Success criteria
- I classify shapes using properties, not appearance or orientation.
- I understand that one shape can belong to several nested groups.
- I justify “always, sometimes, or never” with a property or counterexample.
- I distinguish same size and shape from same shape at a different scale.
Learn and explain
A hierarchy is inclusive: a specific shape keeps every broader name whose rules it satisfies. But the hierarchy branches; it is not one chain.
| Broader family | Branches or overlap |
|---|---|
| Quadrilateral | Trapezium (US: trapezoid), using the inclusive rule below |
| Trapezium | Parallelogram |
| Parallelogram | Rectangle branch; rhombus branch |
| Rectangle and rhombus | Square—the overlap, so every square belongs to both branches |
This lesson uses trapezium (US: trapezoid) for a quadrilateral with at least one pair of parallel sides. Some materials instead require exactly one pair, so always state which rule is being used. With the inclusive rule, the helper confirms that a square belongs to the rectangle and rhombus groups: both branches.
A square does not stop being a rectangle when it also satisfies the rhombus rule. It is the intersection of those two sibling branches.
Create property cards for: number of sides, parallel side pairs, right angles, and equal sides. Sort paper quadrilaterals, then explain why turning a shape does not change its family.
Congruent and similar figures
Congruent figures have the same size and shape even after a translation, reflection, or rotation. Similar figures have the same shape and proportional corresponding lengths, but may have different sizes. A congruent pair is the same-shape special case; a merely similar pair has been scaled.
| Comparison with 4 × 6 | Relationship |
|---|---|
| 6 × 4 | congruent |
| 8 × 12 | similar |
| 8 × 10 | neither |
Model each pair on squared paper and mark corresponding sides. Orientation alone does not decide the relationship.
Independent practice
Which statement is always true?
A square meets every rectangle rule and has an additional equal-side property.
A shape has four sides, two pairs of parallel sides, and four right angles, but not four equal sides. Which most specific name is guaranteed?
Its memberships include quadrilateral, trapezium (US: trapezoid), parallelogram, rectangle; rectangle is the most specific guaranteed name here.
A 4 by 6 rectangle is compared with an 8 by 12 rectangle. Which relationship is most precise?
The helper classifies the pair as similar: both side lengths use the same scale, but the sizes differ.
Unfamiliar transfer — fix the museum labels
A museum sign says, “Squares are not rectangles because rectangles have two long sides.” Write a two-sentence correction. Then draw one congruent pair, one similar-but-not-congruent pair, and one pair that is neither; label the evidence for each judgment.
Delayed check
One week later, draw the branching hierarchy from memory. Show rectangle and rhombus as sibling branches below parallelogram, then link square to both. Add a separate comparison showing how a congruent figure and a scaled similar figure preserve properties.
Evidence path
Save the corrected sign and hierarchy to jeremy/portfolio/math/unit-5/lesson-15-shape-hierarchy/.
Next step
- Repair: sort physical shape cards using one property at a time.
- Continue: if hierarchy and figure comparisons can be explained without prompts, transform a coordinate design.
- Stretch: compare the inclusive and exclusive definitions of trapezium (US: trapezoid) and redraw only the relationships that change.