Numerical Expressions and Order of Operations

Active learning works because numerical expressions and order of operations are about logic, not just computation. Students need to see, discuss, and physically manipulate grouping symbols to build lasting understanding. Movement and collaboration make abstract rules visible and memorable.

• 5th Grade
• Mathematics
• 3 activities
• 15 min–25 min

Learning Objectives

1. Analyze how the placement of parentheses and brackets changes the outcome of a numerical expression.
2. Create a numerical expression that accurately represents a given verbal phrase involving multiple operations.
3. Justify the necessity of a standard order of operations for consistent mathematical problem solving.
4. Calculate the value of complex numerical expressions using the order of operations, showing all steps.

Inquiry Circle

⏱ 25 min·Small Groups

Inquiry Circle: The Parentheses Power-Up

Give groups a string of numbers and operations (e.g., 4 + 6 x 2). Challenge them to place parentheses in different spots to create as many different final values as possible. Groups present their 'highest' and 'lowest' possible values to the class.

Prepare & details

Analyze how grouping symbols alter the meaning of a mathematical expression.

Facilitation Tip: During Collaborative Investigation: The Parentheses Power-Up, circulate and ask groups to explain their grouping choices before calculating to uncover hidden assumptions.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Role Play

⏱ 20 min·Whole Class

Role Play: Order of Operations Line-Up

Students wear cards with numbers and operation symbols. A 'Director' (student) arranges them into an expression. The class must then 'perform' the expression by having the students in parentheses step forward first to solve their part, followed by multiplication/division, and then addition/subtraction.

Prepare & details

Justify the necessity of a standard order of operations for universal mathematical communication.

Facilitation Tip: During Role Play: Order of Operations Line-Up, stand back after giving the first command to let students self-correct mistakes—this builds ownership of the left-to-right rule.

Setup: Open space or rearranged desks for scenario staging

Materials: Character cards with backstory and goals, Scenario briefing sheet

• Apply
• Analyze
• Evaluate
• Social Awareness
• Self-Awareness

Think-Pair-Share

⏱ 15 min·Pairs

Think-Pair-Share: Translating Math Talk

Provide a list of word phrases (e.g., 'triple the sum of five and nine'). Students work in pairs to write the numerical expression. They then swap with another pair to see if they can translate the expression back into words accurately.

Prepare & details

Translate a written phrase into an accurate numerical expression.

Facilitation Tip: During Think-Pair-Share: Translating Math Talk, listen for students who connect the phrase ‘divide by 2 then multiply by 3’ to writing 12 ÷ 2 x 3 instead of 12 ÷ (2 x 3).

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Teaching This Topic

Teach order of operations by anchoring it in clear, everyday language rather than mnemonics like PEMDAS. Use color-coding and physical movement to show that grouping symbols act like traffic signals, stopping computation until the group is resolved. Avoid teaching multiplication before division or addition before subtraction as separate steps—emphasize that these operations share the same priority and must be read left to right.

What to Expect

Successful learning looks like students explaining why parentheses change outcomes, following left-to-right rules for multiplication and division, and translating word phrases into correct expressions without hesitation. They should justify each step using precise mathematical language.

Watch Out for These Misconceptions

Common Misconception

During Role Play: Order of Operations Line-Up, watch for students who freeze when they hear ‘divide and multiply’ because they recall ‘M comes before D’ in PEMDAS.

What to Teach Instead

Pause the line and have students physically reorder themselves to show that division and multiplication are partners. Ask them to walk through the expression 12 ÷ 3 x 2, stepping forward in pairs to model left-to-right calculation.

Common Misconception

During Collaborative Investigation: The Parentheses Power-Up, watch for students who ignore parentheses at the end of an expression because they look like they’re not needed.

What to Teach Instead

Have students use yellow highlighters to circle every set of grouping symbols before writing anything. Ask them to explain why the circled part becomes the first thing they solve, even if it sits at the tail end of the expression.

Assessment Ideas

Quick Check

After Collaborative Investigation: The Parentheses Power-Up, present two expressions on the board: 9 + 6 ÷ 3 and (9 + 6) ÷ 3. Ask students to calculate both, then write a sentence explaining why the answers differ.

Exit Ticket

After Think-Pair-Share: Translating Math Talk, display the phrase ‘Add 7 and 5, then divide the total by 2.’ Ask students to write the expression (7 + 5) ÷ 2 and solve it, circling the step they completed first.

Discussion Prompt

During Role Play: Order of Operations Line-Up, ask students why everyone should solve 20 - 8 ÷ 2 the same way. Facilitate a 2-minute turn-and-talk before taking volunteers to share their reasoning.

Extensions & Scaffolding

• Challenge students who finish early to create a real-world scenario that requires a complex expression with braces, brackets, and parentheses, then exchange with a partner to solve.
• For students who struggle, provide expressions with only one type of grouping symbol at a time before introducing mixed symbols.
• Allow extra time for students to design a short comic strip that explains why 8 ÷ 2 x 4 ≠ 8 ÷ (2 x 4), using speech bubbles to label each step.

Key Vocabulary

Parentheses	Curved symbols used to group numbers and operations, indicating that the enclosed operations should be performed first.
Brackets	Square symbols used to group expressions, often enclosing parentheses, to further clarify the order of operations.
Order of Operations	A set of rules that dictates the sequence in which mathematical operations should be performed to ensure a consistent result.
Numerical Expression	A mathematical phrase that contains numbers, operation symbols, and grouping symbols, but no variables.