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Developmental Mathematics and Its Applications
(DevMap) Project


Students who attend two-year colleges often arrive on campus lacking the basic skills necessary to succeed in college-level work. This is a particularly serious problem for students in Science, Mathematics, Engineering, and Technology (SMET) programs, who may have significant deficits in their mathematics backgrounds but high aspirations for their future success. Students in this situation need to build both their mathematical skills and confidence in their ability to solve challenging problems in order to succeed in the mathematics-intensive programs they have selected.

At the 1997 National Center for Research in Vocational Education (NCRVE) workshop, "Beyond Eighth Grade," industry representatives emphasized the need for "systems thinking" that enables employees to recognize complexities inherent in situations subject to multiple inputs and diverse constraints. In addition, science-based fields such as agricultural biotechnology require technicians who are able to formulate a problem in terms of relevant factors and design an experiment to determine the influence of those various factors. Yet, most developmental programs in mathematics, at both two- and four-year colleges, offer students what amounts to a replication of the high school mathematics curriculum they either studied and forgot or never studied.

We now have a chance for real and positive change. Both the National Council of Teachers of Mathematics (NCTM) Standards, published in 1989, and Crossroads in Mathematics: Standards for Introductory College Mathematics Before Calculus, published by the American Mathematical Association of Two-Year Colleges (AMATYC) in 1995, advocate an integrated approach to mathematics content. Given students' needs in developmental mathematics, the vision of the AMATYC Standards, and COMAP's experience, we are creating a one-year sequence, Developing Mathematics through Applications. We believe there are several benefits to our approach.

  • The program will not be divided into topics called algebra, geometry, intermediate algebra and trigonometry, although all the major concepts in those courses will be covered.
  • The applications-based curriculum will appeal to mature students who are choosing to pursue postsecondary education; the applications can be drawn from areas in which students may find themselves working or from situations that they recognize from their daily lives.
  • Solving the problems posed in DevMap will call for integrating technology in a natural way as compared to the "drill-and-practice" use of technology currently found in many developmental mathematics programs.

An advisory board informed the project. Members of this board were drawn from mathematics faculty at two- and four-year colleges, leaders of AMATYC, and business and industry organizations.


Dr. Solomon Garfunkel --- Executive Director of COMAP
Dr. Garfunkel has served as project director for Undergraduate Mathematics and its Applications (UMAP) and High School Mathematics and its Applications (HiMAP). He was the project director and host for the series For All Practical Purposes: Introduction to Contemporary Mathematics. Most recently he was Co-Principal Investigator on the ARISE project that developed the secondary standards based series, Mathematics: Modeling Our World and is currently Principal Investigator on the Developmental Mathematics and its Applications (DevMap) project. Dr. Garfunkel taught at the University of Connecticut and Cornell University before founding COMAP in 1980.

Dr. Susan L. Forman --- Professor of Mathematics, Bronx Community College,
The City University of New York
Dr. Forman served as Senior Program Officer for Education at the Charles A. Dana Foundation and as Director of College and University Programs for the Mathematical Sciences Education Board of the National Academy of Sciences. Her other experiences include positions as Coordinator of Academic Computing at the City University of New York and at the Fund for the Improvement of Post-Secondary Education. She is currently the Co-Principal Investigator on the Developmental Mathematics and its Applications (DevMap) project at COMAP and teaching at the Bronx Community College.


The DevMap Project is committed to developing a two-semester program that offers an alternative approach to the elementary and intermediate algebra courses currently taught at most two- and four-year colleges. The project is producing materials that present the mathematical content from an integrated point of view through the use of authentic applications, many of which are drawn from the technological workplace. Students completing this program will be prepared to enter a course in precalculus.

The project's principal investigators and writers have worked hard to create a curriculum that aligns with the new vision of both pedagogy and content for effective mathematics education set forth in Crossroads in Mathematics.


Nancy Crisler --- COMAP, faculty member Washington, University, St. Louis, MO
Ms. Crisler was a high school mathematics teacher, and for ten years served as the K–12 Mathematics supervisor for the Pattonville School District in St. Louis County, MO. She was a member of the ARISE writing team and provided staff development to teachers for that project. She is co-author of Discrete Mathematics Through Applications. Currently she is a primary author for the Developmental Mathematics and its Applications (DevMap) project at COMAP and teaches at Washington University in St. Louis, MO.

Gary Simundza --- faculty member, Applied Mathematics & Sciences Department,
Wentworth Institute of Technology, Boston, MA
Professor Simundza has been teaching physics and mathematics for the past 25 years. He was director and principal investigator for the National Science Foundation-funded Mathematics for Technology-Laboratory Investigations project. He is member of the writing team for the Developmental Mathematics and its Applications (DevMap) project at COMAP and teaches at Wentworth Institute of Technology in Boston, MA.

Developing Mathematics through Applications © 2003
Developed and published by COMAP


This book is a different kind of developmental mathematics text. We believe that any mathematics worth learning is best learned in the context of its use by real people in real jobs. Without neglecting necessary practice with fundamental skills, we emphasize connections between mathematics and the workplace, both in the body of the text and in the exercises. Interesting applications motivate students to learn, and guided discovery helps them succeed in their learning by developing habits of persistence and achievement. The AMATYC Standards have been thoroughly embraced by this book: Active investigation of mathematics is emphasized throughout, and it is expected that students will conduct much of this investigation in small, collaborative groups. The content integrates data analysis, measurement, and geometry with traditional topics in algebra. A modeling theme runs throughout the text, as students learn to construct a variety of models, including equations, graphs, tables, arrow diagrams, narrative descriptions, and geometric and statistical models. Access to computing technology is assumed, although the text can be used successfully without it. Many of the contextual situations in the text require students to read for mathematical content. Indeed, an essential part of acquiring the ability to model the world with mathematics is learning to see how mathematics is embedded in those situations. We hope students will be encouraged to increase their reading level as they see that careful reading helps them achieve mathematics competence. Vocabulary is kept accessible and unfamiliar concepts are clearly explained in terms that students will be able to understand.


  • Focuses on mathematical modeling Uses a discovery approach with active involvement of the students Provides connections to real-world applications through investigations, activities, and projects Uses technology throughout the curriculum Emphasizes functions Uses multiple representations in concept development Provides opportunities to collect and analyze real data Promotes problem solving by providing a wide variety of open-ended problems
  • Suggests a variety of teaching methods through the use of Teacher Notes


  • Preparation Reading --- Each chapter begins with a brief introduction that sets the stage for the mathematical development to come, and may include contextual connections or discussions of modeling aspects. It also includes a list of the goals of the chapter.
  • Chapter Sections
    • What You Need to Know --- A list of necessary prerequisite knowledge.
    • What You Will Learn --- A list of the mathematical objectives of the section.
    • Activities --- The first section of each chapter contains an activity that requires students to work in groups and perform an experiment or otherwise collect data. They are then guided in a step-by-step manner to explore a core mathematical concept that will recur throughout the chapter.
    • Discoveries --- Later sections frequently contain explorations that are similar to activities in that they are intended to be done in groups. A discovery is usually a more focused examination of a single mathematical topic and does not necessarily involve active experimentation. If data are needed, they are often provided rather than collected.
    • Examples --- Numerous examples with detailed solutions are provided in the body of each section.
    • Exercises
      • Investigations --- These should be considered as adjuncts to the body of each section and may present new material. They contain in-depth examinations of particular subtopics or applications and some contain guided explorations that are similar to discoveries. Investigations provide an opportunity for each instructor to customize the text for his or her own purposes and students' needs.
      • Projects and Group Activities --- These are similar to the activities that begin each chapter. They require students to work cooperatively and often involve research or experimentation of some sort.
      • Additional Practice --- These provide necessary skill practice with some problems being contextual. They are not necessarily in order from easy to difficult, partly so that students will not just quit after trying a few easy problems.
    • Chapter Review --- This section summarizes the key concepts of the chapter and contains a sampling of exercises from each section similar to the additional practice exercises.
    • Glossary --- A summary of the new terms introduced in the chapter.



COURSE 1 (Book 1)

Chapter 1


Section 1.1

Measuring Length

Section 1.2

Linear Measurement and the Pythagorean Theorem

Section 1.3

Describing the Results of Measurement

Section 1.4

Measuring a Surface: Area

Section 1.5

Volume, Surface Area, and Geometric Models

Section 1.6

Other Kinds of Measurement

Chapter 2

Linear Models

Section 2.1

Measuring Indirectly

Section 2.2

Exploring Algebraic Expressions

Section 2.3

Solving Linear Equations

Section 2.4

Functions and Their Representations

Section 2.5

Linear Functions

Section 2.6

Creating Linear Models

Chapter 3

Modeling Behavior from Data

Section 3.1

Collecting Data and Determining a Model

Section 3.2

Scatter plots and Data-Driven Models

Section 3.3

Lines of Best Fit

Section 3.4

Using Models to Make Predictions

Chapter 4


Section 4.1

Introduction to Polynomials

Section 4.2


Section 4.3

Operations with Polynomials

Section 4.4

Factoring Polynomials

Section 4.5

Modeling with Polynomial Functions

COURSE 2 (Book 2)

Chapter 5

Quadratic Functions and Radicals

Section 5.1


Section 5.2

Quadratic Functions and Their Graphs

Section 5.3

Modeling Data with Quadratic Functions

Section 5.4

Roots and Radicals

Section 5.5

Fractional Exponents and Radical Equations

Section 5.6

Distance in the Plane; Circles

Chapter 6

Rational Expressions and Systems of Equations

Section 6.1

Introduction to Rational Functions

Section 6.2

Modeling with Rational Functions

Section 6.3

Multiplying and Dividing Rational Expressions

Section 6.4


Solving Rational Equations

Section 6.5

Systems of Equations

Chapter 7


Section 7.1

The Meaning of Probability

Section 7.2

Probabilities for Compound Events

Section 7.3

Finding Probabilities from Data

Section 7.4

Combining Probabilities

Section 7.5

Probability Distributions

Section 7.6

Drawing Conclusions from Data

Section 7.7

Modeling Through Simulations


Chapter 8


Section 8.1

Similar Triangles

Section 8.2

Properties of Triangles

Section 8.2

Trigonometric Ratios

Section 8.3

Modeling with Right Triangles

Section 8.4

Trigonometric Ratios for Non-Acute Angles