Evaluating Expressions

Active learning works well for evaluating expressions because students must repeatedly practice substitution and order of operations, two skills that require muscle memory and immediate feedback. By engaging in collaborative tasks and error analysis, students confront their own misunderstandings in real time and build confidence through shared problem-solving.

• 6th Grade
• Mathematics
• 3 activities
• 20 min–35 min

Learning Objectives

1. Calculate the value of algebraic expressions involving whole-number exponents and variables.
2. Justify the necessity of the order of operations for consistent expression evaluation.
3. Analyze the impact of changing variable values on the outcome of an expression.
4. Critique common errors in applying the order of operations, specifically with exponents and multi-step calculations.
5. Compare the results of evaluating an expression before and after applying the order of operations.

Think-Pair-Share

⏱ 20 min·Pairs

Think-Pair-Share: Evaluate and Justify

Give pairs an expression with at least one exponent and a value to substitute (e.g., evaluate 2x² + 3x − 1 for x = 4). Each student evaluates independently, then partners compare their step-by-step work to find where (if anywhere) they diverge. The focus is on identifying the step where errors occur, not just getting the final answer.

Prepare & details

Justify why a standard order of operations is necessary for universal communication.

Facilitation Tip: During Think-Pair-Share, circulate and listen for students using the language 'x squared means x times itself' to clarify their reasoning.

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

Gallery Walk

⏱ 30 min·Small Groups

Gallery Walk: Spot the Mistake

Post 6-8 worked examples of expression evaluation around the room, with one error deliberately introduced in each. Students circulate, identify the error, write the correct step on a sticky note, and explain what rule was violated. The class debrief ranks the most common error types.

Prepare & details

Analyze how changing the value of a variable impacts the value of an expression.

Facilitation Tip: In the Error Analysis Gallery Walk, place a timer near each poster so students move efficiently and focus on identifying errors rather than debating them.

Setup: Wall space or tables arranged around room perimeter

Materials: Large paper/poster boards, Markers, Sticky notes for feedback

• Understand
• Apply
• Analyze
• Create
• Relationship Skills
• Social Awareness

Stations Rotation

⏱ 35 min·Small Groups

Collaborative Task: Expression Value Table

Groups receive an expression and a table with five different values of the variable. Each group member evaluates the expression for one or two values, then the group assembles the complete table and looks for patterns (e.g., does the expression increase or decrease as x increases?). Groups present their pattern observations to the class.

Prepare & details

Critique common errors in applying the order of operations.

Facilitation Tip: For the Expression Value Table, assign each pair a different starting value so the class collects multiple examples to discuss together.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

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

Teaching This Topic

Teach this topic by anchoring substitution in concrete examples first, then connecting it to the symbolic form. Use worked examples that deliberately include common errors so students recognize patterns in missteps. Research shows that students benefit most when they explain their reasoning aloud before writing it down, so incorporate verbal rehearsal before formal work.

What to Expect

Successful learning shows when students consistently substitute values correctly, apply operations in the proper order, and justify their steps with clear reasoning. They should be able to explain why exponentiation comes before multiplication and why left-to-right isn’t enough.

Watch Out for These Misconceptions

Common Misconception

During Think-Pair-Share, watch for students calculating 3² as 3 × 2 = 6 instead of 3 × 3 = 9.

What to Teach Instead

During Think-Pair-Share, ask students to expand the exponent first by writing out 3 × 3 before computing the value. Have partners check that the written expansion matches the symbolic form before calculating.

Common Misconception

During Error Analysis Gallery Walk, watch for students applying operations left to right without following PEMDAS.

What to Teach Instead

During Error Analysis Gallery Walk, direct students to highlight the first operation done correctly and the first error. Then, ask them to rewrite the solution in the correct order using PEMDAS as a guide.

Assessment Ideas

Exit Ticket

After Think-Pair-Share, collect each student’s written evaluation of 3x² + 5 when x = 4. Check that they substitute correctly, expand 4² as 4 × 4, multiply by 3, and add 5. Look for a sentence explaining why exponents come before multiplication.

Quick Check

During the Error Analysis Gallery Walk, pause the class after 3 minutes and ask volunteers to share one error they spotted and why it breaks the order of operations.

Peer Assessment

After the Expression Value Table, have partners swap papers and check each other’s evaluations. Ask them to identify one correct step and one step that could use clearer explanation, such as labeling the operation order.

Extensions & Scaffolding

• Challenge: Provide an expression with two variables, like 2x² + 3y when x = 3 and y = 2, and ask students to create a new expression using the same variables that evaluates to a different result.
• Scaffolding: Give students a partially filled evaluation table with blanks after each operation step so they can focus on the order, not the arithmetic.
• Deeper exploration: Introduce nested parentheses and exponents (e.g., (2x + 1)² when x = 2) and ask students to create a visual representation of the order of operations.

Key Vocabulary

Exponent	A number written as a superscript to indicate how many times the base number is multiplied by itself. For example, in 5², the exponent is 2.
Base	The number that is being multiplied by itself in an expression with an exponent. In 5², the base is 5.
Order of Operations	A set of rules that tells us the correct sequence for performing operations in a mathematical expression, often remembered by the acronym PEMDAS or BODMAS.
Evaluate	To find the numerical value of an expression by substituting given values for variables and performing the indicated operations.