7.1 Paper Folding: Reverse the Folds and Track Layers

Begin with the published task

Riverside describes Paper Folding as pictures of paper being folded and punched, followed by choices showing how the paper looks when unfolded. You must reverse the depicted folds and determine the resulting holes. The direction of each fold, the region covered by a flap, and the punch location all matter.

The diagrams and coordinates below are independent teaching models. During home practice you may use real paper to verify them. Current Riverside proctoring guidance permits scratch paper for every CogAT subtest except Paper Folding. A drawing that helps you learn at home does not grant permission to sketch during that subtest.

Fix the square and coordinate system

Use a unit square with bottom-left corner $(0,0)$ and top-right corner $(1,1)$. Horizontal coordinates increase rightward; vertical coordinates increase upward. This convention makes each worked punch location checkable.

If the left half folds onto the right half along $x=0.5$, a point mirrors across that line. If the bottom half folds onto the top along $y=0.5$, it mirrors across the horizontal line.

Use the actual crease shown. A diagonal reflection is not the same operation as a ninety-degree rotation.

Work two full half-folds

Fold the left half onto the right, then fold the bottom half of that packet onto the top. The final packet occupies the upper-right quarter. Punch through all four layers at $(0.8,0.7)$, away from every edge and crease.

Undo the last, horizontal fold first. The punch locations become $(0.8,0.7)$ and $(0.8,0.3)$. Undo the vertical fold next, adding $(0.2,0.7)$ and $(0.2,0.3)$. The unfolded sheet has four distinct holes.

The pair of horizontal and vertical symmetries is a useful final check. A choice with four holes all on one side has the correct count but the wrong positions.

Work a diagonal fold

Fold the lower-right triangle, where $y<x$, onto the upper-left triangle, where $y\ge x$, along the diagonal $y=x$. The folded packet has vertices $(0,0)$, $(0,1)$, and $(1,1)$. Punch at $(0.2,0.8)$, an interior point of that packet.

Unfolding adds its mirror $(0.8,0.2)$. There are two distinct holes, one on each side of the diagonal. A candidate with $(0.2,0.2)$ would reflect across the wrong line. The coordinate convention prevents calling the folded region lower-left when it is actually upper-left.

Count layers at the punch location

Two folds do not always imply four holes. That count worked above because each fold halved the entire packet and the punch passed through all resulting layers at an interior location.

For a partial-fold example, fold the left quarter strip of the sheet onto the adjacent strip along $x=0.25$. The flap covers the region from $x=0.25$ to $x=0.5$. A punch at $(0.4,0.6)$ passes through two layers and produces a second hole at $(0.1,0.6)$. A punch at $(0.8,0.6)$ lies outside the flap and passes through only one layer. It produces one hole.

The covered region, not the number of arrows in the fold sequence, determines which layers are punched. Track only the layers present at the location.

Handle crease and edge coincidences

For an idealized point exactly on a crease, reflection gives the same point. Do not count the duplicate location twice. A finite circular punch centered on a crease can produce a joined opening; its boundary requires attention to the actual depiction. Avoid applying the “double each time” shortcut blindly.

A punch at an outer edge can create a notch rather than a complete interior hole. Its reflected shape depends on which layers and edges were cut. The answer choices may differ in hole shape as well as count and location.

Reverse multiple folds in order

Write or mentally hold the last folded packet first. Undo the most recent fold, then the preceding fold, continuing back to the original sheet. For partial folds, reflect only locations on the moving flap's image; do not reflect an uncovered point simply because a crease exists elsewhere.

Under a stated sequence of three full packet half-folds with an interior punch through all layers and no coincident locations, eight distinct holes can result. Those conditions matter. Three arbitrary partial or diagonal folds do not guarantee eight.

Verify during learning

Use a plain sheet at home, mark a fold sequence, punch or mark the folded layers, and unfold to compare your prediction. Start with one middle fold, then two, then partial folds. Record whether a miss involved the crease, reversal order, local layer count, or duplicate locations.

The standard complete Paper Folding subtest has 16 items in 10 minutes, subject to approved administration adjustments. Practice mental reversal after physical demonstrations, then follow the actual proctor's material directions.

Official task description and current scratch-paper rule.