Introduction to Order of Operations (Parentheses First) - Third Grade Mathematics

When an expression contains more than one mathematical operation, the order in which you calculate matters immensely. If two people calculate the exact same expression in different orders, they can end up with completely different answers! To ensure that mathematics is universal and unambiguous, mathematicians created the Order of Operations. In this chapter, you will discover the power of parentheses as the ultimate mathematical traffic sign that shouts: "Solve Me First!"

Why Do We Need Rules for Order?

Consider this simple mathematical expression: 3 + 4 x 2

Look at the two different answers you could get depending on your order:

Person A's Way (Left to Right):
First do addition: 3 + 4 = 7
Then do multiplication: 7 x 2 = 14
Answer = 14

Person B's Way (Multiplication First):
First do multiplication: 4 x 2 = 8
Then do addition: 3 + 8 = 11
Answer = 11

Both people performed their arithmetic correctly, but one got 14 and the other got 11! In mathematics, there can only be ONE true answer. This is why mathematicians agreed upon a clear set of rules called the Order of Operations.

The Number One Rule: Parentheses First!

Parentheses are curved brackets ( ) placed around parts of an expression.

The Golden Rule:
ALWAYS calculate whatever is inside the parentheses FIRST!

Parentheses act like a spotlight or VIP pass in mathematics. No matter what operation is inside—addition, subtraction, multiplication, or division—the operations inside parentheses get done first!

Look at how parentheses change the outcome:

Problem 1: (3 + 4) x 2
- Inside parentheses first: 3 + 4 = 7
- Then multiply: 7 x 2 = 14!

Problem 2: 3 + (4 x 2)
- Inside parentheses first: 4 x 2 = 8
- Then add: 3 + 8 = 11!

Notice that the numbers and symbols are in the exact same order, but the parentheses completely changed the result!

Multiplication and Division Before Addition and Subtraction

When an expression does NOT have parentheses, the standard mathematical order of operations rules apply:
1. First Priority: Operations inside Parentheses.

2. Second Priority: Multiplication and Division (from left to right).

3. Third Priority: Addition and Subtraction (from left to right).


Priority Ladder:
+---------------------------------------------------------------+
| Level 1 (Highest):  Parentheses ( )                           |
+---------------------------------------------------------------+
| Level 2:            Multiplication (x) and Division (÷)       |
+---------------------------------------------------------------+
| Level 3 (Lowest):   Addition (+) and Subtraction (-)          |
+---------------------------------------------------------------+

Example: Solving 15 - 2 x 4

Multiplication has higher priority than subtraction!
- Step 1: Multiply: 2 x 4 = 8.

- Step 2: Subtract: 15 - 8 = 7.
Answer = 7. If you had subtracted 15 - 2 = 13 first and then multiplied 13 x 4 = 52, you would have violated the order of operations!

Worked Step-by-Step Examples

Example 1: Combining Parentheses and Division

Solve: 24 ÷ (2 + 6)
- Step 1: Parentheses first! 2 + 6 = 8.

- Step 2: Divide: 24 ÷ 8 = 3.
Answer: 3.

Example 2: Multiple Operations

Solve: (5 x 4) - (12 ÷ 3)
- Step 1: First parentheses: 5 x 4 = 20.

- Step 2: Second parentheses: 12 ÷ 3 = 4.

- Step 3: Subtract the two results: 20 - 4 = 16.
Answer: 16.

Chapter Practice Exercises

  1. Calculate each expression by solving the parentheses first: a. (8 + 2) x 5 b. 8 + (2 x 5) c. (15 - 7) x 3 d. 15 - (7 x 2)
  2. Solve: a. 36 ÷ (4 + 5) b. (36 ÷ 4) + 5 c. (6 x 3) + (8 ÷ 2) d. 40 - (5 x 6)
  3. Insert parentheses to make each equation true: a. 4 + 3 x 2 = 14 b. 20 - 8 ÷ 2 = 6 c. 5 x 6 - 2 = 20
  4. Leo calculates 10 - 3 x 2 and gets 14 because he did 10 - 3 = 7 and 7 x 2 = 14. Explain Leo's mistake and find the correct answer.
  5. Write an expression with parentheses that represents this story: "Maya bought 2 notebooks for 4 dollars each and 1 pen for 3 dollars."

Solutions and Step-by-Step Answers

  1. Parentheses practice: a. (8 + 2) x 5 = 10 x 5 = 50. b. 8 + (2 x 5) = 8 + 10 = 18. c. (15 - 7) x 3 = 8 x 3 = 24. d. 15 - (7 x 2) = 15 - 14 = 1.
  2. Multi-step calculations: a. 36 ÷ (4 + 5) = 36 ÷ 9 = 4. b. (36 ÷ 4) + 5 = 9 + 5 = 14. c. (6 x 3) + (8 ÷ 2) = 18 + 4 = 22. d. 40 - (5 x 6) = 40 - 30 = 10.
  3. Placing parentheses: a. (4 + 3) x 2 = 7 x 2 = 14. b. (20 - 8) ÷ 2 = 12 ÷ 2 = 6. c. 5 x (6 - 2) = 5 x 4 = 20.
  4. Leo calculated from left to right instead of following the order of operations. Multiplication must be calculated before subtraction! Correct calculation: 3 x 2 = 6, then 10 - 6 = 4.
  5. Expression: (2 x 4) + 3 = 8 + 3 = 11 dollars.