Problem
The longest side sets the cube's edge
Cubes are added to a shape until it becomes the smallest cube it can. We need how many are added.
GeometryNumber system
Your answer
How to solve
Strategy Visualize Spatial Relationships — The finished cube has to swallow the shape whole, so its edge cannot be shorter than the shape's longest side. Take that edge, count what the cube holds, and subtract what is already there.
1STEP 1
Find the shape's longest side
The footprint reaches 3 cubes across and the tallest stack is 3 cubes high, so no side is longer than 3.
The cube must be at least 3 along each edge.
5.MD.C.3Visualize Spatial RelationshipsThe finished cube's edge cannot be shorter than the shape's longest side.
Why?
Cubes may only be added, so the shape has to sit inside the finished cube exactly where it already is.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
A stack three cubes long needs three cubes of room in that direction, and a box only two cubes wide has no such room.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
2STEP 2
Count what the smallest cube holds
A cube 3 along each edge holds three edges multiplied together.
3 × 3 × 3 = 27
The finished cube uses 27 cubes.
6.G.A.2Draw A Diagram3STEP 3
Count what is already there
Add the three layers of the given shape.
6 + 2 + 1 = 9
The shape has 9 cubes.
5.MD.C.3Identify Subproblems4STEP 4
Take the difference
What has to be added is what the cube holds minus what is there.
27 - 9 = 18
18 more cubes are needed.
6.G.A.2Identify SubproblemsAnswer
18 cubes
Check the layers of the finished cube: it needs 9 on each of three layers, and the shape supplies 6, 2 and 1, so 3 + 7 + 8 = 18 are added.
Takeaway
The smallest box that swallows a shape is decided by the shape's longest side, not by how many cubes it has.
- Find the shape's longest side
- Count what the smallest cube holds
- Count what is already there
- Take the difference