Reasoning · Grade 6-1 Dual and Semiregular Polyhedra
Problem
The methods' names match the left column at once.
Matching questions often hand you one of the two links for free. Doing the free one first means all your attention is left for the part that actually needs geometry.
4.G.A.2Make A Systematic ListCutting a corner creates a new face there.
Slice the corner off a cube of cheese or a lump of clay and look at the fresh flat surface: it is a triangle, and you can see one of its edges for each of the three sides of the cube that met there.
7.G.A.3Solve An Easier Related ProblemCutting a corner off turns that one vertex into a whole new face, one for every corner cut.
Each cut produces exactly one flat polygon and each new face came from exactly one corner, so the two counts match up one for one.
The finished solid is the trimmed original faces together with the new corner faces, with no gap and no overlap between them.
Cutting at third-points doubles the face's sides.
It is easiest to see on one flat face drawn on paper: mark the third-points and snip the corners off a paper triangle and you are holding a hexagon; mark the midpoints and snip and the triangle is gone, only a smaller triangle-shaped hole pattern remains around the middle.
7.G.A.3Solve An Easier Related ProblemThe tetrahedron gives hexagons and triangles.
Once you know 'faces double their sides, corners become triangles', you can name the answer before you can draw it - and naming the two face shapes is all the matching question asks for.
7.G.A.3Visualize Spatial RelationshipsThe octahedron gives hexagons and squares.
The tetrahedron and the octahedron get the same treatment and both have triangular faces, so the only thing that can tell their answers apart is how many triangles crowd round a vertex. That single number, 3 against 4, is what turns triangles into squares.
7.G.A.3Visualize Spatial RelationshipsCutting the cube at midpoints gives squares and triangles.
This is the one method that says 2 equal parts instead of 3, and that difference is the whole reason the answer has squares rather than octagons. Reading the number in the box carefully is doing real mathematical work.
7.G.A.3Visualize Spatial RelationshipsThe remaining method is the dodecahedron's, matching the last picture.
With a one-to-one matching, the last pair is forced once the other three are settled - but it is still worth checking that the forced pair genuinely makes sense, because that is how you catch a mistake made higher up.
4.G.A.2Eliminate PossibilitiesThe three columns pair up one to one.
In a matching question the check is not 'does my line look right' but 'is every item used exactly once'. Writing the four descriptions in a list makes that instant to see.
4.G.A.2Make A Systematic ListSlice a corner off and you get a new face with one side for every face that met there - so counting what meets at one corner tells you what the whole new solid is made of.