← The longest side decides the box, not the cube count · Cube Stacks and Their Views

The longest side decides the box, not the cube count · 12 practice problems

6.G.A.26.NS.B.4

Generated variants — 12

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

Variant 1 easy answer: 18 cubes

More unit cubes are to be added to the shape at the right to build the smallest possible cube. How many more cubes are needed?

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

A stack of 9 unit cubes is to be grown into the smallest cube that will contain it. We want how many cubes must be added.

Givens
  • The footprint is 3 cells deep and 3 across.
  • The tallest stack is 3 cubes.
  • There are 9 cubes altogether.
Unknowns
  • How many more unit cubes are needed.
Constraints
  • The finished solid must be a cube and must contain the shape as it stands.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#16 Count the Complement

The cube count is a distraction: it says how much is there, not how big the box must be. Measure the shape in each direction, let the longest set the edge, and subtract.

3 · Execute4 carry out the plan

1Measure the shape three ways

#17 Visualize Spatial Relationships 6.G.A.2
It is 3 across, 3 deep and 3 tall.
3, 3, 33,\ 3,\ 3
Three numbers, one of them largest.

2Let the longest set the cube's edge

#16 Count the Complement 6.NS.B.4
A smaller cube could not contain the shape's longest direction, so this is the smallest one that works.
edge=3\text{edge} = 3
A 3 by 3 by 3 cube.

3Count what that cube holds

#7 Identify Subproblems 6.G.A.2
Edge times edge times edge.
3×3×3=273 \times 3 \times 3 = 27
27 cubes in the finished solid.

4Subtract what is already there

#7 Identify Subproblems 6.G.A.2
The stack accounts for 9 of them.
27−9=1827 - 9 = 18
18 more cubes needed.
Answer: 18 cubes
4 · Reviewdoes it hold up?

The next cube down, 2 along each edge, holds 8 cubes -- but it is too short to contain a shape 3 long in one direction, so 3 really is the smallest.

Another way: Filling the cube layer by layer needs 9 cubes per layer minus what each layer already has, which adds to the same 18.

Standardsmin grade 6
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Counting the unit cubes in the shape and in the finished cube.
  • 6.NS.B.4 Find GCF and LCM; distributive property with a common factor — Taking the largest of the three measurements.
💡Takeaway. The box a shape needs is set by its longest side. How many cubes it is made of has nothing to do with it.
Variant 2 medium answer: 17 cubes

More unit cubes are to be added to the shape at the right to build the smallest possible cube. How many more cubes are needed?

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

A stack of 10 unit cubes is to be grown into the smallest cube that will contain it. We want how many cubes must be added.

Givens
  • The footprint is 3 cells deep and 3 across.
  • The tallest stack is 3 cubes.
  • There are 10 cubes altogether.
Unknowns
  • How many more unit cubes are needed.
Constraints
  • The finished solid must be a cube and must contain the shape as it stands.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#16 Count the Complement

The cube count is a distraction: it says how much is there, not how big the box must be. Measure the shape in each direction, let the longest set the edge, and subtract.

3 · Execute4 carry out the plan

1Measure the shape three ways

#17 Visualize Spatial Relationships 6.G.A.2
It is 3 across, 3 deep and 3 tall.
3, 3, 33,\ 3,\ 3
Three numbers, one of them largest.

2Let the longest set the cube's edge

#16 Count the Complement 6.NS.B.4
A smaller cube could not contain the shape's longest direction, so this is the smallest one that works.
edge=3\text{edge} = 3
A 3 by 3 by 3 cube.

3Count what that cube holds

#7 Identify Subproblems 6.G.A.2
Edge times edge times edge.
3×3×3=273 \times 3 \times 3 = 27
27 cubes in the finished solid.

4Subtract what is already there

#7 Identify Subproblems 6.G.A.2
The stack accounts for 10 of them.
27−10=1727 - 10 = 17
17 more cubes needed.
Answer: 17 cubes
4 · Reviewdoes it hold up?

The next cube down, 2 along each edge, holds 8 cubes -- but it is too short to contain a shape 3 long in one direction, so 3 really is the smallest.

Another way: Filling the cube layer by layer needs 9 cubes per layer minus what each layer already has, which adds to the same 17.

Standardsmin grade 6
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Counting the unit cubes in the shape and in the finished cube.
  • 6.NS.B.4 Find GCF and LCM; distributive property with a common factor — Taking the largest of the three measurements.
💡Takeaway. The box a shape needs is set by its longest side. How many cubes it is made of has nothing to do with it.
Variant 3 easy answer: 18 cubes

More unit cubes are to be added to the shape at the right to build the smallest possible cube. How many more cubes are needed?

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

A stack of 9 unit cubes is to be grown into the smallest cube that will contain it. We want how many cubes must be added.

Givens
  • The footprint is 3 cells deep and 3 across.
  • The tallest stack is 3 cubes.
  • There are 9 cubes altogether.
Unknowns
  • How many more unit cubes are needed.
Constraints
  • The finished solid must be a cube and must contain the shape as it stands.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#16 Count the Complement

The cube count is a distraction: it says how much is there, not how big the box must be. Measure the shape in each direction, let the longest set the edge, and subtract.

3 · Execute4 carry out the plan

1Measure the shape three ways

#17 Visualize Spatial Relationships 6.G.A.2
It is 3 across, 3 deep and 3 tall.
3, 3, 33,\ 3,\ 3
Three numbers, one of them largest.

2Let the longest set the cube's edge

#16 Count the Complement 6.NS.B.4
A smaller cube could not contain the shape's longest direction, so this is the smallest one that works.
edge=3\text{edge} = 3
A 3 by 3 by 3 cube.

3Count what that cube holds

#7 Identify Subproblems 6.G.A.2
Edge times edge times edge.
3×3×3=273 \times 3 \times 3 = 27
27 cubes in the finished solid.

4Subtract what is already there

#7 Identify Subproblems 6.G.A.2
The stack accounts for 9 of them.
27−9=1827 - 9 = 18
18 more cubes needed.
Answer: 18 cubes
4 · Reviewdoes it hold up?

The next cube down, 2 along each edge, holds 8 cubes -- but it is too short to contain a shape 3 long in one direction, so 3 really is the smallest.

Another way: Filling the cube layer by layer needs 9 cubes per layer minus what each layer already has, which adds to the same 18.

Standardsmin grade 6
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Counting the unit cubes in the shape and in the finished cube.
  • 6.NS.B.4 Find GCF and LCM; distributive property with a common factor — Taking the largest of the three measurements.
💡Takeaway. The box a shape needs is set by its longest side. How many cubes it is made of has nothing to do with it.
Variant 4 easy answer: 17 cubes

More unit cubes are to be added to the shape at the right to build the smallest possible cube. How many more cubes are needed?

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

A stack of 10 unit cubes is to be grown into the smallest cube that will contain it. We want how many cubes must be added.

Givens
  • The footprint is 3 cells deep and 3 across.
  • The tallest stack is 3 cubes.
  • There are 10 cubes altogether.
Unknowns
  • How many more unit cubes are needed.
Constraints
  • The finished solid must be a cube and must contain the shape as it stands.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#16 Count the Complement

The cube count is a distraction: it says how much is there, not how big the box must be. Measure the shape in each direction, let the longest set the edge, and subtract.

3 · Execute4 carry out the plan

1Measure the shape three ways

#17 Visualize Spatial Relationships 6.G.A.2
It is 3 across, 3 deep and 3 tall.
3, 3, 33,\ 3,\ 3
Three numbers, one of them largest.

2Let the longest set the cube's edge

#16 Count the Complement 6.NS.B.4
A smaller cube could not contain the shape's longest direction, so this is the smallest one that works.
edge=3\text{edge} = 3
A 3 by 3 by 3 cube.

3Count what that cube holds

#7 Identify Subproblems 6.G.A.2
Edge times edge times edge.
3×3×3=273 \times 3 \times 3 = 27
27 cubes in the finished solid.

4Subtract what is already there

#7 Identify Subproblems 6.G.A.2
The stack accounts for 10 of them.
27−10=1727 - 10 = 17
17 more cubes needed.
Answer: 17 cubes
4 · Reviewdoes it hold up?

The next cube down, 2 along each edge, holds 8 cubes -- but it is too short to contain a shape 3 long in one direction, so 3 really is the smallest.

Another way: Filling the cube layer by layer needs 9 cubes per layer minus what each layer already has, which adds to the same 17.

Standardsmin grade 6
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Counting the unit cubes in the shape and in the finished cube.
  • 6.NS.B.4 Find GCF and LCM; distributive property with a common factor — Taking the largest of the three measurements.
💡Takeaway. The box a shape needs is set by its longest side. How many cubes it is made of has nothing to do with it.
Variant 5 easy answer: 18 cubes

More unit cubes are to be added to the shape at the right to build the smallest possible cube. How many more cubes are needed?

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

A stack of 9 unit cubes is to be grown into the smallest cube that will contain it. We want how many cubes must be added.

Givens
  • The footprint is 3 cells deep and 3 across.
  • The tallest stack is 3 cubes.
  • There are 9 cubes altogether.
Unknowns
  • How many more unit cubes are needed.
Constraints
  • The finished solid must be a cube and must contain the shape as it stands.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#16 Count the Complement

The cube count is a distraction: it says how much is there, not how big the box must be. Measure the shape in each direction, let the longest set the edge, and subtract.

3 · Execute4 carry out the plan

1Measure the shape three ways

#17 Visualize Spatial Relationships 6.G.A.2
It is 3 across, 3 deep and 3 tall.
3, 3, 33,\ 3,\ 3
Three numbers, one of them largest.

2Let the longest set the cube's edge

#16 Count the Complement 6.NS.B.4
A smaller cube could not contain the shape's longest direction, so this is the smallest one that works.
edge=3\text{edge} = 3
A 3 by 3 by 3 cube.

3Count what that cube holds

#7 Identify Subproblems 6.G.A.2
Edge times edge times edge.
3×3×3=273 \times 3 \times 3 = 27
27 cubes in the finished solid.

4Subtract what is already there

#7 Identify Subproblems 6.G.A.2
The stack accounts for 9 of them.
27−9=1827 - 9 = 18
18 more cubes needed.
Answer: 18 cubes
4 · Reviewdoes it hold up?

The next cube down, 2 along each edge, holds 8 cubes -- but it is too short to contain a shape 3 long in one direction, so 3 really is the smallest.

Another way: Filling the cube layer by layer needs 9 cubes per layer minus what each layer already has, which adds to the same 18.

Standardsmin grade 6
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Counting the unit cubes in the shape and in the finished cube.
  • 6.NS.B.4 Find GCF and LCM; distributive property with a common factor — Taking the largest of the three measurements.
💡Takeaway. The box a shape needs is set by its longest side. How many cubes it is made of has nothing to do with it.
Variant 6 medium answer: 17 cubes

More unit cubes are to be added to the shape at the right to build the smallest possible cube. How many more cubes are needed?

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

A stack of 10 unit cubes is to be grown into the smallest cube that will contain it. We want how many cubes must be added.

Givens
  • The footprint is 3 cells deep and 3 across.
  • The tallest stack is 3 cubes.
  • There are 10 cubes altogether.
Unknowns
  • How many more unit cubes are needed.
Constraints
  • The finished solid must be a cube and must contain the shape as it stands.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#16 Count the Complement

The cube count is a distraction: it says how much is there, not how big the box must be. Measure the shape in each direction, let the longest set the edge, and subtract.

3 · Execute4 carry out the plan

1Measure the shape three ways

#17 Visualize Spatial Relationships 6.G.A.2
It is 3 across, 3 deep and 3 tall.
3, 3, 33,\ 3,\ 3
Three numbers, one of them largest.

2Let the longest set the cube's edge

#16 Count the Complement 6.NS.B.4
A smaller cube could not contain the shape's longest direction, so this is the smallest one that works.
edge=3\text{edge} = 3
A 3 by 3 by 3 cube.

3Count what that cube holds

#7 Identify Subproblems 6.G.A.2
Edge times edge times edge.
3×3×3=273 \times 3 \times 3 = 27
27 cubes in the finished solid.

4Subtract what is already there

#7 Identify Subproblems 6.G.A.2
The stack accounts for 10 of them.
27−10=1727 - 10 = 17
17 more cubes needed.
Answer: 17 cubes
4 · Reviewdoes it hold up?

The next cube down, 2 along each edge, holds 8 cubes -- but it is too short to contain a shape 3 long in one direction, so 3 really is the smallest.

Another way: Filling the cube layer by layer needs 9 cubes per layer minus what each layer already has, which adds to the same 17.

Standardsmin grade 6
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Counting the unit cubes in the shape and in the finished cube.
  • 6.NS.B.4 Find GCF and LCM; distributive property with a common factor — Taking the largest of the three measurements.
💡Takeaway. The box a shape needs is set by its longest side. How many cubes it is made of has nothing to do with it.
Variant 7 medium answer: 16 cubes

More unit cubes are to be added to the shape at the right to build the smallest possible cube. How many more cubes are needed?

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

A stack of 11 unit cubes is to be grown into the smallest cube that will contain it. We want how many cubes must be added.

Givens
  • The footprint is 3 cells deep and 3 across.
  • The tallest stack is 3 cubes.
  • There are 11 cubes altogether.
Unknowns
  • How many more unit cubes are needed.
Constraints
  • The finished solid must be a cube and must contain the shape as it stands.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#16 Count the Complement

The cube count is a distraction: it says how much is there, not how big the box must be. Measure the shape in each direction, let the longest set the edge, and subtract.

3 · Execute4 carry out the plan

1Measure the shape three ways

#17 Visualize Spatial Relationships 6.G.A.2
It is 3 across, 3 deep and 3 tall.
3, 3, 33,\ 3,\ 3
Three numbers, one of them largest.

2Let the longest set the cube's edge

#16 Count the Complement 6.NS.B.4
A smaller cube could not contain the shape's longest direction, so this is the smallest one that works.
edge=3\text{edge} = 3
A 3 by 3 by 3 cube.

3Count what that cube holds

#7 Identify Subproblems 6.G.A.2
Edge times edge times edge.
3×3×3=273 \times 3 \times 3 = 27
27 cubes in the finished solid.

4Subtract what is already there

#7 Identify Subproblems 6.G.A.2
The stack accounts for 11 of them.
27−11=1627 - 11 = 16
16 more cubes needed.
Answer: 16 cubes
4 · Reviewdoes it hold up?

The next cube down, 2 along each edge, holds 8 cubes -- but it is too short to contain a shape 3 long in one direction, so 3 really is the smallest.

Another way: Filling the cube layer by layer needs 9 cubes per layer minus what each layer already has, which adds to the same 16.

Standardsmin grade 6
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Counting the unit cubes in the shape and in the finished cube.
  • 6.NS.B.4 Find GCF and LCM; distributive property with a common factor — Taking the largest of the three measurements.
💡Takeaway. The box a shape needs is set by its longest side. How many cubes it is made of has nothing to do with it.
Variant 8 hard answer: 16 cubes

More unit cubes are to be added to the shape at the right to build the smallest possible cube. How many more cubes are needed?

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

A stack of 11 unit cubes is to be grown into the smallest cube that will contain it. We want how many cubes must be added.

Givens
  • The footprint is 3 cells deep and 3 across.
  • The tallest stack is 3 cubes.
  • There are 11 cubes altogether.
Unknowns
  • How many more unit cubes are needed.
Constraints
  • The finished solid must be a cube and must contain the shape as it stands.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#16 Count the Complement

The cube count is a distraction: it says how much is there, not how big the box must be. Measure the shape in each direction, let the longest set the edge, and subtract.

3 · Execute4 carry out the plan

1Measure the shape three ways

#17 Visualize Spatial Relationships 6.G.A.2
It is 3 across, 3 deep and 3 tall.
3, 3, 33,\ 3,\ 3
Three numbers, one of them largest.

2Let the longest set the cube's edge

#16 Count the Complement 6.NS.B.4
A smaller cube could not contain the shape's longest direction, so this is the smallest one that works.
edge=3\text{edge} = 3
A 3 by 3 by 3 cube.

3Count what that cube holds

#7 Identify Subproblems 6.G.A.2
Edge times edge times edge.
3×3×3=273 \times 3 \times 3 = 27
27 cubes in the finished solid.

4Subtract what is already there

#7 Identify Subproblems 6.G.A.2
The stack accounts for 11 of them.
27−11=1627 - 11 = 16
16 more cubes needed.
Answer: 16 cubes
4 · Reviewdoes it hold up?

The next cube down, 2 along each edge, holds 8 cubes -- but it is too short to contain a shape 3 long in one direction, so 3 really is the smallest.

Another way: Filling the cube layer by layer needs 9 cubes per layer minus what each layer already has, which adds to the same 16.

Standardsmin grade 6
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Counting the unit cubes in the shape and in the finished cube.
  • 6.NS.B.4 Find GCF and LCM; distributive property with a common factor — Taking the largest of the three measurements.
💡Takeaway. The box a shape needs is set by its longest side. How many cubes it is made of has nothing to do with it.
Variant 9 medium answer: 16 cubes

More unit cubes are to be added to the shape at the right to build the smallest possible cube. How many more cubes are needed?

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

A stack of 11 unit cubes is to be grown into the smallest cube that will contain it. We want how many cubes must be added.

Givens
  • The footprint is 3 cells deep and 3 across.
  • The tallest stack is 3 cubes.
  • There are 11 cubes altogether.
Unknowns
  • How many more unit cubes are needed.
Constraints
  • The finished solid must be a cube and must contain the shape as it stands.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#16 Count the Complement

The cube count is a distraction: it says how much is there, not how big the box must be. Measure the shape in each direction, let the longest set the edge, and subtract.

3 · Execute4 carry out the plan

1Measure the shape three ways

#17 Visualize Spatial Relationships 6.G.A.2
It is 3 across, 3 deep and 3 tall.
3, 3, 33,\ 3,\ 3
Three numbers, one of them largest.

2Let the longest set the cube's edge

#16 Count the Complement 6.NS.B.4
A smaller cube could not contain the shape's longest direction, so this is the smallest one that works.
edge=3\text{edge} = 3
A 3 by 3 by 3 cube.

3Count what that cube holds

#7 Identify Subproblems 6.G.A.2
Edge times edge times edge.
3×3×3=273 \times 3 \times 3 = 27
27 cubes in the finished solid.

4Subtract what is already there

#7 Identify Subproblems 6.G.A.2
The stack accounts for 11 of them.
27−11=1627 - 11 = 16
16 more cubes needed.
Answer: 16 cubes
4 · Reviewdoes it hold up?

The next cube down, 2 along each edge, holds 8 cubes -- but it is too short to contain a shape 3 long in one direction, so 3 really is the smallest.

Another way: Filling the cube layer by layer needs 9 cubes per layer minus what each layer already has, which adds to the same 16.

Standardsmin grade 6
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Counting the unit cubes in the shape and in the finished cube.
  • 6.NS.B.4 Find GCF and LCM; distributive property with a common factor — Taking the largest of the three measurements.
💡Takeaway. The box a shape needs is set by its longest side. How many cubes it is made of has nothing to do with it.
Variant 10 hard answer: 16 cubes

More unit cubes are to be added to the shape at the right to build the smallest possible cube. How many more cubes are needed?

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

A stack of 11 unit cubes is to be grown into the smallest cube that will contain it. We want how many cubes must be added.

Givens
  • The footprint is 3 cells deep and 3 across.
  • The tallest stack is 3 cubes.
  • There are 11 cubes altogether.
Unknowns
  • How many more unit cubes are needed.
Constraints
  • The finished solid must be a cube and must contain the shape as it stands.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#16 Count the Complement

The cube count is a distraction: it says how much is there, not how big the box must be. Measure the shape in each direction, let the longest set the edge, and subtract.

3 · Execute4 carry out the plan

1Measure the shape three ways

#17 Visualize Spatial Relationships 6.G.A.2
It is 3 across, 3 deep and 3 tall.
3, 3, 33,\ 3,\ 3
Three numbers, one of them largest.

2Let the longest set the cube's edge

#16 Count the Complement 6.NS.B.4
A smaller cube could not contain the shape's longest direction, so this is the smallest one that works.
edge=3\text{edge} = 3
A 3 by 3 by 3 cube.

3Count what that cube holds

#7 Identify Subproblems 6.G.A.2
Edge times edge times edge.
3×3×3=273 \times 3 \times 3 = 27
27 cubes in the finished solid.

4Subtract what is already there

#7 Identify Subproblems 6.G.A.2
The stack accounts for 11 of them.
27−11=1627 - 11 = 16
16 more cubes needed.
Answer: 16 cubes
4 · Reviewdoes it hold up?

The next cube down, 2 along each edge, holds 8 cubes -- but it is too short to contain a shape 3 long in one direction, so 3 really is the smallest.

Another way: Filling the cube layer by layer needs 9 cubes per layer minus what each layer already has, which adds to the same 16.

Standardsmin grade 6
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Counting the unit cubes in the shape and in the finished cube.
  • 6.NS.B.4 Find GCF and LCM; distributive property with a common factor — Taking the largest of the three measurements.
💡Takeaway. The box a shape needs is set by its longest side. How many cubes it is made of has nothing to do with it.
Variant 11 hard answer: 49 cubes

More unit cubes are to be added to the shape at the right to build the smallest possible cube. How many more cubes are needed?

seen from above 4 2 1 1 1 1 2 1 1 1 front
Show solution
1 · Understandwhat's really being asked

A stack of 15 unit cubes is to be grown into the smallest cube that will contain it. We want how many cubes must be added.

Givens
  • The footprint is 4 cells deep and 4 across.
  • The tallest stack is 4 cubes.
  • There are 15 cubes altogether.
Unknowns
  • How many more unit cubes are needed.
Constraints
  • The finished solid must be a cube and must contain the shape as it stands.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#16 Count the Complement

The cube count is a distraction: it says how much is there, not how big the box must be. Measure the shape in each direction, let the longest set the edge, and subtract.

3 · Execute4 carry out the plan

1Measure the shape three ways

#17 Visualize Spatial Relationships 6.G.A.2
It is 4 across, 4 deep and 4 tall.
4, 4, 44,\ 4,\ 4
Three numbers, one of them largest.

2Let the longest set the cube's edge

#16 Count the Complement 6.NS.B.4
A smaller cube could not contain the shape's longest direction, so this is the smallest one that works.
edge=4\text{edge} = 4
A 4 by 4 by 4 cube.

3Count what that cube holds

#7 Identify Subproblems 6.G.A.2
Edge times edge times edge.
4×4×4=644 \times 4 \times 4 = 64
64 cubes in the finished solid.

4Subtract what is already there

#7 Identify Subproblems 6.G.A.2
The stack accounts for 15 of them.
64−15=4964 - 15 = 49
49 more cubes needed.
Answer: 49 cubes
4 · Reviewdoes it hold up?

The next cube down, 3 along each edge, holds 27 cubes -- but it is too short to contain a shape 4 long in one direction, so 4 really is the smallest.

Another way: Filling the cube layer by layer needs 16 cubes per layer minus what each layer already has, which adds to the same 49.

Standardsmin grade 6
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Counting the unit cubes in the shape and in the finished cube.
  • 6.NS.B.4 Find GCF and LCM; distributive property with a common factor — Taking the largest of the three measurements.
💡Takeaway. The box a shape needs is set by its longest side. How many cubes it is made of has nothing to do with it.
Variant 12 hard answer: 48 cubes

More unit cubes are to be added to the shape at the right to build the smallest possible cube. How many more cubes are needed?

seen from above 4 1 1 1 2 1 1 2 1 1 1 front
Show solution
1 · Understandwhat's really being asked

A stack of 16 unit cubes is to be grown into the smallest cube that will contain it. We want how many cubes must be added.

Givens
  • The footprint is 4 cells deep and 4 across.
  • The tallest stack is 4 cubes.
  • There are 16 cubes altogether.
Unknowns
  • How many more unit cubes are needed.
Constraints
  • The finished solid must be a cube and must contain the shape as it stands.
2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #7 Identify Subproblems#16 Count the Complement

The cube count is a distraction: it says how much is there, not how big the box must be. Measure the shape in each direction, let the longest set the edge, and subtract.

3 · Execute4 carry out the plan

1Measure the shape three ways

#17 Visualize Spatial Relationships 6.G.A.2
It is 4 across, 4 deep and 4 tall.
4, 4, 44,\ 4,\ 4
Three numbers, one of them largest.

2Let the longest set the cube's edge

#16 Count the Complement 6.NS.B.4
A smaller cube could not contain the shape's longest direction, so this is the smallest one that works.
edge=4\text{edge} = 4
A 4 by 4 by 4 cube.

3Count what that cube holds

#7 Identify Subproblems 6.G.A.2
Edge times edge times edge.
4×4×4=644 \times 4 \times 4 = 64
64 cubes in the finished solid.

4Subtract what is already there

#7 Identify Subproblems 6.G.A.2
The stack accounts for 16 of them.
64−16=4864 - 16 = 48
48 more cubes needed.
Answer: 48 cubes
4 · Reviewdoes it hold up?

The next cube down, 3 along each edge, holds 27 cubes -- but it is too short to contain a shape 4 long in one direction, so 4 really is the smallest.

Another way: Filling the cube layer by layer needs 16 cubes per layer minus what each layer already has, which adds to the same 48.

Standardsmin grade 6
  • 6.G.A.2 Volume of right rectangular prisms with fractional edges — Counting the unit cubes in the shape and in the finished cube.
  • 6.NS.B.4 Find GCF and LCM; distributive property with a common factor — Taking the largest of the three measurements.
💡Takeaway. The box a shape needs is set by its longest side. How many cubes it is made of has nothing to do with it.