AMC 10 · 2012 · #2
Easy mode Grade 3You have a square that is 8 long on every side. You cut it straight across the middle into two matching rectangles. How long and how wide is one of these rectangles?
Pick an answer.
AMC 10 2012 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 measuring $8$ by $8$ is cut straight across into two matching (congruent) rectangles. Find the length and width of one of those rectangles.
Givens: The starting shape is a square with side length $8$; It is cut in half into two congruent rectangles; Answer choices: (A) $2$ by $4$, (B) $2$ by $6$, (C) $2$ by $8$, (D) $4$ by $4$, (E) $4$ by $8$
Unknowns: The two side lengths (dimensions) of one of the resulting rectangles
Understand
Restated: A square measuring $8$ by $8$ is cut straight across into two matching (congruent) rectangles. Find the length and width of one of those rectangles.
Givens: The starting shape is a square with side length $8$; It is cut in half into two congruent rectangles; Answer choices: (A) $2$ by $4$, (B) $2$ by $6$, (C) $2$ by $8$, (D) $4$ by $4$, (E) $4$ by $8$
Plan
Primary tool: #1 Draw a Diagram
Secondary: #17 Visualize Spatial Relationships, #3 Eliminate Possibilities
The whole question is about a shape being cut, so Tool #1 (Draw a Diagram) makes the situation visible: sketch the $8$ by $8$ square and the single cut. Tool #17 (Visualize Spatial Relationships) keeps track of which side gets shortened and which stays the same. Tool #3 (Eliminate Possibilities) guards against the trap choice (D) $4$ by $4$, which comes from wrongly halving both sides.
Execute — Answer: E
3.G.A.2 Step 1 Draw the square and its cut
- Sketch the square, every side $8$.
- Cutting it in half means one straight cut through the middle, splitting it into two equal rectangles.
- That cut runs parallel to one pair of sides, so it slices across the square rather than shrinking it on all sides.
💡 A single straight cut can only shorten the shape in one direction, not both.
3.OA.C.7 Step 2 Halve one side, keep the other
- The cut lands on the middle of one pair of sides, splitting a side of length $8$ into two equal pieces: $8 \div 2 = 4$.
- The pair of sides running along the cut is untouched, so those still measure $8$.
- So each rectangle has one dimension $4$ and one dimension $8$.
💡 Cutting in half divides one length by $2$ while the crosswise length rides along unchanged.
2.G.A.3 Step 3 Read off and match the dimensions
- One rectangle is $4$ by $8$.
- Check the traps: $4$ by $4$ (D) would need both sides halved, and $2$ by $8$ (C) would need the side split into four, neither of which one straight half-cut does.
- The dimensions $4$ by $8$ match choice (E).
💡 Two equal shares of the square must together rebuild it, and $4 \times 8$ twice gives back the $8 \times 8$ area.
3.G.A.2 Sketch the square, every side $8$. Cutting it in half means one straight cut thr 3.OA.C.7 The cut lands on the middle of one pair of sides, splitting a side of length $8$ 2.G.A.3 One rectangle is $4$ by $8$. Check the traps: $4$ by $4$ (D) would need both sid Review
Reasonableness: Check by area: the whole square is $8 \times 8 = 64$, so each half should have area $32$. A $4$ by $8$ rectangle has area $4 \times 8 = 32$ — exactly half, which fits. The trap $4$ by $4$ has area $16$ (a quarter, too small) and $2$ by $8$ has area $16$ as well, so only $4$ by $8$ gives the correct half-area, confirming (E).
Alternative: Skip the picture and reason from area alone: each piece must have area $64 \div 2 = 32$, and it must still share a full side of length $8$ with the square (the uncut side). Then the other side is $32 \div 8 = 4$, giving $4$ by $8$, choice (E).
CCSS standards used (min grade 3)
3.G.A.2Partition shapes into equal parts with equal areas (Understanding that cutting the square in half produces two equal-area rectangles from one straight cut.)3.OA.C.7Fluently multiply and divide within 100 (Halving the cut side, $8 \div 2 = 4$, and checking areas like $4 \times 8 = 32$.)2.G.A.3Partition circles and rectangles into two, three, or four equal shares (Recognizing that two congruent halves reassemble into the original square, ruling out the trap dimensions.)
⭐ One straight cut in half shortens just one side by half — here $8$ becomes $4$ while the other side stays $8$, so the piece is $4$ by $8$.
⭐ One straight cut in half shortens just one side by half — here $8$ becomes $4$ while the other side stays $8$, so the piece is $4$ by $8$.
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