Mastering Mathematical Reasoning · 6th-class

Active learning ideas

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.

• NCCA Curriculum Specifications
• NCCA: Primary - Problem Solving

• 15–30 min
• Pairs → Whole Class
• 4 activities

Activity 01

• 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.

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.'

What to look for

Present students with a problem like: 'Sarah had some sweets. She gave half to her brother, then ate 3. She has 5 sweets left. How many did she start with?' Ask students to write down the reverse operation for each step and the final answer.

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

Activity 02

• 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.

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.

What to look for

Pose a problem where working backwards is not the best strategy. For example: 'John buys 5 apples at €0.50 each. How much does he spend?' Ask students: 'Why is working backwards not the best approach here? What strategy would you use instead and why?'

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

Activity 03

• 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.

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.

What to look for

Give students a problem where they must work backwards. For example: 'A number is multiplied by 3, then 5 is added, resulting in 26. What was the original number?' Ask them to write the original number and list the steps they took in reverse order to find it.

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

Activity 04

• 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.

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

What to look for

Present students with a problem like: 'Sarah had some sweets. She gave half to her brother, then ate 3. She has 5 sweets left. How many did she start with?' Ask students to write down the reverse operation for each step and the final answer.

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

A few notes on teaching this unit

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.

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.

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Watch Out for These Misconceptions

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

  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?'

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

  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.'

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

  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?'

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Methods used in this brief

• Inquiry Circle