Each of the three years of MATH Connections is built around a general theme which serves as a unifying thread for the topics covered. Each year is divided into two half-year books consisting of three or four large chapters. Every chapter has a unifying conceptual theme that connects to the general theme of the year.

Year 1 — Data, Numbers, and Patterns

MATH Connections 1a
begins and ends with data analysis. It starts with hands-on data gathering, presentation, and analysis, then poses questions about correlating two sets of data. This establishes the goal of the term—that students be able to use the linear regression capabilities of a graphing calculator to do defensible forecasting in real-world settings. Students reach this goal by mastering the algebra of first-degree equations and the coordinate geometry of straight lines, gaining familiarity with graphing calculators.

• Chapter 1. Turning Facts into Ideas
• Chapter 2. Welcome to Algebra
• Chapter 3. The Algebra of Straight Lines
• Chapter 4. Graphical Estimation

MATH Connections 1b
generalizes and expands the ideas of Book 1a. It begins with techniques for solving two linear equations in two unknowns and interpreting such solutions in real-world contexts. Functional relationships in everyday life are identified, generalized, brought into mathematical focus, and linked with the algebra and coordinate geometry already developed. These ideas are then linked to an examination of the fundamental counting principle of discrete mathematics and to the basic ideas of probability. Along the way, Book 1b poses questions about correlating two sets of data.

• Chapter 5. Using Lines and Equations
• Chapter 6. How Functions Function
• Chapter 7. Counting Beyond 1, 2, 3
• Chapter 8. Introduction to Probability: What Are the Chances?

Year 2 — Shapes in Space

starts with the most basic ways of measuring length and area. It uses
symmetries of planar shapes to ask and answer questions about polygonal figures. Algebraic ideas from Year 1 are elaborated by providing them with geometric
interpretations. Scaling opens the door to similarity and then to angular measure, which builds on the concept of slope from Year 1. Extensive work with angles and
triangles, of interest in its own right, also lays the groundwork for right angle trigonometry, the last main topic of this book. Standard principles of congruence and triangulation of polygons are developed and employed in innovative ways to make clear their applicability to real-world problems.

• Chapter 1. The Building Blocks of Geometry: Making and Measuring Polygons
• Chapter 2. Similarity and Scaling: Growing and Shrinking Carefully
• Chapter 3. Introduction to Trigonometry: Tangles with Angles

MATH Connections 2b
begins by exploring the role of circles in the world of spatial relationships.
It then generalizes the two-dimensional ideas and thought patterns of Book 2a to three dimensions, starting with fold up patterns and contour lines on topographical maps. This leads to some fundamental properties of three-dimensional shapes. Coordinate geometry connects this spatial world of three dimensions to the powerful tools of algebra. That two-way connection is then used to explore systems of equations in three variables, extending the treatment of two variable equations in Year 1. In addition, matrices are shown to be a convenient way to organize, store, and manipulate information.

• Chapter 4. Circles and Disks
• Chapter 5. Shapes in Space
• Chapter 6. Linear Algebra and Matrices

Year 3 — Mathematical Modeling

• Chapter 1. Algebraic Functions
• Chapter 2. Exponential Functions and Logarithms
• Chapter 3. The Trigonometric Functions
• Chapter 4. Counting, Probability, and Statistics

begins by extending the modeling theme to Linear Programming,
optimization, and topics from graph theory. Then the idea of modeling itself is
examined in some depth by considering the purpose of axioms and axiomatic
systems, logic, and mathematical proof. Various forms of logical arguments, already used informally throughout Years 1 and 2, are explained and used to explore small axiomatic systems, including the group axioms. These logical tools then provide
guidance for a mathematical exploration of infinity, an area in which commonsense intuition is often unreliable. The final chapter explores Euclid’s plane geometry, connecting his system with many geometric concepts from Year 2. It culminates in a brief historical explanation of Euclidean and non-Euclidean geometries as alternative models for the spatial structure of our universe.

• Chapter 5. Optimization: Math Does It Better
• Chapter 6. Playing By the Rules: Logic and Axiomatic Systems
• Chapter 7. Infinity—The Final Frontier?
• Chapter 8. Axioms, Geometry, and Choice

MATH Connections Ancillaries

Appendix A. Using a TI-84 Plus (TI-83 Plus) Graphing Calculator. This appendix appears in all the books because graphing calculators are important tools for virtually every chapter. It provides a gentle introduction to these machines, and also serves as a convenient student reference for the commonly used elementary procedures.

Appendix B. Using a Spreadsheet. This appendix also appears in all the books. Although a spreadsheet is not explicitly required anywhere in these books, it is very handy for doing many problems or Explorations. It should be considered as a legitimate, optional tool for anyone with access to one.

Appendix C. Programming the TI-82 (TI-83). This appendix also appears in all the books. Students can use it to learn useful general principles of programming, as well as techniques specific to the TI-82 (TI-83).

Appendix D. Linear Programming with Excel. This appendix, which appears in Years 2 and 3, is not primarily a tool for doing problems within the chapters. Rather, it describes a technological approach to ideas that come up from time to time in various chapters. Linear Programming itself is discussed in detail in Chapter 5 of Book 3b. This appendix can be used either as a precursor to that discussion, or as an extension of it.