← Count the change a cut makes, not the finished picture · Net and Solid Structure

Count the change a cut makes, not the finished picture · 4 practice problems

7.G.A.36.G.A.4

Generated variants — 4

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

Variant 1 medium answer: 18

The solid shown is a rectangular prism with two of its corners cut off, each cut taking away a triangular pyramid. How many edges does this solid have?

Show solution
1 · Understandwhat's really being asked

A rectangular prism has two of its corners sliced off, each slice taking away a small triangular pyramid. We want the number of edges the remaining solid has.

Givens
  • The solid starts as a rectangular prism.
  • Two corners are cut off.
  • Each cut removes a triangular pyramid and leaves a triangular face.
Unknowns
  • The number of edges of the cut solid.
Constraints
  • The cuts do not meet each other, so no cut removes another cut's work.
2 · Planchoose the strategy

#16 Count the Complement · also uses: #17 Visualize Spatial Relationships#9 Solve an Easier Related Problem

Counting the finished picture is unreliable -- some of the cut corners face away from us. Count what one cut changes instead, then apply it as many times as there are cuts.

3 · Execute4 carry out the plan

1Start from the uncut box

#9 Solve an Easier Related Problem 6.G.A.4
A rectangular prism has 4 edges on the top, 4 on the bottom and 4 uprights.
4+4+4=124 + 4 + 4 = 12
12 edges before any cutting.

2See what one cut destroys

#17 Visualize Spatial Relationships 7.G.A.3
The blade passes across three edges near a corner. Each of those is shortened, but none of them disappears.
−0-0
Nothing is lost, only trimmed.

3See what one cut adds

#17 Visualize Spatial Relationships 7.G.A.3
The cut leaves a flat triangular face behind, and that triangle's three sides are new edges.
+3+3
Every cut is worth exactly three more.

4Apply it two times

#16 Count the Complement 6.G.A.4
The cuts are at different corners and do not interfere, so each one adds its own three.
12+3×2=1812 + 3 \times 2 = 18
The solid has 18 edges.
Answer: 18
4 · Reviewdoes it hold up?

Euler's rule agrees: 8 faces minus 18 edges plus 12 vertices is 2.

Another way: Counting the picture directly gives 18 too, if none of the 6 new edges on the far side is missed -- which is the part that is easy to get wrong.

Standardsmin grade 7
  • 7.G.A.3 Describe the two-dimensional figures that result from slicing three-dimensional figures — Reading the flat face a slice through a solid leaves behind.
  • 6.G.A.4 Surface area using nets of 3D figures — Counting the prism's own edges before anything is cut.
💡Takeaway. When a shape is changed a bit at a time, count what each change does instead of recounting the whole thing.
Variant 2 medium answer: 21

The solid shown is a rectangular prism with three of its corners cut off, each cut taking away a triangular pyramid. How many edges does this solid have?

Show solution
1 · Understandwhat's really being asked

A rectangular prism has three of its corners sliced off, each slice taking away a small triangular pyramid. We want the number of edges the remaining solid has.

Givens
  • The solid starts as a rectangular prism.
  • Three corners are cut off.
  • Each cut removes a triangular pyramid and leaves a triangular face.
Unknowns
  • The number of edges of the cut solid.
Constraints
  • The cuts do not meet each other, so no cut removes another cut's work.
2 · Planchoose the strategy

#16 Count the Complement · also uses: #17 Visualize Spatial Relationships#9 Solve an Easier Related Problem

Counting the finished picture is unreliable -- some of the cut corners face away from us. Count what one cut changes instead, then apply it as many times as there are cuts.

3 · Execute4 carry out the plan

1Start from the uncut box

#9 Solve an Easier Related Problem 6.G.A.4
A rectangular prism has 4 edges on the top, 4 on the bottom and 4 uprights.
4+4+4=124 + 4 + 4 = 12
12 edges before any cutting.

2See what one cut destroys

#17 Visualize Spatial Relationships 7.G.A.3
The blade passes across three edges near a corner. Each of those is shortened, but none of them disappears.
−0-0
Nothing is lost, only trimmed.

3See what one cut adds

#17 Visualize Spatial Relationships 7.G.A.3
The cut leaves a flat triangular face behind, and that triangle's three sides are new edges.
+3+3
Every cut is worth exactly three more.

4Apply it three times

#16 Count the Complement 6.G.A.4
The cuts are at different corners and do not interfere, so each one adds its own three.
12+3×3=2112 + 3 \times 3 = 21
The solid has 21 edges.
Answer: 21
4 · Reviewdoes it hold up?

Euler's rule agrees: 9 faces minus 21 edges plus 14 vertices is 2.

Another way: Counting the picture directly gives 21 too, if none of the 9 new edges on the far side is missed -- which is the part that is easy to get wrong.

Standardsmin grade 7
  • 7.G.A.3 Describe the two-dimensional figures that result from slicing three-dimensional figures — Reading the flat face a slice through a solid leaves behind.
  • 6.G.A.4 Surface area using nets of 3D figures — Counting the prism's own edges before anything is cut.
💡Takeaway. When a shape is changed a bit at a time, count what each change does instead of recounting the whole thing.
Variant 3 medium answer: 24

The solid shown is a rectangular prism with four of its corners cut off, each cut taking away a triangular pyramid. How many edges does this solid have?

Show solution
1 · Understandwhat's really being asked

A rectangular prism has four of its corners sliced off, each slice taking away a small triangular pyramid. We want the number of edges the remaining solid has.

Givens
  • The solid starts as a rectangular prism.
  • Four corners are cut off.
  • Each cut removes a triangular pyramid and leaves a triangular face.
Unknowns
  • The number of edges of the cut solid.
Constraints
  • The cuts do not meet each other, so no cut removes another cut's work.
2 · Planchoose the strategy

#16 Count the Complement · also uses: #17 Visualize Spatial Relationships#9 Solve an Easier Related Problem

Counting the finished picture is unreliable -- some of the cut corners face away from us. Count what one cut changes instead, then apply it as many times as there are cuts.

3 · Execute4 carry out the plan

1Start from the uncut box

#9 Solve an Easier Related Problem 6.G.A.4
A rectangular prism has 4 edges on the top, 4 on the bottom and 4 uprights.
4+4+4=124 + 4 + 4 = 12
12 edges before any cutting.

2See what one cut destroys

#17 Visualize Spatial Relationships 7.G.A.3
The blade passes across three edges near a corner. Each of those is shortened, but none of them disappears.
−0-0
Nothing is lost, only trimmed.

3See what one cut adds

#17 Visualize Spatial Relationships 7.G.A.3
The cut leaves a flat triangular face behind, and that triangle's three sides are new edges.
+3+3
Every cut is worth exactly three more.

4Apply it four times

#16 Count the Complement 6.G.A.4
The cuts are at different corners and do not interfere, so each one adds its own three.
12+3×4=2412 + 3 \times 4 = 24
The solid has 24 edges.
Answer: 24
4 · Reviewdoes it hold up?

Euler's rule agrees: 10 faces minus 24 edges plus 16 vertices is 2.

Another way: Counting the picture directly gives 24 too, if none of the 12 new edges on the far side is missed -- which is the part that is easy to get wrong.

Standardsmin grade 7
  • 7.G.A.3 Describe the two-dimensional figures that result from slicing three-dimensional figures — Reading the flat face a slice through a solid leaves behind.
  • 6.G.A.4 Surface area using nets of 3D figures — Counting the prism's own edges before anything is cut.
💡Takeaway. When a shape is changed a bit at a time, count what each change does instead of recounting the whole thing.
Variant 4 medium answer: 15

The solid shown is a rectangular prism with one of its corners cut off, each cut taking away a triangular pyramid. How many edges does this solid have?

Show solution
1 · Understandwhat's really being asked

A rectangular prism has one of its corners sliced off, each slice taking away a small triangular pyramid. We want the number of edges the remaining solid has.

Givens
  • The solid starts as a rectangular prism.
  • One corner is cut off.
  • Each cut removes a triangular pyramid and leaves a triangular face.
Unknowns
  • The number of edges of the cut solid.
Constraints
  • The cuts do not meet each other, so no cut removes another cut's work.
2 · Planchoose the strategy

#16 Count the Complement · also uses: #17 Visualize Spatial Relationships#9 Solve an Easier Related Problem

Counting the finished picture is unreliable -- some of the cut corners face away from us. Count what one cut changes instead, then apply it as many times as there are cuts.

3 · Execute4 carry out the plan

1Start from the uncut box

#9 Solve an Easier Related Problem 6.G.A.4
A rectangular prism has 4 edges on the top, 4 on the bottom and 4 uprights.
4+4+4=124 + 4 + 4 = 12
12 edges before any cutting.

2See what one cut destroys

#17 Visualize Spatial Relationships 7.G.A.3
The blade passes across three edges near a corner. Each of those is shortened, but none of them disappears.
−0-0
Nothing is lost, only trimmed.

3See what one cut adds

#17 Visualize Spatial Relationships 7.G.A.3
The cut leaves a flat triangular face behind, and that triangle's three sides are new edges.
+3+3
Every cut is worth exactly three more.

4Apply it one time

#16 Count the Complement 6.G.A.4
The cuts are at different corners and do not interfere, so each one adds its own three.
12+3×1=1512 + 3 \times 1 = 15
The solid has 15 edges.
Answer: 15
4 · Reviewdoes it hold up?

Euler's rule agrees: 7 faces minus 15 edges plus 10 vertices is 2.

Another way: Counting the picture directly gives 15 too, if none of the 3 new edges on the far side is missed -- which is the part that is easy to get wrong.

Standardsmin grade 7
  • 7.G.A.3 Describe the two-dimensional figures that result from slicing three-dimensional figures — Reading the flat face a slice through a solid leaves behind.
  • 6.G.A.4 Surface area using nets of 3D figures — Counting the prism's own edges before anything is cut.
💡Takeaway. When a shape is changed a bit at a time, count what each change does instead of recounting the whole thing.