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Which is the solution?

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.

01 / Foundation#Your turn

Which is the solution?

−(x−1)(x−3)>0-(x-1)(x-3)>0
  • All inputs are real unless a smaller domain is stated.
Skills and prerequisite lessons

Scope: JM01 / JM02

  1. Outside [1,3]
  2. (1,3)
  3. [1,3]
  4. {1,3}
Working and explanation
BUILD THE REASONING
Hint 1
Account for the leading minus.
Hint 2
The product before negation must be negative.
Worked solution
  1. Reverse the target sign.
    (x−1)(x−3)<0(x-1)(x-3)<0
  2. Choose between the roots.
    1<x<31<x<3
B: (1,3).
Checks and common pitfalls: A memorised outside rule fails when the leading sign changes.
Think first. Reveal a hint when the class is ready.
Focus on one question
Teacher preparation and assessment
40-minute lesson
MinutesFocusTeaching and student action
5Diagnose regionsUse the negative-leading and repeated-root diagnostics.
8Build signsStudents multiply factor signs on three intervals.
10Endpoints and domainsPairs contrast strict, weak, radical and rational restrictions.
12Model and quantifyGroups solve rectangle feasibility and then the all-real parameter condition.
5Exit intervalRequire 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−r1)(x−r2)a(x-r_1)(x-r_2)
  • Intersect with original restrictions.
    Svalid=Ssign∩DS_{\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.
No sign-in. Work stays in this browser. Export before clearing browser data. Written reasoning is assessed with a checklist.
Curriculum and source notes ↗