Algebra is a combination of:
- A notation system for representing quantitative relationships, and
- A set of rules for manipulating notation without changing the underlying quantitative relationship that it represents.
Why is algebra needed? Because:
- The notation provides a concise and commonly accepted way of accurately communicating quantitative relationships, and
- Changing the appearance of the notation describing a relationship, often repeatedly, allows us to develop insights into the relationship and/or determine the answer to quantitative problems.
While algebra may be “only a set of rules”, applying the rules effectively and efficiently in complex problems can often seem to be as much art as procedure. Having multiple approaches that can work sometimes makes a task more complex.
When students are first introduced to algebra, it is often with the short term goal of learning to work with linear equations. Yet, for all the time students have probably spent taking “pre-Algebra”, many students who are starting to study linear situations are still challenged by the task of converting a problem expressed verbally into algebraic notation, or by the task of converting notation into words. This skill takes some time to master, and is well worth greater attention over time than it seems to receive in many classrooms.
Once one or more notational elements are second-nature to students, manipulating them becomes the next task to master. For example:
can be simplified a bit by “combining like terms”:
However “combining like terms” is a learned process, one which is not always fully understood by the students who are using it successfully. So, it can often be helpful to show that this same simplification could have been achieved by taking a different algebraic route, one which relies on fundamental properties that a student may intuitively understand better: the commutative property of addition, followed by factoring a common factor out of a sum or difference:
Manipulations of an expression or equation using the rules of algebra provide ways to change its appearance without changing the “truth” that it represents. This allows the “type” of problem to be changed to something more useful. For example:
is a sum of three terms (two of which are negative). I call this a “sum” because addition (sometimes of negative quantities) is the final operation that will need to be performed in evaluating this expression for a specific value of x, and therefore the first operation that I would have to “undo” in trying to isolate one term on one side of the equation.
Yet, by factoring the above into
I can convert the “sum” into a “product” where multiplication is the final operation that will need to be performed in evaluating this expression for a specific value of x. Multiplication will also be the first operation that will have to be “undone” in trying to isolate one factor on one side of the equation. This is a critical step when solving quadratics that are equal to zero, as without it we could not use the “zero product property” to set both factors equal to zero and solve them as linear equations.
Without the ability to change a sum into a product by factoring, or a product into a sum by distributing, working with many equations would be more difficult. Yet, this is only one of the many manipulations that the study of algebra persuades us can be performed without changing “the answer” to a problem.
Students should think of algebra not as “a set of rules for solving things”, but instead as “a set of rules for manipulating the appearance of an expression or equation, without changing the quantitative relationship that it represents”. We need to change the appearance of expressions and equations frequently as we work with them, either to solve them for a particular variable, or to get them into a form that is easier to analyze.
This perspective on what algebra is may not seem worth its picky distinctions when students are new to it, but will become increasingly useful over time. It helps students keep the big picture in mind as they learn an ever-growing list of formulas, techniques, rules, and applications.
The four Lovejoy boys (and I include myself in this group) took Algebra I for 8 years, combined.
Where were you when we needed you?