Mathematics · 5th Grade

Active learning ideas

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.

• Common Core State Standards
• CCSS.Math.Content.5.OA.A.1
• CCSS.Math.Content.5.OA.A.2

• 15–25 min
• Pairs → Whole Class
• 3 activities

Activity 01

• 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.

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.

What to look for

Present students with two versions of the same expression, one with parentheses and one without, e.g., 5 + 3 x 2 and (5 + 3) x 2. Ask students to calculate both and explain in writing why the answers are different.

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

Activity 02

• 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.

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.

What to look for

Write the phrase 'Subtract 5 from 12, then multiply the result by 3.' on the board. Ask students to write the numerical expression and then solve it, showing their steps.

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

Activity 03

• 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.

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).

What to look for

Pose the question: 'Why is it important that everyone in math class solves 10 - 4 ÷ 2 the same way?' Facilitate a discussion where students explain the need for a universal order of operations.

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

A few notes on teaching this unit

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.

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

• 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.

  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.

• 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.

  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.

---

Methods used in this brief

• Inquiry Circle
• Role Play
• Think-Pair-Share