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

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

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Generated variants — 12

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

Variant 1 easy

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?

Your answer

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,\ 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.

\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 \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 = 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

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?

Your answer

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,\ 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.

\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 \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 = 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

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?

Your answer

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,\ 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.

\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 \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 = 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

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?

Your answer

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,\ 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.

\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 \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 = 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

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?

Your answer

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,\ 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.

\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 \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 = 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

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?

Your answer

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,\ 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.

\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 \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 = 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

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?

Your answer

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,\ 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.

\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 \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 = 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

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?

Your answer

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,\ 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.

\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 \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 = 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

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?

Your answer

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,\ 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.

\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 \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 = 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

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?

Your answer

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,\ 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.

\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 \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 = 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

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?

Your answer

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,\ 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.

\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 \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 = 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

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?

Your answer

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,\ 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.

\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 \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 = 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.

Where next?

← Back to the problem: The longest side decides the box, not the cube count