1. Read for intent
The Word document supplied the syllabus objective, mathematical language and question evidence. Each item was analysed for the concept, representation and likely misconception it could reveal.
Conjugate roots of a polynomial equation with real coefficients
What is groundbreaking here?
A theorem becomes a consistency-checking tool. Learners use a visible missing mirror root to reconstruct factors, infer coefficients and audit claims about real-coefficient polynomials.
The AI did not merely generate answers. It translated a static assessment document into an instrumented learning progression: learners predict, attempt, receive misconception-specific feedback, open a visual tutorial, retry, and leave semantic evidence that a teacher can inspect.
The six-stage learning journey
- Complete the pairA real-coefficient polynomial has root 2+i. Which root must also occur?
- Find a coefficientGiven 2+i is a root of z³−2z²+kz+10=0 with real k, find k.
- Factor fullyGiven 2 is a root of z³+2z²−3z−10=0, find the other roots.
- Quartic structureOne root of 2z⁴+5z²−3z+5=0 is 1/2−(√3/2)i. Which set gives the other three roots?
- Unknown coefficientsGiven 1−2i is a root of z⁴+z³+mz²+17z+n=0, where m,n are real, which conclusion is correct?
- Test the theoremA polynomial with real coefficients lists 3+2i as a simple root but not 3−2i. What follows?
From a Word document to an expert learning experience
2. Add a learning-design prompt
Frame the conjugate-root theorem as a polynomial detective sequence. Show the missing reflection on the Argand plane, connect it to a real quadratic factor, preserve multiplicity and exact forms, and culminate in checking whether a proposed root list is logically compatible with real coefficients.
3. Build for the SLS frame
The payload is self-contained and responsive, with touch targets, keyboard access, read-aloud support, closed overlays on launch and no dependency on a network library.
4. Preserve proven xAPI know-how
The supplied reference ZIP was treated as a contract. lib/xapiwrapper.min.js, lib/xAPI.js and the injected xAPI head block were copied byte-for-byte; only the learning payload was redesigned.
How a teacher can use it
Give an incomplete root list and ask learners what evidence would prove whether the list or the real-coefficient claim is wrong.
- Use stages 1–2 for diagnosis before formal teaching.
- Ask students to narrate what changes in the visual model before they calculate.
- Use the tutorial and retry trail as evidence of self-correction, not just the final score.
- Upload the accompanying ZIP directly as an SLS interactive resource when xAPI evidence is required.
Quality evidence
Every activity was checked at desktop, mobile and narrow-phone sizes; incorrect-feedback, visual-tutorial and correct-retry paths were exercised; and the package validator confirmed six stages, root-level ZIP entries, offline assets and byte-identical xAPI integration.
MathematicsComplex NumbersAI GeneratedSLSPolynomials