Lesson subsection
Read the explanation, try the on-paper prompts, then explain the idea in your own words. Use AI feedback as a mentor, not a shortcut.
Best flow: read → think on paper → write a short explanation → refine with feedback.
Many mathematical statements look true because they work for the first few examples.
A false conjecture often has patterns that break only in edge cases.
Strategies for spotting false conjectures:
Test boundary values.
Small numbers, negative numbers, 0, or 1 often break patterns.
Test extreme values.
Very large numbers or unusual cases often reveal failures.
Look for hidden assumptions.
Does the conjecture assume positivity? Integrality? Distinct values?
Try a structural contradiction.
Ask: What would need to happen for this to fail?
Example false conjecture:
Every number of the form n² + n + 17 is prime.
It works for many values, but fails at n = 17.
Spotting false conjectures is a skill that improves with pattern recognition.
TL;DR — key idea
False conjectures often fail at boundary or extreme values—testing these systematically reveals counterexamples quickly.
Don’t skip this – writing proofs or explanations on paper is where most of the learning actually happens.
Test the conjecture: "All numbers of the form n² + 3n + 2 are prime." Find a value of n that gives a composite number, and explain why it is a counterexample.
Once you’ve sketched some ideas, summarize the main insight in the reflection box on the right.
In 3–6 sentences, explain the core idea of this subsection as if you were teaching a friend who hasn’t seen it. Focus on the logic, not just the final statements.
AI is optional. Use it to spot gaps and sharpen your wording, not to replace your own thinking.