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.
Strong direct proofs are not just correct—they are readable.
Helpful structuring techniques:
State the goal clearly.
"We want to prove: If P, then Q."
Introduce the assumption explicitly.
"Let n be an integer such that P holds."
Work in a logical sequence.
Each line should follow from previous ones using definitions or known results.
Name important objects.
Instead of "it", write "this integer k" or "this real number x".
Close the proof clearly.
End with a sentence like: "Therefore, Q holds, so the statement is proved."
You can think of a direct proof as a mini-story:
TL;DR — key idea
Good direct proofs have a clear opening (assumptions), a logically ordered middle, and an explicit closing that states the conclusion has been reached.
Don’t skip this – writing proofs or explanations on paper is where most of the learning actually happens.
Take a direct proof you wrote earlier (even/odd, divisibility, or inequalities) and rewrite it focusing only on structure and clarity: - Add an explicit opening sentence stating the theorem. - Make the assumption line very clear ("Let n be ..."). - Add a final sentence that clearly signals the conclusion. Reflect: Did rewriting for structure make the argument easier to follow?
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.