g10-05-12 · Core topics
When does the larger exponent give less?
For positive x, does x³ always exceed x²?
1 MIN · PREDICT ON PAPER
Predict the ordering for x=0.5, 1 and 2 before moving the probe.
Discuss or write first. No typed answer is required to continue.
Positive inputs only. x² = 0.25; x³ = 0.125; difference = 0.125.
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
x³−x²=x²(x−1). Since x²>0 for x>0, the sign is determined by x−1: less below 1, equal at 1, greater above 1.
Conditions: The comparison here assumes x>0 and compares powers of the same base. Comparing two bases is a different question.
Mathematical justification: Factor the difference rather than extrapolating three samples. For positive a<b, b²−a²=(b−a)(b+a)>0.
1 MIN · EXIT CHECK
Compare 0.8² and 0.8³ without decimal multiplication.
Check after answering
0.8²>0.8³ because multiplication by 0.8 reduces a positive value.
Teacher guidance and extension
Before: Read function inputs, outputs and the stated domain; write your first prediction on paper.
Ask: Keep the equality case x=1 between the two opposite inequalities.
Watch for: A larger exponent always produces a larger positive value.
Challenge: In same-power mode, compare a² and b² for positive a<b using factorisation.
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.
Optional local notebook
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