Order of Operations

Students learn best when they move beyond memorization to experience the purpose of the order of operations. Active tasks let them see how grouping symbols and operation hierarchy change outcomes, making abstract rules concrete. Relays, puzzles, and builder activities turn what could feel like a set of steps into a problem-solving challenge they own.

• Grade 5
• Mathematics
• 4 activities
• 20 min–35 min

Learning Objectives

1. Evaluate numerical expressions using the order of operations (PEMDAS/BODMAS) with parentheses, brackets, and braces.
2. Analyze how the placement of grouping symbols (parentheses, brackets, braces) affects the outcome of a numerical expression.
3. Construct a numerical expression that results in a specific target value, applying the order of operations.
4. Explain the necessity of a consistent order of operations for unambiguous mathematical communication.
5. Identify and correct errors in the evaluation of numerical expressions that arise from violating the order of operations.

Escape Room

⏱ 30 min·Small Groups

Relay Race: Order of Ops Dash

Divide class into teams of four. Write expressions with grouping symbols on the board. First student solves and tags the next, who checks and solves the next. Include 8-10 expressions per round. Debrief as a class on common errors.

Prepare & details

Explain why a consistent order of operations is necessary for evaluating expressions.

Facilitation Tip: During the Relay Race, have students write each step on a separate index card so peers can follow the chain of reasoning in real time.

Setup: Group tables with puzzle envelopes, optional locked boxes

Materials: Puzzle packets (4-6 per group), Lock boxes or code sheets, Timer (projected), Hint cards

• Remember
• Apply
• Analyze
• Relationship Skills
• Self-Management

Escape Room

⏱ 25 min·Pairs

Pairs Puzzle: Grouping Switcheroo

Give pairs cards with expressions missing parentheses or brackets and a target result. They insert symbols in different spots, calculate each version, and explain which works. Switch partners midway to share strategies.

Prepare & details

Analyze how the placement of parentheses can change the value of an expression.

Facilitation Tip: In Pairs Puzzle, require students to sketch the expression with arrows showing which operation they evaluate next, making hidden steps visible.

Setup: Group tables with puzzle envelopes, optional locked boxes

Materials: Puzzle packets (4-6 per group), Lock boxes or code sheets, Timer (projected), Hint cards

• Remember
• Apply
• Analyze
• Relationship Skills
• Self-Management

Escape Room

⏱ 35 min·Small Groups

Small Groups: Target Expression Builder

Provide groups with numbers and operations cards plus a target value. They arrange into expressions using order of operations to match it. Groups present one solution; class verifies using the rules.

Prepare & details

Construct a numerical expression that yields a specific result using the order of operations.

Facilitation Tip: For Target Expression Builder, circulate with a clipboard to listen for students’ explanations of why their parentheses placement matters.

Setup: Group tables with puzzle envelopes, optional locked boxes

Materials: Puzzle packets (4-6 per group), Lock boxes or code sheets, Timer (projected), Hint cards

• Remember
• Apply
• Analyze
• Relationship Skills
• Self-Management

Escape Room

⏱ 20 min·Whole Class

Whole Class: Error Detective Hunt

Project expressions with deliberate mistakes. Students signal correct order with thumbs up/down, then vote on fixes. Tally results and discuss why the standard order prevents confusion.

Prepare & details

Explain why a consistent order of operations is necessary for evaluating expressions.

Facilitation Tip: During the Error Detective Hunt, assign each wrong answer to a team for correction and presentation to the class.

Setup: Group tables with puzzle envelopes, optional locked boxes

Materials: Puzzle packets (4-6 per group), Lock boxes or code sheets, Timer (projected), Hint cards

• Remember
• Apply
• Analyze
• Relationship Skills
• Self-Management

Teaching This Topic

Teachers know students resist grouping symbols until they see them change the problem’s outcome. Start with low-floor expressions, then layer in nested symbols and multi-step problems. Avoid teaching PEMDAS as a chant; instead, build from the concrete to the abstract through visual models and collaborative reasoning. Research shows that letting students debate incorrect solutions in small groups corrects misconceptions faster than direct instruction alone.

What to Expect

By the end of these activities, students should apply the order of operations fluently in expressions up to three-digit numbers and nested grouping symbols. They will justify their steps using the hierarchy and use parentheses strategically to hit numerical targets. Missteps become visible through peer discussion and live feedback, leading to lasting understanding.

Watch Out for These Misconceptions

Common Misconception

During Relay Race: Order of Ops Dash, watch for students scanning left to right and ignoring grouping symbols. When you see this, pause the relay and have the student demonstrate both the left-to-right and correct order side by side on the board for the team to discuss.

What to Teach Instead

During Pairs Puzzle: Grouping Switcheroo, watch for students treating parentheses, brackets, and braces as interchangeable. Ask them to build the same expression first with parentheses, then rewrite it with brackets, and finally with braces, comparing the results to see why nesting order matters.

Common Misconception

During Relay Race: Order of Ops Dash, watch for students performing addition before multiplication when it appears first in the expression. Redirect them by shading the multiplication and division areas in their expressions and solving those sections before addressing addition and subtraction.

What to Teach Instead

During Error Detective Hunt, watch for students who default to addition before multiplication. Assign them a problem like 8 + 2 x 5 and ask them to draw an area model to show why multiplication must come first, then revise their original approach.

Common Misconception

During Target Expression Builder, watch for students skipping grouping symbols or placing them randomly. Ask them to explain why their expression equals the target without parentheses, then have them add minimal parentheses to prove the necessity of grouping.

What to Teach Instead

During Small Groups: Target Expression Builder, watch for students who think all grouping symbols work the same. Provide mixed-nesting examples (e.g., 4 x [5 + (3 - 1)]) and ask groups to trace which operation gets evaluated first, second, and third to internalize the hierarchy.

Assessment Ideas

Quick Check

After Relay Race: Order of Ops Dash, give students a complex expression like 5 + [ (3 x 4) - 2 ] / 2 and ask them to write only the first step they would perform and explain why in one sentence.

Exit Ticket

After Small Groups: Target Expression Builder, hand each student a target number (e.g., 10) and a set of numbers and operations (e.g., 2, 3, 4, +, x). Ask them to build a valid expression using all numbers and operations that equals the target, using parentheses as needed.

Discussion Prompt

During Error Detective Hunt, pose the question: 'Two people solve 6 + 2 x 3 and get different answers. One gets 24, the other 12. How could this happen?' Circulate to listen for students’ references to operation hierarchy and grouping symbols in their explanations.

Extensions & Scaffolding

• Challenge early finishers to create a five-number expression with three sets of nested grouping symbols that equals zero.
• Scaffolding for struggling students: provide two-column grids where one side lists operations and the other side holds the result after each step.
• Deeper exploration: invite students to invent a new grouping symbol and write instructions for its use, then trade with peers to solve each other’s expressions.

Key Vocabulary

Order of Operations	A set of rules that dictates the sequence in which mathematical operations should be performed to ensure a consistent result. It is often remembered by acronyms like PEMDAS or BODMAS.
Parentheses	Curved symbols ( ) used to group parts of a mathematical expression. Operations within parentheses are performed first.
Brackets	Square symbols [ ] used to group parts of a mathematical expression, often nested within parentheses. Operations within brackets are performed after operations within inner parentheses.
Braces	Curly symbols { } used to group parts of a mathematical expression, often nested within brackets. Operations within braces are performed after operations within inner brackets.
Numerical Expression	A mathematical phrase that contains numbers, operations, and sometimes variables, which can be evaluated to a single numerical value.