Poster Plan

Poster Plan: Solving Equations from Key Stage 3 to 5

Poster Structure: A1 Portrait (59.4cm × 84.1cm)

┌───────────────────────────────────────────────────────────┐
│  TITLE: "Solving Equations: A Progressive Journey         │
│          Through Key Stages 3-5"                          │
│  Your Name | Pedagogy Module | University of Lincoln      │
│  (5% height, ~4cm)                                        │
├───────────────────────────────────────────────────────────┤
│  1. INTRODUCTION (15% height, ~12cm)                      │
│  ┌─────────────┬──────────────┬──────────────────────┐    │
│  │ Background  │  Personal    │ Literature-Supported │    │
│  │ & Context   │  Motivation  │ Motivation           │    │
│  │ • Definition│  • Tutoring  │ • Research gaps      │    │
│  │ • Importance│    experience│ • Student struggles  │    │
│  │ • Curriculum│  • Observed  │ • Learning benefits  │    │
│  │   centrality│    disconnect│   of progression     │    │
│  │\[1-2 cites\]│ \[personal\] │    \[3-5 cites\]     │    │
│  └─────────────┴──────────────┴──────────────────────┘    │
├───────────────────────────────────────────────────────────┤
│  2. PROGRESSION MAP: Y7-13 (40% height, ~33cm)            │
│                                                           │
│  ┌─────────────────────┬──────────────┬───────────────┐   │
│  │ What's Learned      │ Knowledge,   │ Graphical     │   │
│  │ (Curriculum)        │ Skills &     │ Understanding │   │
│  │ (40%)               │ Understanding│ (25%)         │   │
│  │                     │ (35%)        │               │   │
│  ├─────────────────────┼──────────────┼───────────────┤   │
│  │ ▲ KS5 (Y12-13)      │      ▲       │   \[Sketch:   │   │
│  │                     │  DEVELOPS:   │   Exponential │   │
│  │ • Exp/log eqns      │ • Inverse ops│   curves,     │   │
│  │ • Trig equations    │ • Numerical  │   intersect\] │   │
│  │ • Diff equations    │   methods    │               │   │
│  │ • Newton-Raphson    │ • Modelling  │ "Solution =   │   │
│  │ • Inequalities      │ • Precision  │  intersection"│   │
│  │                     │              │               │   │
│  │ Example: e^x = 10   │ "+1: Can't   │ \[Cite: NC\]  │   │
│  │ Solve: x=ln(10)     │  use formula"│               │   │
│  │ \[Cite: NC KS5\]    │              │               │   │
│  ├─────────────────────┼──────────────┼───────────────┤   │
│  │ ▲ KS4 (Y10-11)      │      ▲       │   \[Sketch:   │   │
│  │                     │  DEVELOPS:   │   Parabola    │   │
│  │ • Quadratic eqns    │ • Factorise  │   with roots\]│   │
│  │ • Completing sq     │ • Structures │               │   │
│  │ • Quad formula      │ • Multiple   │ "Roots = x    │   │
│  │ • Simultaneous      │   methods    │  intercepts"  │   │
│  │   (lin/quad)        │ • Non-linear │               │   │
│  │ • Iteration         │   thinking   │ \[Discriminant│   │
│  │ • Inequalities      │              │  visual\]     │   │
│  │ • Eq vs identity    │ "+1: Not     │               │   │
│  │                     │  linear"     │ \[Cite: NC\]  │   │
│  │ Example: x²-5x+6=0  │              │               │   │
│  │ Solve: (x-2)(x-3)=0 │              │               │   │
│  │ \[Cite: NC KS4\]    │              │               │   │
│  ├─────────────────────┼──────────────┼───────────────┤   │
│  │ ▲ KS3 (Y7-9)        │  DEVELOPS:   │   \[Sketch:   │   │
│  │                     │              │   Two lines   │   │
│  │ • Algebraic manip   │ • Variables  │   intersect\] │   │
│  │ • Substitution      │ • Inverse ops│               │   │
│  │ • Linear equations  │ • Equivalence│ "Point on     │   │
│  │ • Rearranging       │ • Isolation  │  line is a    │   │
│  │ • Simultaneous      │ • Graphical  │  solution"    │   │
│  │   (graphical)       │   link       │               │   │
│  │ • Inequalities      │              │ "Intersection │   │ 
│  │                     │ FOUNDATION   │  = simult.    │   │
│  │ Example: 3x+7=22    │              │  solution"    │   │
│  │ Solve: x=5          │              │               │   │
│  │ \[Cite: NC KS3\]    │              │ \[Cite: NC\]  │   │
│  └─────────────────────┴──────────────┴───────────────┘   │
│                                                           │
│  Vertical arrows showing "building up" progression        │
├───────────────────────────────────────────────────────────┤
│  3. MISCONCEPTIONS & LEARNING THEORY (20%, ~16cm)         │
│  ┌───────────────────────────┬─────────────────────────┐  │
│  │ Common Misconceptions     │ Learning Theories &     │  │
│  │ (50% width)               │ Pedagogical Support     │  │
│  │                           │ (50% width)             │  │
│  │ By Key Stage:             │                         │  │
│  │                           │ 1. Schema Theory        │  │
│  │ KS3:                      │ • Equations as linked   │  │
│  │ • "=" means "makes" not   │   schemas, not isolated │  │
│  │   equivalence \[cite\]    │ • Implication: Show     │  │
│  │ • x+3=7 as "put together" │   connections \[cite\]  │  │
│  │   not addition \[cite\]   │                         │  │
│  │ • Can't rearrange         │ 2. Variation Theory     │  │
│  │   with negatives \[cite\] │ • Learn via variation   │  │
│  │                           │   within invariance     │  │
│  │ KS4:                      │ • "+1 technique"        │  │
│  │ • Forgetting ± in √       │   embodies this \[cite\]│  │
│  │ • Equations vs identities │                         │  │
│  │   confusion \[cite\]      │ 3. ZPD (Vygotsky)       │  │
│  │ • All quadratics          │ • Each level builds in  │  │
│  │   factorise \[cite\]      │   ZPD of previous       │  │
│  │ • Divide by variable      │ • Implication: Check    │  │
│  │   (might be 0) \[cite\]   │   mastery first \[cite\]│  │
│  │                           │                         │  │
│  │ KS5:                      │ 4. Dual Coding (Mayer)  │  │
│  │ • Domain restrictions     │ • Algebraic + visual    │  │
│  │   with logs \[cite\]      │ • Implication: Always   │  │
│  │ • When numerical methods  │   link graphs \[cite\]  │  │
│  │   needed \[cite\]         │                         │  │
│  │ • Iteration divergence    │ 5. Constructivism       │  │
│  │   ignored \[cite\]        │ • Build on prior        │  │
│  │                           │   knowledge explicitly  │  │
│  └───────────────────────────┴─────────────────────────┘  │
├───────────────────────────────────────────────────────────┤
│  4. CONCLUSION & TEACHING IMPLICATIONS (15%, ~12cm)       │
│                                                           │
│  Key Implications for Effective Teaching & Learning:      │
│                                                           │
│  1. Progressive not Disconnected: Teach equations as a    │
│     journey, not isolated topics. Schema theory shows     │
│     connected knowledge is retained better \[cite\]       │
│                                                           │
│  2. "+1 Technique" Pedagogy: Each new equation type adds  │
│     ONE new complexity. Variation theory supports this    │
│     approach \[cite\]. Makes progression explicit.        │
│                                                           │
│  3. Dual Representation Essential: Always connect         │
│     algebraic and graphical. Dual coding theory shows     │
│     this deepens understanding \[cite\]                   │
│                                                           │
│  4. Address Misconceptions Explicitly: Particularly at    │
│     transitions (KS3→4, KS4→5). Research shows targeted   │
│     intervention prevents calcification \[cite\]          │
│                                                           │
│  5. Secure Foundations First: ZPD theory indicates rushed │
│     progression leads to gaps. Ensure mastery before      │
│     advancing complexity \[cite\]                         │
│                                                           │
│  Conclusion: Understanding equations as a coherent        │
│  progression—rather than disconnected procedures—enables  │
│  teachers to scaffold learning effectively and helps      │
│  students see mathematics as an interconnected whole.     │
│                                                           │
├───────────────────────────────────────────────────────────┤
│  5. REFERENCES (5%, ~4cm)                                 │
│  IEEE Format, two columns:                                │
│  \[1\] DfE, "Maths programmes of study: KS3," 2013.       │
│  \[2\] DfE, "Maths programmes of study: KS4," 2014.       │
│  \[3\] DfE, "GCE AS/A level mathematics content," 2016.   │
│  \[4\] C. Kieran, "Learning and teaching algebra at the…  │
│  \[5\] L. Booth, "Algebra: Children's strategies and…     │
│  \[6\] M. Swan, "Dealing with misconceptions in…          │
│  \[7\] J. Piaget, "The psychology of intelligence," …     │
│  \[8\] L. Vygotsky, "Mind in society," 1978.              │
│  \[9\] R. Mayer, "Multimedia learning," 2009.             │
│  \[10\] J. Sweller, "Cognitive load theory," 1998.        │
│  \[11-15\] Additional algebra education research sources  │
└───────────────────────────────────────────────────────────┘

Poster Content

Key Stage 3 Section (Year 7 to 9) - Bottom Box

What’s Learned (from National Curriculum)

• “use and interpret algebraic notation”
• “substitute numerical values into formulae and expressions”
• “simplify and manipulate algebraic expressions… by collecting like terms, multiplying a single term over a bracket, taking out common factors, expanding products”
• “understand and use standard mathematical formulae; rearrange formulae to change the subject”
• “use algebraic methods to solve linear equations in one variable (including all forms that require rearrangement)”
• “use linear and quadratic graphs to estimate values… and to find approximate solutions of simultaneous linear equations”
• “understand and use the concepts and vocabulary of expressions, equations, inequalities, terms, and factors”

Knowledge, Skills & Understanding Developed

• Knowledge: Variables represent unknowns; equations express equivalence; solutions make equations true
• Skills: Isolating variables; inverse operations; substitution
• Understanding: “Solving” means finding values; graphical solutions = coordinate points
• Foundation: All future equation work builds on linear thinking

Graphical

• Sketch of $y=mx+c$ line with point marked
• Sketch of two intersecting lines
• Captions: “Point ON line is a solution” / “Intersection = simultaneous solution”

Key Stage 4 Section (Year 10 to 11) - Middle Box

What’s Learned (from National Curriculum)

• “extend fluency with expressions and equations from key stage 3, to include quadratic equations, simultaneous equations and inequalities”​
• “factorising quadratic expressions of the form x² + bx + c, including the difference of two squares; factorising quadratic expressions of the form $a{x}^2+bx+c$”​
• “identify and interpret roots, intercepts, and turning points of quadratic functions graphically; deduce roots algebraically and turning points by completing the square”​
• “solve quadratic equations… algebraically by factorising, by completing the square and by using the quadratic formula; find approximate solutions using a graph”​
• “solve two simultaneous equations in two variables (linear/linear or linear/quadratic) algebraically; find approximate solutions using a graph”​
• “find approximate solutions to equations numerically using iteration”​
• “know the difference between an equation and an identity”​
• “solve linear inequalities in one or two variables, and quadratic inequalities in one variable; represent the solution set on a number line, using set notation and on a graph”​

Knowledge, Skills & Understanding Developed

• Knowledge: Quadratic structure ($a{x}^2+bx+c$); discriminant indicates root count; equations ≠ identities
• Skills: Factorising; completing square; quadratic formula; simultaneous solving; iterative approximation
• Understanding: Multiple solution methods exist; graphs reveal root count; non-linear requires new techniques
• “+1 Technique”: What if equation isn’t linear? Need factorising, formula, or approximation

Graphical

• Sketch of parabola with marked x-intercepts (roots)
• Three small parabolas showing 0, 1, 2 roots (discriminant)
• Caption: “Roots are where curve crosses x-axis” / “Discriminant tells us how many”

Key Stage 5 Section (Year 12 to 13) - Top Box

What’s Learned (from National Curriculum)

• “Work with quadratic functions and their graphs; the discriminant… completing the square; solution of quadratic equations including solving quadratic equations in a function of the unknown​”
• “Solve simultaneous equations in two variables by elimination and by substitution, including one linear and one quadratic equation​”
• “Solve equations of the form $a^x=b$​”
• “Solve simple trigonometric equations in a given interval, including quadratic equations in sin, cos, and tan and equations involving multiples of the unknown angle​”
• “Locate roots of $f(x)=0$ by considering changes of sign… Understand how the change of sign methods can fail​”
• “Solve equations using the Newton-Raphson method and other recurrence relations… Understand how such methods can fail​”
• “Evaluate the analytical solution of simple first order differential equations with separable variables​”
• “Solve linear and quadratic inequalities in a single variable and interpret such inequalities graphically​”
• “Understand and use parametric equations of curves​“

Knowledge, Skills & Understanding Developed

• Knowledge: Exponential/log inverse relationship; trigonometric patterns; differential equations model change; numerical limits
• Skills: Logarithmic solving; iterative methods; Newton-Raphson; separating variables; parametric conversion
• Understanding: Not all equations have closed-form solutions; approximation is powerful; equations can model real-world change
• “+1 Technique”: What if we can’t factorise OR use a formula? Need logs, trig identities, or numerical methods

Graphical

• Sketch of exponential curve intersecting horizontal line
• Small Newton-Raphson tangent iteration diagram
• Caption: “Solution = where f(x) = g(x)” / “Numerical methods approximate via tangents”

Poster Misconceptions Content

Key Stage 3 Misconceptions

1. “Treating ’=’ as ‘makes’ or ‘the answer is’ rather than equivalence” [Swan, 2001]
2. “Believing x + 3 = 7 means ‘x and 3 makes 7’ (concatenation thinking)” [Booth, 1984]
3. “Cannot work with negative numbers in equations” [Common diagnostic]
4. “Letter-number reversal: $2x=x2$” [Küchemann, 1981]

Key Stage 4 Misconceptions

1. “Forgetting ± when solving x² = 9” [Common error analysis]
2. “Confusing equations (solve for x) with identities (true for all x)” [Kieran, 2007]
3. “Thinking all quadratics factorise nicely” [Watson, 2009]
4. “Dividing both sides by a variable that might be zero” [Tall & Vinner, 1981]
5. “Believing ‘completing the square’ is separate from quadratic formula (not seeing connection)“

Key Stage 5 Misconceptions

1. “Forgetting domain restrictions when using logarithms (x > 0)” [Common calculus error]
2. “Not recognising when numerical methods are necessary vs. analytical solutions”
3. “Ignoring convergence/divergence in iterative methods” [Numerical methods research]
4. “Treating $dy/dx$ as a fraction that can be separated arbitrarily” [Tall, 1992]

Poster Learning Theories Content

1. Schema Theory (Piaget)

• Theory: Knowledge organised in connected schemas; new learning integrates into existing structures
• Application: Equations shouldn’t be taught as disconnected procedures but as linked progression
• Implication: Explicitly show how quadratics build on linears; how exponentials extend algebra
• Citation: Piaget, J. (1952). The origins of intelligence in children

2. Variation Theory (Marton)

• Theory: Learning occurs through experiencing variation against invariance
• Application: “+1 technique” embodies this—what stays same (solving process) and what changes (complexity)
• Implication: Design tasks that vary equation type while keeping solving structure constant
• Citation: Marton, F. (2015). Necessary conditions of learning

3. Zone of Proximal Development (Vygotsky)

• Theory: Learning occurs in zone between what student can do alone vs. with support
• Application: Each equation level should be in ZPD of previous; Key Stage 4 quadratics in ZPD of Key Stage 3 linears
• Implication: Don’t rush progression; ensure mastery before advancing complexity
• Citation: Vygotsky, L. S. (1978). Mind in society

4. Dual Coding Theory (Paivio/Mayer)

• Theory: Information encoded both verbally and visually is better retained and understood
• Application: Teaching equations requires both algebraic manipulation AND graphical representation
• Implication: Always show graph alongside equation; make connections explicit
• Citation: Mayer, R. E. (2009). Multimedia learning (2nd ed.)

5. Constructivism (Bruner)

• Theory: Learners construct knowledge by building on prior understanding
• Application: Equation solving is cumulative—each new type constructed from previous
• Implication: Diagnose gaps in prior knowledge before teaching new equation types
• Citation: Bruner, J. S. (1966). Toward a theory of instruction

Poster Design Specifications

Colour Palette

• Introduction boxes: Light blue (#E3F2FD)
• Key Stage 3 box: Light blue (#BBDEFB)
• Key Stage 4 box: Medium blue (#64B5F6)
• Key Stage 5 box: Dark blue (#1976D2)
• Skills column: Light green (#C8E6C9)
• Graph column: Light grey (#F5F5F5)
• Misconceptions: Light orange (#FFE0B2)
• Learning theory: Light purple (#E1BEE7)
• Conclusion: Teal (#B2DFDB)
• References: White (#FFFFFF)
• Text: All dark grey/black (#212121) for contrast

Typography

• Main title: 36pt bold, sans-serif
• Section headings: 24pt bold
• Subheadings: 16pt bold
• Body text: 12-13pt regular
• Captions: 11pt italic
• References: 10pt regular
• Font: Arial or Helvetica throughout

Spacing

• Margins: 2.5cm all sides
• Gutters: 1.5cm between major sections
• Internal padding: 1cm within boxes
• Line spacing: 1.2-1.5 for readability

Poster Key Reference to Include

Curriculum Documents (Primary sources)

• Department for Education, “Mathematics programmes of study: key stage 3,” National Curriculum in England, 2013.
• Department for Education, “Mathematics programmes of study: key stage 4,” National Curriculum in England, 2014.​
• Department for Education, “GCE AS and A level subject content for mathematics,” 2016.

Learning Theory

• L. S. Vygotsky, Mind in Society. Harvard University Press, 1978.
• J. Piaget, The Psychology of Intelligence. Routledge, 1950.
• R. E. Mayer, Multimedia Learning, 2nd ed. Cambridge University Press, 2009.
• F. Marton, Necessary Conditions of Learning. Routledge, 2015.
• J. S. Bruner, Toward a Theory of Instruction. Harvard University Press, 1966.

Algebra Education Research

• C. Kieran, “Learning and teaching algebra at the middle school through college levels,” in Second Handbook of Research on Mathematics Teaching and Learning, 2007, pp. 707-762.
• L. R. Booth, “Algebra: Children’s strategies and errors,” NFER-Nelson, 1984.
• M. Swan, “Dealing with misconceptions in mathematics,” in Issues in Mathematics Teaching, 2001, pp. 147-165.
• D. Küchemann, “Algebra,” in Children’s Understanding of Mathematics: 11-16, 1981, pp. 102-119.

Cognitive Science

• J. Sweller, “Cognitive load during problem solving,” Cognitive Science, vol. 12, no. 2, pp. 257-285, 1988.
• D. Tall and S. Vinner, “Concept image and concept definition in mathematics,” Educational Studies in Mathematics, vol. 12, pp. 151-169, 1981.

Additional (as Needed for Specific points)

• [15-20] More specific algebra misconceptions, pedagogy, or assessment research