Connected Mathematics 2 provides eight student units for each grade. One additional unit is offered for each grade from the first edition, ©2004. This allows some flexibility in meeting individual state expectations by allowing the choice of an additional unit when needed. Each unit is organized around an important mathematical idea or cluster of related ideas, such as area and perimeter, operations on fractions, ratio and proportion, linear relationships, or quadratic relationships. The format of the student books promotes student engagement with an exploration of important mathematical concepts and related skills and procedures. Since students develop strategies and conceptual understanding by solving problems and discussing their solutions in class, the books do not contain worked-out examples. Instead the students record their work and explanations as well as their growing understanding of definitions and rules in their notebooks.
The organization and features of each student unit are described below.
Each unit opens with a set of three focusing questions that reflect the major mathematical goal(s) of the unit. These questions are intended to draw students into the unit, pique their curiosity, and point to the kinds of ideas they will investigate. As the students move through the unit they will encounter these questions either as a problem to explore in class or as homework.
Unit Opener: Covering and Surrounding
Next, the unit provides a set of goals, or Mathematical Highlights, that preview the important ideas of the unit. The highlights help students track their progress through the unit and provide parents and guardians with an overview of the mathematical concepts, processes, and ways of thinking developed in the unit.
Mathematical Highlights: Covering and Surrounding
The Investigations form the core of a Connected Mathematics unit. It is by working through the Investigations that the students develop conceptual understanding, reasoning, and procedural skill. Each Investigation builds toward the mathematical goals. Each unit includes three to five Investigations with the following key elements:
An Investigation includes two to five carefully sequenced Problems. Each Problem is launched by the teacher; then the students explore the Problem individually, in groups, or as a whole class. As students solve the Problems, they uncover important mathematical relationships and develop problem-solving strategies and skills. A summary occurs at the end of each Problem. The teacher pulls the class together and helps students explicitly describe the mathematics of the Problem, ideas, patterns, relationships, and strategies they found and used.
Problem 5.1: Covering and Surrounding
This feature occurs occasionally before a problem. It is intended to be used as part of the launch for the problem. It reviews or introduces the mathematical ideas needed in the problem.
Getting Ready: Covering and Surrounding
Did You Know?
This feature occasionally occurs to present interesting facts related to the context of an investigation.
Did You Know? Covering and Surrounding
Applications - Connections- Extensions (ACE)
The Problems in each Investigation are followed by a set of exercises meant to be used as homework at the end of each Problem. Students are asked to compare, visualize, model, measure, count, reason, connect, and/or communicate their ideas in writing. Totruly own an idea, strategy, or concept, a student must apply it, connect it to what he or she already knows or has experienced, and seek ways to extend or generalize it.
These exercises help students solidify their understanding by providing practice with ideas and strategies that were in the Investigation. Applications contain contexts both similar to and different from those in the Investigation.
Applications: Covering and Surrounding
A powerful learning strategy is to connect new knowledge to prior learning. The Connections section of the homework provides this opportunity. This section also provides continued review of concepts and skills across the grades. For example, the Connections in Covering and Surrounding, a unit on measurement, contain practice with operations on decimals and fractions. Connections can also connect to "real-world problems." Often these are problems that contain original data sets. For example, in Moving Straight Ahead, a unit on linear relationships, there are connections to sports records.
Connections: Covering and Surrounding
These exercises may provide a challenge for students to think beyond what is covered in the Problems in class, provide an interesting excursion "side ways" that looks at related mathematical ideas, foreshadow mathematics in future units or pursue an interesting application.
Extensions: Covering and Surrounding
At the end of each Investigation, students are asked to reflect on what they have learned. A set of questions helps students organize their thoughts and summarize important concepts and strategies. After thinking about the questions and sketching their own ideas, students discuss the questions with their teacher and their classmates and then write a summary of their findings.
Mathematical Reflections: Covering and Surrounding
At least four units at each grade level include projects. Projects are typically introduced at the beginning of a unit and formally assigned at the end. A list of projects is given on page 53. Projects are open-ended tasks that provide opportunities for students to engage in independent work and to demonstrate their broad understanding of the mathematics of the unit.
Unit Project: Covering And Surrounding
Looking Back and Looking Ahead
This feature provides a review of the "big" ideas and connections in the unit. It includes problems that allow students to demonstrate their understanding, explain their reasoning, summarizing and connecting what they have learned within and across units.
Looking Back and Looking Ahead: Covering and Surrounding
Although students are encouraged to develop their own definitions and examples for key terms, a glossary in English and Spanish is provided at the back of each student book. Glossaries can serve as a guide for the student, the teacher, and parents as students develop understanding of key ideas and strategies.
Glossary: Covering and Surrounding
Technology and other Resources
Connected Mathematics was developed with the belief that calculators should be available to students, and that students should know when and how to use them. In grade 6, students need standard, four-function calculators. Students use four-function calculators to simplify complicated calculations and explore patterns in computations.
For some units in grades 7 and 8, students need access to graphing calculators with table and statistical-display capabilities. Graphing calculators are used to investigate functions and as a tool for solving problems. Students use graphing calculators to explore the shape and features of graphs of linear, exponential, and quadratic functions as well as the patterns of change in the tables of such functions. And in addition to using symbolic solution methods, students use graphing-calculator tables and graphs to solve equations.
Although computers are not required for any of the investigations, applets are provided for many units. Some applets are designed to be used during the Launch, Explore, Summarize sequence and some can be used at various stages of the instruction, including additional practice with the ideas in the unit. The applets are provided on the Student Activities CD-ROM. Resources in Components of CMP
In Connected Mathematics, manipulatives are used only when they can help students develop understanding of mathematical ideas. For example, in Filling and Wrapping, students find all the different rectangular arrangements possible for a given number of cubes. They find the surface area of each arrangement by creating a net (covering) for the arrangement that exactly fits, with no overlap or underlap. They then identify the arrangements that require the least and the most material to wrap. This activity sets the stage for developing the ideas of surface area and volume of rectangular prisms. Most of the manipulatives used in Connected Mathematics are commonly available, and many schools may already have them. Included are rulers, protractors, angle rulers, cubes, square tiles, counters, spinners, and dice.
The two manipulatives described below are unique to Connected Mathematics.
Polystrips are plastic strips that can be pieced together with brass fasteners to form polygons. These manipulatives are used in grade 6 to investigate the relationship among the side lengths of triangles and quadrilaterals. They also are useful in the eighth grade geometry unit, Kaleidoscopes, Hubcaps, and Mirrors.
The CMP Shapes Set® is a set of polygons used in grade 6 to explore sides, angles, and tilings.
Blackline Masters are provided for teachers who do not have Polystrips or the CMP Shapes Set.
A list of materials for each unit is found in the Unit Introduction of each Teacher's Guide.
These materials may all be ordered from the publisher via the online catalog