AMC 8 · 2023 · #2
Grade 4 geometry-2dProblem
A square piece of paper is folded twice into four equal quarters, as shown below, then cut along the dashed line. When unfolded, the paper will match which of the following figures?
Pick an answer.
AMC 8 2023 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: A square sheet is folded in half horizontally and then in half vertically, producing a smaller square that is one quarter the size of the original and is four layers thick. A straight cut is made from the midpoint of the small square's right edge to the midpoint of its bottom edge, removing a tiny triangular corner. When the paper is unfolded, which of the five pictures matches it?
Givens: The square is folded twice, producing four equal quarters stacked on top of each other; A straight cut joins the midpoints of two adjacent edges of the folded square, removing a small right-triangular corner; The cut corner of the folded square corresponds to the exact center of the original paper; Five answer choices (A)-(E) show different cut patterns on the unfolded square
Unknowns: The shape of the hole in the unfolded paper and which choice (A)-(E) it matches
Understand
Restated: A square sheet is folded in half horizontally and then in half vertically, producing a smaller square that is one quarter the size of the original and is four layers thick. A straight cut is made from the midpoint of the small square's right edge to the midpoint of its bottom edge, removing a tiny triangular corner. When the paper is unfolded, which of the five pictures matches it?
Givens: The square is folded twice, producing four equal quarters stacked on top of each other; A straight cut joins the midpoints of two adjacent edges of the folded square, removing a small right-triangular corner; The cut corner of the folded square corresponds to the exact center of the original paper; Five answer choices (A)-(E) show different cut patterns on the unfolded square
Plan
Primary tool: #10 Create a Physical Representation
Secondary: #17 Visualize Spatial Relationships, #3 Eliminate Possibilities
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.
Execute — Answer: E
2.G.A.3 Step 1 - Take an ordinary square scrap of paper and reproduce the folds: fold once across the horizontal midline, then once across the vertical midline.
- The result is a smaller square that is four layers thick.
- The two folds are the original square's two center lines and act as mirrors for whatever cut we make.
💡 Partitioning a square into four equal parts is the Grade 2 "equal shares" idea, applied physically.
K.G.B.6 Step 2 - Identify the corner of the folded square that corresponds to the center of the original paper (the corner where all four layers meet on the inside).
- From the midpoints of the two edges meeting at that corner, cut straight across.
- Because the paper is four layers thick, this single cut detaches four identical small right triangles at once.
💡 When stacked paper is cut once, you get as many identical pieces as layers — the Kindergarten "compose and decompose shapes" intuition.
4.G.A.3 Step 3 - Now unfold the last fold (the vertical one).
- The little triangular notch on the folded square is reflected across the vertical fold line, creating an identical notch on the other half.
- Together they form a single small triangular hole pointing UP from the bottom edge of the top half of the paper.
💡 A fold line acts as a line of symmetry — exactly the Grade 4 line-of-symmetry standard.
4.G.A.3 Step 4 - Unfold the first fold (the horizontal one).
- The upward-pointing triangular hole on the top half gets reflected across the horizontal fold line, producing a matching downward-pointing triangular hole on the bottom half.
- The two triangles meet point-to-point along the center line, sharing their long (hypotenuse) edges right at the middle of the paper.
💡 The second fold line is also a line of symmetry, so we apply the same Grade 4 "fold = mirror" idea once more.
K.G.A.2 Step 5 - Examine the combined hole at the center.
- The two right triangles fit together to form a four-sided figure whose four edges are all equal in length and whose diagonals are horizontal and vertical.
- That is a small square — but rotated 45 degrees, i.e.
- a diamond — sitting exactly at the center of the paper.
💡 A square is still a square when turned on its side — the Kindergarten "name shapes regardless of orientation" standard.
4.G.A.3 Step 6 - Compare to the five answer choices.
- Choice (A) has jagged notches all along the edges of the paper, (B) has notches only along the top edge, (C) has an axis-aligned (non-rotated) square hole in the middle, and (D) has only a single upward-pointing triangle.
- Only (E) shows a single small square rotated 45 degrees, centered exactly at the middle of the paper — matching our diamond-shaped hole.
💡 Checking which choice has both a horizontal AND vertical line of symmetry through the center isolates (E).
2.G.A.3 Take an ordinary square scrap of paper and reproduce the folds: fold once across K.G.B.6 Identify the corner of the folded square that corresponds to the center of the o 4.G.A.3 Now unfold the last fold (the vertical one). The little triangular notch on the 4.G.A.3 Unfold the first fold (the horizontal one). The upward-pointing triangular hole K.G.A.2 Examine the combined hole at the center. The two right triangles fit together to 4.G.A.3 Compare to the five answer choices. Choice (A) has jagged notches all along the Review
Reasonableness: Two sanity checks. (1) Size: the cut removed only a tiny triangle from a quarter-sized folded square, so the final hole should be small relative to the whole paper — (E)'s tiny center diamond fits, while (A)'s edge-wide damage does not. (2) Symmetry: the cut was located at the center of the original paper, and both fold lines are symmetry axes of the original square, so the final hole must lie at the very center AND be symmetric across both the horizontal and vertical midlines. (E) is the only choice satisfying both conditions.
Alternative: Tool #1 (Draw a Diagram) is a clean backup when no paper is available. Sketch the original square divided into four quadrants by its center lines, then draw the cut triangle in the top-left quadrant. Reflect that triangle across the vertical center line, then reflect both copies across the horizontal center line. The four little triangles meet at the middle and outline the 45-degree-rotated square (the diamond). Same answer (E), no scissors required.
CCSS standards used (min grade 4)
K.G.A.2Correctly name shapes regardless of their orientations or overall size (Recognizing that the central diamond-shaped hole is still a square even though it is rotated 45 degrees.)K.G.B.6Compose simple shapes to form larger shapes (Seeing how four congruent right triangles, freed by a single cut through four layers, combine into one square hole.)2.G.A.3Partition circles and rectangles into two, three, or four equal shares (Understanding the two folds as partitioning the original square into four equal quarters.)4.G.A.3Recognize a line of symmetry for a two-dimensional figure (Treating each fold line as a line of symmetry so the cut is mirrored across it during unfolding, and using both symmetry axes to verify the matching answer choice.)
⭐ This AMC 8 problem only needs the Grade 4 idea that "a fold line is a line of symmetry" you already know!
⭐ This AMC 8 problem only needs the Grade 4 idea that "a fold line is a line of symmetry" you already know!
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