Problem
Spot the piece added onto a shape
A shape was built from two of four given pieces. We need which two.
Geometry
Your answer
How to solve
Strategy Eliminate Possibilities — Counting cubes settles the sizes first: the shape holds eight, so it is two four-cube pieces. Then place the most rigid piece -- the flat square, which needs a 2 × 2 patch of floor -- and see whether what is left is one of the others.
1STEP 1
Count the cubes in the finished shape
Five cells on the floor and three of them carrying a second cube.
5 + 3 = 8
The shape has 8 cubes, so it is two four-cube pieces.
5.MD.C.3Visualize Spatial Relationships2STEP 2
Place the piece with the least freedom
The flat square needs four cells forming a 2 × 2 patch, and the floor has exactly one such patch -- the two front cells with the two back cells beside them.
Piece A can only go one place.
6.G.A.4Eliminate PossibilitiesThe flat square piece can only go in one place.
Why?
That piece needs four cells meeting in a by block on the floor, and the shape's floor has exactly one such block.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Why?
Once it is placed, what is left has to be the other piece, because the two pieces together are the whole solid with nothing over.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
3STEP 3
See what is left
What remains is the third cell of the back row on the floor, plus the whole second layer -- a row of three with one cube below one end.
That is the same shape as D, turned over.
6.G.A.4Make A Systematic List4STEP 4
Check the fit
The two pieces cover all eight cubes with nothing left over and nothing counted twice.
4 + 4 = 8
A and D build the shape.
5.MD.C.3Eliminate PossibilitiesAnswer
A and D
Take the two pieces out again: the flat square is the front 2 × 2 block on the floor, and everything else -- the second layer's three plus one floor cube under its end -- is exactly piece D.
Takeaway
Place the piece with the fewest possible positions first -- it decides the rest.
- Count the cubes in the finished shape
- Place the piece with the least freedom
- See what is left
- Check the fit