Problem
Per-cell counts give the front and side views
The top and front views of a ten-cube stack are given. We need the view from the side.
Geometry
Your answer
How to solve
Strategy Organize Information in More Ways — The front view caps each column but does not fill it in, so some stacks are still unknown. The total of ten is the extra fact that closes the gap -- once every height is known, the side view reads straight off.
1STEP 1
Fill in the column that the front view settles
Column 1 is only 1 high from the front, so none of its three stacks can be taller than one -- all three are exactly 1.
1 + 1 + 1 = 3
Column 1 accounts for 3 cubes.
6.G.A.4Eliminate PossibilitiesEvery stack in the left column is exactly one cube tall.
Why?
From the front, all the stacks in one column line up behind each other, so the view shows only the tallest of them.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Why?
That column shows only one cube, so none of its stacks reaches higher, and each occupied cell holds at least 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.
2STEP 2
Fill in the column with a single stack
Column 2 has only one occupied cell, so the front view's 3 must be that stack's height.
Row 2 column 2 is 3 cubes tall.
6.G.A.4Eliminate Possibilities3STEP 3
Use the total for the last column
Column 3 has two stacks, neither taller than 2, and they must make up what is left of the ten.
10 - 3 - 3 = 4, 4 - 2 = 2
Both stacks in column 3 are 2 tall.
5.MD.C.3Organize Information In More Ways4STEP 4
Read the side view along the rows
From the right the front row appears leftmost, and each row shows only its tallest stack.
1 + 0 = 1, 3 + 0 = 3, 2 + 0 = 2
Front row 1, middle row 3, back row 2.
6.G.A.4Visualize Spatial RelationshipsAnswer
Columns 1, 3 and 2 cells high, front to back
Add the heights: 1 and 2 in row 1, 1, 3 and 2 in row 2, 1 in row 3 -- ten cubes, and the front view still reads 1, 3, 2.
Takeaway
A view caps each line but does not fill it in -- the total number of cubes is what settles the rest.
- Fill in the column that the front view settles
- Fill in the column with a single stack
- Use the total for the last column
- Read the side view along the rows