LEARN · EXPLAIN · REVISE

Five points have no three collinear and no four coplanar. How many different planes do their triples determine?

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

高一必修第二册(A版).pdf · 8.4 · PDF 131 / printed page 124

TOPIC 01

Positions of points, lines and planes in space

Build understanding of positions of points, lines and planes in space through definitions, contrasting cases and justified applications.

Use definitions, valid spatial reasoning and metric checks for positions of points, lines and planes in space.
Connect representations and justify the steps, including boundary cases.
Explain a solution and apply the idea to a changed situation.

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

01 / Standard#Your turn

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Count unordered triples.

Hint 2

No-four-coplanar ensures distinct triples yield distinct planes.

Worked solution

Count unordered triples.
Calculate or simplify this relation.
$\binom53=5\cdot4\cdot3/(3\cdot2\cdot1)=10$
Both general-position conditions are needed for this count.

The requested value is 10.

Checks and common pitfalls: Both general-position conditions are needed for this count.

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

Teacher preparation and assessment

What must be true before using the main rule for positions of points, lines and planes in space?
Which representation makes this task easier, and why?
Change one assumption. Does the conclusion survive?

Use definitions, valid spatial reasoning and metric checks for positions of points, lines and planes in space.
Plane determination: Three noncollinear points determine one plane.
Line positions: Spatial lines may intersect, be parallel, or be skew.
Close with: conditions → representation → reasoning → check.