Math Background

Expressions and Equations: Overview

The terms expression and equation have different mathematical meanings. It is important that your students understand the difference. Students will increasingly work with expressions and equations as they advance in school. The work they do this year will help lay the proper foundation for algebra in eighth grade.

An expression consists of a combination of numbers, operation symbols, and grouping symbols such as parentheses ( ).  An algebraic expression is an expression that contains one or more variables.

Examples of expressions and algebraic expressions are as follows

3 + 4
3 x (c + 8)
2 x a
x

Notice that an expression does not contain an equals sign (=).

An equation is a statement which shows that two expressions or values are equal. Examples of equations are

5 + 2 = 7
12 = 4 x a
y = 8
4 = 16 − (9 + c)

Students begin working with expressions and equations before third grade. When learning basic addition facts, they learn that 5 + 3 = 8. This is an example of an equation. When students use flash cards to learn these facts, one side of the card shows the expression 5 + 3.

When solving equations, students can think of an equation as a balance scale. Each pan on the scale holds an expression, and the center support of the scale represents an equals sign. For the scale to be balanced, both sides must be of equal value. For example, the scale below is balanced because the expression on each side of the scale has the same value, 5.

balance scale

Students can think of the expression 4 + 1 as another way of representing 5.

To simplify an expression, one performs all of the operations shown and produces a result. When parentheses are shown with more than one operation or step, such as (8 − 4) + 6, the operation within the parentheses must be performed first.

Help students realize that they use and simplify expressions in everyday life. For example, earning and spending money might be shown as ($20 − $5) − $10.

Cite other examples of simplifying from students' everyday lives, such as an overall golf score from individual holes, measuring and cooking instructions, or discounts and taxes taken at the cash register from the stated purchase price.

A variable is any symbol (most frequently a letter) that represents a number or a value that is unknown. Any letter can be used as a variable to represent any number. An expression that includes a variable is called an algebraic expression. Help students understand the meaning of the word variable by making the connection to words such as various or varied. Have students use various values for n in the algebraic expression n + 5 and record the varied results in a table. This should help dispel the common misconception of some students who tend to think that once n is defined as 5, it will always be 5.


Value of n  

  n + 5
2
3
12
7
8
17

To evaluate an expression is to substitute different values or numbers for a variable, or to find the value. Help students remember the meaning by pointing out the “value” in the word evaluate.

When moving from understanding expressions and equations to using them in word problems, the first step is turning an algebraic expression into words. This may seem to be a straightforward reading of notation, yet there are a variety of ways to write algebraic expression and equations. Various synonyms for operations can be used and all are technically correct. “Plus,” “added to,” and “more than” are all different ways to indicate addition. “Minus,” “take away,” and “subtracted from” can represent subtraction.

The phrase “some number” defines any variable. When students see an expression, such as x − 8, students can write this as “some number minus eight,” “eight less than some number,” or “take away eight from a number.” Be watchful that students don't reverse the meaning when writing 8 − x. This would be written as “eight minus some number” or “some number less than eight.” The mathematical syntax of an algebraic expression should be carried over to the words used.

Writing an equation from a word problem requires similar accuracy. Too often, students misinterpret the words used, and many students are intimidated by the bulk of information presented in paragraph form. A step process often helps in tackling a word problem.

  1. Find the question (generally at the end of the problem).
  2. Define variable as “the unknown number.”
  3. Go back through the problem and look for “operations” words.

Help students to break down the words, word phrases, and sentences into individual parts as shown above. Students tend to focus on the first numbers presented and often subtract or add them without regard to the words used in the text. By underlining and identifying terms such as combined, four more than, and decreased by, students will pinpoint the operations required. Careful reading is also necessary to identify relationships among numbers.

One way to solve equations that students will be familiar with is to find a missing addend. Inverse operations can also be used to solve equations. Addition and subtraction undo each other. Remind students of the balance scale. The amount being subtracted or added from one side of the equation must also be subtracted or added to the other side of the equation to produce a result that “balances,” or is equal.

Students can check their answers by substituting their result back into the original equation.

x + 12 = 17
x + 12 − 12 = 17 − 12
x = 17 − 12
x = 5
x + 12 = 17
5 + 12 = 17
17 = 17

Houghton Mifflin Math Grade 5