Advanced Synthesis
📋 What it is
A full proof combines the tools: state the claim, pick the technique, justify every step, and conclude.
🗣️ Coach says
This is the whole craft in one place. Read the claim, choose your tool, build a gap-free chain of justified steps, and land the conclusion so cleanly that a skeptic runs out of “why?”s. That’s a proof.
🧠 Memory hook
Claim → choose tool → gap-free steps → conclusion no skeptic can dent.
😂 Giggle
What's the difference between a lemma and a theorem?
A lemma is the opening act; a theorem is the headliner!
😲 Whoa!
A finished proof is judged on being both CORRECT and CLEAR — mathematicians prize an elegant short proof over a correct-but-messy long one.
✅ Quick check: You’ve proven every step but your conclusion doesn’t quite match the original claim. Is the proof done?
Say your answer out loud first — then reveal.
No — a proof must land exactly on the stated claim; a conclusion that’s close-but-different leaves the claim unproven.
A valid proof’s final line must be the exact statement you set out to prove.
🧪 Try it! (2 minutes)
Prove “the sum of two even numbers is even” fully: state it, write each even as 2×something, add, and conclude. Check the last line matches the claim.