6.3 Figure Matrices: Combination Rules and Distractor Audits

Distinguish transformation from combination

A transformation changes one figure into another. A combination rule uses features from two input figures to form a result. Original matrix exercises may combine component sets, counts, or fills. State exactly which components participate; otherwise a rule such as “remove overlap” can accidentally remove a frame that the illustration keeps fixed.

The examples below explicitly declare the participating elements and operation. They teach checking procedures without claiming that the publisher requires a named set operation in a particular proportion of items.

Represent components as sets

Suppose one input contains a vertical line and a dot, while another contains a horizontal line and a dot. Treat the dot, vertical line, and horizontal line as identifiable components in fixed positions.

The set interpretation assumes that overlapping components have been identified consistently. Two dots in different corners are not the same component merely because both are dots. Their location can be part of their identity for the exercise.

Use a small truth table

For a component at one location, record whether it is present in the first input and the second input.

The last row distinguishes OR from XOR. A component present in both inputs remains under union but disappears under symmetric difference. An example with no overlapping components cannot distinguish those two operations, because their results are then identical.

Work a fixed-frame example

Each cell has a circular frame that stays unchanged. The first input contains a vertical diameter; the second contains a horizontal diameter. The stated operation combines only internal lines by XOR. Because the diameters are distinct components, the result contains both, forming a plus inside the unchanged circle.

If both inputs contained the same vertical diameter, XOR on internal lines would remove that line, leaving the frame alone. The circular frame is excluded from the operation by the prompt. Including it would cancel the frame and solve a different task.

In a visual question, infer which features the displayed examples treat as fixed and which are combined. Do not assume every visible boundary participates in the same operation.

Check count rules independently

A count relationship may add two quantities, subtract one from another, or follow another stated pattern. If one input contains two dots and the other three, a result of five supports addition for that example. A result of one could support a difference in counts, but does not itself establish whether the result's dot position matters.

A smaller result does not automatically mean XOR. Intersection can also reduce components. To distinguish them, inspect which specific features remain and which disappear. The component truth table supplies the required evidence.

Separate fill from component identity

A figure may keep a square boundary while cycling its fill from white to black and back to white. Another may replace a circle with a triangle while keeping the fill. List those changes separately before combining them.

Do not infer an unshown pattern such as white, black, hatched merely from the first two cells. Many continuations can fit two observations. An original exercise should supply enough evidence for the intended cycle, or explicitly state it. When writing a solution, identify the displayed cells that support each feature prediction.

Audit compound changes

Suppose a practice instruction says “rotate the arrow forty-five degrees clockwise and change it from outline to solid.” If the original arrow points up, the result points diagonally up-right and is solid. It does not point right, which would require ninety degrees, and it does not remain an outline.

A regular hexagon may have rotational symmetries in sixty-degree increments, but that fact does not change an explicitly specified forty-five-degree operation. Track the actual instruction rather than substituting a familiar symmetry angle.

When two changes act on the same feature, check their order. A reflection followed by a rotation can differ from the reverse. When they affect independent attributes, such as orientation and an explicitly assigned fill, calculate each and then verify the combined candidate.

Use an answer-choice checklist

Before looking at choices, predict the participating components, count, fill, orientation, and position. Compare each offered figure with that prediction. Cross out a choice only when you can name its mismatch.

Review with a discriminating example

After a miss, design a fresh pair that distinguishes the competing rules. To separate OR and XOR, include a shared component. To separate rotation and reflection, use an asymmetric mark. To separate count from position, hold the count fixed while moving an element.

This practice makes your rule testable. A confident label is not enough: the operation must predict the actual figure. If two simple rules remain indistinguishable from the evidence, describe that limit rather than claiming that one is automatically official.