4.3 Spatial & Diagrammatic Transformations as Transferable Practice

Spatial and diagrammatic transformations

Why practise diagrams

The current MFS public outline names problem solving but does not publish a separate “abstract reasoning” or “spatial reasoning” domain. Diagrammatic drills are included here because they train rule discovery with unfamiliar information. They should not be presented as a guaranteed live item type.

The key is to convert pictures into features. For each figure, record:

• shape type;
• number of elements;
• position in a frame;
• orientation;
• fill or shading;
• line style;
• size;
• movement direction; and
• interaction between elements.

Analyse one feature at a time. A sequence may rotate an arrow while moving a dot and alternating the fill. Trying to perceive the whole animation at once overloads working memory.

Rotation

A rotation turns a figure around a point. Clockwise quarter-turns change directions:

up → right → down → left → up.

A 180-degree turn reverses direction. A 270-degree clockwise turn is the same as a 90-degree anticlockwise turn.

Rotation preserves handedness. If an L-shaped figure has a dot inside its short arm, the dot remains on the same relative side of the figure as the entire object turns. Mark a distinctive corner and follow it.

For coordinate-style movement around a 3×3 border, write positions in order rather than relying on mental imagery: top-left, top-middle, top-right, middle-right, bottom-right, bottom-middle, bottom-left, middle-left.

Reflection

A reflection flips a figure across a mirror line.

• Vertical mirror: left and right exchange; top and bottom stay.
• Horizontal mirror: top and bottom exchange; left and right stay.
• Diagonal mirror: coordinates exchange according to the diagonal.

Reflection reverses handedness. Text and asymmetric symbols appear mirrored. A common distractor rotates the object instead, preserving handedness.

Use coordinates. In a square, label corners TL, TR, BR, BL. Under a vertical reflection, TL ↔ TR and BL ↔ BR. Under a horizontal reflection, TL ↔ BL and TR ↔ BR.

Translation and combined operations

Translation moves a figure without changing orientation. If a question applies reflection and then rotation, preserve the order: transformations generally do not commute. Draw or imagine the intermediate state.

Example: an arrow points right with a dot above it. Reflect vertically: the arrow points left and the dot remains above. Then rotate 90 degrees clockwise: the arrow points up and the dot, which rotates with the figure, lies to its right.

Counting and progression rules

Some sequences change count rather than orientation:

• add one side each step;
• double the number of dots;
• alternate +1 and +2;
• move one filled segment clockwise while another segment is removed.

Count visible and hidden features separately when overlap occurs. An apparent decrease may result from two elements occupying the same position.

Odd-one-out questions

Define the shared rule among three or more figures. A choice is not odd merely because it looks different. Possible shared properties include number of sides, symmetry axes, rotation relationship, equal counts, enclosure, or line intersections.

Test multiple features. If three figures are rotations of the same asymmetric shape and one is its mirror image, handedness—not orientation—is the decisive property.

Matrices

A 2×2 or 3×3 matrix applies a rule across rows, columns, or both. Common operations include:

• overlay: combine all marks;
• subtraction: remove shared marks or remove the first from the second;
• exclusive-or: keep marks that appear in exactly one input;
• intersection: keep only shared marks;
• progression: rotate, add, or move by a fixed amount;
• distribution: each row and column contains each symbol once.

Use completed rows to infer the rule, then test it on a completed column. A rule that explains only one line is weak. For a Latin-square distribution, check that no symbol repeats within a row or column.

Nets and folding

For cube nets, identify opposite faces. Faces sharing an edge in the folded cube are adjacent; opposite faces can never touch. Rather than spin the whole cube mentally, anchor one face as the base and fold neighbouring faces upward.

For paper folds with holes, reverse the folds in reverse order. Each unfold mirrors existing holes across the fold line unless a hole lies on the fold itself. A fold can also cause multiple layers to overlap, so track layer count.

Verification routine

Describe the proposed rule aloud: “The triangle rotates 90 degrees clockwise, the dot moves one corner anticlockwise, and shading alternates.” Apply it to every transition. Reject a choice if even one feature violates the rule. When two answers remain, inspect handedness, element count, and whether the question applies transformations in a stated sequence.