Reasoning · Grade 4-2 Figure Puzzles
Problem
The seven pieces total 32.
Areas of pieces that do not overlap simply add up, and a right triangle sitting on a grid is always half of the rectangle around it — both are Grade 3 area ideas.
3.MD.C.7Draw A DiagramThe three target shapes also have area 32.
Area is a count of unit squares, and rearranging pieces neither creates nor destroys squares — so the total is an invariant you can lean on.
3.MD.C.7Draw A DiagramEvery shape in the chain has the same area 32, because only one piece is ever moved and nothing is added or lost.
The shape is its pieces put together with no gap and no overlap, so the total is fixed by which pieces are present.
Sliding, turning or flipping a piece lays it onto a copy of itself, so its area travels with it unchanged.
Overlaying shows the area to move is 8.
Overlaying the "before" and "after" pictures shows the hole and the overhang at a glance; matching them is just comparing two areas.
3.MD.C.7Draw A DiagramOnly the two large triangles have area 8.
With only seven candidates you can test them one by one, and a single number — the area — settles five of them instantly.
4.G.A.2Eliminate PossibilitiesMoving the same triangle again gives the parallelogram.
A parallelogram is defined by having two pairs of parallel sides; checking "is it a rectangle?" is just checking whether any corner is square, which a Grade 4 solver can do with the corner of a card.
5.G.B.4Draw A DiagramOnce more gives the trapezoid.
Cutting the seven pieces out of paper and actually sliding the big triangle to three different spots makes "one pair parallel" versus "two pairs parallel" something you can see rather than compute.
5.G.B.4Create A Physical RepresentationSmaller pieces are too small, so it is the large right triangle.
Comparing sizes rules out impossible moves before any of them has to be tried, which is much faster than hunting for a placement.
4.G.A.2Eliminate PossibilitiesLay the new shape on top of the old one — the leftover bit tells you exactly which piece has to move, and here only a big triangle is large enough.