Course Title and Description:

MTH 152 - Mathematics for the Liberal Arts II

Presents topics in functions, combinatorics, probability, statistics and algebraic systems.

Lecture 3 hours per week. 3 credits

Prerequisites: Competency in Math Essentials MTE 1-5 as demonstrated through the placement and diagnostic tests, or by satisfactorily completing the required MTE units or equivalent.
3 creditsTopics in the MTE 1-5 courses include the following:

MTE 1 - Operations with Positive Fractions
MTE 2 - Operations with Positive Decimals and Percents
MTE 3 - Algebra Basics
MTE 4 - First Degree Equations and Inequalities in One Variable
MTE 5 - Linear Equations, Inequalities and Systems of Linear Equations in Two Variables

This course is offered via the Internet and via distributed DVD's in an asynchronous mode. The student will receive instructional information and assignments via these modes and will respond to assignments by submitting work through web forms.

The student must have standard access to the Internet and must have the ability to access the content on the DVD's.  The material on the DVD's is accessible using a variety of media players (e.g., Windows Media Player). 

The instructor is available via web forms (to which students will be introduced at the very beginning of the course), and will normally respond by the end of the day following your submission (and more typically on the same day) with answers to properly posed questions, feedback on your efforts, and other information. Exceptions may occur in the event of Internet problems or other technical events. 

Students with Disabilities

Students with documented disabilities may be eligible for assistance and various accommodations. Please check the Student Support Services link on your Blackboard page.

Note that the course videos as distributed on DVD's do not currently have transcripts, though a version can be provided in which videos are embedded within screen-readable documents. If you require this version of the videos, or transcripts, due to a documented disability, please notify the instructor immediately.

The main goals of this course are to develop both intuitive and analytical competence with combinatorics, probability, statistics and geometry.

The student will solve every problem in the assigned chapters. The student will therefore accomplish most of the objectives implicit in the text.

The student will also read the remaining problems in the problem sets, and take whatever measures are necessary to ensure that he or she can perform adequately on a test existing of problems chosen either at random or by the instructor from the text, or paraphrased from such a problem.

Each problem therefore constitutes a specific objective, which is to understand the problem the principles underlying the solution, and the ramifications of the problem and its relationships to other problems.

Problem assignments are posted on the course Web site. 

The homepage for the course includes statements of specific objectives corresponding to each assignment.  They are included below as well:

Module 1: Counting

Text Chapter 11

1.  Use systematic listing, product tables and tree diagrams to represent the set of possible results of a single-step or multi-step task.

2. For a multi-step task

• Determine if the uniformity criterion applies.
• If the uniformity criterion applies use the fundamental counting theorem to determine the number of ways in which the task can be accomplished.

3. Determine the number of orders in which n objects can be arranged.

4. Recognize whether a given task in which a given number of objects are chosen from a given set involves permutations or combinations.

5. Explain how to reason out the number of permutations of r objects chosen from a set of n objects.

6. Explain how to reason out the number of combinations of r objects chosen from a set of n objects.

7. Explain how to reason out the number of r-element subsets of an n-element set.

8. Use Pascal's Triangle to find C(n, r) for given n and r.

9. Use the binomial theorem to expand a given binomial to a given power.

10. Recognize and verify patterns in Pascal's Triangle.

Module 2: Probability

Text Chapter 12

1. Explain the concept and give the definition of probability.

2. State and apply the formula for theoretical probability.

3. State and apply the formula for empirical probability.

4. State and give an example to illustrate the Law of Large Numbers.

5. Distinguish between odds and probability, and given one of these quantities find the other.

6. Know, determine the applicability of, explain in terms of examples and if applicable use to solve problems:

• the complements rule of probability
• the general addition rule of probability
• the special addition rule of probability

7. Use the probability distribution for a given random variable to determine probabilities of various events.

8.  Know and apply the definition of conditional probability.

9. Explain in terms of Venn diagrams the meaning of and formula for conditional probability.

10. Know and apply the definition of independent events.

11. Know and apply the rule for the probability that two independent events will both occur.

12.  For the binomial probability formula explain, generally or in terms of a nontrivial example, the meaning of each:

• C(n, x)
• p
• q
• p^x
• q^(n-x)
• p^x * q^(n - x)
• why the probability of exactly x successes in n attempts is not p^x * q^(n - x)

13. Apply the concept of and/or formula for binomial probability to solve mathematical or real-world problems.

14.  Explain, generally or in terms of a nontrivial example, the meaning of mathematical expectation.

15. Apply the concept of and/or formula for mathematical expectation to solve mathematical or real-world problems.

16.  Simulate a given process using coin flips or a table of random numbers.

Module 3: Statistics

Text Chapter 12

1.  Read and interpret frequency distributions using the following as requested, and express data given in one format in the other formats:

• listing
• grouped frequency distribution
• stem-and-leaf display
• bar graphs
• pie charts

2.  Calculate and interpret the mean, median and mode of a set of data items.

3. Calculate and interpret the weighted mean, median and mode of a frequency distribution.

4. Recognize when a frequency distribution is symmetric, skewed to the left, skewed to the right and/or bimodal.

5.  For a given set of data calculate and interpret the range and standard deviation.

6. For a given set of data verify Chebyshev's Theorem.

7.  For a given value of a random variable, given the mean and standard deviation of its distribution, find its z value.

8. Calculate percentiles, deciles and quartiles for a set of data items.

9. Construct a box-and-whisker plot of a given set of data.

10. Know the probabilities that a normally distributed random variable will take a value within 1, 2 and 3 standard deviations of the mean.

11. Sketch the standard normal curve and relabel for a given mean and standard deviation.

12. Using a sketch of the normal curve estimate the probabilities of various events.

13. Using a z-table determine the probabilities of various events.

14. Given a set of data, hand-sketch a scatter plot and estimate the slope and y-intercept of the best-fit straight line, as estimate the correlation coefficient.

15. Given a set of data and the necessary formulas, calculate the slope and y-intercept of the best-fit straight line as well as the correlation coefficient, and write the equation of the best-fit straight line.

Module 4: Geometry

Text Chapter 9

1. Know the meanings of points, lines and planes, and the associated notations.

2. Know and apply the definitions of

• line segments
• parallel lines
• intersecting lines
• parallel planes
• intersecting planes
• skew lines
• angle
• vertex
• acute, right and obtuse angles
• perpendicular lines
• vertical angles
• complementary and supplementary angles
• transversals and vertical angles

3. Define, sketch, recognize and apply:

• curves which are and are not closed
• curves which are and are not simple
• regions which are and are not convex
• closed figures which are polygons
• closed figures which are regular polygons
• acute, right, obtuse, equilateral, isosceles and scalene triangles
• various quadrilaterals:  trapezoids, parallelograms, rectangles, squares, rhombi
• circles

4. Know, apply and be able to demonstrate that sum of the angles of a triangle is 360 degrees.

5. Given the angles of a triangle find the measure of any specified exterior angle.

6. Know the theorem about an angle inscribed in a semicircle.

7. Know and apply the definitions of perimeter (or if more appropriate circumference) and area to solve problems involving triangles, the various quadrilaterals and circles.

8. Know and apply

• the congruence properties of triangles
• the properties of isosceles triangles
• properties of similarity of triangles
• the Pythagorean Theorem and its converse

9. Know and apply the definitions and properties of

• the regular polyhedra
• pyramids
• prisms
• cylinders
• cones
• tori (plural of 'torus')

10. Use the formulas for the area and circumference of a circle and the definition of volume to find the area and circumference of a right circular cylinder, and to verify the standard formulas for the area and volume.

11. Explain how to use the area of the base and the altitude of a prism or cylinder to find its volume.

12. Explain how to use the '1/3 principle' to find the volume of a pyramid or a cone, given its altitude and the area of its base.

13. Know and apply the formulas for the surface area and volume of a sphere.

14.  Explain in terms of an example why any pair of straight lines on a sphere must intersect, while it is possible to construct two non-intersecting straight lines on a plane.

15. Explain in terms of an example why the sum of the angles of a triangle constructed on a spherical surface is greater that 180 degrees.

16. Sketch a surface on which the sum of the angles of a triangle would be less than 180 degrees.

17. Identify the genus of a given object.

18. Explain why it is impossible to traverse a network having more than two odd vertices.

19. Given a network determine whether it is traversible, and if it is sketch a path that traverses it.

20.  Show how succesive iterations of the logistic equation can lead to convergence or chaos depending on the value of the parameter m.

21.  For a given construction of self-similar objects of increasing length, identify the scale factor for size that results from a convenient ratio of lengths.

Regular communication is required of the student. This includes turning in assignments in a timely fashion and responding in a timely manner to feedback on these assignments. Any deviation of more than three days from the chosen schedule of the course must be approved in advance by the instructor. Exceptions will of course be made in the event of documented illness or other unexpected emergencies, but the instructor should be informed of such situations within a reasonable time of occurrence.

The text will be Mathematical Ideas, by Miller etc., current edition (see Textbook Information as previously provided) published by Addison-Wesley.

The student will be required to purchase the DVD's, which are sold at low cost through the VHCC bookstore.

Course materials are detailed at the site Course Materials , which is also on the main menu of your homepage.

Units to be covered include Chapters 1-5 inclusive, and Chapter 7, inclusive through Section 7.3.  Supplements may be included.   The chapters covered, in order of coverage, are

• Chapter 11: Counting Methods
• Chapter 12:  Probability
• Chapter 13:  Statistics
• Chapter 9:  Geometry

Students will complete and submit the assignments specified on the homepage.

The instructor will respond in a timely fashion to any work submitted, making suggestions where improvement is needed and posing questions designed to enhance the student's learning experience. The student will be required to respond to all critiques, except those designated otherwise.

Questions posed by students and the instructor's responses will be posted to a site, specified in at the beginning of the course, for the student's review.

Students may on occasion be asked to critique work done by other students.  Full student anonymity will be preserved, with no reference  to the identity of any party in this exchange.

The instructor is available via web forms (to which you will be introduced at the very beginning of the course), and will normally respond by the end of the day following your submission (and more typically on the same day) with answers to properly posed questions, feedback on your efforts, and other information. Exceptions may occur in the event of Internet problems or other technical events. 

Use of email:  Prior to registration and receipt of initial instructions students my use Email to communicate with the instructor.  However email is much less reliable than web forms, and after registration and receipt of initial instructions anything sent through email should first be sent using the appropriate form.

A test on each Chapter will be administered, for a total of four tests.  An optional final examination is also available, which can be given the weight of up to two tests.  Weighting of the final, if taken, will be chosen by the instructor to the best advantage of the student. 

Portfolio and Portfolio Grade

A student's portfolio, consisting of instructor responses to assigned work (consisting of all assigned documents which request student responses; in this course these assignments consist mainly of the q_a_ and Query documents specified in the Assignments Table) will at the end of the term be assigned a grade on a 100-point scale. Each portfolio document is worth 1 point toward the portfolio grade, which will be awarded if the document meets acceptable standards. If the document does not meet acceptable standards, the instructor will request a revision. The total score on the 100-point scale will be equal to the number of points awarded as a percent of the total number of documents assigned. Documents counted toward the portfolio grade include qa's, queries and randomized problems. If this score is higher than the average on the other contributions to the final grade, it will be counted as 1/2 of a test grade (typically around 10% of the final grade); if not it will be counted as 1/4 of a test grade (typically around 5% of the final grade). The actual proportion of the final grade determined by the portfolio depends on weighting contingencies as defined previously and a further explained below.

Criteria for Grading of Tests:

Tests will consist of problems designed to measure the level of your achievement of the course goals. 

Each problem is graded on a 10-point scale, with the following guidelines:

• 10 points will be earned for a correct solution which documents the details of the entire solution process, including all reasoning.
• At least 7 points will be earned for a solution which follows a valid reasoning process and is substantially correct, but with some errors in detail.
• 5 points will be given for any reasonable attempt at a solution.
• Some credit will be given for any steps or attempts to express or apply ideas which are related to the solution.
• 0 points will be earned for any solution, whether correct or not, which does not document the reasoning and the solution process.
• If a problem consists of two or more distinct parts, credit will be pro-rated accordingly.

Determining Final Course Grade

Raw test scores will be normalized to the following scale, according to the difficulty of the test, as specified in advance of each test by the instructor:

A: 90 - 100

B: 80 - 90

C: 70 - 80

D: 60 - 70

F: Less than 60.

The final grade will be a weighted average according to the above guidelines. A summary of the weighting is as follows:

Expanded Explanation of Weighting

The above is the simplest way to specify the grading scale for this course. However some students are uncomfortable with fractions, proportional representations and weighted averages and prefer to see the contributions of various components of the course expressed in terms of point values. Given the contingencies defined above, in which the portfolio can have two different weightings and the final exam grade can three, there are six possible ways the algorithm defined above could be applied.

Each column under 'points' defines a possible way of calculating the student's final grade. The number of points for a given assessment will be multiplied by the student's percent score on that assessment to get the points earned on that assessment. These points will be added to get the student's final percent score for the course.

The student's final grade will be based on the weighting is most advantageous to the student. The student does not need to select one weighting or another. The instructor will examine all possible weightings to determine the highest possible final grade for each individual student, and this will be the course grade given to that student.