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.
Direct proofs often start with definitions. For integers:
To prove statements about even/odd numbers, you almost always:
Example:
Prove: The sum of two odd integers is even.
Let a and b be odd integers.
Then a = 2m + 1 and b = 2n + 1 for some integers m, n.
So a + b = (2m + 1) + (2n + 1) = 2m + 2n + 2 = 2(m + n + 1).
Since m + n + 1 is an integer, a + b is even.
The magic is just: translate → compute → translate back.
TL;DR — key idea
For even/odd proofs, always start from n = 2k or n = 2k + 1, do the algebra, and show the result fits the correct even/odd definition.
Don’t skip this – writing proofs or explanations on paper is where most of the learning actually happens.
Using the definitions of even and odd, prove ONE of the following using a direct proof: 1. The product of two odd integers is odd. 2. The sum of an even integer and an odd integer is odd. Write your proof in full sentences, and make sure you explicitly use the definition "n is even/odd ⇔ n = 2k or 2k + 1".
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.