Navigating Through Algebra in Grades 6-8 (Navigations)

This book shows how middle school students can use mathematical models and represent and analyze mathematical situations and structures to explore the concept of function. The activities and problems require students to use representations related to work with functions, and they highlight some of the interactions that may occur among these representations. The supplemental CD-ROM features interactive electronic activities, master copies of activity pages for students, and additional readings for teachers.

In the discussion of the Algebra Standard in Principles and Standards for School Mathematics (National Council of Teachers of Mathematics [NCTM] 2000), NCTM takes the stand that by explicitly working to develop algebra concepts and algebraic thinking from the prekindergarten years, many, in fact most, students can complete a reasonable equivalent of algebra 1 by the end of grade 8. In the elementary grades, algebraic reasoning is developed informally. This initial development provides the background for a more systematic study of algebra in the middle grades. This book focuses on the development of algebraic reasoning and algebra concepts in grades 6-8. Two themes span the content of algebra at this level: (1) using mathematical models to represent and understand quantitative relationships and (2) representing and analyzing mathematical situations and structures. The concept of mathematical function, which encompasses both themes, receives considerable attention at these grade levels.

Explorations that develop from problems that can be solved by using tables, graphs, verbal descriptions, concrete or pictorial representations, or algebraic symbols offer opportunities for students to build their understandings of mathematical functions. The relationships among the five representations can be shown by the framework in the margin.

Each representation highlights some aspect of the concept of function, but no single representation can help students develop the deep understanding of functions that is needed. In fact, it is the processes connecting one representation to another that help students make sense of the concept of function. For example, moving from a table of data to a graph of those data may be done by constructing the graph by hand. The manual process involves such actions as drawing and scaling the x- and y-axes and plotting pairs of points. Alternatively, using a graphing calculator to move from the data to a graph involves choosing an appropriate graphing window and making sense of the axes as seen on the screen of the graphing calculator. Students may not be explicitly aware of the process of plotting points, even though they can identify specific points on the graph by using the trace key, for example. Students' understanding involves an awareness of the similarities and differences in the processes that occur when they construct a graph of tabular data by hand and with technology. Such an awareness helps them develop a richer understanding of the relationship between a table and a graph of a set of data--an important component in building an understanding of the concept of function.

The four chapters in this book consider some of the topics related to integrating the themes of using mathematical models and representing and analyzing mathematical situations and structures. The activities and problems require students to use multiple representations related to work with functions and highlight some of the interactions that may occur among these representations.