g10-03-01 · Core topics
Does the middle always solve it?
What changes when the leading coefficient changes sign?
1 MIN · PREDICT ON PAPER
For a(x−r)(x−s)≤0, keep r,s fixed and compare positive, negative and zero a.
Discuss or write first. No typed answer is required to continue.
Solution: [-2, 3]
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
Two distinct factors change sign at their roots; a negative multiplier reverses the product sign. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.
Conditions: Roots are real. a=0 is a constant-zero expression and is classified separately.
Mathematical justification: Order the roots u≤v. On each interval the signs of x−u and x−v are fixed. Multiplication gives the sign chart, while root substitution decides endpoints.
1 MIN · EXIT CHECK
Solve −(x−1)(x−4)≤0.
Check after answering
x≤1 or x≥4, including both roots.
Teacher guidance and extension
Before: Factor signs and number lines
Ask: Have one pupil explain the graph and another explain factor signs.
Watch for: “Take the middle” assumes positive leading coefficient and two distinct roots.
Challenge: Explain the repeated-root case without appealing to the graph.
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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