Math 110

Mathematical Explorations

 

 

Catalog Description

An introductory course designed to provide the liberal arts major with an opportunity to develop a broader appreciation of mathematics by exploring ways in which the artistic, aesthetic, intellectual, and humanistic aspects of mathematics are as important as its utility. Topics may include: mathematical reasoning, the infinite, topology, chaos and fractals, symmetry, elementary number theory, modern geometry, and the history of mathematics. Prerequisite: High School Algebra II or MATH 0103.

 

Objectives & Requirements

Most college preparatory mathematics courses focus on acquiring new mathematical tools, skills, and techniques. In contrast, the focus of this course is on new mathematical objects, perspectives, ideas, and connections to other areas using tools that the students have already learned. This enables students to develop a broader understanding of mathematics and more positive appreciation of mathematics - one that is not dominated by an ability simply to perform rote procedures.

 

Math 110 is not a remedial, basic skills, nor mathematical literacy course. While it is appropriate to address some basic skills and literacy issues - one must do this in virtually any course - it is not appropriate for this to become a significant focus of the course. Some basic skills and literacy issues may be addressed in the additional class meeting time of Accuplacer sections of this course.

 

The content of Math 110 and Math 111 must be kept as disjoint as possible as they are sister courses that many students will take together to fulfill their core requirement in mathematics. Math 111 satisfies the "Applied Mathematics" sub-area of the mathematics core and is expected to cover "applied" topics. In contrast, Math 110 satisfies the "Traditional Mathematics" sub-area and is meant to cover topics that are not generally considered "applied mathematics".

 

Possible Topics: Possible topics for Math 110 may include: mathematical reasoning, patterns, the infinite, number theory, set theory, topology, chaos and fractals, the history of mathematics, mathematics and the arts (e.g. mathematics in painting, sculpture, architecture, music, perspective drawing), symmetry (e.g. kaleidescopes, the wallpaper groups, tessellations, Friesze patterns), and modern geometry (e.g. non-Euclidean geometries, taxi-cab geometry, the higher dimensions).

 

Topics to Avoid: Topics that should, generally, be left to Math 111 include: linear programming, voting theory, apportionment, interpretive statistics, descriptive statistics, game theory, fair division, graph theory, networks and scheduling, mathematical modeling, population growth, coding and cryptography, orienteering, and financial mathematics.

 

Appropriate Texts: Appropriate texts that have been used in Math 110 include:

The Heart of Mathematics, by Berger and Starbird, Key Curriculum Press, 2000.

A Mathematics Sampler: Topics for the Liberal Arts, by Berlinghoff, Grant, and Skrein, Ardsley House, 2001. [The chapters "Problems and Solutions", "Mathematics of Patterns: Number Theory", "Mathematics of Form: Geometry", "Mathematics of Infinity: Cantor's Theory of Sets", "Mathematics of Symmetry: Finite Groups", "Mathematics of Space and Time: Four-Dimensional Geometry" are more appropriate for Math 110 while the remaining chapters, "Mathematics of Chance: Probability and Statistics", "Mathematics of Machines: Microcomputers and Programming", "Mathematics of Connection: Graph Theory" are more appropriate for Math 111.]

Mathematics: A Human Endeavor, by Harold Jacobs, W.H. Freeman, 1994. [The chapters "Functions and Their Graphs", "Large Numbers and Logarithms", and "Mathematical Curves" are not appropriate for either Math 110 or Math 111. The chapters "Mathematical Ways of Thinking", "Number Sequences", "Symmetry and Regular Figures", and "Topics in Topology" are more appropriate for Math 110. The remaining chapters, "Methods of Counting", "Mathematics of Chance", and "An Introduction to Statistics", are more appropriate for Math 111.]

Discovering the Art of Number Theory: A Topical Guide, by Julian Fleron, 2003.

Excursions in Modern Mathematics, by P. Tannenbaum and R. Arnold, Prentice Hall, 2001. [Part III, "Growth and Symmetry" is more appropriate for Math 110 while the remaining parts, "The Mathematics of Social Choice", "Management Science", and "Statistics" are more appropriate for Math 111.]

 

Math 111

Mathematical Applications

 

 

Catalog Description

An introductory course designed to provide the liberal arts major with opportunities to investigate ways in which mathematics is used to solve real world problems in a variety of disciplines. Applications may include such topics as voting schemes, fair division, networks, scheduling, finance, probability and statistics. Prerequisite: High School Algebra II or MATH 0103.                                                                                      

Objectives & Requirements

Most college preparatory mathematics courses focus on acquiring new mathematical tools, skills, and techniques. In contrast, the focus of this course is on new mathematical objects, perspectives, ideas, and connections to other areas using tools that the students have already learned. This enables students to develop a broader understanding of mathematics and more positive appreciation of mathematics - one that is not dominated by an ability simply to perform rote procedures.

 

Math 111 is not a remedial, basic skills, nor mathematical literacy course. While it is appropriate to address some basic skills and literacy issues - one must do this in virtually any course - it is not appropriate for this to become a significant focus of the course.

 

The content of Math 110 and Math 111 must be kept as disjoint as possible as they are sister courses that many students will take together to fulfill their core requirement in mathematics. Math 110 satisfies the "Traditional Mathematics" sub-area and is meant to cover topics that are not generally considered "applied mathematics". In contrast, Math 111, which also satisfies the "Traditional Mathematics" sub-area of the mathematics core, is expected to cover "applied" topics.

Possible Topics: Possible topics for Math 111 may include: linear programming, voting theory, apportionment, interpretive statistics, descriptive statistics, game theory, fair division, graph theory, networks and scheduling, mathematical modeling, population growth, coding and cryptography, orienteering, financial mathematics.

 

Topics to Avoid: Topics that should, generally, be left to Math 110 include: mathematical reasoning, patterns, the infinite, number theory, topology, chaos and fractals, the history of mathematics, mathematics and the arts, symmetry, and modern geometry.

 

Appropriate Texts: Appropriate texts that have been used in the past include:

Excursions in Modern Mathematics, 4th edition, by Arnold and Tannenbaum, Prentice Hall, 2001.

[Parts I, II, and IV, "The Mathematics of Social Choice", "Management Science", and "Statistics" are more appropriate for Math 111. While the remaining part, Part III, "Growth and Symmetry" is more appropriate for Math 110]

For All Practical Purposes, 5th edition, COMAP (Consortium for Mathematics and its Applications), W.H. Freeman, 2000.

 

 

Math 108

Elementary Statistics

 

 

Catalog Description     

An introduction to basic concepts and techniques of statistics for students needing skills for research techniques in education, business, and the physical, life, and social sciences or to simply understand the mass of information in daily life. Topics included: graphical techniques such as histograms or box plots; measures of location and spread; scatter plots and correlation, sampling and sampling distribution, estimation and statistical inference (confidence intervals and/or hypothesis testing). Pre-requisites: High School Algebra II or Math 0103.

 

Course Objectives

Descriptive stats: Students will develop the skills to interpret and evaluate statistical results. They will be able to judge the validity of an experimental set-up, and design experiments. The abuses of statistics are discussed together with common mistakes and misconceptions. Students will learn how to represent data both graphically and using descriptive statistics. While students will compute parameters such as mean, standard deviation, skewness etc., the main focus will be on the interpretation of these statistical measures. Students will be introduced to the concept of probability and probability distributions and their properties.

 

Inferential stats: Students will learn how to investigate a claim and come up with a conclusion. They gain an appreciation for the power of statistics and an understanding of its limitations. The concepts of confidence and significance are discussed. Recognizing the extensive use of software in the work place today, the focus is again on using proper methods and set-ups, and on the interpretation of results.                                             

 

Instructional Objectives

The students will understand and be able to compute, create, and interpret:

1.     types of samples and data

2.     experiment design

3.     graphical representation of data (histogram, box plots)

4.     measures of center and their implications (mean, median, mode, midrange)

5.     measures of variation and their interpretation (range, standard deviation, variance, min, max)

6.     measures of position (percentiles, quartiles, z-score)

7.     detection of outliers, unusual and extreme values

8.     Chebychev's theorem and the empirical rule

9.     probability and probability distributions, expected values

10.  Law of large numbers

11.  binomial, uniform, and normal distribution

12.  how to use the normal distribution to find probabilities given a data point and a data point given a probability

13.  Rare event rule

14.  Central limit theorem

15.  how to compute confidence intervals for mean, proportion, standard deviation

      meaning of confidence level

      computation of margin of error

      choosing the correct distribution

      determining sample size

16.  student-t distribution and c²- distribution

17.  Correlation and regression

      how to choose an appropriate model

      how to interpret the r-value

18.  hypothesis testing for claims about mean, proportion, standard deviation

            phrasing H_o and H₁

            meaning of the significance of a test

            type 1 and type 2 error

            how to choose a test statistic

            p-value method

            traditional method

            how to phrase conclusions

 

Some of topics 15, 16, 17, 18 may be omitted as time requires. However, an introduction to the general concepts is desirable.

 

 

Math 115

Mathematics for Business and Social Sciences

 

 

Catalog Description

An introduction to algebraic modeling, with an emphasis on applications in business and the social sciences. Topics include: using algebraic models to describe the relationship between variables, using graphs to visualize models, and choosing and interpreting various models. Calculus is introduced and is used as a tool for studying the structure of algebraic models. Prerequisite: High School Algebra II or Math 103.

 

Course Objectives

Algebraic methods are utilized and integrated with realistic situations, with the majority of examples coming from business applications. Students will learn how to construct algebraic models, and how the tools of algebra and calculus can be used to understand and interpret these models. Students will develop both analytic and graphical methods of description and communication. Algebraic concepts and methods developed include: constructing and interpreting functions and graphs; constructing and solving linear and quadratic equations and models; understanding and applying linear model methodology such as linear regression and/or linear programming; understanding and producing the derivative function; and applying the derivative function to curve sketching and optimization problems.

 

Instructional Objectives

The student will understand/know how to:

    1.   manipulate fractions, percents, and exponents.

    2.   evaluate algebraic expressions.

    3.   solve linear equations.

    4.   solve quadratic equations using the quadratic formula.

    5.   (optional) solve quadratic equations using factoring.

    6.   (optional) solve linear and quadratic inequalities in one variable.

    7.   (optional) solve simple rational equations.

    8.   solve equations for a given variable (“formulas”)

    9.   translate realistic problems into algebraic models.

  10.   determine when a given situation can be modeled using a linear equation.

  11.   fit a linear model to data.

  12.   interpret the parameters of a linear model (slope, y–intercept).

  13.   graph a linear model.

  14.   interpret the graph of a linear model (slope, y–intercept, x–intercept).

  15.   graph a system of linear equations.

  16.   applications of fixed cost, variable cost, and selling price.

  17.   applications of cost functions, revenue functions, and profit functions.

  18.   (optional) applications of supply and demand functions.                                  

  19.   basic methodology and application of linear programming and/or linear regression.

  20.   the theory and notation of functions.

  21.   the derivative function.

  22.   (optional) what differentiation tells us about rates of change.

  23.   produce the derivative function from a given function (using the rules of differentiation: constant rule, power rule, constant multiple rule, sum rule).

  24.   graph quadratic functions.

  25.   applications of the quadratic function.               

  26.   graph higher – order polynomial functions using the derivative function.

  27.     use differentiation to solve optimization problems.

  28.     (optional) interpret and perform marginal analysis.

 

Math 150                                                                

Foundations: Mathematical Reasoning

 

 

Course Description

An introductory course. Topics include: finding, analyzing, and describing patterns; sets and classification; functions and relations; inductive and deductive reasoning; problem solving; and logic. Students will develop a conceptual understanding of the course material in a learning environment that models the pedagogical foundations of the Massachusetts Curriculum Frameworks for Mathematics and the National Council of Teachers of Mathematics (NCTM) Standards.

 

Course Objectives

Required topics in MA 150 include:

 

 

•       Problem solving techniques

•       Patterns

•       Inductive and deductive reasoning

•       Logic

•       Sets and classification

•       Functions and relations

      

Instructional Objectives

Upon completion of the course, students will be able to:

 

•       Appropriately select and apply different problem-solving techniques

•       Clearly articulate the problem-solving method(s) used

 

•       Utilize mathematical and logical reasoning

•       Construct simple proofs

•       Use basic set theory to describe different types of sets and their relationships

•       Use functions to describe and solve applied problems

•       Describe mathematical patterns and incorporate them in problem-solving

•       Describe the difference between deductive and inductive reasoning, including the strengths and limits of each

 

Pedagogy

One of the course objectives is to introduce students to some of the teaching pedagogy outlined by the NCTM. Following the NCTM Standards, the course is designed to educate students to become active participants (rather than passive observers) in mathematical thinking, and to encourage them to educate their future students in the same spirit. This may be done using some or all of the following approaches:

 

•       Small group work;

•       Emphasis on student verbal explanation of problem-solving processes rather than just providing answers; verbally explaining or defending solutions can help develop students’ mathematical thinking, as well as articulation skills; 

•       Increasing students’ self-reliance on checking solutions leads to deeper mathematical thinking. They have to think about whether or not their answers make sense. They also have to think about the problem further to devise a way to check the solution. As a result their understanding and thinking about the problem will be deepened. If the professor simply says, “That’s right” or “That’s wrong,” thinking about the problem will immediately cease;

•       The Constructivist approach to learning is emphasized: students discover and build mathematical concepts themselves, rather than just memorizing them without really understanding. According to schema learning theory, knowledge is acquired in greater depth and is more efficiently retained if it can be connected with the learner’s pre-existing knowledge. It cannot be assumed that learners will make the connections without help. The connections of new material to other knowledge structures must be made, and in keeping with the self-reliance issue raised earlier, the connections should be made by the students themselves whenever possible;

•       Using manipulatives as a tool for understanding mathematical concepts and for solving problems;

•       Developing different problem-solving strategies (estimating, drawing a diagram, discovering patterns, constructing a table, etc.) that are applicable to a wide variety of situations.

 

Other objectives vary from instructor to instructor. There is a real concern that many students preparing to be elementary school teachers lack basic arithmetic skills that they need to teach their students. This issue is frequently addressed in the “Accuplacer” sections of MA 150. Another concern is that, as teachers, the attitude they have about mathematics is likely to carry over into their own classroom. Having students keep a journal is one way of helping them deal with this issue.

 

 

Appropriate Texts: Appropriate texts that have been used in the past include:

            Bassarear, T. Mathematics for Elementary School Teachers, 2^nd Ed., Houghton-Mifflin

            Co., 2001.

Bennett and Nelson. Mathematics for Elementary Teachers, 5^th Ed., XanEdu Publishing

            Services, 2003.

            Ohanian, S. Garbage Pizza, Patchwork Quilts, and Math Magic, W. H. Freeman and

            Co., 1992.

            Schifter, Deborah (ed.). Reconstructing Mathematics Education, Teachers College Press,

            1993.

            Ma, Liping. Knowing and Teaching Elementary Mathematics, Lawrence Erlbaum

            Associates, Inc., 1999.

 

            It is strongly recommended that anyone teaching the course for the first time read one or

            more of the books by Ohanian, Schifter, or Ma.

 

Math 104

PreCalculus

 

 

Catalog Description

Topics covered include an in-depth investigation of functions; graphing; exponential and logarithmic functions; and trigonometry. Prerequisite: High School Algebra II or Math 0103.

 

Course Objectives

The objective of this course is to give the students the fundamentals they will need in the calculus courses. In particular, students will be introduced to topics including functions (symbolically, numerically, and graphically), trigonometry, and exponential and logarithmic functions. At the same time, students' ability to think independently while problem solving should be developed. These ideas may or may not be supplemented through the use of graphing calculators (use of a graphing calculator is recommended).

 

Required Topics

As this is a course on functions, students start by learning the concepts of relation, function, and 1-1 function, including domain and range, behavior of the function (increasing/decreasing), continuity, extrema, y- and x-intercepts, asymptotes, even/odd, symmetries.

 

Students will investigate the definition, graphs, and properties of the following functions:

•                Linear functions, slope and y-intercept, solution of linear equations

•                Quadratic functions and other even-powered functions, quadratic formula, vertex, solution of quadratic equations

 

•                Cubic functions and other odd-powered functions

•                Polynomials and rational functions

 

•                Absolute value function

•                Step functions

•                Square root and other roots

•                1/x

•                The three basic trigonometric functions sine, cosine, tangent, degrees and radians, definition of a periodic function

 

•                Exponential function and logarithms, change of base formula

•                Inverse functions

•                Composition of functions

•                Polar Coordinates, complex numbers (optional)

 

Applications are covered as appropriate.

Students gain an understanding of the definition and practical meaning of the slope and rate of change.

Transformations and translations of graphs are investigated.

 

Math 105

Calculus I

 

 

Catalog Description

A standard first semester course in calculus. Topics include limits and continuity, the derivative and its properties, applications of differentiation, introduction to anti-differentiation, the definite integrals, and the Fundamental Theorem of Calculus. Prerequisites: Four years of High School Math, including Algebra I and II and Geometry or MATH 0104.

 

Course Objectives

Math 105 is a course in differential calculus and its applications. As such, the required topics in this course include:

•                Multiple representations of a function (symbolic, numeric, graphic)

•                Mathematical models (both using and inventing)

•                Limits

•                Continuity

•                Tangents, velocities and rates of change

•                The definition of the derivative

•                The derivative as a function

•                Multiple representations of the derivative (symbolic, numeric, graphic)

•                Using graphical, numerical and symbolic representations of the first and second derivative to obtain information about the original function

•                Using graphical, numerical and symbolic representations of the original function’s first and second derivative to obtain information about the first and second derivative.

•                Derivatives of elementary functions. That is, derivatives of polynomial, rational, exponential, trigonometric, inverse trigonometric and logarithmic functions

•                The product, quotient and chain rules                                                                 

•                Using the first and second derivatives to find maximum and minimum values

•                Using the first and second derivatives to determine the shape of curves

•                Using the first and second derivatives to solve optimization problems

•                Antiderivatives

 

Instructional Objectives

It is important that students gain a strong conceptual understanding of calculus and its role, as well as an ability to use the algorithms of calculus to solve problems. Upon successful completion of this course students will be able to:

 

 

1.     Discuss the concepts of function, rate of change, limit of a function, derivative of a function, and antiderivative of a function using multiple representations.

2.     Create and use mathematical models.

3.     Use multiple strategies in problem solving.

4.     Use appropriate technology when necessary in problem solving. In particular, students should be able to use the Voyage 200 or TI-89 appropriately.

5.     Understand the difference between discrete and continuous behavior and the importance of calculus in modeling continuous phenomenon.

6.     Describe the relationship between a function and its derivatives using multiple representations.

7.     Relate tangents, velocities and derivatives to other rates of change (i.e. population growth/decay).

8.     Be able to estimate the first and second derivatives numerically and graphically.

9.     Acquire differential calculus skills of computing derivatives of polynomial, rational, exponential, trigonometric, inverse trigonometric, logarithmic functions, and compositions of these functions.

10.  Determine the shape of curves by locating critical points, maxima, minima, and points of inflection of a function using information given by the first and second derivative.

11.  Analyze and solve optimization and other problems using appropriate technology with numerical, graphical, and analytical capabilities.

 

Required Text

            Stewart, James. (2001). Calculus: Concepts and Contexts (Second Edition). Brooks/Cole.

 

Required Hand-Held Technology

            Texas Instruments TI-89 or Voyage 200.

 

Math 106

Calculus II

 

 

Catalog Description

A continuation of Calculus I. Topics include techniques of integration, applications of the integral, series and sequences, L'Hôpital's Rule, approximation of functions. Prerequisite: MATH 0105 or equivalent.

 

Course Objectives

Math 106 is a course in integral calculus and its applications. As such, the required topics in this course include:

• Antiderivatives
• Riemann Sums
• The definite and indefinite integrals
• Different representations of the integral
• The Fundamental Theorem of Calculus
• Techniques of Integration (e.g. substitution, integration by parts, partial fractions)
• Improper Integrals
• Applications of the Definite Integral (e.g. areas, volumes, work)
• Convergence and divergence of numerical series
• Tests for convergence and divergence, such as the integral test, the ratio test, the comparison test, alternating series test
• Estimating the value of a numerical series to a specified degree of accuracy
• Geometric series
• Finding Power and Taylor Series for functions such as exp (x), sin (x), cos (x), arctan (x), 1/(1-x), etc.
• Intervals and radius of convergence for Power and Taylor Series
• Using Power and Taylor Series to approximate functions. (e.g.: use an approximation of exp (x²) to estimate dx)
• Introductory Differential Equations (e.g.: definitions, slope fields, separation of variables)

Instructional Objectives

It is important that students gain a strong conceptual understanding of calculus and its role, as well as the ability to use the algorithms of calculus to solve problems. Upon successful completion of this course students will be able to:

1.     Discuss the relationship between a function and its antiderivative.

2.     Estimate antiderivatives numerically, graphically and symbolically.

3.     Describe the relation between the definite integral and Riemann sums.

4.     Use appropriate tools for computing antiderivatives such as substitution, integration by parts, partial fractions, technology (specifically the Voyage 200 or the TI-89) and integral tables.

5.     Use integration to solve area, volume, work and other applied problems.

6.     Compute improper integrals and use them to solve applied problems.

7.     Determine the limit of a sequence numerically, graphically and symbolically.

8.     Describe the relationship and differences between sequences and series.

9.     Determine the convergence or divergence of a series using numerical and symbolic techniques.

10.  Estimate the sum of a series numerically to a given degree of accuracy.

11.  Determine the power series expansion for a function about a given point.

12.  Use power series expansions to integrate functions (e.g. ).

13.  Recognize a differential equation.

14.  Determine if a function solves a differential equation.

15.  Solve separable differential equations graphically (e.g. with slope fields) and symbolically.

16.  Use differential equations to solve applied problems.

 

Required Text

Stewart, James. (2001). Calculus: Concepts and Contexts (Second Edition). Brooks/Cole.

 

Required Hand-Held Technology

Texas Instruments TI-89 or Voyage 200