Competition · AMC preparation · step 4 of 4
AMC 8 · 2023 · #2
Grade 4 geometry-2d
Pick an answer.
AMC 8 2023 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Paper-folding-and-cutting problems are exactly what Tool #10 was designed for: fold an actual scrap of paper, make the cut, and unfold it. The answer literally appears in your hand. If no paper is available, walk through the same moves mentally with Tool #17 (Visualize). Either way we finish with Tool #3 (Eliminate) by comparing the resulting shape to the five answer choices.
Fold a real square
Grab a real scrap and fold it the same way — horizontal midline, then vertical — into a four-layer square; both fold lines act as mirrors.
Partitioning a square into four equal parts is the Grade 2 "equal shares" idea, applied physically.
2.G.A.3Create A Physical RepresentationFind the center corner
At the corner that maps to the paper's center, cut between the two edge midpoints — four layers, so one cut frees four identical triangles.
When stacked paper is cut once, you get as many identical pieces as layers — the Kindergarten "compose and decompose shapes" intuition.
K.G.B.6Create A Physical RepresentationUnfold the vertical fold
Unfold the vertical fold: the notch mirrors across that line, forming one small triangular hole on the bottom edge of the top half.
A fold line acts as a line of symmetry — exactly the Grade 4 line-of-symmetry standard.
4.G.A.3Visualize Spatial RelationshipsUnfold the horizontal fold
Unfold the horizontal fold: the hole mirrors again into a matching triangle below, the two meeting at the paper's center.
The second fold line is also a line of symmetry, so we apply the same Grade 4 "fold = mirror" idea once more.
Unfolding the horizontal fold reflects the top half's upward triangular notch across the center line, giving the bottom half a matching downward notch, and the two notches join along the center line into one hole centered on the paper.
▸ Why?
Unfolding reverses the fold, and that fold was a flip that had laid the bottom half onto the top half; flipping the bottom half back down carries the cut it received to the mirror-image spot the same distance below the center line, so the notch reappears identical but pointing the opposite way.
▸ Why?
A flip lays a shape exactly onto a copy across the fold line and keeps every length and angle unchanged, so the returned notch is the same size and shape as the one above, only mirrored.
▸ Why?
The original cut reached all the way to the fold line, so on each half the notch just touches the center line; the flip leaves points sitting on that line unmoved, so the upward notch from above and the downward notch from below share the center line and meet there edge-to-edge.
▸ Why?
A flip maps each half exactly onto the other across the fold line, and any point already on the line maps to itself, so a notch touching the line from one side is met by its mirror touching the same line from the other side.
Examine the center hole
The two triangles join into a four-sided figure with equal sides and upright diagonals — a small square tilted 45 degrees: a diamond.
A square is still a square when turned on its side — the Kindergarten "name shapes regardless of orientation" standard.
K.G.A.2Visualize Spatial RelationshipsMatch to the choices
Match the five choices: only (E) has a single 45-degree-tilted square hole at the paper's center; the rest fail the two-mirror symmetry.
Checking which choice has both a horizontal AND vertical line of symmetry through the center isolates (E).
4.G.A.3Eliminate PossibilitiesThis AMC 8 problem only needs the Grade 4 idea that "a fold line is a line of symmetry" you already know!
- Fold a real square
- Find the center corner
- Unfold the vertical fold
- Unfold the horizontal fold
- Examine the center hole
- Match to the choices
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