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

Variant 1 easy answer: 648,000 cubes

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.)

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 answer: 480,000 cubes

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.)

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 answer: 280,000 cubes

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.)

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 answer: 60,000 cubes

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.)

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 answer: 480,000 cubes

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.)

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 answer: 375,000 cubes

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.)

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 answer: 192,000 cubes

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.)

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 answer: 192,000 cubes

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.)

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 answer: 1,000,000 cubes

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.)

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 answer: 24,000 cubes

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.)

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 answer: 36,000 cubes

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.)

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 answer: 72,000 cubes

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.)

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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