Course Title and Description:

MTH 151 - Mathematics for the Liberal Arts I

Presents topics in sets, logic, numeration systems, geometric systems, and elementary computer concepts.

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 credits

Topics 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 respect to the behavior of logic, numbers, graphs, operations, and mathematical structures, to develop the ability to solve real-world problems and to understand rudimentary mathematical models and the modeling process. Inherent in the approach used in this course will also be the enhancement of communications skills and mathematical confidence.

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 is advised to 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.  Those objectives are repeated below:

Module 1 Objectives (corresponding to text Chapter 2):  Know and apply the following to answer questions and solve problems.

1. Correctly recognize and apply standard symbols and terminology for describing and denoting finite and infinite sets, inclusion in a set, set equality, cardinality.
2. Correctly interpret and properly represent sets using set-builder notation.
3. Correctly recognize and apply standard symbols and terminology for describing and denoting subsets, proper subset, complement of a set.
4. Represent sets and their relationships using Venn diagrams.
5. Correctly recognize and apply standard symbols and terminology for describing and denoting set union, intersection, difference, equality of sets and cartesian products.
6. Describe sets in words, and given a description of a set in words write the set in mathematical notation.
7. Correctly shade Venn diagrams representing various set operations.
8. State and apply deMorgan's Laws, and explain those laws both in words and using Venn diagrams.
9. State the cardinal number formula, and explain it in words or with Venn diagrams.
10. Analyze data from surveys, tables or reports when given sufficient information to reason out the cardinalities of all 8 regions into which 3 sets generally divide the universal set.
11. Where it is possible to do so, define one-to-one correspondences between sets; where this is not possible show why.
12. State and apply the meaning of aleph-null.

Module 2 Objectives (corresponding to text Chapter 1):  Know and apply the following to answer questions and solve problems.

1. Correctly distinguish between inductive and deductive arguments.
2. Solve problems using inductive and deductive reasoning.
3. Apply inductive reasoning to solve problems involving number patterns and figurate numbers.
4. State and apply to the solution of problems a variety of problem-solving strategies.
5. Construct and interpret pie graphs, histograms, point graphs and line graphs.
6. Quickly approximate the results of arithmetic calculations using reasonable estimates.

Module 3 Objectives (corresponding to text Chapter 3):  Know and apply the following to answer questions and solve problems.

1. State definition of the word 'statement' and distinguish between statements and non-statements.
2. Symbolize compound statements formed by conjunctions, disjunctions and negations.
3. Identify and apply universal and existential quantifiers.
4. Negate statements formed from universal ('all do' or 'all do not') or existential ('some do' or 'some do not') quantifiers.
5. State the definitions of real numbers, integers, whole numbers, counting numbers, rational numbers, and irrational numbers, and the relationships among these sets.
6. Apply the rules for determining the truth values of negations, conjunctions and disjunctions.
7. Construct truth tables, in which each column can be evaluated by reference to at most two other columns, to determine the truth values of compound statements formed by negations, conjunctions and disjunctions of simple statements.
8. Validate deMorgan's Laws using truth tables.
9. Symbolize compound verbal statements in terms of simple statements, and evaluate the truth values of those statements.
10. State and be able to explain the reasons for the rule used to evaluate the truth value of a conditional.
11. Translate a verbal or symbolic conditional into a disjunction, and negate it.
12. Use truth tables, in which each column can be evaluated by reference to at most two other columns, to determine the truth values of compound statements formed by negations, conjunctions, conditionals and disjunctions of simple statements.
13. Given a circuit translate it as a compound logical statement, and vice versa.
14. Evaluate a compound statement by evaluating the corresponding circuit.
15. State and apply the definitions of the converse, inverse and contrapositive of a conditional.
16. Evaluate the relative truth values of a statement and its converse, inverse and contrapositive.
17. State, interpret and apply alternative verbal forms of the conditional.
18. Apply the converse, inverse and contrapositive to write equivalent forms of conditional statements.
19. State and apply the definition of the biconditional.
20. State and apply the definition of 'consistent' and 'contrary'.
21. Use Euler diagrams to represent conditionals, negations and quantified statements.
22. Use Euler diagrams to represent and evaluate arguments.
23. Given an argument involving compound statements, write the argument as a conditional, evaluate the conditional and determine the validity of the argument.
24. Given a set of premises, use modus ponens, modus tollens, disjunctive syllogism and/or transitivity, as appropriate, to draw a valid conclusion.  (It is not required that you know the terms ' modus ponens', 'modus tollens', or 'disjunctive syllogism'.  You should know the term 'transitivity' and recognize when you are using it.  It's not required but it's a good idea to understand the meaning of 'direct argument' (vernacular for 'modus ponens') and 'argument by contradiction' (vernacular for 'modus tollens').) 

Module 4 Objectives (corresponding to text Chapter 4):  Know and apply the following to answer questions and solve problems.

1. Given a table defining the symbols for Egyptian, Roman or Chinese numeration systems, write a given Hindu-Arabic number in the notation of that system, and write a number in any of those systems in Hindu-Arabic form.
2. Perform addition or subtraction using Egyptian symbols (as opposed, for example, to translating the Egyptian symbols to Hindu-Arabic notation and doing the arithmetic in the usual fashion before translating back to Egyptian symbols).
3. Perform multiplication using the Egyptian algorithm.
4. Given a table defining the symbols for Babylonian, Mayan or Greek numeration systems, write a given Hindu-Arabic number in the notation of that system, and write a number in any of those systems in Hindu-Arabic form.
5. Perform arithmetic in the Hindu-Arabic system by first expanding each number in powers of 10.
6. Interpret numbers represented on an abacus.
7. Apply the Russian Peasant Method for multiplication.
8. Apply the above to solve problems.
9. Convert a base-10 number to another specified base (including hexadecimal).
10. Convert a number in a specified base (including hexadecimal) to a base-10 number.
11. Convert between binary and hexadecimal representations of a number.
12. Perform 'clock arithmetic' on a clock with a given number of hours.
13. Perform modular arithmetic for a given modulus.

Module 5 Objectives (corresponding to text Chapter 5):  Know and apply the following to answer questions and solve problems.

1. State and apply the definitions of prime and composite numbers.
2. Apply divisibility tests for divisibility by 2, 3, 4, 5, 6, 8, 9, 10 and 12 to a given number.
3. Calculate the prime factorization of a given number.
4. Calculate all the natural-number factors of a given number.
5. Apply the Sieve of Erasthones to find the primes up to number n, where n is a specified natural number.
6. Explain how to prove that there are infinitely many primes.
7. Check the Mersenne numbers M_n = 2^n - 1 for values of n up through 10 and identify corresponding Mersenne primes.
8. Generate Fermat numbers 2^(2^n) + 1 for values of n up to 3, and identify those which are prime.
9. Identify a given number as perfect, deficient or abundant.
10. List the perfect numbers less than 100.
11. Verify Goldbach's Conjecture for a given number by showing how it is the sum of two primes.
12. Verify whether two given numbers are amicable.
13. Identify twin primes up to 100.
14. State Fermat's Last Theorem and explain how it is related to the Pythagorean Theorem.
15. Find the greatest common factor or least common multiple of two or more given numbers based on their prime factorizations.
16. Find the greatest common factor or least common multiple of two or more given using division by primes.
17. Find the greatest common factor of two numbers using the Euclidean Algorithm
18. State and identify the pattern of the Fibonacci sequence.
19. State and apply the recursion formula for the Fibonacci sequence.
20. Prove and/or verify stated characteristic of the Fibonacci sequence.
21. Demonstrate how the successive ratios of the Fibonacci sequence approach the golden ratio.

Module 6 Objectives (corresponding to text Chapter 7):  Know and apply the following to answer questions and solve problems.

1. State the properties of equality and use them to solve linear equations.
2. Write the linear equation that models a given linear phenomenon.
3. Apply the above to solve problems, especially problems involving geometrical quantities, mixtures, interest and rates.
4. Write and reduce to lowest terms the ratio of two given quantities.
5. Correctly apply the word 'per' to the ratio of two quantities.
6. Write and solve statements of proportion.
7. State the forms of the statements of direct variation, inverse variation, and variation with a power.
8. Apply direct variation to solve problems.
9. Write equations for, apply and solve problems related to joint or combined variation.

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 2:  The Basic Concepts of Set Theory
• Chapter 1:  The Art of Problem Solving
• Chapter 3:  Introduction to Logic
• Chapter 4:  Numeration and Mathematical Systems
• Chapter 5:  Number Theory
• Chapter 7: The Basic Concepts of Algebra

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 six 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 with the exception of labs, which are graded in a different category) 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.