Using Variables Basic Algebra for Natural Numbers Basic Algebra for Integers

ORDINARY ALGEBRA

Most people who study what is called algebra do so primarily through drill and practice, so much so, that they may think of algebra as a collection of techniques. If a problem is posed without showing how it can be solved, they may respond by saying “How do you expect me to do that?” Because of this, few adults have the kind of understanding of algebraic concepts that allows for easy application of these concepts to a variety of situations. Nor can they help their children understand these concepts. To help rectify this we are designing a collection of resources for adults entitled "Understanding Ordinary Algebra".

I wish these resources would be used by everyone who can imagine some need to deepen his or her understanding of algebraic concepts. However I realize that while a desire to learn is an essential part of a person’s motivation, an incentive beyond this is also useful. The incentive I propose is that a deeper understanding of algebraic concepts is essential for adults who want to help children in constructing these concepts. Because of this, I include the types of materials, activities, and questions that might be used as a resource for helping children acquire such concepts. However these resources are directed much more to those adults who feels some need for a deeper understanding of algebra. They contain much more discussion about ideas than books written for secondary or middle school students. These resources are not designed to be a course in learning algebraic skills. To learn such skills, a person needs to devote time to solving more problems than have been included.

The three resources below can be accessed via link or downloaded as Microsoft Word Files.

Using Variables.doc Many people think of the letter ‘x’ when they think about algebra. They might even think of algebra as a bunch of rules for manipulating x. This is a very superficial way of thinking about algebra, but it is what many students experience when they study algebra in school. They may not realize that algebra is both a specialized language and a way of reasoning. It is a language that uses letters of the alphabet as variables and other concise symbols for talking about numbers. Algebraic language can be used to describe numerical patterns. It can be used to formulate numerical questions. The primary purpose of this first resource is to illustrate these descriptive aspects of algebraic language. However the language of algebra also lends itself to reasoning about these patterns and questions, and that is the primary reason for its power. This type of reasoning will be mostly left to other resources. Using Variables is merely the first in a series that provides materials and ideas that relate such concepts to ideas about language and reasoning.

Basic Algebra for Natural Numbers The two main topics of algebra are finding roots to conditional equations and finding equivalent expressions. Furthermore understanding the interplay between these two topics is essential to understanding ordinary algebra. In particular, finding equivalent expressions allows us to find equivalent equations, and this is one of the main aids in finding roots to equations. This idea will be further developed in this unit by relating it to the basic laws for the algebra of natural numbers. In addition to a study of laws for equivalent expressions, this unit will introduce some rules that can be used for finding equivalent equations without using equivalent expressions. These will be called cancellation rules.

Basic Algebra for Integers Subtraction and division are only partial operations on natural numbers. This unit will focus on an extension of the natural numbers in which subtraction will be a total operation. This is done by inventing a negative integer to be paired with each positive integer. Each element of a pair is called the opposite of the other. This suggests a new operation that assigns integer to its opposite. Crucial to understanding the algebra for the integers is to realize that altho extending the operations of addition and multiplication to the integers could have been done in other ways, there are good reasons for extending them exactly in the way it was done. While some of these reasons are due to applied considerations, the most fundamental considerations were conceptual. To appreciate the algebra of integers it is necessary to be aware of this. In particular, it is essential to understand the conceptual reasons for our decision about the product of two negatives. Many students never understand why this is a reasonable conceptual decision. To adequately appreciate this decision, a person must realize that the way we define the operations for integers is motivated primarily by the following consideration. While going beyond the conceptual limitations of the natural numbers, these extended operations will satisfy the laws they satisfied for natural numbers. To understand this, a person must acquire a concept of multiplication that goes beyond the idea that multiplication is repeated addition. There is one essential idea below about the concept of multiplication that is used in contemporary mathematics. Multiplication is conceptualized as an operation that distributes over addition.