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How to Decode the Structure of an Algebraic Expression

When you first look at an advanced algebraic expression, it can be a little overwhelming. And no wonder! You're basically looking at a complex grammatical sentence in a different language. 

If you're a newbie to the language of math, you're probably not going to be able to understand very much, unless you break the expression down into little parts that you can understand. 

When I was in 6th grade, my teacher taught us how to diagram sentences. I loved learning how to break up a complicated English sentence into prepositional phrases, conjunctions, and all the other parts of speech. 

Decoding the structure of algebraic expressions is the math version of diagramming sentences in English.

You don't have to do it the exact same way that I do it, but if you practice breaking down the structure of expressions into smaller pieces, it will make it much easier to understand the language of math. 

Helpful Vocabulary Words

You have to know the parts of speech before you diagram an English sentence. Similarly, you need know some math vocabulary before you can decode the structure of an algebraic expression.

If you're not familiar with any of these vocabulary words, quickly review them before continuing.

Addend

Argument

Augend

Base

Coefficient

Difference

Dividend

Divisor

Equation

Exponent

Expression

Factor 

Minuend

Product

Power

Quotient

Roots

Subtrahend

Sum

Term

Variable

How to Decode the
Structure of Algebraic Expressions

These are the basic steps I follow when I decode algebraic expressions. 

  1. If you are given an equation (or inequality), identify the expressions on both sides of the equal/inequality sign. 
  2. Notice which operations (\(+,-,\times,\div, x^{...}, \sqrt{...}\)) are happening in the expression(s).
  3. Pay attention to which operations are layered within other operations. 

These steps are a little vague and generic because the easiest way to learn how to decode algebraic expressions is to see examples.

So, I've decoded 4 famous math expressions on this page. Feel free to try them on your own and then see the way that I decoded them. 

Slope Intercept Form of a Line

\(y=mx+b\)

Standard Equation of a Circle

\((x-h)^{2}+(y-k)^{2}=r^{2}\)

Difference of Squares

\(a^{2}-b^{2}=(a+b)(a-b)\)

Quadratic Formula

\(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\)

Algebraic Structure of the 
Slope-Intercept Form of a Line
\(y=mx+b\)

The slope-intercept form of a line is an equation with two expressions.

The expression on the left side of the equal sign (\(y\)) is a variable.

The expression on the right side of the equal sign (\(mx+b\)) is a sum.

There are two terms in the sum. The first term (\(mx\)) is the product of two variables. The second term (\(b\)) is a variable.

So, the overall structure of the slope intercept form of a line is...

Algebraic Structure of the
Standard Equation of a Circle
\((x-h)^{2}+(y-k)^{2}=r^{2}\)

The standard equation of a circle is an equation with two expressions.

The expression on the left side of the equal sign (\((x-h)^{2}+(y-k)^{2}\)) is a sum.

The expression on the right side of the equal sign (\(r^{2}\)) is a power.

On the right side, the power has a variable (\(r\)) in the base and a number (\(2\)) in the exponent. 

On the left side, the sum has two terms (\((x-h)^{2}\) and \((y-k)^{2}\)). Each of these terms is a power.

Both of these powers have an exponent of \(2\) and bases that are differences (\((x-h)\) and \((y-k)\)) of variables.

So, the overall structure for the standard equation of a circle is...

Algebraic Structure of the
Difference of Squares Relationship
\(a^{2}-b^{2}=(a+b)(a-b)\)

The difference of squares relationship is an equation with two expressions.

The expression on the left side of the equal sign (\(a^{2}-b^{2}\)) is a difference.

The expression on the right side of the equal sign (\((a+b)(a-b)\)) is a product.

On the left side, the difference has two terms (\(a^{2}\) and \(b^{2}\)).

Each of these terms is a power with a variable (\(a\) and \(b\)) in the base and a number (\(2\)) in the exponent.

On the right side, the first factor in the product (\((a+b)\)) is a sum of two variables (\(a\) and \(b\)).

The second factor (\((a-b)\)) is the difference of two variables. 

So, the overall structure of the difference of squares relationship is...

Algebraic Structure of the
Quadratic Formula
\(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\)

The quadratic formula is an equation with two expressions.

The expression on the left side of the equal sign (\(x\)) is a variable.

The expression on the right side of the equal sign (\(\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\)) is a quotient.

The dividend of the quotient (\(-b\pm\sqrt{b^{2}-4ac}\)) is a sum (or difference) of a variable (\(-b\)) and a root (\(\sqrt{b^{2}-4ac}\)).

The divisor of the quotient (\(2a\)) is the product of a number (\(2\)) and a variable (\(a\)). 

The argument of the root (\(b^{2}-4ac\)) is a difference.

The subtrahend of the difference (\(b^{2}\)) is a power with a variable (\(b\)) in the base and a number (\(2\)) in the exponent.

The minuend of the difference (\(4ac\)) is the product of a number (\(4\)) and two variables (\(a\) and \(c\)) .

So, the overall structure of the quadratic formula is...

Helpful Questions to Decode
the Structure of an Algebraic Expression

Is it an expression, equation, or inequality?

Equations and inequalities have TWO expressions connected with an equal sign or an inequality sign. 

If you do not have an equal sign or an inequality sign, then you are dealing with just one expression. 

For example,  \(a^{2}+b^{2}=c^{2}\), is an equation with two expressions (\(a^{2}+b^{2}\) and  \(c^{2}\)).

But \(\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) is just one expression because there is no equal sign or inequality sign. 

What is the outer-most operation?

If you have a lot of parentheses (or grouping symbols) in your algebraic expression, you can go to the "farthest out" set of parentheses to see what the outer-most operation is. 

It can be helpful to temporarily ignore what is going on inside of the parentheses and just look at what is being done to the parentheses.

Are they being raised to an exponent? Are you adding something to the parentheses? Are there two sets of parentheses being multiplied by each other? 

For example, if you were decoding the structure of \((x-a)(x+b)\), you could temporarily ignore the stuff inside of the parentheses and think about it like this...

\((....)(....)\)

This makes it really easy to see that the outer-most operation is multiplication. So, that makes this expression a product. 

Once you know what the outer-most operation is, what are the inner operations?

Many algebraic expressions are multi-layered.

A product will have at least two factors and each of those factors could have another operation (or multiple operations) inside of it. 

Sums, powers, differences, roots, and quotients work the same way. The may have other operations layered inside of their terms, bases, exponents, arguments, dividends, etc.

In \((x-a)(x+b)\), the outer-most operation is multiplication but there's also subtraction and addition within the multiplication.

The factors that are multiplied together are  \((x-a)\) and \((x+b)\). The first factor is a difference and the second factor is a sum.

So, the entire expression is a product of a difference and a sum. 

If you evaluated the expression with order of operations, what is the last operation you would complete?

It can be helpful to write a list of all the things you would do if you used order of operations to simplify the expression (assuming you knew what numbers the variables represented). 

Once you have that list, you can work backwards through the list to identify the structure of the algebraic expression. 

For example, if you wanted to decode the structure of \(mx+b\), you could make the following list of steps based on the order of operations:

  1. Multiply m by x.
  2. Add b to the result of Step 1. 

When you work backwards through the list, you can see that the last step is addition, so that means that \(mx+b\) is a sum. 

The second-to-last step was multiplication so that means that \(mx\) is a product.

The entire expression is the sum of a product (\(mx\)) and a variable (\(b\)). 

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