Depth · one concept from first principles · step 2 of 4
A cube is built from unit cubes, as shown at the right. A straight line of cubes is removed through the middle of each face, so that each face has a hole through it. How many unit cubes are left?
Each line runs the full width of the cube, so it holds 5 cubes, and there are three lines.
15 -- but not all of them are different cubes.
5.MD.C.3Visualize Spatial RelationshipsAll three lines pass through the cube at the very centre, so that one cube was counted three times when it should be counted once.
The count is 2 too many.
5.MD.C.3Change Focus Count The ComplementThe centre cube was counted three times but can only be removed once.
All three lines run through the middle of the cube, so that one cube belongs to every one of them.
Counting the removed cubes means counting each cube once, so a cube belonging to three lines still contributes one to the count.
Take those two extra countings off.
13 different cubes come out.
5.MD.C.3Identify SubproblemsSubtract the removed cubes from the 125 the cube started with.
112 cubes are left.
6.G.A.2Identify SubproblemsWhen things you are counting overlap, count them all and then take the overlaps back off.
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