## Using Inverse Operations to Solve Equations: When Students Ask

• Why should I bother learning this?
Discuss how multiplication and division equations can help you calculate the amount you might earn on a daily, weekly, monthly or annual basis. You use equations to find how long it will take you to travel a given distance at a certain rate or how to equally divide items among a group of people.
Explain to students that the simple expressions and equations that they are learning are the beginnings of algebra, which they will learn more fully in 8th grade and beyond. Discuss how algebra and equations are used extensively in science and how many questions about the universe have been answered by mathematical equations rather than by direct observation. Ask the students if they know of any scientific equations, such as E = mc2.
• Is an expression the same as an equation?
An expression is a number, a variable, or any combination of numbers, variables, operation signs, and grouping symbols.
Examples of expressions are:
3 + 4
3 x (c + 8)
2 − a

An equation is a statement that shows that two mathematical expressions are equal. Examples of equations are:

5 + 2 = 7
12 = 4 x a
y = 8
14 = 6 + (9 − c)

Discuss how expressions are like phrases, whereas equations are like sentences.

• What's an inverse operation?
An inverse operation is any operation that undoes another operation. Inverse operations are opposites. Have six students stand at the front of the room. Ask four more to come up. Ask which operation you just performed and then write the equation 6 + 4 = 10 on the chalkboard. Ask students what operation would undo what you just did, so that the original number of students will be at the front of the room. Explain that addition and subtraction are inverse operations, as are multiplication and division. Remind students that they frequently use inverse operations to check their computations.
• How can a word problem be translated into an algebraic expression or equation?
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 words and phrases that signify multiplication and division, such as “times,” “the product,” “separated into equal groups,” “divided evenly,” and so on. Often the solution required is stated as a question, such as “How many groups were there?” or “How much did she earn in all?” 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 have to be done in a certain order? Why can't I just do them in order from left to right?
Just like written and spoken languages, mathematical language has certain rules called the Order of Operations. Write this on the board: “Dog cat tree chased up the.” Ask students what the sentence tells you and what is wrong with it. Explain that just as we have rules of grammar for the proper order of words and phrases in a sentence, we have rules that tell us how to “read” a mathematical sentence, such as an equation or an expression. Show students the expression 20 × (4 − 2) x 3
Have half the class solve it using the correct order of operations:
20 × (2) x 3
10 x 3
30

Have the other half of the class ignore the parentheses and solve from left to right.

5 − 2 x 3
3 x 3
9

Discuss the difference between the results and the need to follow the Order of Operations.