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A rectangle has perimeter 20 m. For which positive side lengths is its area at least 21 m²? State all possible shorter-side lengths too.

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

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高一必修第一册(A版).pdf · 2.3 · PDF 57 / printed page 50

Revisit first: Roots and completeness of solutions

TOPIC 01

Quadratic signs and solution sets

Use roots to partition a line, then retain endpoint and domain conditions.

What you will be able to explain

Justify the sign on each interval instead of memorising an unqualified inside/outside rule.
Intersect an algebraic solution set with modelling and domain restrictions.

Try it. Leave your reasoning visible.

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

Examples 1–3: study a full demonstration; finish a solution from its first step; then choose a method independently before opening the solution. The remaining exercises may continue after class.

01 / Transfer#Your turn

A rectangle has perimeter 20 m. For which positive side lengths is its area at least 21 m²? State all possible shorter-side lengths too.

2(x+y)=20,\quad xy\ge21

Positive real side lengths, all four sides fenced.

Skills and prerequisite lessons

Scope: JM01 / JM02

Working and explanation

BUILD THE REASONING

Hint 1

Eliminate y, then distinguish an arbitrary side from the shorter side.

Hint 2

The area condition becomes (x−3)(x−7)≤0.

Worked solution

Combine the model with positive lengths.
$y=10-x,\quad x\in[3,7]$
The shorter side is min(x,10−x).
$3\le\min(x,y)\le5$

Sides x and 10−x with 3≤x≤7; shorter side lies in [3,5].

Checks and common pitfalls: A second interpretation is required after solving the algebraic inequality.

Reasoning checklist · self / teacher assessment

Combine the model with positive lengths.
The shorter side is min(x,10−x).

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

Teacher preparation and assessment

40-minute lesson

Minutes	Focus	Teaching and student action
5	Diagnose regions	Use the negative-leading and repeated-root diagnostics.
8	Build signs	Students multiply factor signs on three intervals.
10	Endpoints and domains	Pairs contrast strict, weak, radical and rational restrictions.
12	Model and quantify	Groups solve rectangle feasibility and then the all-real parameter condition.
5	Exit interval	Require a sign reason and correct endpoint notation.

Question sequence

What information is missing if we only solve the equation?
Does “for every x” permit checking only three values?

Board development

Multiply signs before drawing the solution.
$a(x-r_1)(x-r_2)$
Intersect with original restrictions.
$S_{\mathrm{valid}}=S_{\mathrm{sign}}\cap D$

Expected correct and incorrect responses

“The square is nonnegative, so strict negativity is impossible” is a complete reason.
“Include −2 because ≥ allows equality” overlooks the denominator.

Follow-up questions

Which endpoint would survive after adding a denominator?

Remediation

For inside/outside errors, rebuild a two-factor sign chart instead of recalling a slogan.

Exit check

Solve (x−1)²≥0 and (x−1)²>0; explain the single-point difference.

Assessment checklist and sample responses

2 marks: sign regions and equality.
Full: (2,3] after intersection; incomplete: [2,3].
2 marks: model interpretation.
Full: arbitrary side [3,7], shorter side [3,5]; partial: no shorter-side restriction.