Working Backwards Strategy

Active learning suits this topic because reversing operations requires physical manipulation of steps to build durable mental models. When students pair forward actions with their reverses, they internalize patterns rather than memorize procedures. Movement and collaboration also reduce cognitive load, allowing working memory to focus on logical sequencing rather than recalling rules.

• 6th-class
• Mastering Mathematical Reasoning
• 4 activities
• 15 min–30 min

Learning Objectives

1. Analyze word problems to determine if the working backwards strategy is the most efficient method for finding the initial value.
2. Apply the working backwards strategy by reversing operations to solve problems with a known end result.
3. Explain each step taken when using the working backwards strategy to verify the accuracy of a solution.
4. Calculate the initial quantity in a multi-step problem by systematically reversing each operation.
5. Compare the working backwards strategy with other problem-solving methods for a given scenario.

Ready-to-Use Activities

Inquiry Circle

⏱ 20 min·Pairs

Pairs: Reversal Card Match

Pairs receive cards showing end results and forward operations, like 'ends with 20 after +5 and x2'. They match or create reverse steps: divide by 2, subtract 5. Partners verify by working forward from their start and discuss why each reverse works.

Prepare & details

Analyze when the working backwards strategy is most helpful for solving a problem.

Facilitation Tip: During Reversal Card Match, circulate and listen for students verbalizing the connection between forward and reverse operations, such as 'If I doubled the marbles, I halve them to go back.'

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

Inquiry Circle

⏱ 30 min·Small Groups

Small Groups: Relay Reverse

In groups of 4, assign a multi-step problem with known end. Last student reverses final operation and passes result; previous reverses next, until first finds start. Group checks solution forward and explains to class.

Prepare & details

Apply the working backwards strategy to solve a problem by starting from the known end result.

Facilitation Tip: In Relay Reverse, stand at the finish line to time each group’s final step and prompt them to verify their answer forward before declaring success.

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

Inquiry Circle

⏱ 25 min·Whole Class

Whole Class: Build the Path

Display a problem on board. Students think alone for 2 minutes, pair-share reverses for 4 minutes, then whole class contributes steps to build solution on chart paper. Vote on best explanation.

Prepare & details

Explain each step taken when using the working backwards strategy to check a solution.

Facilitation Tip: For Build the Path, step in as students reach dead ends to ask guiding questions like 'What would undo the last action you took?' without solving it for them.

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

Inquiry Circle

⏱ 15 min·Individual

Individual: Puzzle Journal

Students get 3 tiered problems in journals. They solve backwards solo, draw arrows for each reverse step, then note when strategy fits best. Share one with partner for feedback.

Prepare & details

Analyze when the working backwards strategy is most helpful for solving a problem.

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

Teaching This Topic

Teach this strategy by modeling the language of reversal aloud. Use think-alouds to show how each step undoes the previous one, emphasizing precision in operation choice. Avoid rushing to shortcuts; students benefit from slow, deliberate reversal even in multi-step problems. Research suggests kinesthetic pairing accelerates comprehension, so prioritize movement over worksheets in early lessons.

What to Expect

Successful learning looks like students confidently identifying reverse operations without hesitation and explaining each step aloud. They should recognize when working backwards is appropriate and justify their process to peers. Persistent reversal errors should decrease as activities progress, with students self-correcting using their matched pairs or relay logs.

Watch Out for These Misconceptions

Common Misconception

During Reversal Card Match, watch for students pairing + only with minus without considering the operation’s context.

What to Teach Instead

Guide pairs to sort cards by operation first, then match forward actions with their true reverses. Ask, 'What undoes doubling? Halving or subtracting? How do you know?'

Common Misconception

During Relay Reverse, watch for groups skipping verification steps or accepting answers without forward checks.

What to Teach Instead

Require each group to demonstrate their solution forward before recording it on the board, using peer questioning like 'Show me how you’d get back to the start with these numbers.'

Common Misconception

During Build the Path, watch for students assuming working backwards only applies to subtraction scenarios.

What to Teach Instead

Use the class-built path to highlight varied operations, such as 'If we added 30% tax to a price, how do we reverse tax to find the original cost?'

Assessment Ideas

Quick Check

After Reversal Card Match, collect matched pairs from three pairs of students and quickly scan for correct reverses. Ask one student per pair to explain their match aloud to verify understanding.

Discussion Prompt

During Relay Reverse, pause halfway to ask groups to justify why working backwards is or isn’t suitable for their assigned problem, noting their reasoning for assessment.

Exit Ticket

After Puzzle Journal, collect journals and review the reverse steps and final answer. Look for logical sequences and correct operation reversals, noting patterns of error for targeted review.

Extensions & Scaffolding

• Challenge students to create their own multi-step working backwards problems for peers to solve, including a variety of operations.
• Scaffolding: Provide partially completed reverse operation strips during Puzzle Journal for students to fill in missing steps.
• Deeper exploration: Introduce problems with fractions or decimals during Build the Path to extend beyond whole numbers.

Key Vocabulary

Working Backwards Strategy	A problem-solving technique where you start with the final answer and reverse the steps to find the initial condition.
Reverse Operation	The opposite mathematical operation that undoes another operation, such as addition undoing subtraction or multiplication undoing division.
Initial Value	The starting number or quantity in a problem before any operations have been applied.
End Result	The final number or quantity in a problem after all operations have been completed.