Nested Grouping: Inside Out

Lesson 2 of 2: Brackets, Braces, and Nesting

In this lesson:

• Meet all three grouping symbols: ( ) [ ] { }
• Evaluate nested expressions from the inside out
• Write expressions using grouping symbols

Learning Objectives

By the end of this lesson, you should be able to:

4. Evaluate expressions with nested grouping symbols by working from the innermost group outward
5. Write a numerical expression using appropriate grouping symbols to match a described calculation or a given target value

Three Grouping Symbols — One Rule

All three mean the same thing: evaluate what is inside me first

Inside Out: [4 × (2 + 3)] − 5

Start with the innermost group — work outward one layer at a time.

Let's Practice Together: [3 × (5 − 2)] + 4

Step 1: Innermost group — the parentheses
(5 − 2) = 3 → expression becomes: [3 × 3] + 4

Step 2: Next group — the brackets
[3 × 3] = 9 → expression becomes: 9 + 4

Step 3: Evaluate what remains
9 + 4 = 13

Your Turn: [6 + (4 × 2)] − 7

Evaluate step by step — write every rewrite:

Step 1: Innermost group ( ) → _____ = _____

→ expression becomes: [6 + _____] − 7

Step 2: Brackets [ ] → _____ = _____

→ expression becomes: _____ − 7

Step 3: Final answer → _____

Quick Check

In the expression {[(2 + 3) × 4] + 1}, which group do you evaluate first?

• A: The braces { }
• B: The brackets [ ]
• C: The parentheses ( )

Which group has nothing else nested inside it?

Three Levels: {3 × [2 + (4 × 1)]} + 6

Level 1 — innermost parentheses:
(4 × 1) = 4 → {3 × [2 + 4]} + 6

Level 2 — brackets:
[2 + 4] = 6 → {3 × 6} + 6

Level 3 — braces:
{3 × 6} = 18 → 18 + 6 = 24

Evaluation Template

For any nested expression, fill in each line:

Original → {[5 × (3 + 2)] − 8}

After ( ) → _______________

After [ ] → _______________

After { } → _______________

Final answer → _______________

Use this template whenever you face a nested expression

Key Takeaways

✓ ( ), [ ], and { } are all grouping symbols — same rule for all three
✓ With nested grouping: start with the innermost group, work outward
✓ Rewrite the expression after each group is evaluated — show every step

Watch out: Start with the innermost ( ) — not the outermost { }
Watch out: [ ] and { } follow the same rule as ( ) — no extra rules for different symbols

Well Done!

You can now:

• Evaluate any expression with grouping symbols — including nested levels
• Work through nested layers from the inside out
• Write expressions with grouping symbols to match a description

Next up: writing and interpreting numerical expressions (5.OA.A.2)