Problem
Start where the front and side views pin the count down
Three views are given and cubes are to be stacked to match all of them. We need how many different stacks do.
Geometry
Your answer
How to solve
Strategy Make a Systematic List — Four stacks, and each view gives the largest of a pair. Some of those largest values force a single height; where they do not, count the choices left. There are only four stacks, so listing beats guessing.
1STEP 1
Name the four stacks
The top view has two rows and two columns, so there are four stacks: back left, back right, front left and front right.
Four unknown heights.
6.G.A.4Visualize Spatial Relationships2STEP 2
Cap the left column at 2
The front view's left column is only 2 high, so neither of the two left stacks can be taller than 2.
Both left stacks are 1 or 2.
6.EE.B.8Eliminate Possibilities3STEP 3
Force the two right stacks to 3
The side view says each row reaches 3, and the left stacks cannot, so both right stacks must be 3.
Two of the four heights are settled.
6.G.A.4Eliminate PossibilitiesBoth right-hand stacks have to be three cubes tall.
Why?
The side view says each row reaches three, and a view shows the tallest stack in the line it looks along.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Why?
The left stacks are capped at two by the front view, so in each row the three can only come from the right-hand one.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
4STEP 4
List the choices left
The two left stacks are each 1 or 2, and the front view needs at least one of them to reach 2, so the pairs are 2 and 1, 1 and 2, or 2 and 2.
4 - 1 = 3
Three stacks are possible.
6.EE.B.8Make A Systematic ListAnswer
3
Check the one that was dropped: with both left stacks at 1, the front view's left column would only be 1 high, not 2.
Takeaway
Each view gives the largest height along a line -- use the ones that force a single value first, then list what is left.
- Name the four stacks
- Cap the left column at 2
- Force the two right stacks to 3
- List the choices left