← A cell's number is how many layers it appears in · Cube Stacks and Their Views

A cell's number is how many layers it appears in · 12 practice problems

6.G.A.4

Generated variants — 12

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy answer: Row 2: -, 2, 2, -; row 3: 1, 2, 1, 3; row 4: 3, -, -, - (14 cubes in all)

The pictures show a stack of unit cubes, one layer at a time. Draw the view from above and write in each cell how many cubes are stacked there.

Show solution

1 · Understandwhat's really being asked

A stack of unit cubes is drawn one layer at a time. We want the view from above with each cell's number of cubes written in.

Givens

Layer 1 covers 7 cells.
Layer 2 covers 5 cells.
Layer 3 covers 2 cells.

Unknowns

The number of cubes at each cell of the top view.

Constraints

Every stack rests on the floor, so nothing floats above a gap.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #15 Organize Information in More Ways#2 Make a Systematic List

The layer pictures cut the stack sideways and the top view counts straight down, so the two are different slices of the same thing. Translate by asking, for each cell, how many layers it appears in.

3 · Execute4 carry out the plan

1Read layer 1 as the footprint

#17 Visualize Spatial Relationships 6.G.A.4

Every stack starts on the floor, so a cell holding any cubes at all appears in layer 1.

\text{layer 1} = 7\text{ cells}

That is the whole top view's shape.

2Count each cell's layers

#15 Organize Information in More Ways 6.G.A.4

A cell in all the layers has that many cubes; one in layer 1 alone has just the one.

\text{cell} = \text{layers it is in}

Now every cell has a number.

3Write them into the grid

#2 Make a Systematic List 6.G.A.4

Reading rows from the back, the numbers are as follows.

0 0 0 0 \quad 0 2 2 0 \quad 1 2 1 3 \quad 3 0 0 0

The top view is filled in.

4Check against the layer counts

#2 Make a Systematic List 6.G.A.4

The numbers should add to the same total as the layers do.

7 + 5 + 2 = 14

14 cubes either way.

Answer: Row 2: -, 2, 2, -; row 3: 1, 2, 1, 3; row 4: 3, -, -, - (14 cubes in all)

4 · Reviewdoes it hold up?

The largest number in the grid is 3, which is exactly how many layers were drawn -- as it has to be.

Another way: Counting the cubes layer by layer gives 7 + 5 + 2 = 14, the same total reached without ever writing the grid.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Translating between layer pictures and a numbered top view.

💡Takeaway. Layers cut a stack sideways; the top view counts downwards. A cell's number is how many layers it shows up in.

Variant 2 easy answer: Row 1: 1, 1, -, -; row 2: -, 3, 2, 1; row 3: 2, -, 1, 3; row 4: -, 1, -, - (15 cubes in all)

The pictures show a stack of unit cubes, one layer at a time. Draw the view from above and write in each cell how many cubes are stacked there.

Show solution

1 · Understandwhat's really being asked

A stack of unit cubes is drawn one layer at a time. We want the view from above with each cell's number of cubes written in.

Givens

Layer 1 covers 9 cells.
Layer 2 covers 4 cells.
Layer 3 covers 2 cells.

Unknowns

The number of cubes at each cell of the top view.

Constraints

Every stack rests on the floor, so nothing floats above a gap.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #15 Organize Information in More Ways#2 Make a Systematic List

The layer pictures cut the stack sideways and the top view counts straight down, so the two are different slices of the same thing. Translate by asking, for each cell, how many layers it appears in.

3 · Execute4 carry out the plan

1Read layer 1 as the footprint

#17 Visualize Spatial Relationships 6.G.A.4

Every stack starts on the floor, so a cell holding any cubes at all appears in layer 1.

\text{layer 1} = 9\text{ cells}

That is the whole top view's shape.

2Count each cell's layers

#15 Organize Information in More Ways 6.G.A.4

A cell in all the layers has that many cubes; one in layer 1 alone has just the one.

\text{cell} = \text{layers it is in}

Now every cell has a number.

3Write them into the grid

#2 Make a Systematic List 6.G.A.4

Reading rows from the back, the numbers are as follows.

1 1 0 0 \quad 0 3 2 1 \quad 2 0 1 3 \quad 0 1 0 0

The top view is filled in.

4Check against the layer counts

#2 Make a Systematic List 6.G.A.4

The numbers should add to the same total as the layers do.

9 + 4 + 2 = 15

15 cubes either way.

Answer: Row 1: 1, 1, -, -; row 2: -, 3, 2, 1; row 3: 2, -, 1, 3; row 4: -, 1, -, - (15 cubes in all)

4 · Reviewdoes it hold up?

The largest number in the grid is 3, which is exactly how many layers were drawn -- as it has to be.

Another way: Counting the cubes layer by layer gives 9 + 4 + 2 = 15, the same total reached without ever writing the grid.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Translating between layer pictures and a numbered top view.

💡Takeaway. Layers cut a stack sideways; the top view counts downwards. A cell's number is how many layers it shows up in.

Variant 3 easy answer: Row 1: -, 2, -, 1; row 2: 1, 1, 3, -; row 3: 2, -, 1, 2; row 4: -, 1, -, 1 (15 cubes in all)

The pictures show a stack of unit cubes, one layer at a time. Draw the view from above and write in each cell how many cubes are stacked there.

Show solution

1 · Understandwhat's really being asked

A stack of unit cubes is drawn one layer at a time. We want the view from above with each cell's number of cubes written in.

Givens

Layer 1 covers 10 cells.
Layer 2 covers 4 cells.
Layer 3 covers 1 cells.

Unknowns

The number of cubes at each cell of the top view.

Constraints

Every stack rests on the floor, so nothing floats above a gap.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #15 Organize Information in More Ways#2 Make a Systematic List

The layer pictures cut the stack sideways and the top view counts straight down, so the two are different slices of the same thing. Translate by asking, for each cell, how many layers it appears in.

3 · Execute4 carry out the plan

1Read layer 1 as the footprint

#17 Visualize Spatial Relationships 6.G.A.4

Every stack starts on the floor, so a cell holding any cubes at all appears in layer 1.

\text{layer 1} = 10\text{ cells}

That is the whole top view's shape.

2Count each cell's layers

#15 Organize Information in More Ways 6.G.A.4

A cell in all the layers has that many cubes; one in layer 1 alone has just the one.

\text{cell} = \text{layers it is in}

Now every cell has a number.

3Write them into the grid

#2 Make a Systematic List 6.G.A.4

Reading rows from the back, the numbers are as follows.

0 2 0 1 \quad 1 1 3 0 \quad 2 0 1 2 \quad 0 1 0 1

The top view is filled in.

4Check against the layer counts

#2 Make a Systematic List 6.G.A.4

The numbers should add to the same total as the layers do.

10 + 4 + 1 = 15

15 cubes either way.

Answer: Row 1: -, 2, -, 1; row 2: 1, 1, 3, -; row 3: 2, -, 1, 2; row 4: -, 1, -, 1 (15 cubes in all)

4 · Reviewdoes it hold up?

The largest number in the grid is 3, which is exactly how many layers were drawn -- as it has to be.

Another way: Counting the cubes layer by layer gives 10 + 4 + 1 = 15, the same total reached without ever writing the grid.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Translating between layer pictures and a numbered top view.

💡Takeaway. Layers cut a stack sideways; the top view counts downwards. A cell's number is how many layers it shows up in.

Variant 4 medium answer: Row 1: 1, -, 2, -; row 2: -, 3, 1, 1; row 3: 2, 1, -, 3; row 4: -, -, 1, - (15 cubes in all)

The pictures show a stack of unit cubes, one layer at a time. Draw the view from above and write in each cell how many cubes are stacked there.

Show solution

1 · Understandwhat's really being asked

A stack of unit cubes is drawn one layer at a time. We want the view from above with each cell's number of cubes written in.

Givens

Layer 1 covers 9 cells.
Layer 2 covers 4 cells.
Layer 3 covers 2 cells.

Unknowns

The number of cubes at each cell of the top view.

Constraints

Every stack rests on the floor, so nothing floats above a gap.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #15 Organize Information in More Ways#2 Make a Systematic List

The layer pictures cut the stack sideways and the top view counts straight down, so the two are different slices of the same thing. Translate by asking, for each cell, how many layers it appears in.

3 · Execute4 carry out the plan

1Read layer 1 as the footprint

#17 Visualize Spatial Relationships 6.G.A.4

Every stack starts on the floor, so a cell holding any cubes at all appears in layer 1.

\text{layer 1} = 9\text{ cells}

That is the whole top view's shape.

2Count each cell's layers

#15 Organize Information in More Ways 6.G.A.4

A cell in all the layers has that many cubes; one in layer 1 alone has just the one.

\text{cell} = \text{layers it is in}

Now every cell has a number.

3Write them into the grid

#2 Make a Systematic List 6.G.A.4

Reading rows from the back, the numbers are as follows.

1 0 2 0 \quad 0 3 1 1 \quad 2 1 0 3 \quad 0 0 1 0

The top view is filled in.

4Check against the layer counts

#2 Make a Systematic List 6.G.A.4

The numbers should add to the same total as the layers do.

9 + 4 + 2 = 15

15 cubes either way.

Answer: Row 1: 1, -, 2, -; row 2: -, 3, 1, 1; row 3: 2, 1, -, 3; row 4: -, -, 1, - (15 cubes in all)

4 · Reviewdoes it hold up?

The largest number in the grid is 3, which is exactly how many layers were drawn -- as it has to be.

Another way: Counting the cubes layer by layer gives 9 + 4 + 2 = 15, the same total reached without ever writing the grid.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Translating between layer pictures and a numbered top view.

💡Takeaway. Layers cut a stack sideways; the top view counts downwards. A cell's number is how many layers it shows up in.

Variant 5 hard answer: Row 1: 3, -, 1, -; row 2: 1, 2, -, 3; row 3: -, 1, 2, 1; row 4: -, -, 1, - (15 cubes in all)

The pictures show a stack of unit cubes, one layer at a time. Draw the view from above and write in each cell how many cubes are stacked there.

Show solution

1 · Understandwhat's really being asked

A stack of unit cubes is drawn one layer at a time. We want the view from above with each cell's number of cubes written in.

Givens

Layer 1 covers 9 cells.
Layer 2 covers 4 cells.
Layer 3 covers 2 cells.

Unknowns

The number of cubes at each cell of the top view.

Constraints

Every stack rests on the floor, so nothing floats above a gap.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #15 Organize Information in More Ways#2 Make a Systematic List

The layer pictures cut the stack sideways and the top view counts straight down, so the two are different slices of the same thing. Translate by asking, for each cell, how many layers it appears in.

3 · Execute4 carry out the plan

1Read layer 1 as the footprint

#17 Visualize Spatial Relationships 6.G.A.4

Every stack starts on the floor, so a cell holding any cubes at all appears in layer 1.

\text{layer 1} = 9\text{ cells}

That is the whole top view's shape.

2Count each cell's layers

#15 Organize Information in More Ways 6.G.A.4

A cell in all the layers has that many cubes; one in layer 1 alone has just the one.

\text{cell} = \text{layers it is in}

Now every cell has a number.

3Write them into the grid

#2 Make a Systematic List 6.G.A.4

Reading rows from the back, the numbers are as follows.

3 0 1 0 \quad 1 2 0 3 \quad 0 1 2 1 \quad 0 0 1 0

The top view is filled in.

4Check against the layer counts

#2 Make a Systematic List 6.G.A.4

The numbers should add to the same total as the layers do.

9 + 4 + 2 = 15

15 cubes either way.

Answer: Row 1: 3, -, 1, -; row 2: 1, 2, -, 3; row 3: -, 1, 2, 1; row 4: -, -, 1, - (15 cubes in all)

4 · Reviewdoes it hold up?

The largest number in the grid is 3, which is exactly how many layers were drawn -- as it has to be.

Another way: Counting the cubes layer by layer gives 9 + 4 + 2 = 15, the same total reached without ever writing the grid.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Translating between layer pictures and a numbered top view.

💡Takeaway. Layers cut a stack sideways; the top view counts downwards. A cell's number is how many layers it shows up in.

Variant 6 medium answer: Row 1: -, -, 3, 1; row 2: 1, 2, 1, -; row 3: 3, 1, -, 2; row 4: -, 1, -, - (15 cubes in all)

The pictures show a stack of unit cubes, one layer at a time. Draw the view from above and write in each cell how many cubes are stacked there.

Show solution

1 · Understandwhat's really being asked

A stack of unit cubes is drawn one layer at a time. We want the view from above with each cell's number of cubes written in.

Givens

Layer 1 covers 9 cells.
Layer 2 covers 4 cells.
Layer 3 covers 2 cells.

Unknowns

The number of cubes at each cell of the top view.

Constraints

Every stack rests on the floor, so nothing floats above a gap.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #15 Organize Information in More Ways#2 Make a Systematic List

The layer pictures cut the stack sideways and the top view counts straight down, so the two are different slices of the same thing. Translate by asking, for each cell, how many layers it appears in.

3 · Execute4 carry out the plan

1Read layer 1 as the footprint

#17 Visualize Spatial Relationships 6.G.A.4

Every stack starts on the floor, so a cell holding any cubes at all appears in layer 1.

\text{layer 1} = 9\text{ cells}

That is the whole top view's shape.

2Count each cell's layers

#15 Organize Information in More Ways 6.G.A.4

A cell in all the layers has that many cubes; one in layer 1 alone has just the one.

\text{cell} = \text{layers it is in}

Now every cell has a number.

3Write them into the grid

#2 Make a Systematic List 6.G.A.4

Reading rows from the back, the numbers are as follows.

0 0 3 1 \quad 1 2 1 0 \quad 3 1 0 2 \quad 0 1 0 0

The top view is filled in.

4Check against the layer counts

#2 Make a Systematic List 6.G.A.4

The numbers should add to the same total as the layers do.

9 + 4 + 2 = 15

15 cubes either way.

Answer: Row 1: -, -, 3, 1; row 2: 1, 2, 1, -; row 3: 3, 1, -, 2; row 4: -, 1, -, - (15 cubes in all)

4 · Reviewdoes it hold up?

The largest number in the grid is 3, which is exactly how many layers were drawn -- as it has to be.

Another way: Counting the cubes layer by layer gives 9 + 4 + 2 = 15, the same total reached without ever writing the grid.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Translating between layer pictures and a numbered top view.

💡Takeaway. Layers cut a stack sideways; the top view counts downwards. A cell's number is how many layers it shows up in.

Variant 7 easy answer: Row 1: -, 1, -, -; row 2: 2, 3, 1, -; row 3: 1, 1, 2, -; row 4: -, -, 3, 1 (15 cubes in all)

The pictures show a stack of unit cubes, one layer at a time. Draw the view from above and write in each cell how many cubes are stacked there.

Show solution

1 · Understandwhat's really being asked

A stack of unit cubes is drawn one layer at a time. We want the view from above with each cell's number of cubes written in.

Givens

Layer 1 covers 9 cells.
Layer 2 covers 4 cells.
Layer 3 covers 2 cells.

Unknowns

The number of cubes at each cell of the top view.

Constraints

Every stack rests on the floor, so nothing floats above a gap.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #15 Organize Information in More Ways#2 Make a Systematic List

The layer pictures cut the stack sideways and the top view counts straight down, so the two are different slices of the same thing. Translate by asking, for each cell, how many layers it appears in.

3 · Execute4 carry out the plan

1Read layer 1 as the footprint

#17 Visualize Spatial Relationships 6.G.A.4

Every stack starts on the floor, so a cell holding any cubes at all appears in layer 1.

\text{layer 1} = 9\text{ cells}

That is the whole top view's shape.

2Count each cell's layers

#15 Organize Information in More Ways 6.G.A.4

A cell in all the layers has that many cubes; one in layer 1 alone has just the one.

\text{cell} = \text{layers it is in}

Now every cell has a number.

3Write them into the grid

#2 Make a Systematic List 6.G.A.4

Reading rows from the back, the numbers are as follows.

0 1 0 0 \quad 2 3 1 0 \quad 1 1 2 0 \quad 0 0 3 1

The top view is filled in.

4Check against the layer counts

#2 Make a Systematic List 6.G.A.4

The numbers should add to the same total as the layers do.

9 + 4 + 2 = 15

15 cubes either way.

Answer: Row 1: -, 1, -, -; row 2: 2, 3, 1, -; row 3: 1, 1, 2, -; row 4: -, -, 3, 1 (15 cubes in all)

4 · Reviewdoes it hold up?

The largest number in the grid is 3, which is exactly how many layers were drawn -- as it has to be.

Another way: Counting the cubes layer by layer gives 9 + 4 + 2 = 15, the same total reached without ever writing the grid.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Translating between layer pictures and a numbered top view.

💡Takeaway. Layers cut a stack sideways; the top view counts downwards. A cell's number is how many layers it shows up in.

Variant 8 hard answer: Row 1: -, 3, -, 1; row 2: 2, 1, 3, -; row 3: 1, -, 1, 2; row 4: -, 1, -, - (15 cubes in all)

The pictures show a stack of unit cubes, one layer at a time. Draw the view from above and write in each cell how many cubes are stacked there.

Show solution

1 · Understandwhat's really being asked

A stack of unit cubes is drawn one layer at a time. We want the view from above with each cell's number of cubes written in.

Givens

Layer 1 covers 9 cells.
Layer 2 covers 4 cells.
Layer 3 covers 2 cells.

Unknowns

The number of cubes at each cell of the top view.

Constraints

Every stack rests on the floor, so nothing floats above a gap.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #15 Organize Information in More Ways#2 Make a Systematic List

The layer pictures cut the stack sideways and the top view counts straight down, so the two are different slices of the same thing. Translate by asking, for each cell, how many layers it appears in.

3 · Execute4 carry out the plan

1Read layer 1 as the footprint

#17 Visualize Spatial Relationships 6.G.A.4

Every stack starts on the floor, so a cell holding any cubes at all appears in layer 1.

\text{layer 1} = 9\text{ cells}

That is the whole top view's shape.

2Count each cell's layers

#15 Organize Information in More Ways 6.G.A.4

A cell in all the layers has that many cubes; one in layer 1 alone has just the one.

\text{cell} = \text{layers it is in}

Now every cell has a number.

3Write them into the grid

#2 Make a Systematic List 6.G.A.4

Reading rows from the back, the numbers are as follows.

0 3 0 1 \quad 2 1 3 0 \quad 1 0 1 2 \quad 0 1 0 0

The top view is filled in.

4Check against the layer counts

#2 Make a Systematic List 6.G.A.4

The numbers should add to the same total as the layers do.

9 + 4 + 2 = 15

15 cubes either way.

Answer: Row 1: -, 3, -, 1; row 2: 2, 1, 3, -; row 3: 1, -, 1, 2; row 4: -, 1, -, - (15 cubes in all)

4 · Reviewdoes it hold up?

The largest number in the grid is 3, which is exactly how many layers were drawn -- as it has to be.

Another way: Counting the cubes layer by layer gives 9 + 4 + 2 = 15, the same total reached without ever writing the grid.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Translating between layer pictures and a numbered top view.

💡Takeaway. Layers cut a stack sideways; the top view counts downwards. A cell's number is how many layers it shows up in.

Variant 9 medium answer: Row 1: 2, -, -, 1; row 2: 1, 3, 2, -; row 3: -, 1, 1, 3; row 4: 1, -, -, - (15 cubes in all)

The pictures show a stack of unit cubes, one layer at a time. Draw the view from above and write in each cell how many cubes are stacked there.

Show solution

1 · Understandwhat's really being asked

A stack of unit cubes is drawn one layer at a time. We want the view from above with each cell's number of cubes written in.

Givens

Layer 1 covers 9 cells.
Layer 2 covers 4 cells.
Layer 3 covers 2 cells.

Unknowns

The number of cubes at each cell of the top view.

Constraints

Every stack rests on the floor, so nothing floats above a gap.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #15 Organize Information in More Ways#2 Make a Systematic List

The layer pictures cut the stack sideways and the top view counts straight down, so the two are different slices of the same thing. Translate by asking, for each cell, how many layers it appears in.

3 · Execute4 carry out the plan

1Read layer 1 as the footprint

#17 Visualize Spatial Relationships 6.G.A.4

Every stack starts on the floor, so a cell holding any cubes at all appears in layer 1.

\text{layer 1} = 9\text{ cells}

That is the whole top view's shape.

2Count each cell's layers

#15 Organize Information in More Ways 6.G.A.4

A cell in all the layers has that many cubes; one in layer 1 alone has just the one.

\text{cell} = \text{layers it is in}

Now every cell has a number.

3Write them into the grid

#2 Make a Systematic List 6.G.A.4

Reading rows from the back, the numbers are as follows.

2 0 0 1 \quad 1 3 2 0 \quad 0 1 1 3 \quad 1 0 0 0

The top view is filled in.

4Check against the layer counts

#2 Make a Systematic List 6.G.A.4

The numbers should add to the same total as the layers do.

9 + 4 + 2 = 15

15 cubes either way.

Answer: Row 1: 2, -, -, 1; row 2: 1, 3, 2, -; row 3: -, 1, 1, 3; row 4: 1, -, -, - (15 cubes in all)

4 · Reviewdoes it hold up?

The largest number in the grid is 3, which is exactly how many layers were drawn -- as it has to be.

Another way: Counting the cubes layer by layer gives 9 + 4 + 2 = 15, the same total reached without ever writing the grid.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Translating between layer pictures and a numbered top view.

💡Takeaway. Layers cut a stack sideways; the top view counts downwards. A cell's number is how many layers it shows up in.

Variant 10 hard answer: Row 1: -, 1, 2, -; row 2: 1, 3, -, 1; row 3: 2, -, 3, 1; row 4: -, 1, -, - (15 cubes in all)

The pictures show a stack of unit cubes, one layer at a time. Draw the view from above and write in each cell how many cubes are stacked there.

Show solution

1 · Understandwhat's really being asked

A stack of unit cubes is drawn one layer at a time. We want the view from above with each cell's number of cubes written in.

Givens

Layer 1 covers 9 cells.
Layer 2 covers 4 cells.
Layer 3 covers 2 cells.

Unknowns

The number of cubes at each cell of the top view.

Constraints

Every stack rests on the floor, so nothing floats above a gap.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #15 Organize Information in More Ways#2 Make a Systematic List

The layer pictures cut the stack sideways and the top view counts straight down, so the two are different slices of the same thing. Translate by asking, for each cell, how many layers it appears in.

3 · Execute4 carry out the plan

1Read layer 1 as the footprint

#17 Visualize Spatial Relationships 6.G.A.4

Every stack starts on the floor, so a cell holding any cubes at all appears in layer 1.

\text{layer 1} = 9\text{ cells}

That is the whole top view's shape.

2Count each cell's layers

#15 Organize Information in More Ways 6.G.A.4

A cell in all the layers has that many cubes; one in layer 1 alone has just the one.

\text{cell} = \text{layers it is in}

Now every cell has a number.

3Write them into the grid

#2 Make a Systematic List 6.G.A.4

Reading rows from the back, the numbers are as follows.

0 1 2 0 \quad 1 3 0 1 \quad 2 0 3 1 \quad 0 1 0 0

The top view is filled in.

4Check against the layer counts

#2 Make a Systematic List 6.G.A.4

The numbers should add to the same total as the layers do.

9 + 4 + 2 = 15

15 cubes either way.

Answer: Row 1: -, 1, 2, -; row 2: 1, 3, -, 1; row 3: 2, -, 3, 1; row 4: -, 1, -, - (15 cubes in all)

4 · Reviewdoes it hold up?

The largest number in the grid is 3, which is exactly how many layers were drawn -- as it has to be.

Another way: Counting the cubes layer by layer gives 9 + 4 + 2 = 15, the same total reached without ever writing the grid.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Translating between layer pictures and a numbered top view.

💡Takeaway. Layers cut a stack sideways; the top view counts downwards. A cell's number is how many layers it shows up in.

Variant 11 medium answer: Row 1: -, 2, 1, -; row 2: 3, 1, -, 2; row 3: 1, -, 3, 1; row 4: -, 1, -, - (15 cubes in all)

The pictures show a stack of unit cubes, one layer at a time. Draw the view from above and write in each cell how many cubes are stacked there.

Show solution

1 · Understandwhat's really being asked

A stack of unit cubes is drawn one layer at a time. We want the view from above with each cell's number of cubes written in.

Givens

Layer 1 covers 9 cells.
Layer 2 covers 4 cells.
Layer 3 covers 2 cells.

Unknowns

The number of cubes at each cell of the top view.

Constraints

Every stack rests on the floor, so nothing floats above a gap.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #15 Organize Information in More Ways#2 Make a Systematic List

The layer pictures cut the stack sideways and the top view counts straight down, so the two are different slices of the same thing. Translate by asking, for each cell, how many layers it appears in.

3 · Execute4 carry out the plan

1Read layer 1 as the footprint

#17 Visualize Spatial Relationships 6.G.A.4

Every stack starts on the floor, so a cell holding any cubes at all appears in layer 1.

\text{layer 1} = 9\text{ cells}

That is the whole top view's shape.

2Count each cell's layers

#15 Organize Information in More Ways 6.G.A.4

A cell in all the layers has that many cubes; one in layer 1 alone has just the one.

\text{cell} = \text{layers it is in}

Now every cell has a number.

3Write them into the grid

#2 Make a Systematic List 6.G.A.4

Reading rows from the back, the numbers are as follows.

0 2 1 0 \quad 3 1 0 2 \quad 1 0 3 1 \quad 0 1 0 0

The top view is filled in.

4Check against the layer counts

#2 Make a Systematic List 6.G.A.4

The numbers should add to the same total as the layers do.

9 + 4 + 2 = 15

15 cubes either way.

Answer: Row 1: -, 2, 1, -; row 2: 3, 1, -, 2; row 3: 1, -, 3, 1; row 4: -, 1, -, - (15 cubes in all)

4 · Reviewdoes it hold up?

The largest number in the grid is 3, which is exactly how many layers were drawn -- as it has to be.

Another way: Counting the cubes layer by layer gives 9 + 4 + 2 = 15, the same total reached without ever writing the grid.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Translating between layer pictures and a numbered top view.

💡Takeaway. Layers cut a stack sideways; the top view counts downwards. A cell's number is how many layers it shows up in.

Variant 12 hard answer: Row 1: 1, -, -, 3; row 2: -, 2, 1, 1; row 3: 3, 1, 2, -; row 4: -, -, 1, - (15 cubes in all)

The pictures show a stack of unit cubes, one layer at a time. Draw the view from above and write in each cell how many cubes are stacked there.

Show solution

1 · Understandwhat's really being asked

A stack of unit cubes is drawn one layer at a time. We want the view from above with each cell's number of cubes written in.

Givens

Layer 1 covers 9 cells.
Layer 2 covers 4 cells.
Layer 3 covers 2 cells.

Unknowns

The number of cubes at each cell of the top view.

Constraints

Every stack rests on the floor, so nothing floats above a gap.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #15 Organize Information in More Ways#2 Make a Systematic List

The layer pictures cut the stack sideways and the top view counts straight down, so the two are different slices of the same thing. Translate by asking, for each cell, how many layers it appears in.

3 · Execute4 carry out the plan

1Read layer 1 as the footprint

#17 Visualize Spatial Relationships 6.G.A.4

Every stack starts on the floor, so a cell holding any cubes at all appears in layer 1.

\text{layer 1} = 9\text{ cells}

That is the whole top view's shape.

2Count each cell's layers

#15 Organize Information in More Ways 6.G.A.4

A cell in all the layers has that many cubes; one in layer 1 alone has just the one.

\text{cell} = \text{layers it is in}

Now every cell has a number.

3Write them into the grid

#2 Make a Systematic List 6.G.A.4

Reading rows from the back, the numbers are as follows.

1 0 0 3 \quad 0 2 1 1 \quad 3 1 2 0 \quad 0 0 1 0

The top view is filled in.

4Check against the layer counts

#2 Make a Systematic List 6.G.A.4

The numbers should add to the same total as the layers do.

9 + 4 + 2 = 15

15 cubes either way.

Answer: Row 1: 1, -, -, 3; row 2: -, 2, 1, 1; row 3: 3, 1, 2, -; row 4: -, -, 1, - (15 cubes in all)

4 · Reviewdoes it hold up?

The largest number in the grid is 3, which is exactly how many layers were drawn -- as it has to be.

Another way: Counting the cubes layer by layer gives 9 + 4 + 2 = 15, the same total reached without ever writing the grid.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Translating between layer pictures and a numbered top view.

💡Takeaway. Layers cut a stack sideways; the top view counts downwards. A cell's number is how many layers it shows up in.