Similarity and Three-dimensional Coordinates
Relate length, area and volume scale factors and read coordinates from three-dimensional diagrams.
How to study National 5 Mathematics
Secure one operational skill at a time, connect it to neighbouring techniques, then apply it in unfamiliar contexts with complete working and a checked conclusion.
Core concepts
Concept 1
For similar shapes, a length scale factor k produces an area factor k² and a volume factor k³.
Exam cue: State whether the given ratio concerns length, area or volume.
Concept 2
Corresponding dimensions must be matched before forming a scale factor.
Exam cue: Take a square or cube root when recovering a length factor.
Concept 3
Three-dimensional coordinates are ordered consistently along the diagram’s labelled axes and must be read from the represented point.
Exam cue: Trace each coordinate parallel to its axis in the diagram.
Risk pitfalls and guardrails
Using the length factor unchanged for area or volume.
Guardrail: Check signs, coefficients and both sides of every algebraic transformation.
Comparing non-corresponding dimensions.
Guardrail: Check signs, brackets, restrictions, units, accuracy and whether the answer needs a reason.
Reading 3D coordinates in the order that looks visually convenient.
Guardrail: Check signs, brackets, restrictions, units, accuracy and whether the answer needs a reason.
Memory anchors
Length factor
k.
Area factor
k².
Volume factor
k³.
Recover length from area
Take the square root of the area factor.
Recover length from volume
Take the cube root of the volume factor.
3D coordinate
Read the ordered axes exactly as labelled.
Checkpoint rule
Do the check-up only after you can summarize each concept in one sentence and identify one dangerous pitfall from memory.
Knowledge Check (after reading)
Short check-up to confirm understanding of this module.
Check-up Questions
Two similar shapes have length scale factor 3 from smaller to larger. Find the area scale factor.
Two similar solids have length scale factor 4. Find the volume scale factor.
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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