g10-14-01 · Additional mathematics
Not intersecting does not imply parallel
Can two space lines neither intersect nor run parallel?
1 MIN · PREDICT ON PAPER
Compare AB with an upper parallel edge,an adjacent edge and a noncoplanar vertical edge.
Discuss or write first. No typed answer is required to continue.
Rotate the view; the mathematical relation stays fixed.
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
Space lines can intersect,be parallel,or be skew. Skew lines are nonparallel and noncoplanar, even if their projections cross.
Conditions: The objects are full lines with nonzero directions, not only the visible edge segments.
Mathematical justification: Nonparallel lines are coplanar exactly when(q−p)·(d×e)=0; if the scalar triple product is nonzero, no common plane or intersection exists.
1 MIN · EXIT CHECK
Does “two lines do not meet” prove they are parallel in space?
Check after answering
No; they may be skew.
Teacher guidance and extension
Before: Points, lines, planes and vector directions.
Ask: Does rotating the picture preserve the spatial relation while moving its apparent crossing?
Watch for: Inferring an actual intersection from a crossing in the flat picture.
Challenge: Name a pair of cube edges that are skew.
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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