LEARN · EXPLAIN · REVISE

Disprove: a positive product forces both factors to be positive.

Read the idea, work independently, then explain what changed.

高一必修第一册(A版).pdf · 1.4 · PDF 24 / printed page 17

TOPIC 01

Sufficient and necessary conditions

Build understanding of sufficient and necessary conditions through definitions, contrasting cases and justified applications.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Transfer#Worked example

ab>0

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Test both directions and seek a counterexample when a direction fails.

Hint 2

Two negative factors give a positive product.

Worked solution

Choose a=-2, b=-3.

Checks and common pitfalls: A single admissible counterexample disproves the universal claim.

Reasoning checklist · self / teacher assessment

Think first. Reveal a hint when the class is ready.

Teacher preparation and assessment

What must be true before using the main rule for sufficient and necessary conditions?
Which representation makes this task easier, and why?
Change one assumption. Does the conclusion survive?

Apply sufficient and necessary conditions with explicit conditions.
Implication: If p implies q, p is sufficient for q and q is necessary for p.
$p\Rightarrow q$
Equivalence: Both directions are required for a necessary and sufficient condition.
$p\iff q$
Close with: conditions → representation → reasoning → check.

Expected reasoning: If p implies q, p is sufficient for q and q is necessary for p.
Expected reasoning: Both directions are required for a necessary and sufficient condition.
Expected correction: A converse is not automatic.