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A device samples only at integer times and always records 2. Construct a constant model and a nonconstant sinusoidal model fitting every record, then suggest one extra measurement that distinguishes them.

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

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高一必修第一册(A版).pdf · 5.6 · PDF 238 / printed page 231

Revisit first: Unit-circle coordinates and signs Identity choice and domain preservation

TOPIC 01

Amplitude, period and phase in models

Read parameters as observable changes and reconstruct models without hiding ambiguity.

What you will be able to explain

Distinguish amplitude, vertical shift, angular frequency and horizontal shift.
Reconstruct a sinusoid from sufficient observations and recognise when samples are insufficient.

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 device samples only at integer times and always records 2. Construct a constant model and a nonconstant sinusoidal model fitting every record, then suggest one extra measurement that distinguishes them.

y(n)=2,\quad n\in\mathbb Z

Readings are exact; no frequency restriction is assumed.

Skills and prerequisite lessons

Scope: JM01 / JM02

Working and explanation

BUILD THE REASONING

Hint 1

A full-cycle oscillation can vanish at every integer.

Hint 2

Try adding sin(2πt).

Worked solution

Construct two compatible models.
$f(t)=2,\quad g(t)=2+\sin(2\pi t)$
Verify every integer sample.
$g(n)=2+\sin(2\pi n)=2$
Choose an unsampled quarter-cycle time.
$f(1/4)=2,\quad g(1/4)=3$

Models f=2 and g=2+sin(2πt); measure at t=1/4.

Checks and common pitfalls: Equal samples do not prove a constant continuous signal; the new measurement is a discriminating test.

Reasoning checklist · self / teacher assessment

Construct two compatible models.
Verify every integer sample.
Choose an unsampled quarter-cycle time.

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

Teacher preparation and assessment

40-minute lesson

Minutes	Focus	Teaching and student action
5	Read the displacement	Use the phase diagnostic and require factoring before choosing.
8	Separate parameters	Annotate amplitude, midline and one full cycle on a graph.
10	Reconstruct and qualify	Pairs reconstruct from extrema and discuss why phase remains free.
12	Interpret and sample	Groups solve the tide threshold and devise a measurement to distinguish sampled models.
5	Exit direction	Distinguish an upward crossing from an arbitrary zero.

Question sequence

Which measured feature fixes each parameter?
Could two different continuous curves fit all samples?

Board development

Factor before reading horizontal shift.
$\omega(x+\phi/\omega)$
Retain time units in a model.
$T=12\text{ hours}$

Expected correct and incorrect responses

“Range determines A,D but not phase” is a correct limitation.
“All readings equal, so the function is constant” overclaims the data.

Follow-up questions

What observation fixes the direction of a crossing?

Remediation

For phase errors, solve ωx+φ=0 directly, then compare with the claimed shift.

Exit check

For y=−3sin(2x)+1, give amplitude, range and least period. Expected 3, [−2,4], π.

Assessment checklist and sample responses

2 marks: parameter roles.
Full: divide phase by frequency; partial: use phase alone as shift.
2 marks: evidence and ambiguity.
Full: two fitted models plus t=1/4 distinguishing test; partial: two formulas without checking samples.