← Write the heights down, take the cubes off there, then read the views · Cube Stacks and Their Views

Write the heights down, take the cubes off there, then read the views · 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: Front: columns 1, 2 and 1 high. Side: columns 1, 1 and 2 high, front to back.

A shape is built from 1010 unit cubes. After the 33 red cubes are taken away, draw what it looks like from the front and from the side.

seen from above 2 3 1 1 2 1 front side front
Show solution
1 · Understandwhat's really being asked

A stack of 10 cubes loses 3 of them, each from the top of its own stack. We want the front and side views of what is left.

Givens
  • The stack starts with 10 unit cubes.
  • 3 cubes are removed, each the top of its stack.
Unknowns
  • The views from the front and from the side afterwards.
Constraints
  • Only top cubes are removed, so nothing is left floating.
2 · Planchoose the strategy

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

Redrawing from the picture loses the cubes hidden behind others. Write the heights into a top-view grid instead, do the removal there, and each view is then the largest number along a line.

3 · Execute4 carry out the plan

1Take the cubes off the grid

#15 Organize Information in More Ways 6.G.A.4
Each removed cube is the top of its stack, so its cell drops by one.
10−3=710 - 3 = 7
7 cubes left.

2Read the front view

#17 Visualize Spatial Relationships 6.G.A.4
Looking from the front, each column shows only its tallest stack.
front=1,2,1\text{front} = 1, 2, 1
Left to right.

3Read the side view

#17 Visualize Spatial Relationships 6.G.A.4
Looking from the side, each row shows only its tallest stack, front row first.
side=1,1,2\text{side} = 1, 1, 2
Front to back.

4Check the two agree

#2 Make a Systematic List 6.G.A.4
Both views must reach the same tallest stack.
max⁡=2\max = 2
They do.
Answer: Front: columns 1, 2 and 1 high. Side: columns 1, 1 and 2 high, front to back.
4 · Reviewdoes it hold up?

The heights left add to 7, which is the 10 we started with less the 3 removed.

Another way: Redrawing the solid and looking at it gives the same views, if no cube hidden behind another is forgotten -- which is the part the grid makes safe.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Taking the front and side views of a stack of cubes.
💡Takeaway. Write the heights down before you change anything. Then each view is just the biggest number in its line.
Variant 2 medium answer: Front: columns 2, 2 and 1 high. Side: columns 1, 1 and 2 high, front to back.

A shape is built from 1111 unit cubes. After the 33 red cubes are taken away, draw what it looks like from the front and from the side.

seen from above 2 3 2 1 2 1 front side front
Show solution
1 · Understandwhat's really being asked

A stack of 11 cubes loses 3 of them, each from the top of its own stack. We want the front and side views of what is left.

Givens
  • The stack starts with 11 unit cubes.
  • 3 cubes are removed, each the top of its stack.
Unknowns
  • The views from the front and from the side afterwards.
Constraints
  • Only top cubes are removed, so nothing is left floating.
2 · Planchoose the strategy

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

Redrawing from the picture loses the cubes hidden behind others. Write the heights into a top-view grid instead, do the removal there, and each view is then the largest number along a line.

3 · Execute4 carry out the plan

1Take the cubes off the grid

#15 Organize Information in More Ways 6.G.A.4
Each removed cube is the top of its stack, so its cell drops by one.
11−3=811 - 3 = 8
8 cubes left.

2Read the front view

#17 Visualize Spatial Relationships 6.G.A.4
Looking from the front, each column shows only its tallest stack.
front=2,2,1\text{front} = 2, 2, 1
Left to right.

3Read the side view

#17 Visualize Spatial Relationships 6.G.A.4
Looking from the side, each row shows only its tallest stack, front row first.
side=1,1,2\text{side} = 1, 1, 2
Front to back.

4Check the two agree

#2 Make a Systematic List 6.G.A.4
Both views must reach the same tallest stack.
max⁡=2\max = 2
They do.
Answer: Front: columns 2, 2 and 1 high. Side: columns 1, 1 and 2 high, front to back.
4 · Reviewdoes it hold up?

The heights left add to 8, which is the 11 we started with less the 3 removed.

Another way: Redrawing the solid and looking at it gives the same views, if no cube hidden behind another is forgotten -- which is the part the grid makes safe.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Taking the front and side views of a stack of cubes.
💡Takeaway. Write the heights down before you change anything. Then each view is just the biggest number in its line.
Variant 3 easy answer: Front: columns 2, 1 and 1 high. Side: columns 1, 2 and 2 high, front to back.

A shape is built from 1111 unit cubes. After the 33 red cubes are taken away, draw what it looks like from the front and from the side.

seen from above 3 1 2 2 2 1 front side front
Show solution
1 · Understandwhat's really being asked

A stack of 11 cubes loses 3 of them, each from the top of its own stack. We want the front and side views of what is left.

Givens
  • The stack starts with 11 unit cubes.
  • 3 cubes are removed, each the top of its stack.
Unknowns
  • The views from the front and from the side afterwards.
Constraints
  • Only top cubes are removed, so nothing is left floating.
2 · Planchoose the strategy

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

Redrawing from the picture loses the cubes hidden behind others. Write the heights into a top-view grid instead, do the removal there, and each view is then the largest number along a line.

3 · Execute4 carry out the plan

1Take the cubes off the grid

#15 Organize Information in More Ways 6.G.A.4
Each removed cube is the top of its stack, so its cell drops by one.
11−3=811 - 3 = 8
8 cubes left.

2Read the front view

#17 Visualize Spatial Relationships 6.G.A.4
Looking from the front, each column shows only its tallest stack.
front=2,1,1\text{front} = 2, 1, 1
Left to right.

3Read the side view

#17 Visualize Spatial Relationships 6.G.A.4
Looking from the side, each row shows only its tallest stack, front row first.
side=1,2,2\text{side} = 1, 2, 2
Front to back.

4Check the two agree

#2 Make a Systematic List 6.G.A.4
Both views must reach the same tallest stack.
max⁡=2\max = 2
They do.
Answer: Front: columns 2, 1 and 1 high. Side: columns 1, 2 and 2 high, front to back.
4 · Reviewdoes it hold up?

The heights left add to 8, which is the 11 we started with less the 3 removed.

Another way: Redrawing the solid and looking at it gives the same views, if no cube hidden behind another is forgotten -- which is the part the grid makes safe.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Taking the front and side views of a stack of cubes.
💡Takeaway. Write the heights down before you change anything. Then each view is just the biggest number in its line.
Variant 4 easy answer: Front: columns 3, 2 and 0 high. Side: columns 1, 1 and 3 high, front to back.

A shape is built from 1111 unit cubes. After the 33 red cubes are taken away, draw what it looks like from the front and from the side.

seen from above 3 3 1 2 1 1 front side front
Show solution
1 · Understandwhat's really being asked

A stack of 11 cubes loses 3 of them, each from the top of its own stack. We want the front and side views of what is left.

Givens
  • The stack starts with 11 unit cubes.
  • 3 cubes are removed, each the top of its stack.
Unknowns
  • The views from the front and from the side afterwards.
Constraints
  • Only top cubes are removed, so nothing is left floating.
2 · Planchoose the strategy

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

Redrawing from the picture loses the cubes hidden behind others. Write the heights into a top-view grid instead, do the removal there, and each view is then the largest number along a line.

3 · Execute4 carry out the plan

1Take the cubes off the grid

#15 Organize Information in More Ways 6.G.A.4
Each removed cube is the top of its stack, so its cell drops by one.
11−3=811 - 3 = 8
8 cubes left.

2Read the front view

#17 Visualize Spatial Relationships 6.G.A.4
Looking from the front, each column shows only its tallest stack.
front=3,2,0\text{front} = 3, 2, 0
Left to right.

3Read the side view

#17 Visualize Spatial Relationships 6.G.A.4
Looking from the side, each row shows only its tallest stack, front row first.
side=1,1,3\text{side} = 1, 1, 3
Front to back.

4Check the two agree

#2 Make a Systematic List 6.G.A.4
Both views must reach the same tallest stack.
max⁡=3\max = 3
They do.
Answer: Front: columns 3, 2 and 0 high. Side: columns 1, 1 and 3 high, front to back.
4 · Reviewdoes it hold up?

The heights left add to 8, which is the 11 we started with less the 3 removed.

Another way: Redrawing the solid and looking at it gives the same views, if no cube hidden behind another is forgotten -- which is the part the grid makes safe.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Taking the front and side views of a stack of cubes.
💡Takeaway. Write the heights down before you change anything. Then each view is just the biggest number in its line.
Variant 5 medium answer: Front: columns 2, 2 and 1 high. Side: columns 1, 1 and 2 high, front to back.

A shape is built from 1111 unit cubes. After the 33 red cubes are taken away, draw what it looks like from the front and from the side.

seen from above 3 3 2 1 1 1 front side front
Show solution
1 · Understandwhat's really being asked

A stack of 11 cubes loses 3 of them, each from the top of its own stack. We want the front and side views of what is left.

Givens
  • The stack starts with 11 unit cubes.
  • 3 cubes are removed, each the top of its stack.
Unknowns
  • The views from the front and from the side afterwards.
Constraints
  • Only top cubes are removed, so nothing is left floating.
2 · Planchoose the strategy

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

Redrawing from the picture loses the cubes hidden behind others. Write the heights into a top-view grid instead, do the removal there, and each view is then the largest number along a line.

3 · Execute4 carry out the plan

1Take the cubes off the grid

#15 Organize Information in More Ways 6.G.A.4
Each removed cube is the top of its stack, so its cell drops by one.
11−3=811 - 3 = 8
8 cubes left.

2Read the front view

#17 Visualize Spatial Relationships 6.G.A.4
Looking from the front, each column shows only its tallest stack.
front=2,2,1\text{front} = 2, 2, 1
Left to right.

3Read the side view

#17 Visualize Spatial Relationships 6.G.A.4
Looking from the side, each row shows only its tallest stack, front row first.
side=1,1,2\text{side} = 1, 1, 2
Front to back.

4Check the two agree

#2 Make a Systematic List 6.G.A.4
Both views must reach the same tallest stack.
max⁡=2\max = 2
They do.
Answer: Front: columns 2, 2 and 1 high. Side: columns 1, 1 and 2 high, front to back.
4 · Reviewdoes it hold up?

The heights left add to 8, which is the 11 we started with less the 3 removed.

Another way: Redrawing the solid and looking at it gives the same views, if no cube hidden behind another is forgotten -- which is the part the grid makes safe.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Taking the front and side views of a stack of cubes.
💡Takeaway. Write the heights down before you change anything. Then each view is just the biggest number in its line.
Variant 6 easy answer: Front: columns 2, 2 and 1 high. Side: columns 1, 1 and 2 high, front to back.

A shape is built from 1111 unit cubes. After the 33 red cubes are taken away, draw what it looks like from the front and from the side.

seen from above 3 2 1 2 2 1 front side front
Show solution
1 · Understandwhat's really being asked

A stack of 11 cubes loses 3 of them, each from the top of its own stack. We want the front and side views of what is left.

Givens
  • The stack starts with 11 unit cubes.
  • 3 cubes are removed, each the top of its stack.
Unknowns
  • The views from the front and from the side afterwards.
Constraints
  • Only top cubes are removed, so nothing is left floating.
2 · Planchoose the strategy

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

Redrawing from the picture loses the cubes hidden behind others. Write the heights into a top-view grid instead, do the removal there, and each view is then the largest number along a line.

3 · Execute4 carry out the plan

1Take the cubes off the grid

#15 Organize Information in More Ways 6.G.A.4
Each removed cube is the top of its stack, so its cell drops by one.
11−3=811 - 3 = 8
8 cubes left.

2Read the front view

#17 Visualize Spatial Relationships 6.G.A.4
Looking from the front, each column shows only its tallest stack.
front=2,2,1\text{front} = 2, 2, 1
Left to right.

3Read the side view

#17 Visualize Spatial Relationships 6.G.A.4
Looking from the side, each row shows only its tallest stack, front row first.
side=1,1,2\text{side} = 1, 1, 2
Front to back.

4Check the two agree

#2 Make a Systematic List 6.G.A.4
Both views must reach the same tallest stack.
max⁡=2\max = 2
They do.
Answer: Front: columns 2, 2 and 1 high. Side: columns 1, 1 and 2 high, front to back.
4 · Reviewdoes it hold up?

The heights left add to 8, which is the 11 we started with less the 3 removed.

Another way: Redrawing the solid and looking at it gives the same views, if no cube hidden behind another is forgotten -- which is the part the grid makes safe.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Taking the front and side views of a stack of cubes.
💡Takeaway. Write the heights down before you change anything. Then each view is just the biggest number in its line.
Variant 7 medium answer: Front: columns 2, 2 and 1 high. Side: columns 1, 2 and 2 high, front to back.

A shape is built from 1111 unit cubes. After the 33 red cubes are taken away, draw what it looks like from the front and from the side.

seen from above 2 2 2 3 1 1 front side front
Show solution
1 · Understandwhat's really being asked

A stack of 11 cubes loses 3 of them, each from the top of its own stack. We want the front and side views of what is left.

Givens
  • The stack starts with 11 unit cubes.
  • 3 cubes are removed, each the top of its stack.
Unknowns
  • The views from the front and from the side afterwards.
Constraints
  • Only top cubes are removed, so nothing is left floating.
2 · Planchoose the strategy

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

Redrawing from the picture loses the cubes hidden behind others. Write the heights into a top-view grid instead, do the removal there, and each view is then the largest number along a line.

3 · Execute4 carry out the plan

1Take the cubes off the grid

#15 Organize Information in More Ways 6.G.A.4
Each removed cube is the top of its stack, so its cell drops by one.
11−3=811 - 3 = 8
8 cubes left.

2Read the front view

#17 Visualize Spatial Relationships 6.G.A.4
Looking from the front, each column shows only its tallest stack.
front=2,2,1\text{front} = 2, 2, 1
Left to right.

3Read the side view

#17 Visualize Spatial Relationships 6.G.A.4
Looking from the side, each row shows only its tallest stack, front row first.
side=1,2,2\text{side} = 1, 2, 2
Front to back.

4Check the two agree

#2 Make a Systematic List 6.G.A.4
Both views must reach the same tallest stack.
max⁡=2\max = 2
They do.
Answer: Front: columns 2, 2 and 1 high. Side: columns 1, 2 and 2 high, front to back.
4 · Reviewdoes it hold up?

The heights left add to 8, which is the 11 we started with less the 3 removed.

Another way: Redrawing the solid and looking at it gives the same views, if no cube hidden behind another is forgotten -- which is the part the grid makes safe.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Taking the front and side views of a stack of cubes.
💡Takeaway. Write the heights down before you change anything. Then each view is just the biggest number in its line.
Variant 8 hard answer: Front: columns 2, 1 and 1 high. Side: columns 1, 1 and 2 high, front to back.

A shape is built from 1111 unit cubes. After the 33 red cubes are taken away, draw what it looks like from the front and from the side.

seen from above 3 2 1 1 1 2 1 front side front
Show solution
1 · Understandwhat's really being asked

A stack of 11 cubes loses 3 of them, each from the top of its own stack. We want the front and side views of what is left.

Givens
  • The stack starts with 11 unit cubes.
  • 3 cubes are removed, each the top of its stack.
Unknowns
  • The views from the front and from the side afterwards.
Constraints
  • Only top cubes are removed, so nothing is left floating.
2 · Planchoose the strategy

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

Redrawing from the picture loses the cubes hidden behind others. Write the heights into a top-view grid instead, do the removal there, and each view is then the largest number along a line.

3 · Execute4 carry out the plan

1Take the cubes off the grid

#15 Organize Information in More Ways 6.G.A.4
Each removed cube is the top of its stack, so its cell drops by one.
11−3=811 - 3 = 8
8 cubes left.

2Read the front view

#17 Visualize Spatial Relationships 6.G.A.4
Looking from the front, each column shows only its tallest stack.
front=2,1,1\text{front} = 2, 1, 1
Left to right.

3Read the side view

#17 Visualize Spatial Relationships 6.G.A.4
Looking from the side, each row shows only its tallest stack, front row first.
side=1,1,2\text{side} = 1, 1, 2
Front to back.

4Check the two agree

#2 Make a Systematic List 6.G.A.4
Both views must reach the same tallest stack.
max⁡=2\max = 2
They do.
Answer: Front: columns 2, 1 and 1 high. Side: columns 1, 1 and 2 high, front to back.
4 · Reviewdoes it hold up?

The heights left add to 8, which is the 11 we started with less the 3 removed.

Another way: Redrawing the solid and looking at it gives the same views, if no cube hidden behind another is forgotten -- which is the part the grid makes safe.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Taking the front and side views of a stack of cubes.
💡Takeaway. Write the heights down before you change anything. Then each view is just the biggest number in its line.
Variant 9 hard answer: Front: columns 2, 2 and 1 high. Side: columns 0, 2 and 2 high, front to back.

A shape is built from 1111 unit cubes. After the 33 red cubes are taken away, draw what it looks like from the front and from the side.

seen from above 3 1 1 1 3 1 1 front side front
Show solution
1 · Understandwhat's really being asked

A stack of 11 cubes loses 3 of them, each from the top of its own stack. We want the front and side views of what is left.

Givens
  • The stack starts with 11 unit cubes.
  • 3 cubes are removed, each the top of its stack.
Unknowns
  • The views from the front and from the side afterwards.
Constraints
  • Only top cubes are removed, so nothing is left floating.
2 · Planchoose the strategy

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

Redrawing from the picture loses the cubes hidden behind others. Write the heights into a top-view grid instead, do the removal there, and each view is then the largest number along a line.

3 · Execute4 carry out the plan

1Take the cubes off the grid

#15 Organize Information in More Ways 6.G.A.4
Each removed cube is the top of its stack, so its cell drops by one.
11−3=811 - 3 = 8
8 cubes left.

2Read the front view

#17 Visualize Spatial Relationships 6.G.A.4
Looking from the front, each column shows only its tallest stack.
front=2,2,1\text{front} = 2, 2, 1
Left to right.

3Read the side view

#17 Visualize Spatial Relationships 6.G.A.4
Looking from the side, each row shows only its tallest stack, front row first.
side=0,2,2\text{side} = 0, 2, 2
Front to back.

4Check the two agree

#2 Make a Systematic List 6.G.A.4
Both views must reach the same tallest stack.
max⁡=2\max = 2
They do.
Answer: Front: columns 2, 2 and 1 high. Side: columns 0, 2 and 2 high, front to back.
4 · Reviewdoes it hold up?

The heights left add to 8, which is the 11 we started with less the 3 removed.

Another way: Redrawing the solid and looking at it gives the same views, if no cube hidden behind another is forgotten -- which is the part the grid makes safe.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Taking the front and side views of a stack of cubes.
💡Takeaway. Write the heights down before you change anything. Then each view is just the biggest number in its line.
Variant 10 hard answer: Front: columns 1, 1 and 2 high. Side: columns 1, 1 and 2 high, front to back.

A shape is built from 1111 unit cubes. After the 33 red cubes are taken away, draw what it looks like from the front and from the side.

seen from above 1 2 3 2 1 1 1 front side front
Show solution
1 · Understandwhat's really being asked

A stack of 11 cubes loses 3 of them, each from the top of its own stack. We want the front and side views of what is left.

Givens
  • The stack starts with 11 unit cubes.
  • 3 cubes are removed, each the top of its stack.
Unknowns
  • The views from the front and from the side afterwards.
Constraints
  • Only top cubes are removed, so nothing is left floating.
2 · Planchoose the strategy

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

Redrawing from the picture loses the cubes hidden behind others. Write the heights into a top-view grid instead, do the removal there, and each view is then the largest number along a line.

3 · Execute4 carry out the plan

1Take the cubes off the grid

#15 Organize Information in More Ways 6.G.A.4
Each removed cube is the top of its stack, so its cell drops by one.
11−3=811 - 3 = 8
8 cubes left.

2Read the front view

#17 Visualize Spatial Relationships 6.G.A.4
Looking from the front, each column shows only its tallest stack.
front=1,1,2\text{front} = 1, 1, 2
Left to right.

3Read the side view

#17 Visualize Spatial Relationships 6.G.A.4
Looking from the side, each row shows only its tallest stack, front row first.
side=1,1,2\text{side} = 1, 1, 2
Front to back.

4Check the two agree

#2 Make a Systematic List 6.G.A.4
Both views must reach the same tallest stack.
max⁡=2\max = 2
They do.
Answer: Front: columns 1, 1 and 2 high. Side: columns 1, 1 and 2 high, front to back.
4 · Reviewdoes it hold up?

The heights left add to 8, which is the 11 we started with less the 3 removed.

Another way: Redrawing the solid and looking at it gives the same views, if no cube hidden behind another is forgotten -- which is the part the grid makes safe.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Taking the front and side views of a stack of cubes.
💡Takeaway. Write the heights down before you change anything. Then each view is just the biggest number in its line.
Variant 11 medium answer: Front: columns 1, 2 and 1 high. Side: columns 1, 1 and 2 high, front to back.

A shape is built from 1111 unit cubes. After the 33 red cubes are taken away, draw what it looks like from the front and from the side.

seen from above 1 3 2 2 1 1 1 front side front
Show solution
1 · Understandwhat's really being asked

A stack of 11 cubes loses 3 of them, each from the top of its own stack. We want the front and side views of what is left.

Givens
  • The stack starts with 11 unit cubes.
  • 3 cubes are removed, each the top of its stack.
Unknowns
  • The views from the front and from the side afterwards.
Constraints
  • Only top cubes are removed, so nothing is left floating.
2 · Planchoose the strategy

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

Redrawing from the picture loses the cubes hidden behind others. Write the heights into a top-view grid instead, do the removal there, and each view is then the largest number along a line.

3 · Execute4 carry out the plan

1Take the cubes off the grid

#15 Organize Information in More Ways 6.G.A.4
Each removed cube is the top of its stack, so its cell drops by one.
11−3=811 - 3 = 8
8 cubes left.

2Read the front view

#17 Visualize Spatial Relationships 6.G.A.4
Looking from the front, each column shows only its tallest stack.
front=1,2,1\text{front} = 1, 2, 1
Left to right.

3Read the side view

#17 Visualize Spatial Relationships 6.G.A.4
Looking from the side, each row shows only its tallest stack, front row first.
side=1,1,2\text{side} = 1, 1, 2
Front to back.

4Check the two agree

#2 Make a Systematic List 6.G.A.4
Both views must reach the same tallest stack.
max⁡=2\max = 2
They do.
Answer: Front: columns 1, 2 and 1 high. Side: columns 1, 1 and 2 high, front to back.
4 · Reviewdoes it hold up?

The heights left add to 8, which is the 11 we started with less the 3 removed.

Another way: Redrawing the solid and looking at it gives the same views, if no cube hidden behind another is forgotten -- which is the part the grid makes safe.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Taking the front and side views of a stack of cubes.
💡Takeaway. Write the heights down before you change anything. Then each view is just the biggest number in its line.
Variant 12 hard answer: Front: columns 1, 2 and 2 high. Side: columns 1, 2 and 2 high, front to back.

A shape is built from 1212 unit cubes. After the 33 red cubes are taken away, draw what it looks like from the front and from the side.

seen from above 2 1 3 1 3 1 1 front side front
Show solution
1 · Understandwhat's really being asked

A stack of 12 cubes loses 3 of them, each from the top of its own stack. We want the front and side views of what is left.

Givens
  • The stack starts with 12 unit cubes.
  • 3 cubes are removed, each the top of its stack.
Unknowns
  • The views from the front and from the side afterwards.
Constraints
  • Only top cubes are removed, so nothing is left floating.
2 · Planchoose the strategy

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

Redrawing from the picture loses the cubes hidden behind others. Write the heights into a top-view grid instead, do the removal there, and each view is then the largest number along a line.

3 · Execute4 carry out the plan

1Take the cubes off the grid

#15 Organize Information in More Ways 6.G.A.4
Each removed cube is the top of its stack, so its cell drops by one.
12−3=912 - 3 = 9
9 cubes left.

2Read the front view

#17 Visualize Spatial Relationships 6.G.A.4
Looking from the front, each column shows only its tallest stack.
front=1,2,2\text{front} = 1, 2, 2
Left to right.

3Read the side view

#17 Visualize Spatial Relationships 6.G.A.4
Looking from the side, each row shows only its tallest stack, front row first.
side=1,2,2\text{side} = 1, 2, 2
Front to back.

4Check the two agree

#2 Make a Systematic List 6.G.A.4
Both views must reach the same tallest stack.
max⁡=2\max = 2
They do.
Answer: Front: columns 1, 2 and 2 high. Side: columns 1, 2 and 2 high, front to back.
4 · Reviewdoes it hold up?

The heights left add to 9, which is the 12 we started with less the 3 removed.

Another way: Redrawing the solid and looking at it gives the same views, if no cube hidden behind another is forgotten -- which is the part the grid makes safe.

Standardsmin grade 6
  • 6.G.A.4 Surface area using nets of 3D figures — Taking the front and side views of a stack of cubes.
💡Takeaway. Write the heights down before you change anything. Then each view is just the biggest number in its line.