g10-02-02 · Core topics
Necessary or sufficient?
Which interval is contained in which?
1 MIN · PREDICT ON PAPER
For p:x>a and q:x>b, predict both implication directions when a<b, a=b and a>b.
Discuss or write first. No typed answer is required to continue.
Sufficient, not necessary
2 MIN · EXPLAIN AND REVISE
Which condition does your claim need?
Compare at least two cases. Write a reason, then check the explanation together.
Reveal explanation and conditions
Sufficiency is containment of the p-solution set in the q-solution set; necessity reverses this containment. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.
Conditions: Both inequalities are strict and share the real-number universe.
Mathematical justification: If a≥b, x>a implies x>b. If a<b, the midpoint (a+b)/2 satisfies p and not q. Thus the comparison completely classifies the directions.
1 MIN · EXIT CHECK
x>3 is what condition for x>1?
Check after answering
Sufficient but not necessary; x=2 refutes the converse.
Teacher guidance and extension
Before: Implications and intervals
Ask: Require a general interval argument and a witness for the failing direction.
Watch for: A single common solution proves neither implication universally.
Challenge: Replace one strict inequality by a non-strict one and examine equal thresholds.
Suggested timing: predict 1 min, compare 3 min, explain 2 min, exit check 1 min. Observations can suggest a claim; they do not prove it for every input.
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