Parentheses, Brackets, and Braces: Fifth Grade Mathematics

Nested grouping symbols—parentheses, brackets, and braces—establish layers of calculation that must be evaluated systematically from the innermost layer to the outermost layer.

The Hierarchy of Nested Grouping Symbols

When an expression contains multiple layers of grouping symbols, three different shapes are used to organize the visual structure:
- Parentheses: ( ) are the innermost symbols.

- Brackets: [ ] enclose parentheses.

- Braces: { } enclose brackets and parentheses.

+--------------------------------------------------------------------------+
|                  NESTED GROUPING SYMBOLS STRUCTURE                       |
|                                                                          |
|       {  Braces  [  Brackets  (  Parentheses  )  ]  }                    |
|                                                                          |
|       Order of Evaluation:                                               |
|       1. Innermost:  ( Parentheses )                                     |
|       2. Middle:     [ Brackets ]                                        |
|       3. Outermost:  { Braces }                                          |
+--------------------------------------------------------------------------+

The Inside-Out Evaluation Algorithm

Always locate the innermost parentheses first. Evaluate the operations within them according to PEMDAS. Once the inside value is a single number, remove the parentheses and move outward to the brackets.

Evaluating:  2 * { 15 + [ 4 * (8 - 3) ] }

Step 1: Innermost parentheses: (8 - 3) = 5
        2 * { 15 + [ 4 * 5 ] }

Step 2: Brackets: [ 4 * 5 ] = 20
        2 * { 15 + 20 }

Step 3: Braces: { 15 + 20 } = 35
        2 * 35

Step 4: Final outer operation:
        70

How Grouping Symbols Alter Numerical Values

Grouping symbols override standard operational precedence. Observe how the same digits and operations change values depending on where grouping symbols are placed:

Without Symbols:     4 + 6 * 2 - 3   =  4 + 12 - 3  = 13
With Parentheses:   (4 + 6) * (2 - 3)=  10 * (-1)   = -10
With Brackets:      [ (4 + 6) * 2 ] - 3 = [ 10 * 2 ] - 3 = 20 - 3 = 17

Exercises and Solutions

Exercise 1: Evaluating Nested Expressions. Evaluate the expression { 45 - [ 3 * (6 + 4) ] } + 12. Show each layer of evaluation from inside to outside.

Solution 1: Step 1: Evaluate the innermost parentheses: (6 + 4) = 10. Rewrite: { 45 - [ 3 * 10 ] } + 12. Step 2: Evaluate the brackets: [ 3 * 10 ] = 30. Rewrite: { 45 - 30 } + 12. Step 3: Evaluate the braces: { 45 - 30 } = 15. Rewrite: 15 + 12. Step 4: Perform the final addition: 15 + 12 = 27. The value of the expression is 27.

Exercise 2: Inserting Grouping Symbols for a Target Value. Insert parentheses into the expression 5 * 8 - 4 + 2 so that the expression equals 22. Explain your reasoning.

Solution 2: Without parentheses, 5 * 8 - 4 + 2 = 40 - 4 + 2 = 38. Test placing parentheses around (8 - 4): 5 * (8 - 4) + 2 = 5 * 4 + 2 = 20 + 2 = 22. This matches the target value of 22. The correctly grouped expression is 5 * (8 - 4) + 2.