New Jersey Mathematics Curriculum Framework

## STANDARD 13 - ALGEBRA

### K-12 Overview

 All students will develop an understanding of algebraic concepts and processes and will use them to represent and analyze relationships among variable quantities and to solve problems.

### Descriptive Statement

Algebra is a language used to express mathematical relationships. Students need to understand how quantities are related to one another, and how algebra can be used to concisely express and analyze those relationships. Modern technology provides tools for supplementing the traditional focus on algebraic techniques, such as solving equations, with a more visual perspective, with graphs of equations displayed on a screen. Students can then focus on understanding the relationship between the equation and the graph, and on what the graph represents in a real-life situation.

### Meaning and Importance

Algebra is the language of patterns and relationships through which much of mathematics is communicated. It is a tool which people can and do use to model real situations and answer questions about them. It is also a way of operating with concepts at an abstract level and then applying them, often leading to the development of generalizations and insights beyond the original context. The use of algebra should begin in the primary grades and should be developed throughout the elementary and secondary grades.

The algebra which is appropriate for all students in the twenty-first century moves away from a focus on manipulating symbols to include a greater emphasis on conceptual understanding, on algebra as a means of representation, and on algebraic methods as problem-solving tools. These changes in emphasis are a result of changes in technology and the resulting changes in the needs of society.

The vision proposed by this Framework stresses the need to prepare students for a world that is rapidly changing in response to technological advances. Throughout history, the use of mathematics has changed with the growing demands of society as human interaction extended to larger groups of people. In the same way that increased trade in the fifteenth century required businessmen to replace Roman numerals with the Hindu system and teachers changed what they taught, today's education must reflect the changes in content required by today's society. More and more, the ability to use algebra in describing and analyzing real-world situations is a basic skill. Thus, this standard calls for algebra for all students.

What will students gain by studying algebra? In a 1993 conference on Algebra for All, the following points were made in response to the commonly asked question, "Why study algebra?"

• Algebra provides methods for moving from the specific to the general. It involves discovering the patterns among items in a set and developing the language needed to think about and communicate it to others.

• Algebra provides procedures for manipulating symbols to allow for understanding of the world around us.

• Algebra provides a vehicle for understanding our world through mathematical models.

• Algebra is the science of variables. It enables us to deal with large bodies of data by identifying variables (quantities which change in value) and by imposing or finding structures within the data.

• Algebra is the basic set of ideas and techniques for describing and reasoning about relations between variable quantities.

Standard 8 (Numerical Operations) addressed the need for us to rethink our approach to paper-and-pencil computation in light of the availability of calculators; the need to examine the dominance of paper-and-pencil symbolic manipulation in algebra is just as important. The development of manipulatives, graphing calculators, and computers have made a more intuitive view of algebra accessible to all students, regardless of their previous mathematical performance. These tools permit and encourage visual representations which are more readily understood. No longer need students struggle with abstract concepts presented with very few ties to real-life situations. Rather, the new view of algebra offers real situations for students to examine, to generalize, and to represent in ways which facilitate the asking and answering of meaningful questions. Moreover, inexpensive symbolic processors perform algebraic manipulations, such as factoring, quickly and easily, reducing the need for drill and mastery of paper-and-pencil symbol manipulation.

### K-12 Development and Emphases

Algebra is so significant a part of mathematics that its foundation must begin to be built in the very early grades. It must be a part of an entire curriculum which involves creating, representing, and using quantitative relationships. In such a curriculum, algebraic concepts should be introduced in conjunction with the study of patterns and developed throughout each student's mathematical education. The earlier students are exposed to informal algebraic experiences, the more willing they will be to use algebra to represent patterns.

The concept of representing unknown quantities begins with using symbols such as pictures, boxes, or blanks (i.e., 3 + = 7). It is vital that students recognize that the symbol that is used to represent an unknown quantity has meaning. The only way this can be accomplished is to consistently relate the use of unknowns to actual situations; otherwise, students lack the ability to judge whether their answers make sense.

As students develop their understanding of arithmetic operations, they need to investigate the patterns which arise. Some of these patterns (which are commonly called properties) should be initially expressed in words. As the students develop more facility with variables, the properties can be expressed in symbolic form.

In the middle grades, problem situations should provide opportunities to generalize patterns and use additional symbols such as names and literal variables (letters). This development should continue throughout the remainder of the program, ensuring that the relationship between the variables (unknowns) and the quantities they represent is consistently stressed. Middle school students should extend their ability to use algebra to generalize patterns by exploring different types of relationships and by formalizing some of those relationships as functions. They should explore and generalize patterns which arise from nature, including non-linear relationships. As students move into the secondary grades, the graphing calculator and graphing software provide tools for examining relationships between x-intercepts and roots, between turning points and maximum or minimum values, and between the slope of a curve and its rate of change. As the student continues through high school, similar experiences should be provided for other functions, such as exponential and polynomial functions; these functions should be introduced using situations to which students can relate.

The use of algebra as a tool to model real world situations requires the ability to represent data in tables, pictures, graphs, equations or inequalities, and rules. Through exploration of problems and patterns, students are provided with opportunities to develop the ability to use concrete materials as well as the representations mentioned above. Having students use multiple representations for the same situation helps them develop an understanding of the connections among them. The opportunity to verbally explain these different representations and their connections provides the foundation for more formal expressions.

A fundamental skill in algebra is the evaluation of expressions and the solution of equations and inequalities. This process will be easier to understand if it is related to situations which give them meaning. Expressions, equations, and inequalities should arise from students' exploration in a variety of areas such as statistics, probability, and geometry. Elementary students begin constructing and solving open sentences. The use of concrete materials and calculators allow them to explore solutions to real-life situations. Gradually, students are led to expand these informal methods to include graphical solutions and formal methods. The relationship between the solutions of equations and the graphs of the related functions must be stressed regularly.

In summary, there are algebraic concepts and skills which all students must know and apply confidently regardless of their ultimate career. To assure that all children have access to such learning, algebraic thinking must be woven throughout the entire fabric of the mathematics curriculum.

Note: Although each content standard is discussed in a separate chapter, it is not the intention that each be treated separately in the classroom. Indeed, as noted in the Introduction to this Framework, an effective curriculum is one that successfully integrates these areas to present students with rich and meaningful cross-strand experiences.