Problem
Height on a plan makes it a solid
Three grids show a stack of cubes one layer at a time. We need the view from above with the number of cubes written in each cell.
Geometry
Your answer
How to solve
Strategy Visualize Spatial Relationships — The layer pictures are horizontal slices; the top view is a vertical count. So go cell by cell and ask how many of the three slices that cell appears in -- that number is the height of the stack there.
1STEP 1
Read the footprint off layer 1
Every stack rests on the floor, so layer 1 is exactly the set of cells the top view will use.
2 + 4 + 1 = 7
Seven cells are occupied.
5.MD.C.3Visualize Spatial Relationships2STEP 2
Count how many layers each cell appears in
Row 4 column 1 and row 3 column 4 appear in all three layers; row 2 columns 2 and 3 and row 3 column 2 appear in two; row 3 columns 1 and 3 appear in layer 1 only.
Each cell's count is how many slices it shows up in.
6.G.A.4Organize Information In More WaysA cell's number is how many layers it appears in.
Why?
Each layer picture is one horizontal slice, so a cell that is shaded there holds exactly one cube at that height.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Why?
The cubes above one cell are stacked with no gaps, so counting the slices that cell appears in counts the whole stack.
🧱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
Write the counts into the top view
Row 2 gets 2 and 2 in columns 2 and 3; row 3 gets 1, 2, 1, 3; row 4 gets 3 in column 1.
Seven numbers fill the top view.
6.G.A.4Organize Information In More Ways4STEP 4
Check the total
The numbers written down should add to the same total as the three layers counted separately.
7 + 5 + 2 = 14
Both ways give 14 cubes.
5.MD.C.3Identify SubproblemsAnswer
Row 2: -, 2, 2, -; row 3: 1, 2, 1, 3; row 4: 3, -, -, - (14 cubes in all)
Add the written numbers: 2 + 2 + 1 + 2 + 1 + 3 + 3 = 14, and the layers give 7 + 5 + 2 = 14 as well.
Takeaway
Layer pictures slice a stack sideways; the top view counts straight down.
- Read the footprint off layer 1
- Count how many layers each cell appears in
- Write the counts into the top view
- Check the total