7.2 Figure Classification: Visual Categories and Invariants

Find a feature shared by all three figures

Riverside describes Figure Classification as three figures belonging to a group and a choice that completes that group. The task resembles verbal classification, but the category is expressed visually. It may concern shape, count, symmetry, position, or a relationship among components.

An invariant is a feature that stays unchanged under a specified transformation. For example, rotating a triangle preserves its number of sides. It does not preserve the direction of a marked arrow inside it. Say what transformation you mean before calling something invariant.

These original examples isolate clear visual categories. They do not claim that an official item must use a mathematical vocabulary term such as topology or rotational group.

Start with all three examples

Consider the visible group △, □, ⬠: a triangle, square, and pentagon. A shared property is being a closed polygon with straight sides. A hexagon belongs; a circle does not. The differing side counts show that “exactly four sides” cannot be the category.

The figures may also share outline fill. If the options include several outline polygons, that property alone will not discriminate. A good classification explanation identifies the shared feature or combination that separates the choices.

Distinguish rotation from reflection categories

Three arrows pointing up, right, and down can belong to a group of rotations of the same marked arrow. If an asymmetric internal feature stays attached consistently, a reflected arrow may fail that group even though its outer boundary resembles a rotated version.

For a symmetric unmarked shape, rotation and reflection may be visually indistinguishable. Do not claim that every directional triangle is chiral. A shape is chiral when its mirror image cannot be superimposed by rotations and translations in the relevant plane. Use an asymmetric arrangement if your practice item intends to test that distinction.

If the three examples vary in orientation, orientation alone is unlikely to be the unchanged category. Check the boundary and the relationship of internal marks instead.

Check symmetry of the whole figure

An unmarked square has four reflection axes. An ordinary non-square rectangle has two. A general oblique parallelogram with unequal adjacent sides has no reflection axis, although it has a half-turn rotational symmetry. Special parallelograms, such as rectangles and rhombi, can have reflection symmetry, so “all parallelograms have none” is false.

An interior dot away from the center can destroy some or all symmetries of an otherwise symmetric boundary. Count symmetries of the complete marked figure, not only the outline. If a dot is centered in a square, it preserves the square's reflection axes; if it is at a generic off-center position, those axes may no longer preserve the figure.

Keep geometric properties distinct

Convexity means that the straight segment between any two points of the region stays within the region. A polygon with an inward notch is nonconvex. Convexity is a geometric property, not a topological invariant: bending or deforming a shape can change it without cutting or joining components.

Connectivity concerns whether parts are joined. Containment concerns whether one feature lies inside another. These are different from side count and convexity. A category explanation should use the property actually shown rather than calling every visual feature “topological.”

Work a containment example

Imagine three different closed polygons, each with one dot strictly inside its boundary. The correct addition is another closed polygon with a dot strictly inside. A dot lying in the open notch of a concave polygon may be outside the polygon even if it appears near its center.

To make an independent practice question unambiguous, explicitly distinguish “inside the polygon's region” from “inside its bounding rectangle.” A surrounding rectangle is a different boundary. Choose figures with clearly placed dots, away from edges, so the intended rule is visible.

Reject a partial match

Suppose every given figure has a solid triangle inside an outline circle. A choice with a solid square inside an outline circle matches fill and containment but fails the inner shape. A choice with an outline triangle matches shape but fails fill. A complete prediction tracks each shared feature.

Do not add a condition that the three figures do not share. If one dot is above center and another below, “dot above center” is not the group rule. Use variation among the examples to rule out over-specific categories.

Learn from classification errors

Record the category you proposed and test it against each given figure. If one example fails, revise the category. If the proposed category covers multiple options, inspect a more specific shared relation or acknowledge that the exercise is ambiguous.

Create a fresh transfer example by changing an irrelevant attribute, such as orientation, while preserving the defining property. Then create a nonmember by changing only that property. This comparison shows whether you understand the category boundary.

The complete standard Figure Classification subtest has 22 items in 10 minutes. Accuracy in identifying the full category should come before optional timed practice. A single missed visual category is evidence about that exercise, not a diagnosis of a student's ability or health.