Peel and Solve method for solving linear equations

Peel and Solve (P&S) is a structured, error-reducing method for solving linear equations that teaches students to remove the outermost operation first, improving inverse operation sequencing and reducing transposition mistakes. This page publishes the full Peel and Solve paper in HTML so that maths teachers, students, and mathematics education researchers searching for linear equation methods, reverse BIDMAS alternatives, or Cognitive Load Theory-informed approaches can read, search, and reference the method directly.

Author: Jai Verma · DOI: 10.5281/zenodo.15021729

Abstract

Objective: This paper introduces Peel and Solve (P&S), a structured method for solving linear equations designed to help students correctly apply inverse operations.

Methods: Unlike traditional reverse BIDMAS approaches, which assume students can intuitively determine the order of operations to undo, P&S explicitly directs students to identify and remove the outermost operation first, reducing errors in sequencing and transposition. The method integrates Cognitive Load Theory (Mayer, 2019) and aligns with evidence-based strategies for addressing common algebraic misconceptions documented in mathematics education research.

Results: P&S systematises the removal of operations, eliminating ambiguities present in traditional reverse BIDMAS methods. It aligns with NCTM’s Principles to Actions (2014) on procedural-conceptual balance. Preliminary classroom observations indicate that P&S may help reduce transposition errors, but controlled trials are required to measure its impact objectively.

Conclusion: P&S provides a structured, step-by-step approach to solving linear equations, offering a potential scaffold for students struggling with inverse operations. However, it should be used alongside conceptual teaching strategies such as Singapore CPA frameworks. Further empirical research is required to validate its impact on student error rates and long-term algebraic retention.

1. Introduction

Linear equation-solving errors often arise from misapplying inverse operations or misinterpreting structural relationships between terms. Common pitfalls include:

  • Sign errors when subtracting or moving terms across the equals sign.
  • Fraction misinterpretations, such as treating numerators and denominators as independent integers.
  • Transposing terms without maintaining balance in the equation.

Traditional reverse BIDMAS (SAMDIB) methods fail to address these issues because they assume students will automatically recognise which operation to undo first. While BIDMAS provides a precise order for applying operations, reverse BIDMAS simply reverses this order, it does not explicitly teach students how to determine the correct first step within an equation’s structure. This often leads to mistakes, particularly in more complex cases.

For example, in the equation:

3(x - 3) / 4 + 2 = 10

many students misapply reverse BIDMAS (SAMDIB) by focusing on subtraction first. Since subtraction appears first in SAMDIB, they assume they must undo −3 first by adding 3 to both sides. However, the actual outermost operation is +2, which should be removed first. Later, they often multiply by 3 prematurely, further complicating the solution.

Similarly, in:

3x + 2 = 7

if students are encouraged to be flexible in their approach (which is correct in theory but often misapplied), they may:

  • Divide first, producing a fraction early and requiring additional steps to simplify later.
  • Apply division to all terms, but then misapply subtraction in a different equation by subtracting from every term incorrectly: 3x - 2 + 2 - 2 = 7 - 2.

This leads to confusion about why division applies to everything, but subtraction does not. Without explicit guidance, students are left guessing the correct order or relying on reverse BIDMAS alongside the Onion Skin method to determine the first step. Peel and Solve (P&S) removes this ambiguity by explicitly teaching students how to identify the first step, rather than expecting them to intuitively reverse BIDMAS correctly.

While Onion Skin/Backtracking relies on visual layering, P&S is a structured decision-making framework that works without diagrams, making it easier to apply consistently across different types of equations. By guiding students through a step-by-step identification of the outermost operation, P&S provides a repeatable method for solving equations efficiently.

This paper outlines the Peel and Solve (P&S) method as a structured procedural tool for reducing algebra errors. While informal classroom experiences inform its development, empirical studies are needed to validate its effectiveness across diverse learning environments.

2. Theoretical Framework: Cognitive Load Theory

P&S applies cognitive load theory (Mayer, 2019) by:

  1. Reducing extraneous load: Providing a clearer path to solving equations without mentally reconstructing BIDMAS order.
  2. Chunking complex tasks: Breaking equation-solving into four manageable steps.
  3. Guiding attention: Helping students focus on one operation at a time.

For example, solving the equation (x - 3) / 5 + 7 = 10 becomes a sequence of simple steps (subtract 7 → multiply by 5 → add 3), rather than requiring students to track multiple steps simultaneously.

3. The Peel and Solve Method

3.1 Full Methodological Framework

Goal: Isolate x by systematically removing operations in reverse order of their application.

Note: This method assumes students understand BIDMAS (order of operations). A quick review may help ensure correct identification of the last operation applied. A step-by-step breakdown of how to determine the outermost operation is covered in Section 3.4.

Step 1: Identify the Outermost Operation

  • Focus on the side containing x.
  • Ask any of these questions:
    • "If x were a number, what operation would I perform last?"
    • "What is the furthest thing from x on this side of the equation?"
    • "What is the last thing you would do to x when calculating its value?"

Example: For the equation (x - 3) / 5 + 7 = 10 , replacing x with 18 shows that +7 is the last operation applied.

Visual example:

(x - 3) / 5 + 7 = 10

  1. Outermost operation: +7
  2. Inverse: −7
  3. Apply to both sides.
  • "If there are variables on both sides, what should I focus on first?"

If x appears on both sides, the focus shifts to isolating the numerical term first rather than immediately moving x-terms, preventing students from eliminating an x-term too early before simplifying.

Example: For 9x = 3x + 66 , ask: "What is the furthest thing from 66 on this side of the equation?"

By treating 66 as the focal point, students correctly identify the outermost operation in relation to it, preventing missteps when balancing equations with variables on both sides. This approach ensures that constants and variables are correctly combined, allowing students to simplify the equation by getting all variable terms onto one side before solving for x.

Step 2: Determine the Inverse Operation

Map each operation to its inverse/opposite, for example:

  • Addition ↔ Subtraction
  • Multiplication ↔ Division
  • Denominator ↔ Multiply by denominator
  • Squared ↔ Square root
  • Reciprocal ↔ Reciprocal

Note: Some students understand ‘opposite’ better than ‘inverse’ when first learning the concept. The terms ‘inverse’ and ‘opposite’ are interchangeable in P&S, and students should use whichever makes the concept clearer for them.

Step 3: Apply the Inverse to Both Sides

  • Maintain balance by ensuring symmetric operations.
  • Example: Subtracting 7 from (x - 3) / 5 + 7 = 10 yields (x - 3) / 5 = 3 .

Step 4: Repeat Until x Is Isolated

  • Reapply Steps 1-3 until x = _.

3.2 Simplified Protocol for Classroom Use

Table 1: P&S Protocol Summary

Step Action Example (Equation: 3x + 5 = 17)
1 Identify outermost layer +5 (addition)
2 Select inverse/opposite operation Subtract 5
3 Apply inverse to both sides 3x = 12
4 Repeat Divide by 3 → x = 4

3.3 Numbered Steps for Procedural Clarity

Explicitly numbering each step helps students track their progress, reducing errors and cognitive load.

A simple format to follow:

  1. Identify the outermost operation.
  2. Write the inverse operation.
  3. Apply it to both sides.

For example, when solving 3x + 5 = 17 , students would write:

  1. +5
  2. −5
  3. bs (both sides)

This structured approach reinforces consistency, making equation-solving easier to follow and recall. By externalising the process, students avoid common mistakes and build a clear, repeatable framework for applying inverse operations.

3.4 Teaching Students to Identify the Outermost Layer

Many students struggle to recognise the outermost operation at first, so I introduce it progressively using increasingly complex equations.

Approach:

  • Start with a simple equation and ask students guiding questions.
  • If they are unsure, substitute a number for x and walk through the steps forward to see what was applied last.
  • Gradually introduce more complex structures while reinforcing the pattern.

Example walkthrough:

  1. 3x → Furthest from x?

    • 3x means 3 × x.
    • ×3 is the outermost operation.

  2. 3x + 2 → Furthest from x?

    • First, ×3, then +2.
    • +2 is the outermost operation.

  3. (3x + 2) / 3 → Furthest from x?

    • First, ×3, then +2, then ÷3.
    • ÷3 is the outermost operation.

  4. 5((3x + 2) / 3) + 4 → Furthest from x?

    • First, ×3, then +2, then ÷3, then ×5, then +4.
    • +4 is the outermost operation.

  5. (5(3x + 2) + 4) / (-2z) → Furthest from x?

    • First, ×3, then +2, then ×5, then +4, then ÷(−2z).
    • ÷(−2z) is the outermost operation.

By explicitly practising this process, students develop the ability to consistently spot the correct first step without relying on reverse BIDMAS, which can be confusing in complex algebraic equations. P&S also trains students to efficiently identify the outermost layer without needing diagrams for visual layering, as required by Backtracking or Onion Skin methods. This makes it easier to apply across different equation types, improving consistency and reducing unnecessary cognitive load.

4. Addressing Documented Misconceptions

4.1 Sign Errors

P&S reduces sign errors by:

  • Explicitly isolating negatives: Students treat subtraction as adding a negative, e.g., −3 becomes +(−3), making sign reversal easier to track.
  • Avoiding term transposition: By focusing on inverse operations rather than "moving terms", P&S avoids common errors in handling negative values.

Booth et al. (2014) identified sign-related errors as 'highly prevalent in equation-solving activities' compared to other types of algebraic errors, making them a critical focus area for intervention strategies. Similarly, Deringöl (2019) highlighted that misconceptions about fractions, particularly regarding their operational structure, persist among primary and secondary school students, contributing to difficulties in algebraic problem-solving.

4.2 Fraction Misinterpretations

P&S addresses fraction related misconceptions by:

  • Denominator as single operation: P&S treats denominators (e.g., 4x in 9 / (4x) ) as indivisible units, preventing errors where students mistakenly apply operations to only part of a fraction. In such cases, the correct step is to multiply by the denominator on both sides.

  • Reciprocal as an operation: If 1 / x = 3 , the inverse operation is to take the reciprocal of both sides, which gives x = 1 / 3 .

    This prevents common errors where students mistakenly multiply both sides by x instead of isolating it. In cases where the denominator is a product (e.g., 1 / (4x) ), taking the reciprocal (4x ) is equivalent to multiplying by 4x , so either approach works.

  • Equivalence reinforcement: By requiring operations on both sides, P&S avoids mistakes like x / 5 = 3 → x = 3 / 5 , ensuring that fractions remain balanced across the equation. This aligns with Deringöl's (2019) findings that students struggle with understanding fraction equivalence, particularly when variables are involved in algebraic contexts.

Booth et al. (2014) found that students often struggle with fractions in algebra, with many incorrectly applying operations to parts of fractions rather than treating them as unified mathematical entities.

4.3 Structural Balance

P&S eliminates transposition errors by:

  • Rejecting "move across the equals sign" metaphors, which can lead to unbalanced operations.
  • Enforcing symmetric operations, e.g., multiplying both sides by 4x in the equation 9 / (4x) = y .

Kieran (1992) documented that preserving equation balance is a persistent challenge for algebra learners, with many incorrectly applying operations to just one side of an equation.

5. Integrating Conceptual Frameworks

5.1 Singapore CPA Methodology

P&S can be integrated with Singapore CPA as follows:

  • Concrete: Use algebra tiles to model equation layers. For 3x + 5 = 17 , students remove five unit tiles before splitting remaining tiles into three groups.
  • Pictorial: Transition to visual workflows mirroring P&S steps (EEF, 2019).
  • Abstract: Apply P&S algorithmically after tactile/visual mastery.

5.2 Digital Tools (EEF, 2019)

P&S aligns with recommendations for digital tools by the EEF:

  • Interactive platforms: Tools like Desmos or GeoGebra can simulate equation layers, allowing students to "peel" operations digitally.
  • Real-time feedback: Educational apps can provide instant error correction during P&S practice.

6. Comparative Analysis

Compared to traditional Reverse BIDMAS, the Balance Model, and Algebra Tiles, Peel and Solve offers a structured and systematic method that explicitly guides students in selecting the first inverse operation.

Unlike reverse BIDMAS, which assumes students can recognise the correct operation to undo without guidance, and Onion Skin/Backtracking, which relies on visual layering, P&S provides a step-by-step procedural framework that explicitly guides students in selecting the first step and eliminating ambiguity.

While Algebra Tiles offer concrete visual representation, P&S complements this with a procedural framework that can be applied even when manipulatives are not available. The Balance Method's emphasis on equation equilibrium aligns with P&S's insistence on applying operations to both sides, but P&S adds clarity on operation sequencing.

However, the effectiveness of P&S compared to these existing methods requires further empirical research.

7. Limitations and Future Research

While P&S addresses procedural gaps, its reliance on rote application risks superficial understanding. To bridge this:

  1. Controlled trials: Compare P&S error rates against reverse BIDMAS in diverse classrooms, drawing on frameworks like those outlined by Angrist et al. (2023), which emphasise evidence-based interventions for improving educational outcomes.
  2. Long-term retention: Track whether P&S-trained students maintain accuracy in later algebra topics.
  3. Hybrid implementation: Test P&S paired with CPA/digital tools, as advocated by EEF (2019).

Data Availability Statement: This paper presents a theoretical proposal rather than an experimental study. No new data were collected for this research. Future researchers may contact the author for collaboration on controlled trials to validate the effectiveness of the Peel and Solve method.

8. Conclusion

Peel and Solve provides a structured procedural framework for potentially reducing common errors in linear equation solving. By systematising inverse operation sequencing, it addresses challenges documented in algebraic literature. This paper presents P&S as a theoretical framework and practical teaching tool that requires empirical validation through controlled studies. Its full potential depends on integration with conceptual methods like CPA and rigorous classroom testing across diverse learning environments.

9. Next Steps

Empirical studies comparing P&S with traditional methods are critical to validate its efficacy. Researchers should design controlled trials measuring error reduction rates and retention, incorporating CPA tools to assess conceptual-procedural balance.

Declarations

  • Ethics Approval: Not applicable (theoretical study).
  • Funding: No funding was received for this research.
  • Competing Interests: The author declares no competing interests.

References

  1. Mayer, R. E. (2019). Cognitive load theory. Educational Psychology Review, 31(2), 1-18.
  2. National Council of Teachers of Mathematics. (2014). Principles to actions. NCTM.
  3. Booth, J. L., Newton, K. J., & Twiss-Garrity, L. K. (2014). The impact of fraction magnitude knowledge on algebra performance and learning. Journal of Experimental Child Psychology, 118, 110-118. https://doi.org/10.1016/j.jecp.2013.09.001
  4. Kieran, C. (1992). The learning and teaching of school algebra. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 390-419). Macmillan.
  5. Education Endowment Foundation. (2019). Using Digital Technology to Improve Learning. https://educationendowmentfoundation.org.uk/education-evidence/guidance-reports/digital
  6. Deringöl, Y. (2019). Misconceptions of primary school students about fractions. International Journal of Evaluation and Research in Education, 8 (1), 29-38. http://doi.org/10.11591/ijere.v8i1.16290
  7. Angrist, N., Djankov, S., Goldberg, P. K., & Patrinos, H. A. (2023). How to improve education outcomes most efficiently: A comparison of 150 interventions using the new learning-adjusted years of schooling metric. World Bank Policy Research Working Paper. https://hdl.handle.net/10986/34658