feat(studios): add Line Studio and Function Studio interactive learning modules

[Jatin-nicon]

📌 Summary

This PR introduces two complete, interactive mathematical discovery studios to Tenali:

1. Line Studio (Cluster 2: Embodied Line Geometry & The Line Rule)
2. Function Studio (Cluster 3: Rules Beyond the Line & Function Notation)

Both modules provide an inquiry-first learning environment where students discover mathematical principles organically through dynamic GeoGebra canvases before formal algebraic names and notations are introduced.

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🚀 Key Modules Added

1. Line Studio (client/src/modules/LineStudio)

A 10-step pedagogical journey uncovering straight-line geometry and the invariant $y = ax + b$:

• Dynamic GeoGebra Integration (GeoGebraLineLab):
  • Live coordinate grid with interactive point placement, dynamic guidelines, and line plotting.
  • Smart viewport bounding: automatically calculates the farthest plotted coordinate across all 5 verification points ($A$ through $E$) to zoom and frame the canvas cleanly.
• Inquiry Progression:
  • Explores steepness/slope ratio ($\Delta y / \Delta x$) and y-intercept intuitively.
  • Step 7 refined with accessible language ("slowly" instead of "gradually").
  • Unobstructed display for steepness and intercept values in Question 9.
  • Concludes with the Naming Handover Ceremony, formally naming the slope-intercept rule only after the student has earned the concept.
• Robust Answer Evaluation (answerEvaluator.js):
  • Validates equations, points, and slope inputs with tolerance for algebraic equivalents.
  • Complete unit test suite (answerEvaluator.test.js).

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2. Function Studio (client/src/modules/FunctionStudio)

A 10-step inquiry module transitioning from straight lines to generalized functions and functional notation $f(x)$:

• Rules Beyond the Line:
  • Investigates non-linear curves ($y = x^2$) and absolute value shapes ($y = |x|$).
  • Contrasts single-valued rules with rules that violate the one-input-one-output pattern ($y^2 = x$).
  • Step 6 designed as a purely conceptual checkpoint comparing well-formed vs. broken rules.
• Formal Notation & Evaluation ($f(x)$):
  • Strict line plotting and functional rule definition ($f(x) = ax + b$).
  • GeoGebra Algebra View panel simulation for function evaluation ($f(2)$, $f(4)$).
  • Inverse rule exploration ($f(a) = y$, finding input $a$ given output $y$) across 3 progressive stages.
• Clean Input UX & Anti-Spoil Design:
  • Placeholders guide input format without leaking answers (e.g. e.g. a = ? or number, e.g. f(...), and generic '?' placeholders).
  • Sanitized validation messages in equationParser.js so errors never reveal target values.
• Algebra Engine (equationParser.js):
  • Multi-format parser supporting $y = ax + b$, $f(x) = ax + b$, points $(x, y)$, and inverse queries ($a = k$).

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3. Application Integration

• Registered both LineStudio and FunctionStudio within client/src/App.jsx and tile catalog (features/tiles.js).
• Warm parchment and Tenali Teal visual design system consistency applied across all trays, formula bars, and dialogs.

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🧪 Verification & Quality Checks

• Client Build: npm run build completed cleanly in client/ (0 errors, 0 warnings).
• Local Execution: Tested full flows end-to-end on localhost:5173 backed by localhost:4000.
• Responsiveness: Verified layout and GeoGebra canvas behavior across different viewport sizes.
• Unit Tests: Ran answerEvaluator.test.js for Line Studio evaluator checks.

[github-actions]

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