← Senior Mathematics Studio

LEARN · EXPLAIN · REVISE

Choose c² for x²/9−y²/16=1.

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

Return to the lesson / paper ↗

高二選擇性必修第一册(A版).pdf · 3.2 · PDF 123 / printed page 118

Revisit first: Ellipse: definition, axes and constraints

Teacher preparation and assessment

40-minute lesson

Minutes	Focus	Teaching and student action
5	Prediction and diagnosis	Locate the sign of the focal difference at each vertex.
8	Concept derivation	Derive and sketch asymptote slopes with horizontal and vertical axes.
10	Guided application	Calculate a point and test whether an asymptote intersects.
12	Transfer and explanation	Classify the offset-line and focal-difference boundary cases.
5	Exit evidence	Explain why a single intersection is insufficient for tangency.

Question sequence

Which focus is farther on the right branch?
Can a line have one intersection without being tangent?

Board development

Tabulate absolute difference 0, between 0 and 2c, equal to 2c and greater than 2c.
Substitute y=x+k before mentioning any discriminant.

Expected correct and incorrect responses

Students set the discriminant to zero even after the x² term disappears. Circle the actual leading coefficient first.
Correct: “The quadratic coefficient cancels, so one intersection need not be tangency.”
Incorrect: “A line with exactly one intersection must be tangent.”

Follow-up questions

How does the unique intersection move as the parallel line offset tends to zero?

Remediation

Use the vertices to decide which signed focal difference is positive.

Exit check

Give an equation of a one-intersection non-tangent line and explain the degree loss.

Assessment checklist and sample responses

Require branch sign, standard-form parameters and a separate tangency justification. Sample: substituting y=x+1 gives −2x−1=1, so x=−1; this linear equation alone does not imply tangency.
Correct: “The quadratic coefficient cancels, so one intersection need not be tangency.”