Reasoning · Grade 5-2 Rectangular Prisms
Problem
The six faces split into three opposite pairs.
Hold any box and look at it from opposite sides: what you see is the same rectangle both times, so a box can never need five of one shape and one of another.
K.G.B.4Visualize Spatial RelationshipsThe six faces of a box come in three matching pairs, so a box can never need five of one shape and one of another.
Looking at a box from opposite sides shows the very same rectangle, because sliding one face across to the other lands it exactly.
Three edge lengths taken two at a time give exactly three different pairings, which is why there are three kinds of face and no more.
Three edges at one corner decide everything.
Three numbers taken two at a time give exactly three different pairings, which is why a prism has exactly three kinds of face and not more.
6.G.A.4Visualize Spatial RelationshipsSet (1) has edges 3, 2 and 1 cm.
Each number counted exactly twice is the whole test: if some number turned up only once, no face could be glued to it along that side and the box would never close.
6.G.A.4Make A Systematic ListThe six pieces fit together.
Cut the six pieces out and tape them: a wrong pairing shows up at once because two sides that have to be taped together are different lengths.
6.G.A.4Create A Physical RepresentationSketch it and label the twelve edges.
An oblique sketch is only a rectangle, a copy of it slid over, and lines joining matching corners; dashing the three hidden edges is what makes it read as a solid.
2.G.A.1Draw A DiagramSet (2) has a square face: edges 3, 3 and 2 cm.
Two of the three edge lengths being equal is allowed; it just means two of the three kinds of face turn out identical, so four of the six rectangles look the same.
6.G.A.4Make A Systematic ListSketch and label it the same way.
Marking one edge of each of the three directions is enough, because parallel edges of a prism are always equal — writing 3 cm on all four top edges says nothing extra.
2.G.A.1Draw A DiagramThe total face area also checks out.
Taping pieces together neither loses nor gains any card, so the areas have to agree exactly — a cheap arithmetic check on a spatial answer.
6.G.A.4Make A Systematic ListA box is built from just three numbers, and every one of them has to show up on exactly two of the three different faces - that is all you need to read the box off its rectangles!