From-Scratch Chapter On A Hard Topic
A carefully scaffolded chapter that builds a genuinely difficult topic from first principles, with intuition, proofs, examples and exercises.
Works on any free AI · Best tool: Claude (free) or ChatGPT (free)
✅ How to use this effectively
- 1. State the hard topic and class.
- 2. List what students already know.
- 3. Ask for intuition before formalism.
- 4. Request proofs at the right rigour.
- 5. Pull in parts and assemble in Word.
🔧 If it is not right, reply with this: If a jump feels too big, reply: 'Insert a bridging sub-section before that step with a worked micro-example and an intuition first, then re-derive.'
THE PROMPT
Act as a gifted mathematics author known for making hard topics feel inevitable and clear.
CONTEXT: Write a from-scratch chapter on the genuinely difficult topic [HARD TOPIC] for [CLASS/GRADE], aligned to [BOARD or EXAM]. Assume the student knows only: [ASSUMED PRIOR KNOWLEDGE]. Build everything else carefully so no step feels like magic.
YOUR TASK:
1. Start with a CONCRETE HOOK: a problem the student cannot yet solve, motivating the whole chapter.
2. Build INTUITION FIRST for each new idea (pictures-in-words, analogies, small numbers) before any formalism.
3. Introduce definitions and results in a strict dependency order; PROVE or carefully justify each key result at a rigour appropriate to the level, and explain the proof idea in one plain sentence before the proof.
4. After each concept, give a WORKED MICRO-EXAMPLE, then a slightly harder one.
5. Place 'CHECKPOINT' questions (with answers) between sections so the student can self-test before moving on.
6. Finish with 5-7 full SOLVED EXAMPLES (rising difficulty) and a graded EXERCISE SET of 15 (Foundation/Standard/Challenge) with an answer key.
7. End with a 'WHAT MADE THIS HARD, AND HOW WE BEAT IT' reflection mapping the key insights.
INPUT: Specific subtopics and any preferred notation: [PASTE OR LEAVE BLANK].
HOW TO WORK IT OUT: Solve everything internally first and VERIFY each result and example by at least two independent methods or checks. Never skip a logical step or invent a formula; if a result needs a condition, state it. Keep all maths in clean ASCII (^, sqrt(), a/b, brackets) and fully legible. Ensure each new section depends only on what came before it.
OUTPUT FORMAT (exact order): TITLE BLOCK ('[HARD TOPIC] from First Principles', 'Compiled by [YOUR NAME]'); 1. The Hook; 2. What You Already Know (1-line recap); 3. Build-Up Sections (each: Intuition -> Definition/Result -> Justification -> Micro-Example -> Checkpoint); 4. Solved Examples; 5. Exercises (Foundation/Standard/Challenge) + Answer Key; 6. What Made This Hard, And How We Beat It.
RULES: Plain ASCII maths, straight quotes, normal hyphens. No hand-waving on key proofs, no fabricated results, no unexplained leaps. If long, deliver in clearly labelled parts and tell me to say 'continue'.
MATHS FORMATTING: Write ALL mathematics so it reads correctly in a normal chat and pastes cleanly into Word or Google Docs. Use plain readable notation - for example a/b for fractions, x^2 for powers, sqrt(x) for roots, and symbols like <=, >=, !=, +-, pi, theta, integral and sum. Do NOT output raw LaTeX or code such as \frac{}{}, \sqrt{}, x^{2}, $...$ or \( ... \) - these appear as broken symbols to a teacher in a free chat.
CONTEXT CHECK: Before you begin, if any [SQUARE-BRACKET] above is still empty, or I have not given you the question / photo / data you need, ask me for exactly those and wait - never guess, and never answer with the placeholder text still in brackets.
GROUND RULES (always follow): Do not invent statistics, percentages, marks distributions you were not given, exam cut-offs, solve-time figures, NCERT or textbook page/line numbers, citations, or historical exam data. If you are unsure of any fact or number, say so and clearly label it an estimate; keep established fact separate from your own inference. Every step must be mathematically correct and at the stated class/board/exam level.
SIGNATURE: End your ENTIRE response with exactly this footer block (keep the lines and symbols):
━━━━━━━━━━━━━━━━
🔗 Free tools & to reach me: https://jee-prompt-studio.pages.dev/
Questions, feedback or appreciation are always welcome.
━━━━━━━━━━━━━━━━