Practice · A view caps each line; the cube count settles the rest · Sensim Math

Variant 1 easy

The views from above and from the front are shown for a shape built from $16$ unit cubes. Draw the view from the side.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A stack of 16 unit cubes is shown from above and from the front. We want what it looks like from the side.

Givens

The view from above shows which cells hold a stack.
The view from the front is 2, 3, 2 cells high, left to right.
There are 16 cubes in all.

Unknowns

The view from the side.

Constraints

Every stack rests on the floor, so the footprint is the whole set of occupied cells.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #3 Eliminate Possibilities#2 Make a Systematic List

The side view cannot be got from the front view: a front view only gives the tallest stack in each column. Work out the individual heights first, using the count to settle what the views leave open, then read the side shadow off them.

3 · Execute3 carry out the plan

1Read what the front view fixes

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

It gives the tallest stack in each column and nothing more; a column showing 1 has every one of its stacks at 1.

\text{front} = 2, 3, 2

Some stacks are already forced.

2Use the count on what is left

#3 Eliminate Possibilities 6.G.A.4

The forced stacks account for part of the 16 cubes, and the rest has to fill the remaining cells without breaking any column's cap.

\text{total} = 16

Every stack is now known.

3Take the shadow from the side

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

Looking along the rows, each row shows only its tallest stack.

\text{side} = 3, 3, 2

Front to back, that is the side view.

Answer: Columns 3, 3 and 2 cells high, front to back

4 · Reviewdoes it hold up?

The tallest stack anywhere is 3, and both the front and the side views reach exactly that -- as any two views of the same stack must.

Another way: Writing the heights into the top view's cells and reading each row's largest is the same work with the numbers written down instead of held in mind.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Relating a stack of cubes to its views from three directions.

💡Takeaway. A view tells you the tallest in each line, not each stack. The cube count fills in the rest.

Variant 2 easy

The views from above and from the front are shown for a shape built from $9$ unit cubes. Draw the view from the side.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A stack of 9 unit cubes is shown from above and from the front. We want what it looks like from the side.

Givens

The view from above shows which cells hold a stack.
The view from the front is 1, 3, 2 cells high, left to right.
There are 9 cubes in all.

Unknowns

The view from the side.

Constraints

Every stack rests on the floor, so the footprint is the whole set of occupied cells.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #3 Eliminate Possibilities#2 Make a Systematic List

The side view cannot be got from the front view: a front view only gives the tallest stack in each column. Work out the individual heights first, using the count to settle what the views leave open, then read the side shadow off them.

3 · Execute3 carry out the plan

1Read what the front view fixes

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

It gives the tallest stack in each column and nothing more; a column showing 1 has every one of its stacks at 1.

\text{front} = 1, 3, 2

Some stacks are already forced.

2Use the count on what is left

#3 Eliminate Possibilities 6.G.A.4

The forced stacks account for part of the 9 cubes, and the rest has to fill the remaining cells without breaking any column's cap.

\text{total} = 9

Every stack is now known.

3Take the shadow from the side

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

Looking along the rows, each row shows only its tallest stack.

\text{side} = 3, 1, 3

Front to back, that is the side view.

Answer: Columns 3, 1 and 3 cells high, front to back

4 · Reviewdoes it hold up?

The tallest stack anywhere is 3, and both the front and the side views reach exactly that -- as any two views of the same stack must.

Another way: Writing the heights into the top view's cells and reading each row's largest is the same work with the numbers written down instead of held in mind.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Relating a stack of cubes to its views from three directions.

💡Takeaway. A view tells you the tallest in each line, not each stack. The cube count fills in the rest.

Variant 3 easy

The views from above and from the front are shown for a shape built from $10$ unit cubes. Draw the view from the side.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A stack of 10 unit cubes is shown from above and from the front. We want what it looks like from the side.

Givens

The view from above shows which cells hold a stack.
The view from the front is 1, 3, 2 cells high, left to right.
There are 10 cubes in all.

Unknowns

The view from the side.

Constraints

Every stack rests on the floor, so the footprint is the whole set of occupied cells.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #3 Eliminate Possibilities#2 Make a Systematic List

The side view cannot be got from the front view: a front view only gives the tallest stack in each column. Work out the individual heights first, using the count to settle what the views leave open, then read the side shadow off them.

3 · Execute3 carry out the plan

1Read what the front view fixes

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

It gives the tallest stack in each column and nothing more; a column showing 1 has every one of its stacks at 1.

\text{front} = 1, 3, 2

Some stacks are already forced.

2Use the count on what is left

#3 Eliminate Possibilities 6.G.A.4

The forced stacks account for part of the 10 cubes, and the rest has to fill the remaining cells without breaking any column's cap.

\text{total} = 10

Every stack is now known.

3Take the shadow from the side

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

Looking along the rows, each row shows only its tallest stack.

\text{side} = 1, 3, 2

Front to back, that is the side view.

Answer: Columns 1, 3 and 2 cells high, front to back

4 · Reviewdoes it hold up?

The tallest stack anywhere is 3, and both the front and the side views reach exactly that -- as any two views of the same stack must.

Another way: Writing the heights into the top view's cells and reading each row's largest is the same work with the numbers written down instead of held in mind.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Relating a stack of cubes to its views from three directions.

💡Takeaway. A view tells you the tallest in each line, not each stack. The cube count fills in the rest.

Variant 4 medium

The views from above and from the front are shown for a shape built from $10$ unit cubes. Draw the view from the side.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A stack of 10 unit cubes is shown from above and from the front. We want what it looks like from the side.

Givens

The view from above shows which cells hold a stack.
The view from the front is 3, 2, 3 cells high, left to right.
There are 10 cubes in all.

Unknowns

The view from the side.

Constraints

Every stack rests on the floor, so the footprint is the whole set of occupied cells.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #3 Eliminate Possibilities#2 Make a Systematic List

The side view cannot be got from the front view: a front view only gives the tallest stack in each column. Work out the individual heights first, using the count to settle what the views leave open, then read the side shadow off them.

3 · Execute3 carry out the plan

1Read what the front view fixes

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

It gives the tallest stack in each column and nothing more; a column showing 1 has every one of its stacks at 1.

\text{front} = 3, 2, 3

Some stacks are already forced.

2Use the count on what is left

#3 Eliminate Possibilities 6.G.A.4

The forced stacks account for part of the 10 cubes, and the rest has to fill the remaining cells without breaking any column's cap.

\text{total} = 10

Every stack is now known.

3Take the shadow from the side

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

Looking along the rows, each row shows only its tallest stack.

\text{side} = 3, 3, 2

Front to back, that is the side view.

Answer: Columns 3, 3 and 2 cells high, front to back

4 · Reviewdoes it hold up?

The tallest stack anywhere is 3, and both the front and the side views reach exactly that -- as any two views of the same stack must.

Another way: Writing the heights into the top view's cells and reading each row's largest is the same work with the numbers written down instead of held in mind.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Relating a stack of cubes to its views from three directions.

💡Takeaway. A view tells you the tallest in each line, not each stack. The cube count fills in the rest.

Variant 5 easy

The views from above and from the front are shown for a shape built from $10$ unit cubes. Draw the view from the side.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A stack of 10 unit cubes is shown from above and from the front. We want what it looks like from the side.

Givens

The view from above shows which cells hold a stack.
The view from the front is 3, 3, 2 cells high, left to right.
There are 10 cubes in all.

Unknowns

The view from the side.

Constraints

Every stack rests on the floor, so the footprint is the whole set of occupied cells.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #3 Eliminate Possibilities#2 Make a Systematic List

The side view cannot be got from the front view: a front view only gives the tallest stack in each column. Work out the individual heights first, using the count to settle what the views leave open, then read the side shadow off them.

3 · Execute3 carry out the plan

1Read what the front view fixes

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

It gives the tallest stack in each column and nothing more; a column showing 1 has every one of its stacks at 1.

\text{front} = 3, 3, 2

Some stacks are already forced.

2Use the count on what is left

#3 Eliminate Possibilities 6.G.A.4

The forced stacks account for part of the 10 cubes, and the rest has to fill the remaining cells without breaking any column's cap.

\text{total} = 10

Every stack is now known.

3Take the shadow from the side

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

Looking along the rows, each row shows only its tallest stack.

\text{side} = 3, 2, 3

Front to back, that is the side view.

Answer: Columns 3, 2 and 3 cells high, front to back

4 · Reviewdoes it hold up?

The tallest stack anywhere is 3, and both the front and the side views reach exactly that -- as any two views of the same stack must.

Another way: Writing the heights into the top view's cells and reading each row's largest is the same work with the numbers written down instead of held in mind.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Relating a stack of cubes to its views from three directions.

💡Takeaway. A view tells you the tallest in each line, not each stack. The cube count fills in the rest.

Variant 6 medium

The views from above and from the front are shown for a shape built from $12$ unit cubes. Draw the view from the side.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A stack of 12 unit cubes is shown from above and from the front. We want what it looks like from the side.

Givens

The view from above shows which cells hold a stack.
The view from the front is 2, 3, 1 cells high, left to right.
There are 12 cubes in all.

Unknowns

The view from the side.

Constraints

Every stack rests on the floor, so the footprint is the whole set of occupied cells.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #3 Eliminate Possibilities#2 Make a Systematic List

The side view cannot be got from the front view: a front view only gives the tallest stack in each column. Work out the individual heights first, using the count to settle what the views leave open, then read the side shadow off them.

3 · Execute3 carry out the plan

1Read what the front view fixes

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

It gives the tallest stack in each column and nothing more; a column showing 1 has every one of its stacks at 1.

\text{front} = 2, 3, 1

Some stacks are already forced.

2Use the count on what is left

#3 Eliminate Possibilities 6.G.A.4

The forced stacks account for part of the 12 cubes, and the rest has to fill the remaining cells without breaking any column's cap.

\text{total} = 12

Every stack is now known.

3Take the shadow from the side

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

Looking along the rows, each row shows only its tallest stack.

\text{side} = 1, 3, 3

Front to back, that is the side view.

Answer: Columns 1, 3 and 3 cells high, front to back

4 · Reviewdoes it hold up?

The tallest stack anywhere is 3, and both the front and the side views reach exactly that -- as any two views of the same stack must.

Another way: Writing the heights into the top view's cells and reading each row's largest is the same work with the numbers written down instead of held in mind.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Relating a stack of cubes to its views from three directions.

💡Takeaway. A view tells you the tallest in each line, not each stack. The cube count fills in the rest.

Variant 7 medium

The views from above and from the front are shown for a shape built from $14$ unit cubes. Draw the view from the side.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A stack of 14 unit cubes is shown from above and from the front. We want what it looks like from the side.

Givens

The view from above shows which cells hold a stack.
The view from the front is 3, 2, 2 cells high, left to right.
There are 14 cubes in all.

Unknowns

The view from the side.

Constraints

Every stack rests on the floor, so the footprint is the whole set of occupied cells.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #3 Eliminate Possibilities#2 Make a Systematic List

The side view cannot be got from the front view: a front view only gives the tallest stack in each column. Work out the individual heights first, using the count to settle what the views leave open, then read the side shadow off them.

3 · Execute3 carry out the plan

1Read what the front view fixes

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

It gives the tallest stack in each column and nothing more; a column showing 1 has every one of its stacks at 1.

\text{front} = 3, 2, 2

Some stacks are already forced.

2Use the count on what is left

#3 Eliminate Possibilities 6.G.A.4

The forced stacks account for part of the 14 cubes, and the rest has to fill the remaining cells without breaking any column's cap.

\text{total} = 14

Every stack is now known.

3Take the shadow from the side

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

Looking along the rows, each row shows only its tallest stack.

\text{side} = 3, 2, 3

Front to back, that is the side view.

Answer: Columns 3, 2 and 3 cells high, front to back

4 · Reviewdoes it hold up?

The tallest stack anywhere is 3, and both the front and the side views reach exactly that -- as any two views of the same stack must.

Another way: Writing the heights into the top view's cells and reading each row's largest is the same work with the numbers written down instead of held in mind.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Relating a stack of cubes to its views from three directions.

💡Takeaway. A view tells you the tallest in each line, not each stack. The cube count fills in the rest.

Variant 8 hard

The views from above and from the front are shown for a shape built from $9$ unit cubes. Draw the view from the side.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A stack of 9 unit cubes is shown from above and from the front. We want what it looks like from the side.

Givens

The view from above shows which cells hold a stack.
The view from the front is 1, 2, 1 cells high, left to right.
There are 9 cubes in all.

Unknowns

The view from the side.

Constraints

Every stack rests on the floor, so the footprint is the whole set of occupied cells.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #3 Eliminate Possibilities#2 Make a Systematic List

The side view cannot be got from the front view: a front view only gives the tallest stack in each column. Work out the individual heights first, using the count to settle what the views leave open, then read the side shadow off them.

3 · Execute3 carry out the plan

1Read what the front view fixes

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

It gives the tallest stack in each column and nothing more; a column showing 1 has every one of its stacks at 1.

\text{front} = 1, 2, 1

Some stacks are already forced.

2Use the count on what is left

#3 Eliminate Possibilities 6.G.A.4

The forced stacks account for part of the 9 cubes, and the rest has to fill the remaining cells without breaking any column's cap.

\text{total} = 9

Every stack is now known.

3Take the shadow from the side

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

Looking along the rows, each row shows only its tallest stack.

\text{side} = 2, 1, 2

Front to back, that is the side view.

Answer: Columns 2, 1 and 2 cells high, front to back

4 · Reviewdoes it hold up?

The tallest stack anywhere is 2, and both the front and the side views reach exactly that -- as any two views of the same stack must.

Another way: Writing the heights into the top view's cells and reading each row's largest is the same work with the numbers written down instead of held in mind.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Relating a stack of cubes to its views from three directions.

💡Takeaway. A view tells you the tallest in each line, not each stack. The cube count fills in the rest.

Variant 9 medium

The views from above and from the front are shown for a shape built from $9$ unit cubes. Draw the view from the side.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A stack of 9 unit cubes is shown from above and from the front. We want what it looks like from the side.

Givens

The view from above shows which cells hold a stack.
The view from the front is 2, 1, 1 cells high, left to right.
There are 9 cubes in all.

Unknowns

The view from the side.

Constraints

Every stack rests on the floor, so the footprint is the whole set of occupied cells.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #3 Eliminate Possibilities#2 Make a Systematic List

The side view cannot be got from the front view: a front view only gives the tallest stack in each column. Work out the individual heights first, using the count to settle what the views leave open, then read the side shadow off them.

3 · Execute3 carry out the plan

1Read what the front view fixes

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

It gives the tallest stack in each column and nothing more; a column showing 1 has every one of its stacks at 1.

\text{front} = 2, 1, 1

Some stacks are already forced.

2Use the count on what is left

#3 Eliminate Possibilities 6.G.A.4

The forced stacks account for part of the 9 cubes, and the rest has to fill the remaining cells without breaking any column's cap.

\text{total} = 9

Every stack is now known.

3Take the shadow from the side

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

Looking along the rows, each row shows only its tallest stack.

\text{side} = 2, 1, 2

Front to back, that is the side view.

Answer: Columns 2, 1 and 2 cells high, front to back

4 · Reviewdoes it hold up?

The tallest stack anywhere is 2, and both the front and the side views reach exactly that -- as any two views of the same stack must.

Another way: Writing the heights into the top view's cells and reading each row's largest is the same work with the numbers written down instead of held in mind.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Relating a stack of cubes to its views from three directions.

💡Takeaway. A view tells you the tallest in each line, not each stack. The cube count fills in the rest.

Variant 10 hard

The views from above and from the front are shown for a shape built from $18$ unit cubes. Draw the view from the side.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A stack of 18 unit cubes is shown from above and from the front. We want what it looks like from the side.

Givens

The view from above shows which cells hold a stack.
The view from the front is 1, 3, 3 cells high, left to right.
There are 18 cubes in all.

Unknowns

The view from the side.

Constraints

Every stack rests on the floor, so the footprint is the whole set of occupied cells.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #3 Eliminate Possibilities#2 Make a Systematic List

The side view cannot be got from the front view: a front view only gives the tallest stack in each column. Work out the individual heights first, using the count to settle what the views leave open, then read the side shadow off them.

3 · Execute3 carry out the plan

1Read what the front view fixes

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

It gives the tallest stack in each column and nothing more; a column showing 1 has every one of its stacks at 1.

\text{front} = 1, 3, 3

Some stacks are already forced.

2Use the count on what is left

#3 Eliminate Possibilities 6.G.A.4

The forced stacks account for part of the 18 cubes, and the rest has to fill the remaining cells without breaking any column's cap.

\text{total} = 18

Every stack is now known.

3Take the shadow from the side

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

Looking along the rows, each row shows only its tallest stack.

\text{side} = 3, 3, 3

Front to back, that is the side view.

Answer: Columns 3, 3 and 3 cells high, front to back

4 · Reviewdoes it hold up?

The tallest stack anywhere is 3, and both the front and the side views reach exactly that -- as any two views of the same stack must.

Another way: Writing the heights into the top view's cells and reading each row's largest is the same work with the numbers written down instead of held in mind.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Relating a stack of cubes to its views from three directions.

💡Takeaway. A view tells you the tallest in each line, not each stack. The cube count fills in the rest.

Variant 11 hard

The views from above and from the front are shown for a shape built from $12$ unit cubes. Draw the view from the side.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A stack of 12 unit cubes is shown from above and from the front. We want what it looks like from the side.

Givens

The view from above shows which cells hold a stack.
The view from the front is 3, 1, 2 cells high, left to right.
There are 12 cubes in all.

Unknowns

The view from the side.

Constraints

Every stack rests on the floor, so the footprint is the whole set of occupied cells.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #3 Eliminate Possibilities#2 Make a Systematic List

The side view cannot be got from the front view: a front view only gives the tallest stack in each column. Work out the individual heights first, using the count to settle what the views leave open, then read the side shadow off them.

3 · Execute3 carry out the plan

1Read what the front view fixes

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

It gives the tallest stack in each column and nothing more; a column showing 1 has every one of its stacks at 1.

\text{front} = 3, 1, 2

Some stacks are already forced.

2Use the count on what is left

#3 Eliminate Possibilities 6.G.A.4

The forced stacks account for part of the 12 cubes, and the rest has to fill the remaining cells without breaking any column's cap.

\text{total} = 12

Every stack is now known.

3Take the shadow from the side

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

Looking along the rows, each row shows only its tallest stack.

\text{side} = 3, 3, 3

Front to back, that is the side view.

Answer: Columns 3, 3 and 3 cells high, front to back

4 · Reviewdoes it hold up?

The tallest stack anywhere is 3, and both the front and the side views reach exactly that -- as any two views of the same stack must.

Another way: Writing the heights into the top view's cells and reading each row's largest is the same work with the numbers written down instead of held in mind.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Relating a stack of cubes to its views from three directions.

💡Takeaway. A view tells you the tallest in each line, not each stack. The cube count fills in the rest.

Variant 12 hard

The views from above and from the front are shown for a shape built from $17$ unit cubes. Draw the view from the side.

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A stack of 17 unit cubes is shown from above and from the front. We want what it looks like from the side.

Givens

The view from above shows which cells hold a stack.
The view from the front is 3, 2, 3 cells high, left to right.
There are 17 cubes in all.

Unknowns

The view from the side.

Constraints

Every stack rests on the floor, so the footprint is the whole set of occupied cells.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #3 Eliminate Possibilities#2 Make a Systematic List

The side view cannot be got from the front view: a front view only gives the tallest stack in each column. Work out the individual heights first, using the count to settle what the views leave open, then read the side shadow off them.

3 · Execute3 carry out the plan

1Read what the front view fixes

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

It gives the tallest stack in each column and nothing more; a column showing 1 has every one of its stacks at 1.

\text{front} = 3, 2, 3

Some stacks are already forced.

2Use the count on what is left

#3 Eliminate Possibilities 6.G.A.4

The forced stacks account for part of the 17 cubes, and the rest has to fill the remaining cells without breaking any column's cap.

\text{total} = 17

Every stack is now known.

3Take the shadow from the side

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

Looking along the rows, each row shows only its tallest stack.

\text{side} = 3, 3, 3

Front to back, that is the side view.

Answer: Columns 3, 3 and 3 cells high, front to back

4 · Reviewdoes it hold up?

The tallest stack anywhere is 3, and both the front and the side views reach exactly that -- as any two views of the same stack must.

Another way: Writing the heights into the top view's cells and reading each row's largest is the same work with the numbers written down instead of held in mind.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Relating a stack of cubes to its views from three directions.

💡Takeaway. A view tells you the tallest in each line, not each stack. The cube count fills in the rest.

A view caps each line; the cube count settles the rest · 12 practice problems

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