## Expressions and Equations: When Students Ask

• Why should I bother learning this?
This is an excellent opportunity to connect mathematics with reading and language arts. In these subjects, students are frequently asked to learn the definition and spelling of new vocabulary. This holds true for mathematics as well. Students need to understand the meaning of expression, equation, and variable in order to further their experiences in mathematics. Writing expressions and equations enables us to translate everyday life experiences into mathematics.
• Is an expression the same as an equation?
No! An expression consists of a combination of numbers, operation symbols, and grouping symbols. Examples of expressions are

3 + 4
3 x (c + 8)
2 − a

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
14 = 6 + (9 − c)

• How can a word problem be translated into an algebraic expression?
Any word problem can be broken down into words or word phrases. These individual elements have a counterpart in the language of mathematics. Keep an eye out for obvious addition and subtraction synonyms like “plus,” “the sum,” “the difference between,” and so on. Often the solution required is stated as a question, such as “How many are left?” or “How much did the shoes cost?” The question usually indicates the unknown number that can be represented by a variable. An expression or equation can be created by using the other information given in the word problem.
• Why do operations within parentheses have to be done before other operations shown in an expression or equation?
The solution to an equation or the simplification of a longer expression can be quite different, depending on the order of operation. At this level, students are only doing parentheses first, and then a left-to-right guide as an order of operations, but later when multiplication and division come into play, the order of operations is even more critical. Demonstrate how parentheses affect the result of an equation as shown below.

(13 − 7) + (3 − 2) = ?
(6) + (1) = 7

If parentheses were ignored and the operations were done as 13 − (7 + 3) − 2, after simplifying, it would be 13 − 10 − 2 = 1.

• How does using inverse operations place the variable alone on one side of an equation?
Show how doing the opposite, or “undoing” the expression that includes the variable, results in isolating the variable.
• Why does a variable have to vary? Why can't it be one thing?
The idea of a simple letter doing so many different tasks in so many different forms is often hard for students to accept. They want the comfort of x staying the same after they worked so hard to give it a value. But remind students that equations are extremely helpful in describing a process. A scientist may want a rule that describes how the temperatures of chemicals change under pressure. The temperature or the amount of pressure may change but the equation used has remained the same. In other words, the conditions may vary but the description of the process stays the same. Variables don't have to be letters; they could be * or ∞ or ¶; it doesn't matter.