Common Patterns That Often Fail

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.

10–20 min focusProof-first mindset

Best flow: read → think on paper → write a short explanation → refine with feedback.

Reading

Core explanation

Many mathematical mistakes come from assuming a pattern that looks true but fails in edge cases.

Common failing patterns include:

Assuming “cancelation” works everywhere.
Assuming exponent rules extend to sums. (They often don't.)
Assuming even/odd behavior behaves linearly.
Assuming “working for many examples” means a statement is true.
Assuming properties of ℝ apply to ℤ or modular arithmetic.

Example:

Statement: If x·y = 0, then x = 0 or y = 0.

True over real numbers—false over modular arithmetic.

Pattern failures reveal where intuition must be retrained.

TL;DR — key idea

Many common mathematical “rules” only work in certain structures—counterexamples identify exactly where patterns break.

Try these in your notebook

Don’t skip this – writing proofs or explanations on paper is where most of the learning actually happens.

1
Provide a counterexample to one of these false statements: 1. "If a + b is even, then a and b must both be even." 2. "If ab = ac, then b = c." 3. "If a² = b², then a = b." Explain why the counterexample breaks the pattern you expected.

Once you’ve sketched some ideas, summarize the main insight in the reflection box on the right.

Check your understanding

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.