g10-10-02 · Syllabus supplement
A drawing does not preserve every length
Can equal perpendicular sides look unequal in an oblique drawing?
1 MIN · PREDICT ON PAPER
Draw a square with its depth lengths halved. Predict the displayed side ratio and angle.
Discuss or write first. No typed answer is required to continue.
The drawn depth is scaled; do not measure it as a real length.
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
An oblique representation keeps the chosen front axis and scales/reorients depth. Parallel segments remain parallel, but general lengths and angles do not remain true.
Conditions: The picture uses a specified depth scale q>0 and nonparallel projected axes. A collapsed drawing cannot recover depth.
Mathematical justification: The map(x,y)↦(x+qy cosθ,qy sinθ) is linear and preserves parallel directions, but sends unit axes to lengths1,q and angleθ.
1 MIN · EXIT CHECK
A depth edge of real length6 is drawn with q=1/2. What is its displayed length?
Check after answering
3 drawing units; the real length remains6.
Teacher guidance and extension
Before: Plane area, Pythagoras and units.
Ask: Which labels state real measurements even when the drawing is foreshortened?
Watch for: Measuring the paper slant and treating it as an actual space angle or length.
Challenge: Choose a projection that collapses the square and explain what information is lost.
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
Paper responses are enough. Drafts remain in this tab; choose local saving to include them in the existing complete backup.
Complete backup and restore