Teach Through a Counterexample
A sharp counterexample that exposes why a careless claim is false.
Works on any free AI · Best tool: Any AI chat (ChatGPT/Gemini/Claude/Copilot)
✅ How to use this effectively
- 1. Open any free AI chat.
- 2. Paste the prompt.
- 3. Add the concept and class level.
- 4. Send and study the counterexample.
- 5. Use it to sharpen the definition.
🔧 If it is not right, reply with this: If the counterexample is too obscure, reply: "Give a simpler counterexample with small whole numbers that a Class 10 student can check by hand."
THE PROMPT
ROLE: Act as a precise mathematics teacher who uses counterexamples to sharpen understanding.
CONTEXT: I teach [CLASS/GRADE] for [BOARD or EXAM e.g. CBSE / ICSE / State Board / JEE / NEET]. Students over-generalise this concept and believe a claim that is not always true: [WRITE THE CONCEPT AND THE TEMPTING WRONG CLAIM HERE]. I want a clean counterexample to fix this.
YOUR TASK:
1. State the tempting but false claim clearly.
2. Give one simple, concrete counterexample that breaks it.
3. Walk through why the counterexample violates the claim, step by step.
4. State the corrected, precise version of the claim (with its real conditions).
5. Add one practice case for students to test their understanding.
INPUT: Here is the concept and the wrong claim: [PASTE OR TYPE IT HERE]. Class level: [CLASS/GRADE].
HOW TO WORK IT OUT: Construct the counterexample, then VERIFY it two independent ways to be certain it truly breaks the claim. Re-check that the corrected claim is actually correct and not over-restricted. Never invent a fake counterexample or formula. Keep all maths legible in clear plain text.
OUTPUT FORMAT (use these exact headings):
- THE TEMPTING (FALSE) CLAIM
- THE COUNTEREXAMPLE (concrete, with small numbers if possible)
- WHY IT BREAKS THE CLAIM (step by step, with VERIFICATION)
- THE CORRECTED CLAIM (precise, with conditions)
- YOUR TURN (one case for students, with the answer)
RULES: The counterexample must be as simple as possible and verifiable by hand. The corrected claim must be exactly right, neither too loose nor too tight. Define every symbol.
FORMAT THE OUTPUT — PRIMARY DELIVERABLE:
Return one complete, editable document ready to paste into Word or Google Docs. Use a descriptive title, short section headings, readable paragraphs, numbered actions and tables only where they make the material easier to use. Include every requested section; do not return formatting advice instead of the document.
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):
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🔗 Free tools & to reach me: https://jee-prompt-studio.pages.dev/
Questions, feedback or appreciation are always welcome.
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