Practice · Count cubes along each edge, not volume divided by volume · Sensim Math

Variant 1 easy

The box at the right is to be filled completely with cubes whose edges are $4\ \text{cm}$ long. How many cubes are needed? (Ignore the thickness of the box.)

Your answer

Show solution

1 · Understandwhat's really being asked

A box 4.8 m by 2.4 m by 3.6 m is packed full of cubes with 4 cm edges. We want how many cubes.

Givens

The box is 4.8 m long, 2.4 m deep and 3.6 m tall.
Each cube has an edge of 4 cm.

Unknowns

How many cubes fill the box.

Constraints

The cubes fill it completely, with no gaps and nothing sticking out.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #17 Visualize Spatial Relationships#10 Create a Physical Representation

The box is in metres and the cube in centimetres, so convert first -- nothing can be compared until they share a unit. Then count how many cubes fit along each edge, which keeps the picture attached to the arithmetic.

3 · Execute3 carry out the plan

1Put the box in centimetres

#8 Analyze the Units 6.G.A.2

A metre is 100 cm.

480 \times 240 \times 360\ \text{cm}

Now both are in the same unit.

2Count the cubes along each edge

#17 Visualize Spatial Relationships 5.MD.C.5

Each edge divided by the cube's 4 cm.

480 \div 4 = 120,\quad 240 \div 4 = 60,\quad 360 \div 4 = 90

120 by 60 by 90 cubes.

3Multiply the three counts

#10 Create a Physical Representation 5.MD.C.5

One layer is the first two multiplied, and there are as many layers as cubes up the height.

120 \times 60 \times 90 = 648{,}000

648,000 cubes.

Answer: 648,000 cubes

4 · Reviewdoes it hold up?

By volume instead: the box is 41,472,000 cm3 and each cube is 64 cm3, and dividing gives the same 648,000.

Another way: Counting one layer first -- 7,200 cubes -- and multiplying by the 90 layers is the same product in a different order.

Standardsmin grade 6

6.G.A.2 Volume of right rectangular prisms with fractional edges — Filling a prism with unit cubes to get its volume.
5.MD.C.5 Relate volume to the operations of multiplication and addition — Counting the cubes as length times width times height.

💡Takeaway. Make the units match before you do anything else. Then count along each edge and multiply.

Variant 2 easy

The box at the right is to be filled completely with cubes whose edges are $5\ \text{cm}$ long. How many cubes are needed? (Ignore the thickness of the box.)

Your answer

Show solution

1 · Understandwhat's really being asked

A box 5 m by 3 m by 4 m is packed full of cubes with 5 cm edges. We want how many cubes.

Givens

The box is 5 m long, 3 m deep and 4 m tall.
Each cube has an edge of 5 cm.

Unknowns

How many cubes fill the box.

Constraints

The cubes fill it completely, with no gaps and nothing sticking out.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #17 Visualize Spatial Relationships#10 Create a Physical Representation

The box is in metres and the cube in centimetres, so convert first -- nothing can be compared until they share a unit. Then count how many cubes fit along each edge, which keeps the picture attached to the arithmetic.

3 · Execute3 carry out the plan

1Put the box in centimetres

#8 Analyze the Units 6.G.A.2

A metre is 100 cm.

500 \times 300 \times 400\ \text{cm}

Now both are in the same unit.

2Count the cubes along each edge

#17 Visualize Spatial Relationships 5.MD.C.5

Each edge divided by the cube's 5 cm.

500 \div 5 = 100,\quad 300 \div 5 = 60,\quad 400 \div 5 = 80

100 by 60 by 80 cubes.

3Multiply the three counts

#10 Create a Physical Representation 5.MD.C.5

One layer is the first two multiplied, and there are as many layers as cubes up the height.

100 \times 60 \times 80 = 480{,}000

480,000 cubes.

Answer: 480,000 cubes

4 · Reviewdoes it hold up?

By volume instead: the box is 60,000,000 cm3 and each cube is 125 cm3, and dividing gives the same 480,000.

Another way: Counting one layer first -- 6,000 cubes -- and multiplying by the 80 layers is the same product in a different order.

Standardsmin grade 6

6.G.A.2 Volume of right rectangular prisms with fractional edges — Filling a prism with unit cubes to get its volume.
5.MD.C.5 Relate volume to the operations of multiplication and addition — Counting the cubes as length times width times height.

💡Takeaway. Make the units match before you do anything else. Then count along each edge and multiply.

Variant 3 easy

The box at the right is to be filled completely with cubes whose edges are $5\ \text{cm}$ long. How many cubes are needed? (Ignore the thickness of the box.)

Your answer

Show solution

1 · Understandwhat's really being asked

A box 4 m by 2.5 m by 3.5 m is packed full of cubes with 5 cm edges. We want how many cubes.

Givens

The box is 4 m long, 2.5 m deep and 3.5 m tall.
Each cube has an edge of 5 cm.

Unknowns

How many cubes fill the box.

Constraints

The cubes fill it completely, with no gaps and nothing sticking out.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #17 Visualize Spatial Relationships#10 Create a Physical Representation

The box is in metres and the cube in centimetres, so convert first -- nothing can be compared until they share a unit. Then count how many cubes fit along each edge, which keeps the picture attached to the arithmetic.

3 · Execute3 carry out the plan

1Put the box in centimetres

#8 Analyze the Units 6.G.A.2

A metre is 100 cm.

400 \times 250 \times 350\ \text{cm}

Now both are in the same unit.

2Count the cubes along each edge

#17 Visualize Spatial Relationships 5.MD.C.5

Each edge divided by the cube's 5 cm.

400 \div 5 = 80,\quad 250 \div 5 = 50,\quad 350 \div 5 = 70

80 by 50 by 70 cubes.

3Multiply the three counts

#10 Create a Physical Representation 5.MD.C.5

One layer is the first two multiplied, and there are as many layers as cubes up the height.

80 \times 50 \times 70 = 280{,}000

280,000 cubes.

Answer: 280,000 cubes

4 · Reviewdoes it hold up?

By volume instead: the box is 35,000,000 cm3 and each cube is 125 cm3, and dividing gives the same 280,000.

Another way: Counting one layer first -- 4,000 cubes -- and multiplying by the 70 layers is the same product in a different order.

Standardsmin grade 6

6.G.A.2 Volume of right rectangular prisms with fractional edges — Filling a prism with unit cubes to get its volume.
5.MD.C.5 Relate volume to the operations of multiplication and addition — Counting the cubes as length times width times height.

💡Takeaway. Make the units match before you do anything else. Then count along each edge and multiply.

Variant 4 medium

The box at the right is to be filled completely with cubes whose edges are $6\ \text{cm}$ long. How many cubes are needed? (Ignore the thickness of the box.)

Your answer

Show solution

1 · Understandwhat's really being asked

A box 3 m by 2.4 m by 1.8 m is packed full of cubes with 6 cm edges. We want how many cubes.

Givens

The box is 3 m long, 2.4 m deep and 1.8 m tall.
Each cube has an edge of 6 cm.

Unknowns

How many cubes fill the box.

Constraints

The cubes fill it completely, with no gaps and nothing sticking out.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #17 Visualize Spatial Relationships#10 Create a Physical Representation

The box is in metres and the cube in centimetres, so convert first -- nothing can be compared until they share a unit. Then count how many cubes fit along each edge, which keeps the picture attached to the arithmetic.

3 · Execute3 carry out the plan

1Put the box in centimetres

#8 Analyze the Units 6.G.A.2

A metre is 100 cm.

300 \times 240 \times 180\ \text{cm}

Now both are in the same unit.

2Count the cubes along each edge

#17 Visualize Spatial Relationships 5.MD.C.5

Each edge divided by the cube's 6 cm.

300 \div 6 = 50,\quad 240 \div 6 = 40,\quad 180 \div 6 = 30

50 by 40 by 30 cubes.

3Multiply the three counts

#10 Create a Physical Representation 5.MD.C.5

One layer is the first two multiplied, and there are as many layers as cubes up the height.

50 \times 40 \times 30 = 60{,}000

60,000 cubes.

Answer: 60,000 cubes

4 · Reviewdoes it hold up?

By volume instead: the box is 12,960,000 cm3 and each cube is 216 cm3, and dividing gives the same 60,000.

Another way: Counting one layer first -- 2,000 cubes -- and multiplying by the 30 layers is the same product in a different order.

Standardsmin grade 6

6.G.A.2 Volume of right rectangular prisms with fractional edges — Filling a prism with unit cubes to get its volume.
5.MD.C.5 Relate volume to the operations of multiplication and addition — Counting the cubes as length times width times height.

💡Takeaway. Make the units match before you do anything else. Then count along each edge and multiply.

Variant 5 easy

The box at the right is to be filled completely with cubes whose edges are $6\ \text{cm}$ long. How many cubes are needed? (Ignore the thickness of the box.)

Your answer

Show solution

1 · Understandwhat's really being asked

A box 6 m by 3.6 m by 4.8 m is packed full of cubes with 6 cm edges. We want how many cubes.

Givens

The box is 6 m long, 3.6 m deep and 4.8 m tall.
Each cube has an edge of 6 cm.

Unknowns

How many cubes fill the box.

Constraints

The cubes fill it completely, with no gaps and nothing sticking out.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #17 Visualize Spatial Relationships#10 Create a Physical Representation

The box is in metres and the cube in centimetres, so convert first -- nothing can be compared until they share a unit. Then count how many cubes fit along each edge, which keeps the picture attached to the arithmetic.

3 · Execute3 carry out the plan

1Put the box in centimetres

#8 Analyze the Units 6.G.A.2

A metre is 100 cm.

600 \times 360 \times 480\ \text{cm}

Now both are in the same unit.

2Count the cubes along each edge

#17 Visualize Spatial Relationships 5.MD.C.5

Each edge divided by the cube's 6 cm.

600 \div 6 = 100,\quad 360 \div 6 = 60,\quad 480 \div 6 = 80

100 by 60 by 80 cubes.

3Multiply the three counts

#10 Create a Physical Representation 5.MD.C.5

One layer is the first two multiplied, and there are as many layers as cubes up the height.

100 \times 60 \times 80 = 480{,}000

480,000 cubes.

Answer: 480,000 cubes

4 · Reviewdoes it hold up?

By volume instead: the box is 103,680,000 cm3 and each cube is 216 cm3, and dividing gives the same 480,000.

Another way: Counting one layer first -- 6,000 cubes -- and multiplying by the 80 layers is the same product in a different order.

Standardsmin grade 6

6.G.A.2 Volume of right rectangular prisms with fractional edges — Filling a prism with unit cubes to get its volume.
5.MD.C.5 Relate volume to the operations of multiplication and addition — Counting the cubes as length times width times height.

💡Takeaway. Make the units match before you do anything else. Then count along each edge and multiply.

Variant 6 medium

The box at the right is to be filled completely with cubes whose edges are $8\ \text{cm}$ long. How many cubes are needed? (Ignore the thickness of the box.)

Your answer

Show solution

1 · Understandwhat's really being asked

A box 8 m by 6 m by 4 m is packed full of cubes with 8 cm edges. We want how many cubes.

Givens

The box is 8 m long, 6 m deep and 4 m tall.
Each cube has an edge of 8 cm.

Unknowns

How many cubes fill the box.

Constraints

The cubes fill it completely, with no gaps and nothing sticking out.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #17 Visualize Spatial Relationships#10 Create a Physical Representation

The box is in metres and the cube in centimetres, so convert first -- nothing can be compared until they share a unit. Then count how many cubes fit along each edge, which keeps the picture attached to the arithmetic.

3 · Execute3 carry out the plan

1Put the box in centimetres

#8 Analyze the Units 6.G.A.2

A metre is 100 cm.

800 \times 600 \times 400\ \text{cm}

Now both are in the same unit.

2Count the cubes along each edge

#17 Visualize Spatial Relationships 5.MD.C.5

Each edge divided by the cube's 8 cm.

800 \div 8 = 100,\quad 600 \div 8 = 75,\quad 400 \div 8 = 50

100 by 75 by 50 cubes.

3Multiply the three counts

#10 Create a Physical Representation 5.MD.C.5

One layer is the first two multiplied, and there are as many layers as cubes up the height.

100 \times 75 \times 50 = 375{,}000

375,000 cubes.

Answer: 375,000 cubes

4 · Reviewdoes it hold up?

By volume instead: the box is 192,000,000 cm3 and each cube is 512 cm3, and dividing gives the same 375,000.

Another way: Counting one layer first -- 7,500 cubes -- and multiplying by the 50 layers is the same product in a different order.

Standardsmin grade 6

6.G.A.2 Volume of right rectangular prisms with fractional edges — Filling a prism with unit cubes to get its volume.
5.MD.C.5 Relate volume to the operations of multiplication and addition — Counting the cubes as length times width times height.

💡Takeaway. Make the units match before you do anything else. Then count along each edge and multiply.

Variant 7 medium

The box at the right is to be filled completely with cubes whose edges are $8\ \text{cm}$ long. How many cubes are needed? (Ignore the thickness of the box.)

Your answer

Show solution

1 · Understandwhat's really being asked

A box 6.4 m by 3.2 m by 4.8 m is packed full of cubes with 8 cm edges. We want how many cubes.

Givens

The box is 6.4 m long, 3.2 m deep and 4.8 m tall.
Each cube has an edge of 8 cm.

Unknowns

How many cubes fill the box.

Constraints

The cubes fill it completely, with no gaps and nothing sticking out.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #17 Visualize Spatial Relationships#10 Create a Physical Representation

The box is in metres and the cube in centimetres, so convert first -- nothing can be compared until they share a unit. Then count how many cubes fit along each edge, which keeps the picture attached to the arithmetic.

3 · Execute3 carry out the plan

1Put the box in centimetres

#8 Analyze the Units 6.G.A.2

A metre is 100 cm.

640 \times 320 \times 480\ \text{cm}

Now both are in the same unit.

2Count the cubes along each edge

#17 Visualize Spatial Relationships 5.MD.C.5

Each edge divided by the cube's 8 cm.

640 \div 8 = 80,\quad 320 \div 8 = 40,\quad 480 \div 8 = 60

80 by 40 by 60 cubes.

3Multiply the three counts

#10 Create a Physical Representation 5.MD.C.5

One layer is the first two multiplied, and there are as many layers as cubes up the height.

80 \times 40 \times 60 = 192{,}000

192,000 cubes.

Answer: 192,000 cubes

4 · Reviewdoes it hold up?

By volume instead: the box is 98,304,000 cm3 and each cube is 512 cm3, and dividing gives the same 192,000.

Another way: Counting one layer first -- 3,200 cubes -- and multiplying by the 60 layers is the same product in a different order.

Standardsmin grade 6

6.G.A.2 Volume of right rectangular prisms with fractional edges — Filling a prism with unit cubes to get its volume.
5.MD.C.5 Relate volume to the operations of multiplication and addition — Counting the cubes as length times width times height.

💡Takeaway. Make the units match before you do anything else. Then count along each edge and multiply.

Variant 8 medium

The box at the right is to be filled completely with cubes whose edges are $9\ \text{cm}$ long. How many cubes are needed? (Ignore the thickness of the box.)

Your answer

Show solution

1 · Understandwhat's really being asked

A box 7.2 m by 5.4 m by 3.6 m is packed full of cubes with 9 cm edges. We want how many cubes.

Givens

The box is 7.2 m long, 5.4 m deep and 3.6 m tall.
Each cube has an edge of 9 cm.

Unknowns

How many cubes fill the box.

Constraints

The cubes fill it completely, with no gaps and nothing sticking out.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #17 Visualize Spatial Relationships#10 Create a Physical Representation

The box is in metres and the cube in centimetres, so convert first -- nothing can be compared until they share a unit. Then count how many cubes fit along each edge, which keeps the picture attached to the arithmetic.

3 · Execute3 carry out the plan

1Put the box in centimetres

#8 Analyze the Units 6.G.A.2

A metre is 100 cm.

720 \times 540 \times 360\ \text{cm}

Now both are in the same unit.

2Count the cubes along each edge

#17 Visualize Spatial Relationships 5.MD.C.5

Each edge divided by the cube's 9 cm.

720 \div 9 = 80,\quad 540 \div 9 = 60,\quad 360 \div 9 = 40

80 by 60 by 40 cubes.

3Multiply the three counts

#10 Create a Physical Representation 5.MD.C.5

One layer is the first two multiplied, and there are as many layers as cubes up the height.

80 \times 60 \times 40 = 192{,}000

192,000 cubes.

Answer: 192,000 cubes

4 · Reviewdoes it hold up?

By volume instead: the box is 139,968,000 cm3 and each cube is 729 cm3, and dividing gives the same 192,000.

Another way: Counting one layer first -- 4,800 cubes -- and multiplying by the 40 layers is the same product in a different order.

Standardsmin grade 6

6.G.A.2 Volume of right rectangular prisms with fractional edges — Filling a prism with unit cubes to get its volume.
5.MD.C.5 Relate volume to the operations of multiplication and addition — Counting the cubes as length times width times height.

💡Takeaway. Make the units match before you do anything else. Then count along each edge and multiply.

Variant 9 hard

The box at the right is to be filled completely with cubes whose edges are $5\ \text{cm}$ long. How many cubes are needed? (Ignore the thickness of the box.)

Your answer

Show solution

1 · Understandwhat's really being asked

A box 10 m by 5 m by 2.5 m is packed full of cubes with 5 cm edges. We want how many cubes.

Givens

The box is 10 m long, 5 m deep and 2.5 m tall.
Each cube has an edge of 5 cm.

Unknowns

How many cubes fill the box.

Constraints

The cubes fill it completely, with no gaps and nothing sticking out.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #17 Visualize Spatial Relationships#10 Create a Physical Representation

The box is in metres and the cube in centimetres, so convert first -- nothing can be compared until they share a unit. Then count how many cubes fit along each edge, which keeps the picture attached to the arithmetic.

3 · Execute3 carry out the plan

1Put the box in centimetres

#8 Analyze the Units 6.G.A.2

A metre is 100 cm.

1000 \times 500 \times 250\ \text{cm}

Now both are in the same unit.

2Count the cubes along each edge

#17 Visualize Spatial Relationships 5.MD.C.5

Each edge divided by the cube's 5 cm.

1000 \div 5 = 200,\quad 500 \div 5 = 100,\quad 250 \div 5 = 50

200 by 100 by 50 cubes.

3Multiply the three counts

#10 Create a Physical Representation 5.MD.C.5

One layer is the first two multiplied, and there are as many layers as cubes up the height.

200 \times 100 \times 50 = 1{,}000{,}000

1,000,000 cubes.

Answer: 1,000,000 cubes

4 · Reviewdoes it hold up?

By volume instead: the box is 125,000,000 cm3 and each cube is 125 cm3, and dividing gives the same 1,000,000.

Another way: Counting one layer first -- 20,000 cubes -- and multiplying by the 50 layers is the same product in a different order.

Standardsmin grade 6

6.G.A.2 Volume of right rectangular prisms with fractional edges — Filling a prism with unit cubes to get its volume.
5.MD.C.5 Relate volume to the operations of multiplication and addition — Counting the cubes as length times width times height.

💡Takeaway. Make the units match before you do anything else. Then count along each edge and multiply.

Variant 10 hard

The box at the right is to be filled completely with cubes whose edges are $14\ \text{cm}$ long. How many cubes are needed? (Ignore the thickness of the box.)

Your answer

Show solution

1 · Understandwhat's really being asked

A box 5.6 m by 4.2 m by 2.8 m is packed full of cubes with 14 cm edges. We want how many cubes.

Givens

The box is 5.6 m long, 4.2 m deep and 2.8 m tall.
Each cube has an edge of 14 cm.

Unknowns

How many cubes fill the box.

Constraints

The cubes fill it completely, with no gaps and nothing sticking out.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #17 Visualize Spatial Relationships#10 Create a Physical Representation

The box is in metres and the cube in centimetres, so convert first -- nothing can be compared until they share a unit. Then count how many cubes fit along each edge, which keeps the picture attached to the arithmetic.

3 · Execute3 carry out the plan

1Put the box in centimetres

#8 Analyze the Units 6.G.A.2

A metre is 100 cm.

560 \times 420 \times 280\ \text{cm}

Now both are in the same unit.

2Count the cubes along each edge

#17 Visualize Spatial Relationships 5.MD.C.5

Each edge divided by the cube's 14 cm.

560 \div 14 = 40,\quad 420 \div 14 = 30,\quad 280 \div 14 = 20

40 by 30 by 20 cubes.

3Multiply the three counts

#10 Create a Physical Representation 5.MD.C.5

One layer is the first two multiplied, and there are as many layers as cubes up the height.

40 \times 30 \times 20 = 24{,}000

24,000 cubes.

Answer: 24,000 cubes

4 · Reviewdoes it hold up?

By volume instead: the box is 65,856,000 cm3 and each cube is 2744 cm3, and dividing gives the same 24,000.

Another way: Counting one layer first -- 1,200 cubes -- and multiplying by the 20 layers is the same product in a different order.

Standardsmin grade 6

6.G.A.2 Volume of right rectangular prisms with fractional edges — Filling a prism with unit cubes to get its volume.
5.MD.C.5 Relate volume to the operations of multiplication and addition — Counting the cubes as length times width times height.

💡Takeaway. Make the units match before you do anything else. Then count along each edge and multiply.

Variant 11 hard

The box at the right is to be filled completely with cubes whose edges are $15\ \text{cm}$ long. How many cubes are needed? (Ignore the thickness of the box.)

Your answer

Show solution

1 · Understandwhat's really being asked

A box 9 m by 4.5 m by 3 m is packed full of cubes with 15 cm edges. We want how many cubes.

Givens

The box is 9 m long, 4.5 m deep and 3 m tall.
Each cube has an edge of 15 cm.

Unknowns

How many cubes fill the box.

Constraints

The cubes fill it completely, with no gaps and nothing sticking out.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #17 Visualize Spatial Relationships#10 Create a Physical Representation

The box is in metres and the cube in centimetres, so convert first -- nothing can be compared until they share a unit. Then count how many cubes fit along each edge, which keeps the picture attached to the arithmetic.

3 · Execute3 carry out the plan

1Put the box in centimetres

#8 Analyze the Units 6.G.A.2

A metre is 100 cm.

900 \times 450 \times 300\ \text{cm}

Now both are in the same unit.

2Count the cubes along each edge

#17 Visualize Spatial Relationships 5.MD.C.5

Each edge divided by the cube's 15 cm.

900 \div 15 = 60,\quad 450 \div 15 = 30,\quad 300 \div 15 = 20

60 by 30 by 20 cubes.

3Multiply the three counts

#10 Create a Physical Representation 5.MD.C.5

One layer is the first two multiplied, and there are as many layers as cubes up the height.

60 \times 30 \times 20 = 36{,}000

36,000 cubes.

Answer: 36,000 cubes

4 · Reviewdoes it hold up?

By volume instead: the box is 121,500,000 cm3 and each cube is 3375 cm3, and dividing gives the same 36,000.

Another way: Counting one layer first -- 1,800 cubes -- and multiplying by the 20 layers is the same product in a different order.

Standardsmin grade 6

6.G.A.2 Volume of right rectangular prisms with fractional edges — Filling a prism with unit cubes to get its volume.
5.MD.C.5 Relate volume to the operations of multiplication and addition — Counting the cubes as length times width times height.

💡Takeaway. Make the units match before you do anything else. Then count along each edge and multiply.

Variant 12 hard

The box at the right is to be filled completely with cubes whose edges are $20\ \text{cm}$ long. How many cubes are needed? (Ignore the thickness of the box.)

Your answer

Show solution

1 · Understandwhat's really being asked

A box 12 m by 8 m by 6 m is packed full of cubes with 20 cm edges. We want how many cubes.

Givens

The box is 12 m long, 8 m deep and 6 m tall.
Each cube has an edge of 20 cm.

Unknowns

How many cubes fill the box.

Constraints

The cubes fill it completely, with no gaps and nothing sticking out.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #17 Visualize Spatial Relationships#10 Create a Physical Representation

The box is in metres and the cube in centimetres, so convert first -- nothing can be compared until they share a unit. Then count how many cubes fit along each edge, which keeps the picture attached to the arithmetic.

3 · Execute3 carry out the plan

1Put the box in centimetres

#8 Analyze the Units 6.G.A.2

A metre is 100 cm.

1200 \times 800 \times 600\ \text{cm}

Now both are in the same unit.

2Count the cubes along each edge

#17 Visualize Spatial Relationships 5.MD.C.5

Each edge divided by the cube's 20 cm.

1200 \div 20 = 60,\quad 800 \div 20 = 40,\quad 600 \div 20 = 30

60 by 40 by 30 cubes.

3Multiply the three counts

#10 Create a Physical Representation 5.MD.C.5

One layer is the first two multiplied, and there are as many layers as cubes up the height.

60 \times 40 \times 30 = 72{,}000

72,000 cubes.

Answer: 72,000 cubes

4 · Reviewdoes it hold up?

By volume instead: the box is 576,000,000 cm3 and each cube is 8000 cm3, and dividing gives the same 72,000.

Another way: Counting one layer first -- 2,400 cubes -- and multiplying by the 30 layers is the same product in a different order.

Standardsmin grade 6

6.G.A.2 Volume of right rectangular prisms with fractional edges — Filling a prism with unit cubes to get its volume.
5.MD.C.5 Relate volume to the operations of multiplication and addition — Counting the cubes as length times width times height.

💡Takeaway. Make the units match before you do anything else. Then count along each edge and multiply.

Count cubes along each edge, not volume divided by volume · 12 practice problems

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