Depth · one concept from first principles · step 2 of 4
See the shape after cubes are removed
A shape is built from unit cubes. After the red cubes are taken away, draw what it looks like from the front and from the side.
Geometry
Your answer
How to solve
Strategy Organize Information in More Ways — Drawing from the picture invites mistakes, because cubes hide behind cubes. Write the stack heights into a top view instead, take the three cubes off there, and each view is then just the biggest number along a row or a column.
1STEP 1
Take the three cubes off the counts
Each red cube is the top of its stack, so its cell's number drops by one -- and the stack of one at row 1 column 3 empties.
3 - 1 = 2, 2 - 1 = 1, 1 - 1 = 0
The counts become 3, 2 in row 1, 1, 1 in row 2 and 1 in row 3.
5.MD.C.3Organize Information In More WaysRemoving a red cube lowers its cell's count by exactly one.
Why?
Each red cube is the top of its own stack, so taking it away leaves the cubes beneath it standing and does not disturb any other cell.
🧱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 cell's count is the number of cubes stacked on it, so losing one cube lowers that count by one.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
2STEP 2
Read the front view down the columns
From the front, one column of the top view lines up behind itself, so only the tallest stack in that column shows.
3 - 0 = 3, 2 - 0 = 2
Left column 3 high, middle 2, right empty.
6.G.A.4Visualize Spatial Relationships3STEP 3
Read the side view along the rows
From the right, a row lines up behind itself, and the front row appears leftmost.
1 + 0 = 1, 1 + 0 = 1, 3 + 0 = 3
Front row 1, middle row 1, back row 3.
6.G.A.4Visualize Spatial Relationships4STEP 4
Check against the total
The counts left should add to the eleven cubes minus the three taken away.
11 - 3 = 8
3 + 2 + 1 + 1 + 1 = 8, as it should be.
5.MD.C.3Identify SubproblemsAnswer
Front: columns 3, 2 and 0 high. Side: columns 1, 1 and 3 high, front to back.
Both views must agree on the tallest stack: the front view's tallest is 3 and so is the side view's, which is the height of the row 1 column 1 stack.
Takeaway
Write the heights into the top view first -- then each view is just the biggest number along a line.
- Take the three cubes off the counts
- Read the front view down the columns
- Read the side view along the rows
- Check against the total
Where next?
Another one like thissuggested
Step back and solidifysuggested
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