The order of operations is one of those topics that looks easy from a distance and gets surprisingly tricky up close. Most 5th graders learn the PEMDAS acronym in a day. Many of them still get 5.OA.A.1 items wrong on the state test. The reason isn't the rule — it's the way the rule interacts with nested grouping symbols.
This post is about teaching the part that actually causes the lost points.
PEMDAS is fine. The trap is what comes inside the P.
The standard rule — Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right) — is something 5th graders can recite by Friday of the first week of the unit.
What they cannot reliably do, even by week 3, is solve expressions like this:
3 × [5 + (6 − 2)] = ?
Or this:
2 × {3 + [4 × (1 + 1)]} = ?
This is where 5.OA.A.1 lives. The standard explicitly mentions parentheses, brackets, AND braces. And state tests check it.
How to teach nesting so it sticks
The teaching move is to label the levels.
Show students that grouping symbols come in three matching pairs:
- ( ) round parentheses — innermost level
- [ ] square brackets — middle level
- { } curly braces — outermost level
The rule: solve from the innermost level outward. Round parens first, then square brackets, then curly braces.
I draw this on the board as concentric circles. The expression lives in layers, like a Russian nesting doll. Students unwrap one layer at a time and rewrite the expression as a simpler version after each unwrapping.
Worked example: 2 × {3 + [4 × (1 + 1)]}
Layer 1 (innermost parens): 1 + 1 = 2. Rewrite: 2 × {3 + [4 × 2]}
Layer 2 (brackets): 4 × 2 = 8. Rewrite: 2 × {3 + 8}
Layer 3 (braces): 3 + 8 = 11. Rewrite: 2 × 11
Final step: 2 × 11 = 22.
When students rewrite the expression after each step, they stop losing track of what they have left. Without the rewriting, they try to hold three layers in their head, lose one, and arrive at the wrong answer.
The "place the parentheses" puzzle is the secret weapon
Test items often go in reverse. The student gets an expression and a target value and has to place parentheses to make the equation true.
Example: "Place ONE pair of parentheses in 8 + 12 ÷ 2 × 5 so the result equals 50."
Without parentheses: 8 + 6 × 5 = 38. Wrong.
With parens around (8 + 12): (8 + 12) ÷ 2 × 5 = 20 ÷ 2 × 5 = 10 × 5 = 50. Correct.
This puzzle format is the single best way to teach the order of operations because it forces students to think about HOW parentheses change the order, not just memorize the order. Drill three of these per day for a week and students start to see expressions structurally rather than as random sequences of operations.
Multiplication and division go LEFT TO RIGHT (and so do addition and subtraction)
The biggest non-nesting mistake in 5th grade order of operations is treating the M-D and A-S pairs as separate steps. Students learn "M before D" because the letters look like a ranking. They aren't.
Multiplication and division have the SAME priority. When both show up, you do them in the order they appear, LEFT TO RIGHT.
50 ÷ 5 × 2 = ?
Wrong (M before D): 5 × 2 = 10; 50 ÷ 10 = 5. Right (left to right): 50 ÷ 5 = 10; 10 × 2 = 20.
Same rule for addition and subtraction. The fix: teach students that "PEMDAS" is really "P E (MD) (AS)" — two pairs of equal-priority operations done left to right.
The packet
I built a 5th grade test-prep packet with 40 order-of-operations problems covering the full standard from simple two-step problems through 4-level nested brackets and braces, plus word problems and "place the parentheses" puzzles. Every problem has a worked answer key.
5th Grade Math Test Prep: Order of Operations — $4
What's inside:
- Section 1: Two-Operation Problems (10 problems)
- Section 2: Parentheses, Brackets, and Braces with nesting (10 problems)
- Section 3: Real-World Word Problems (10 problems)
- Section 4: Expressions, Equivalence, and Challenge — including "place the parentheses" puzzles (10 problems)
- Complete answer key with step-by-step worked solutions
Standards: 5.OA.A.1, 5.OA.A.2. Single classroom license.
The takeaway
If your students can recite PEMDAS but miss order-of-operations items on the state test, the gap is almost always nesting. Teach the three-pair grouping-symbol hierarchy, drill the "rewrite the expression after each step" habit, and add three "place the parentheses" puzzles per day. The lost points come back fast.