g10-14-02 · Additional mathematics
A parallel line must lie outside the plane
Can the outside-plane condition be omitted?
1 MIN · PREDICT ON PAPER
Keep a line direction parallel to the floor and lower its height to0. Predict its relation to the floor.
Discuss or write first. No typed answer is required to continue.
The direction is parallel and every point remains outside.
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
A line parallel to a direction inside a plane can either lie in that plane or remain outside it. Strict line-plane parallelism requires no intersection.
Conditions: To use the criterion, the line must lie outside the plane and be parallel to a line within it.
Mathematical justification: For plane z=0 and line(t,0,h+kt), k=0,h≠0 means no meeting; k=h=0 means containment; k≠0 gives one meeting at t=−h/k.
1 MIN · EXIT CHECK
A line with points(t,0,1) has what relation to z=0?
Check after answering
Parallel and outside: z remains1 for every t.
Teacher guidance and extension
Before: Points, lines, planes and vector directions.
Ask: Which parameter changes direction, and which only moves the line up or down?
Watch for: Calling a contained line parallel to its own plane.
Challenge: Construct a line parallel to a plane direction but intersecting the plane.
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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