Reasoning · Grade 4-2 Figure Puzzles
Problem
The example falls into 3 pieces.
Counting the pieces is something you can do before drawing anything, and it removes half the possibilities straight away — an idea a fourth grader can apply to any cutting puzzle.
2.G.A.1Solve An Easier Related ProblemThe useful cuts are diagonals and lines parallel to a side.
A quadrilateral earns its name from its parallel sides, so if you want named pieces you should cut in directions that are already parallel to something in the figure.
4.G.A.2Make A Systematic ListOnly cuts running parallel to a side already in the figure are worth making, because a named quadrilateral needs parallel sides.
A trapezoid or a parallelogram earns its name from having sides that never meet, so a cut has to supply such a pair.
The pieces have to fill the figure with no gap and no overlap, so every cut must run right across from edge to edge.
For (1), two diagonals from one vertex give 3 pieces.
Two cuts from one vertex act like a fan: they never meet again, so they leave three pieces, and the outer one keeps four corners while the two inner ones are triangles.
4.G.A.2Draw A DiagramFor (2), a long diagonal and a crossing line give 4 trapezoids.
Cutting a symmetric shape through its middle in two directions at once gives four matching pieces, and here each piece keeps one short parallel side and one long one — the trapezoid shape.
4.G.A.2Draw A DiagramFor (3), two long diagonals give 2 triangles and 2 rhombuses.
The rhombus is the piece that demands the extra fact that the centre is one side-length from every vertex; that single measurement is what upgrades an ordinary parallelogram to a rhombus.
5.G.B.4Identify SubproblemsFor (4), two lines parallel to a side give 3 pieces.
Parallel cuts can never meet, so three pieces are guaranteed; and slicing a corner off with a cut parallel to a side is the standard way to make a trapezoid appear.
4.G.A.2Draw A DiagramFor (5) the cuts give 1 triangle and 3 trapezoids.
Because the diagonal AD runs parallel to two sides at once, every quadrilateral that has part of AD as a side automatically has a parallel partner — so three trapezoids come out almost by themselves.
4.G.A.2Identify SubproblemsFor (6) the four pieces are all different shapes.
A parallelogram appears exactly where a piece is boxed in by two different pairs of parallel lines; away from that corner the same two cuts leave only one pair parallel, which is why the neighbouring piece stays an ordinary trapezoid.
5.G.B.4Identify SubproblemsAll six sheets match their lists.
Counting corners is the quickest check of all, and it catches the commonest slip in this kind of puzzle — a cut that accidentally passes through a vertex and turns a four-sided piece into a triangle.
2.G.A.1Make A Systematic ListDecide first whether the two cuts should cross inside the shape or not, then aim them along the hexagon's own parallel lines — that is what makes the pieces come out with the names you were asked for.