes on all graphs
6. Create graphs where the axes are required to have different scales (e.g. -10 to 10 on the horizontal axis and -1000 to 1000 on the vertical axis).
7. Interpret graphs, intercepts and other points in the context of an application. Express intercepts as ordered pairs.
8. Create tables and graphs with labels that communicate the context of an application problem and its dependent and independent quantities
4. Slope
1. Write and interpret a slope as a rate of change in context (include the unit of the slope)
2. Find the slope of a line from a graph, from two points, and from a table of values.
3. Find the slope from all forms of a linear equation.
4. Given the graph of a line, identify the slope as positive, negative, zero, or undefined. Given two non-vertical lines, identify the line with greater slope.
5. Linear Equations in Two Variables
1. Identify a linear equation in two variables
2. Manipulate a linear equation into slope-intercept form; identify the slope and the vertical intercept given a linear equation
3. Recognize equations of horizontal and vertical lines and identify their slopes as zero or undefined
4. Write the equation of a line in slope-intercept form
5. Write the equation of a line in point-slope form
6. Graphing Linear Equations in Two Variables
1. Graph a line with a known point and slope.
2. Emphasize that the graph of a line is a visual representation of the solution set to a linear equation
3. Given a linear equation, find at least three ordered pairs that satisfy the equation and graph the line using those ordered pairs.
4. Given an equation in slope-intercept, plot its graph using the slope and vertical intercept
5. Given an equation in point-slope, plot its graph using the slope and the suggested point
6. Given an equation in standard form, plot its graph by calculating horizontal and vertical intercepts, and check with a third point
7. Given an equation for a vertical or horizontal line, plot its graph.
8. Create and graph a linear model based on data and make predictions based upon the model
7. Systems of Linear Equations in Two Variables
1. Solve and check systems of equations using the following methods: graphically, using the substitution method, and using the addition/elimination method
2. Create and solve real-world models involving systems of linear equations in two variables
3. Properly define variables; include units in variable definitions.
4. State contextual conclusions using complete sentences
5. Given the equations of two lines, classify them as parallel, perpendicular, or neither

# MTH 65

## Course Description [SAC approved, slightly modified by Curriculum Committee and then approved]

Introduces algebraic concepts and processes with a focus on polynomials, exponents, roots, geometry, dimensional analysis, solving quadratic equations, and graphing parabolas. Emphasizes number-sense, applications, graphs, formulas, and proper mathematical notation. Recommended: MTH 60 or MTH 62 be taken within the past 4 terms. The PCC math department recommends that students take MTH courses in consecutive terms. Audit available.

### Addendum to the Course Description

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.

## Intended Outcomes [SAC approved, Curriculum Committee approved]

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.

## Course Content (Themes, Concepts, Issues and Skills) [Subcommittee approved]

### Themes

• Number Sense
• Algebraic manipulation
• Graphical understanding
• Problem solving
• Effective communication
• Critical thinking
• Applications, formulas, and modeling

### 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
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. $$\sqrt[3]{x^8}$$, $$\sqrt{x}\cdot\sqrt[3]{x}$$)
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
8. [This item may end up in MTH 95] Add, subtract, and multiply complex numbers and write the result in $$a+bi$$ form
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. [This item may end up in MTH 95] Solve quadratic equations by completing the square
4. Solve radical equations that have a single radical term
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 $$x=-\frac{b}{2a}$$), 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. Memorize and apply the perimeter and area formulas for rectangles, circles, and triangles
3. Memorize and apply the volume formula for a rectangular solid and a right circular cylinder
4. Use the appropriate geometric formula(s) for a given shape to find a desired quantity
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-nonmetric 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

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## 1 Comment

1. Hi! I just want to ensure that complex numbers will be in one of the Dev math CCOGs since it's needed for MTH 112. If we vote to take it out of MTH 65 then would that mean it could potentially not be in MTH 95 as well based on the split committee vote?