← Fold a strip of four to find opposite faces · Net and Solid Structure

Fold a strip of four to find opposite faces · 12 practice problems

6.G.A.4

Generated variants — 12

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

Variant 1 medium answer: C

On a die, the two faces that are parallel to each other have pip counts that add up to 77. In the net of the die shown at the right, among the faces labeled A, B, and C, write the label of the face that gets the largest number of pips. (A die is a cube.)

The net is shaped like a cross/staircase. Four faces run in a single horizontal row; from left to right they are A, B, the face with 55 pips, and the face with 66 pips. The face with 44 pips is attached above B, and the face below A is C.

A B C
Show solution
1 · Understandwhat's really being asked

A net of a die has three faces already showing 5, 6 and 4 pips, and three faces labelled A, B and C. We need which of A, B, C ends up with the largest number of pips.

Givens
  • Opposite faces of a die add to 7.
  • In the row of four, the third face shows 5 and the fourth shows 6.
  • The flap above B shows 4.
Unknowns
  • Which of A, B and C has the largest pip count.
Constraints
  • Every face of a die shows between 1 and 6 pips, all different.
2 · Planchoose the strategy

#10 Create a Physical Representation · also uses: #17 Visualize Spatial Relationships#1 Draw a Diagram

Fold the net in your head -- or on paper -- to see which squares land opposite each other. Once the pairs are known, the add-to-7 rule fills in all three unknown faces and they can just be compared.

3 · Execute4 carry out the plan

1Fold the strip of four into a band

#10 Create a Physical Representation 6.G.A.4
The four faces A, B, 5, 6 are in one straight row. A strip of four squares wraps all the way around the cube and its two ends join, so the 1st and 3rd squares land opposite each other, and so do the 2nd and 4th.
A opposite the third square, B opposite the fourth.

2Find A and B using the sum-7 rule

#17 Visualize Spatial Relationships 6.G.A.4
A is opposite the face with 5 pips, so A and 5 must add to 7. That makes A = 2. B is opposite the face with 6 pips, so B = 7 - 6 = 1.
A=7−5=2,B=7−6=1A = 7 - 5 = 2, \quad B = 7 - 6 = 1
A shows 2, B shows 1.

3Find C from the last pair of faces

#1 Draw a Diagram 6.G.A.4
The two flaps that are not in the main strip are the 4-face (above B) and C (below A). Those two become the top and bottom of the cube, so they are the last opposite pair.
C=7−4=3C = 7 - 4 = 3
C shows 3.

4Compare A, B, and C

#17 Visualize Spatial Relationships 6.G.A.4
Now compare the three numbers: A = 2, B = 1, C = 3. The largest is 3, which is on face C.
C=3>A=2>B=1C = 3 > A = 2 > B = 1
Face C it is.
Answer: C
4 · Reviewdoes it hold up?

All six faces together are 5, 6, 4, 2, 1 and 3 -- the numbers 1 to 6 with none missing and none repeated, which is what a die must show.

Another way: Because opposite faces add to 7, the largest of A, B and C is opposite the smallest of the three given faces. The order of A, B, C is C > A > B either way.

Standardsmin grade 6
  • 6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles — Folding the net to find which faces end up opposite.
💡Takeaway. Fold a strip of four squares and the ends meet, so every other square in the strip is opposite its neighbour's neighbour.
Variant 2 medium answer: A

On a die, the two faces that are parallel to each other have pip counts that add up to 77. In the net of the die shown at the right, among the faces labeled A, B, and C, write the label of the face that gets the largest number of pips. (A die is a cube.)

The net is shaped like a cross/staircase. Four faces run in a single horizontal row; from left to right they are A, B, the face with 11 pips, and the face with 22 pips. The face with 33 pips is attached above B, and the face below A is C.

A B C
Show solution
1 · Understandwhat's really being asked

A net of a die has three faces already showing 1, 2 and 3 pips, and three faces labelled A, B and C. We need which of A, B, C ends up with the largest number of pips.

Givens
  • Opposite faces of a die add to 7.
  • In the row of four, the third face shows 1 and the fourth shows 2.
  • The flap above B shows 3.
Unknowns
  • Which of A, B and C has the largest pip count.
Constraints
  • Every face of a die shows between 1 and 6 pips, all different.
2 · Planchoose the strategy

#10 Create a Physical Representation · also uses: #17 Visualize Spatial Relationships#1 Draw a Diagram

Fold the net in your head -- or on paper -- to see which squares land opposite each other. Once the pairs are known, the add-to-7 rule fills in all three unknown faces and they can just be compared.

3 · Execute4 carry out the plan

1Fold the strip of four into a band

#10 Create a Physical Representation 6.G.A.4
The four faces A, B, 1, 2 are in one straight row. A strip of four squares wraps all the way around the cube and its two ends join, so the 1st and 3rd squares land opposite each other, and so do the 2nd and 4th.
A opposite the third square, B opposite the fourth.

2Find A and B using the sum-7 rule

#17 Visualize Spatial Relationships 6.G.A.4
A is opposite the face with 1 pips, so A and 1 must add to 7. That makes A = 6. B is opposite the face with 2 pips, so B = 7 - 2 = 5.
A=7−1=6,B=7−2=5A = 7 - 1 = 6, \quad B = 7 - 2 = 5
A shows 6, B shows 5.

3Find C from the last pair of faces

#1 Draw a Diagram 6.G.A.4
The two flaps that are not in the main strip are the 3-face (above B) and C (below A). Those two become the top and bottom of the cube, so they are the last opposite pair.
C=7−3=4C = 7 - 3 = 4
C shows 4.

4Compare A, B, and C

#17 Visualize Spatial Relationships 6.G.A.4
Now compare the three numbers: A = 6, B = 5, C = 4. The largest is 6, which is on face A.
A=6>B=5>C=4A = 6 > B = 5 > C = 4
Face A it is.
Answer: A
4 · Reviewdoes it hold up?

All six faces together are 1, 2, 3, 6, 5 and 4 -- the numbers 1 to 6 with none missing and none repeated, which is what a die must show.

Another way: Because opposite faces add to 7, the largest of A, B and C is opposite the smallest of the three given faces. The order of A, B, C is A > B > C either way.

Standardsmin grade 6
  • 6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles — Folding the net to find which faces end up opposite.
💡Takeaway. Fold a strip of four squares and the ends meet, so every other square in the strip is opposite its neighbour's neighbour.
Variant 3 medium answer: A

On a die, the two faces that are parallel to each other have pip counts that add up to 77. In the net of the die shown at the right, among the faces labeled A, B, and C, write the label of the face that gets the smallest number of pips. (A die is a cube.)

The net is shaped like a cross/staircase. Four faces run in a single horizontal row; from left to right they are A, B, the face with 44 pips, and the face with 11 pips. The face with 22 pips is attached above B, and the face below A is C.

A B C
Show solution
1 · Understandwhat's really being asked

A net of a die has three faces already showing 4, 1 and 2 pips, and three faces labelled A, B and C. We need which of A, B, C ends up with the smallest number of pips.

Givens
  • Opposite faces of a die add to 7.
  • In the row of four, the third face shows 4 and the fourth shows 1.
  • The flap above B shows 2.
Unknowns
  • Which of A, B and C has the smallest pip count.
Constraints
  • Every face of a die shows between 1 and 6 pips, all different.
2 · Planchoose the strategy

#10 Create a Physical Representation · also uses: #17 Visualize Spatial Relationships#1 Draw a Diagram

Fold the net in your head -- or on paper -- to see which squares land opposite each other. Once the pairs are known, the add-to-7 rule fills in all three unknown faces and they can just be compared.

3 · Execute4 carry out the plan

1Fold the strip of four into a band

#10 Create a Physical Representation 6.G.A.4
The four faces A, B, 4, 1 are in one straight row. A strip of four squares wraps all the way around the cube and its two ends join, so the 1st and 3rd squares land opposite each other, and so do the 2nd and 4th.
A opposite the third square, B opposite the fourth.

2Find A and B using the sum-7 rule

#17 Visualize Spatial Relationships 6.G.A.4
A is opposite the face with 4 pips, so A and 4 must add to 7. That makes A = 3. B is opposite the face with 1 pips, so B = 7 - 1 = 6.
A=7−4=3,B=7−1=6A = 7 - 4 = 3, \quad B = 7 - 1 = 6
A shows 3, B shows 6.

3Find C from the last pair of faces

#1 Draw a Diagram 6.G.A.4
The two flaps that are not in the main strip are the 2-face (above B) and C (below A). Those two become the top and bottom of the cube, so they are the last opposite pair.
C=7−2=5C = 7 - 2 = 5
C shows 5.

4Compare A, B, and C

#17 Visualize Spatial Relationships 6.G.A.4
Now compare the three numbers: A = 3, B = 6, C = 5. The smallest is 3, which is on face A.
B=6>C=5>A=3B = 6 > C = 5 > A = 3
Face A it is.
Answer: A
4 · Reviewdoes it hold up?

All six faces together are 4, 1, 2, 3, 6 and 5 -- the numbers 1 to 6 with none missing and none repeated, which is what a die must show.

Another way: Because opposite faces add to 7, the smallest of A, B and C is opposite the largest of the three given faces. The order of A, B, C is B > C > A either way.

Standardsmin grade 6
  • 6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles — Folding the net to find which faces end up opposite.
💡Takeaway. Fold a strip of four squares and the ends meet, so every other square in the strip is opposite its neighbour's neighbour.
Variant 4 medium answer: B

On a die, the two faces that are parallel to each other have pip counts that add up to 77. In the net of the die shown at the right, among the faces labeled A, B, and C, write the label of the face that gets the smallest number of pips. (A die is a cube.)

The net is shaped like a cross/staircase. Four faces run in a single horizontal row; from left to right they are A, B, the face with 33 pips, and the face with 55 pips. The face with 11 pips is attached above B, and the face below A is C.

A B C
Show solution
1 · Understandwhat's really being asked

A net of a die has three faces already showing 3, 5 and 1 pips, and three faces labelled A, B and C. We need which of A, B, C ends up with the smallest number of pips.

Givens
  • Opposite faces of a die add to 7.
  • In the row of four, the third face shows 3 and the fourth shows 5.
  • The flap above B shows 1.
Unknowns
  • Which of A, B and C has the smallest pip count.
Constraints
  • Every face of a die shows between 1 and 6 pips, all different.
2 · Planchoose the strategy

#10 Create a Physical Representation · also uses: #17 Visualize Spatial Relationships#1 Draw a Diagram

Fold the net in your head -- or on paper -- to see which squares land opposite each other. Once the pairs are known, the add-to-7 rule fills in all three unknown faces and they can just be compared.

3 · Execute4 carry out the plan

1Fold the strip of four into a band

#10 Create a Physical Representation 6.G.A.4
The four faces A, B, 3, 5 are in one straight row. A strip of four squares wraps all the way around the cube and its two ends join, so the 1st and 3rd squares land opposite each other, and so do the 2nd and 4th.
A opposite the third square, B opposite the fourth.

2Find A and B using the sum-7 rule

#17 Visualize Spatial Relationships 6.G.A.4
A is opposite the face with 3 pips, so A and 3 must add to 7. That makes A = 4. B is opposite the face with 5 pips, so B = 7 - 5 = 2.
A=7−3=4,B=7−5=2A = 7 - 3 = 4, \quad B = 7 - 5 = 2
A shows 4, B shows 2.

3Find C from the last pair of faces

#1 Draw a Diagram 6.G.A.4
The two flaps that are not in the main strip are the 1-face (above B) and C (below A). Those two become the top and bottom of the cube, so they are the last opposite pair.
C=7−1=6C = 7 - 1 = 6
C shows 6.

4Compare A, B, and C

#17 Visualize Spatial Relationships 6.G.A.4
Now compare the three numbers: A = 4, B = 2, C = 6. The smallest is 2, which is on face B.
C=6>A=4>B=2C = 6 > A = 4 > B = 2
Face B it is.
Answer: B
4 · Reviewdoes it hold up?

All six faces together are 3, 5, 1, 4, 2 and 6 -- the numbers 1 to 6 with none missing and none repeated, which is what a die must show.

Another way: Because opposite faces add to 7, the smallest of A, B and C is opposite the largest of the three given faces. The order of A, B, C is C > A > B either way.

Standardsmin grade 6
  • 6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles — Folding the net to find which faces end up opposite.
💡Takeaway. Fold a strip of four squares and the ends meet, so every other square in the strip is opposite its neighbour's neighbour.
Variant 5 medium answer: B

On a die, the two faces that are parallel to each other have pip counts that add up to 77. In the net of the die shown at the right, among the faces labeled A, B, and C, write the label of the face that gets the largest number of pips. (A die is a cube.)

The net is shaped like a cross/staircase. Four faces run in a single horizontal row; from left to right they are A, B, the face with 55 pips, and the face with 33 pips. The face with 66 pips is attached above B, and the face below A is C.

A B C
Show solution
1 · Understandwhat's really being asked

A net of a die has three faces already showing 5, 3 and 6 pips, and three faces labelled A, B and C. We need which of A, B, C ends up with the largest number of pips.

Givens
  • Opposite faces of a die add to 7.
  • In the row of four, the third face shows 5 and the fourth shows 3.
  • The flap above B shows 6.
Unknowns
  • Which of A, B and C has the largest pip count.
Constraints
  • Every face of a die shows between 1 and 6 pips, all different.
2 · Planchoose the strategy

#10 Create a Physical Representation · also uses: #17 Visualize Spatial Relationships#1 Draw a Diagram

Fold the net in your head -- or on paper -- to see which squares land opposite each other. Once the pairs are known, the add-to-7 rule fills in all three unknown faces and they can just be compared.

3 · Execute4 carry out the plan

1Fold the strip of four into a band

#10 Create a Physical Representation 6.G.A.4
The four faces A, B, 5, 3 are in one straight row. A strip of four squares wraps all the way around the cube and its two ends join, so the 1st and 3rd squares land opposite each other, and so do the 2nd and 4th.
A opposite the third square, B opposite the fourth.

2Find A and B using the sum-7 rule

#17 Visualize Spatial Relationships 6.G.A.4
A is opposite the face with 5 pips, so A and 5 must add to 7. That makes A = 2. B is opposite the face with 3 pips, so B = 7 - 3 = 4.
A=7−5=2,B=7−3=4A = 7 - 5 = 2, \quad B = 7 - 3 = 4
A shows 2, B shows 4.

3Find C from the last pair of faces

#1 Draw a Diagram 6.G.A.4
The two flaps that are not in the main strip are the 6-face (above B) and C (below A). Those two become the top and bottom of the cube, so they are the last opposite pair.
C=7−6=1C = 7 - 6 = 1
C shows 1.

4Compare A, B, and C

#17 Visualize Spatial Relationships 6.G.A.4
Now compare the three numbers: A = 2, B = 4, C = 1. The largest is 4, which is on face B.
B=4>A=2>C=1B = 4 > A = 2 > C = 1
Face B it is.
Answer: B
4 · Reviewdoes it hold up?

All six faces together are 5, 3, 6, 2, 4 and 1 -- the numbers 1 to 6 with none missing and none repeated, which is what a die must show.

Another way: Because opposite faces add to 7, the largest of A, B and C is opposite the smallest of the three given faces. The order of A, B, C is B > A > C either way.

Standardsmin grade 6
  • 6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles — Folding the net to find which faces end up opposite.
💡Takeaway. Fold a strip of four squares and the ends meet, so every other square in the strip is opposite its neighbour's neighbour.
Variant 6 medium answer: C

On a die, the two faces that are parallel to each other have pip counts that add up to 77. In the net of the die shown at the right, among the faces labeled A, B, and C, write the label of the face that gets the largest number of pips. (A die is a cube.)

The net is shaped like a cross/staircase. Four faces run in a single horizontal row; from left to right they are A, B, the face with 44 pips, and the face with 66 pips. The face with 22 pips is attached above B, and the face below A is C.

A B C
Show solution
1 · Understandwhat's really being asked

A net of a die has three faces already showing 4, 6 and 2 pips, and three faces labelled A, B and C. We need which of A, B, C ends up with the largest number of pips.

Givens
  • Opposite faces of a die add to 7.
  • In the row of four, the third face shows 4 and the fourth shows 6.
  • The flap above B shows 2.
Unknowns
  • Which of A, B and C has the largest pip count.
Constraints
  • Every face of a die shows between 1 and 6 pips, all different.
2 · Planchoose the strategy

#10 Create a Physical Representation · also uses: #17 Visualize Spatial Relationships#1 Draw a Diagram

Fold the net in your head -- or on paper -- to see which squares land opposite each other. Once the pairs are known, the add-to-7 rule fills in all three unknown faces and they can just be compared.

3 · Execute4 carry out the plan

1Fold the strip of four into a band

#10 Create a Physical Representation 6.G.A.4
The four faces A, B, 4, 6 are in one straight row. A strip of four squares wraps all the way around the cube and its two ends join, so the 1st and 3rd squares land opposite each other, and so do the 2nd and 4th.
A opposite the third square, B opposite the fourth.

2Find A and B using the sum-7 rule

#17 Visualize Spatial Relationships 6.G.A.4
A is opposite the face with 4 pips, so A and 4 must add to 7. That makes A = 3. B is opposite the face with 6 pips, so B = 7 - 6 = 1.
A=7−4=3,B=7−6=1A = 7 - 4 = 3, \quad B = 7 - 6 = 1
A shows 3, B shows 1.

3Find C from the last pair of faces

#1 Draw a Diagram 6.G.A.4
The two flaps that are not in the main strip are the 2-face (above B) and C (below A). Those two become the top and bottom of the cube, so they are the last opposite pair.
C=7−2=5C = 7 - 2 = 5
C shows 5.

4Compare A, B, and C

#17 Visualize Spatial Relationships 6.G.A.4
Now compare the three numbers: A = 3, B = 1, C = 5. The largest is 5, which is on face C.
C=5>A=3>B=1C = 5 > A = 3 > B = 1
Face C it is.
Answer: C
4 · Reviewdoes it hold up?

All six faces together are 4, 6, 2, 3, 1 and 5 -- the numbers 1 to 6 with none missing and none repeated, which is what a die must show.

Another way: Because opposite faces add to 7, the largest of A, B and C is opposite the smallest of the three given faces. The order of A, B, C is C > A > B either way.

Standardsmin grade 6
  • 6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles — Folding the net to find which faces end up opposite.
💡Takeaway. Fold a strip of four squares and the ends meet, so every other square in the strip is opposite its neighbour's neighbour.
Variant 7 medium answer: A

On a die, the two faces that are parallel to each other have pip counts that add up to 77. In the net of the die shown at the right, among the faces labeled A, B, and C, write the label of the face that gets the smallest number of pips. (A die is a cube.)

The net is shaped like a cross/staircase. Four faces run in a single horizontal row; from left to right they are A, B, the face with 66 pips, and the face with 44 pips. The face with 55 pips is attached above B, and the face below A is C.

A B C
Show solution
1 · Understandwhat's really being asked

A net of a die has three faces already showing 6, 4 and 5 pips, and three faces labelled A, B and C. We need which of A, B, C ends up with the smallest number of pips.

Givens
  • Opposite faces of a die add to 7.
  • In the row of four, the third face shows 6 and the fourth shows 4.
  • The flap above B shows 5.
Unknowns
  • Which of A, B and C has the smallest pip count.
Constraints
  • Every face of a die shows between 1 and 6 pips, all different.
2 · Planchoose the strategy

#10 Create a Physical Representation · also uses: #17 Visualize Spatial Relationships#1 Draw a Diagram

Fold the net in your head -- or on paper -- to see which squares land opposite each other. Once the pairs are known, the add-to-7 rule fills in all three unknown faces and they can just be compared.

3 · Execute4 carry out the plan

1Fold the strip of four into a band

#10 Create a Physical Representation 6.G.A.4
The four faces A, B, 6, 4 are in one straight row. A strip of four squares wraps all the way around the cube and its two ends join, so the 1st and 3rd squares land opposite each other, and so do the 2nd and 4th.
A opposite the third square, B opposite the fourth.

2Find A and B using the sum-7 rule

#17 Visualize Spatial Relationships 6.G.A.4
A is opposite the face with 6 pips, so A and 6 must add to 7. That makes A = 1. B is opposite the face with 4 pips, so B = 7 - 4 = 3.
A=7−6=1,B=7−4=3A = 7 - 6 = 1, \quad B = 7 - 4 = 3
A shows 1, B shows 3.

3Find C from the last pair of faces

#1 Draw a Diagram 6.G.A.4
The two flaps that are not in the main strip are the 5-face (above B) and C (below A). Those two become the top and bottom of the cube, so they are the last opposite pair.
C=7−5=2C = 7 - 5 = 2
C shows 2.

4Compare A, B, and C

#17 Visualize Spatial Relationships 6.G.A.4
Now compare the three numbers: A = 1, B = 3, C = 2. The smallest is 1, which is on face A.
B=3>C=2>A=1B = 3 > C = 2 > A = 1
Face A it is.
Answer: A
4 · Reviewdoes it hold up?

All six faces together are 6, 4, 5, 1, 3 and 2 -- the numbers 1 to 6 with none missing and none repeated, which is what a die must show.

Another way: Because opposite faces add to 7, the smallest of A, B and C is opposite the largest of the three given faces. The order of A, B, C is B > C > A either way.

Standardsmin grade 6
  • 6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles — Folding the net to find which faces end up opposite.
💡Takeaway. Fold a strip of four squares and the ends meet, so every other square in the strip is opposite its neighbour's neighbour.
Variant 8 medium answer: C

On a die, the two faces that are parallel to each other have pip counts that add up to 77. In the net of the die shown at the right, among the faces labeled A, B, and C, write the label of the face that gets the smallest number of pips. (A die is a cube.)

The net is shaped like a cross/staircase. Four faces run in a single horizontal row; from left to right they are A, B, the face with 11 pips, and the face with 22 pips. The face with 33 pips is attached above B, and the face below A is C.

A B C
Show solution
1 · Understandwhat's really being asked

A net of a die has three faces already showing 1, 2 and 3 pips, and three faces labelled A, B and C. We need which of A, B, C ends up with the smallest number of pips.

Givens
  • Opposite faces of a die add to 7.
  • In the row of four, the third face shows 1 and the fourth shows 2.
  • The flap above B shows 3.
Unknowns
  • Which of A, B and C has the smallest pip count.
Constraints
  • Every face of a die shows between 1 and 6 pips, all different.
2 · Planchoose the strategy

#10 Create a Physical Representation · also uses: #17 Visualize Spatial Relationships#1 Draw a Diagram

Fold the net in your head -- or on paper -- to see which squares land opposite each other. Once the pairs are known, the add-to-7 rule fills in all three unknown faces and they can just be compared.

3 · Execute4 carry out the plan

1Fold the strip of four into a band

#10 Create a Physical Representation 6.G.A.4
The four faces A, B, 1, 2 are in one straight row. A strip of four squares wraps all the way around the cube and its two ends join, so the 1st and 3rd squares land opposite each other, and so do the 2nd and 4th.
A opposite the third square, B opposite the fourth.

2Find A and B using the sum-7 rule

#17 Visualize Spatial Relationships 6.G.A.4
A is opposite the face with 1 pips, so A and 1 must add to 7. That makes A = 6. B is opposite the face with 2 pips, so B = 7 - 2 = 5.
A=7−1=6,B=7−2=5A = 7 - 1 = 6, \quad B = 7 - 2 = 5
A shows 6, B shows 5.

3Find C from the last pair of faces

#1 Draw a Diagram 6.G.A.4
The two flaps that are not in the main strip are the 3-face (above B) and C (below A). Those two become the top and bottom of the cube, so they are the last opposite pair.
C=7−3=4C = 7 - 3 = 4
C shows 4.

4Compare A, B, and C

#17 Visualize Spatial Relationships 6.G.A.4
Now compare the three numbers: A = 6, B = 5, C = 4. The smallest is 4, which is on face C.
A=6>B=5>C=4A = 6 > B = 5 > C = 4
Face C it is.
Answer: C
4 · Reviewdoes it hold up?

All six faces together are 1, 2, 3, 6, 5 and 4 -- the numbers 1 to 6 with none missing and none repeated, which is what a die must show.

Another way: Because opposite faces add to 7, the smallest of A, B and C is opposite the largest of the three given faces. The order of A, B, C is A > B > C either way.

Standardsmin grade 6
  • 6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles — Folding the net to find which faces end up opposite.
💡Takeaway. Fold a strip of four squares and the ends meet, so every other square in the strip is opposite its neighbour's neighbour.
Variant 9 medium answer: B

On a die, the two faces that are parallel to each other have pip counts that add up to 77. In the net of the die shown at the right, among the faces labeled A, B, and C, write the label of the face that gets the smallest number of pips. (A die is a cube.)

The net is shaped like a cross/staircase. Four faces run in a single horizontal row; from left to right they are A, B, the face with 22 pips, and the face with 33 pips. The face with 11 pips is attached above B, and the face below A is C.

A B C
Show solution
1 · Understandwhat's really being asked

A net of a die has three faces already showing 2, 3 and 1 pips, and three faces labelled A, B and C. We need which of A, B, C ends up with the smallest number of pips.

Givens
  • Opposite faces of a die add to 7.
  • In the row of four, the third face shows 2 and the fourth shows 3.
  • The flap above B shows 1.
Unknowns
  • Which of A, B and C has the smallest pip count.
Constraints
  • Every face of a die shows between 1 and 6 pips, all different.
2 · Planchoose the strategy

#10 Create a Physical Representation · also uses: #17 Visualize Spatial Relationships#1 Draw a Diagram

Fold the net in your head -- or on paper -- to see which squares land opposite each other. Once the pairs are known, the add-to-7 rule fills in all three unknown faces and they can just be compared.

3 · Execute4 carry out the plan

1Fold the strip of four into a band

#10 Create a Physical Representation 6.G.A.4
The four faces A, B, 2, 3 are in one straight row. A strip of four squares wraps all the way around the cube and its two ends join, so the 1st and 3rd squares land opposite each other, and so do the 2nd and 4th.
A opposite the third square, B opposite the fourth.

2Find A and B using the sum-7 rule

#17 Visualize Spatial Relationships 6.G.A.4
A is opposite the face with 2 pips, so A and 2 must add to 7. That makes A = 5. B is opposite the face with 3 pips, so B = 7 - 3 = 4.
A=7−2=5,B=7−3=4A = 7 - 2 = 5, \quad B = 7 - 3 = 4
A shows 5, B shows 4.

3Find C from the last pair of faces

#1 Draw a Diagram 6.G.A.4
The two flaps that are not in the main strip are the 1-face (above B) and C (below A). Those two become the top and bottom of the cube, so they are the last opposite pair.
C=7−1=6C = 7 - 1 = 6
C shows 6.

4Compare A, B, and C

#17 Visualize Spatial Relationships 6.G.A.4
Now compare the three numbers: A = 5, B = 4, C = 6. The smallest is 4, which is on face B.
C=6>A=5>B=4C = 6 > A = 5 > B = 4
Face B it is.
Answer: B
4 · Reviewdoes it hold up?

All six faces together are 2, 3, 1, 5, 4 and 6 -- the numbers 1 to 6 with none missing and none repeated, which is what a die must show.

Another way: Because opposite faces add to 7, the smallest of A, B and C is opposite the largest of the three given faces. The order of A, B, C is C > A > B either way.

Standardsmin grade 6
  • 6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles — Folding the net to find which faces end up opposite.
💡Takeaway. Fold a strip of four squares and the ends meet, so every other square in the strip is opposite its neighbour's neighbour.
Variant 10 medium answer: C

On a die, the two faces that are parallel to each other have pip counts that add up to 77. In the net of the die shown at the right, among the faces labeled A, B, and C, write the label of the face that gets the largest number of pips. (A die is a cube.)

The net is shaped like a cross/staircase. Four faces run in a single horizontal row; from left to right they are A, B, the face with 66 pips, and the face with 55 pips. The face with 33 pips is attached above B, and the face below A is C.

A B C
Show solution
1 · Understandwhat's really being asked

A net of a die has three faces already showing 6, 5 and 3 pips, and three faces labelled A, B and C. We need which of A, B, C ends up with the largest number of pips.

Givens
  • Opposite faces of a die add to 7.
  • In the row of four, the third face shows 6 and the fourth shows 5.
  • The flap above B shows 3.
Unknowns
  • Which of A, B and C has the largest pip count.
Constraints
  • Every face of a die shows between 1 and 6 pips, all different.
2 · Planchoose the strategy

#10 Create a Physical Representation · also uses: #17 Visualize Spatial Relationships#1 Draw a Diagram

Fold the net in your head -- or on paper -- to see which squares land opposite each other. Once the pairs are known, the add-to-7 rule fills in all three unknown faces and they can just be compared.

3 · Execute4 carry out the plan

1Fold the strip of four into a band

#10 Create a Physical Representation 6.G.A.4
The four faces A, B, 6, 5 are in one straight row. A strip of four squares wraps all the way around the cube and its two ends join, so the 1st and 3rd squares land opposite each other, and so do the 2nd and 4th.
A opposite the third square, B opposite the fourth.

2Find A and B using the sum-7 rule

#17 Visualize Spatial Relationships 6.G.A.4
A is opposite the face with 6 pips, so A and 6 must add to 7. That makes A = 1. B is opposite the face with 5 pips, so B = 7 - 5 = 2.
A=7−6=1,B=7−5=2A = 7 - 6 = 1, \quad B = 7 - 5 = 2
A shows 1, B shows 2.

3Find C from the last pair of faces

#1 Draw a Diagram 6.G.A.4
The two flaps that are not in the main strip are the 3-face (above B) and C (below A). Those two become the top and bottom of the cube, so they are the last opposite pair.
C=7−3=4C = 7 - 3 = 4
C shows 4.

4Compare A, B, and C

#17 Visualize Spatial Relationships 6.G.A.4
Now compare the three numbers: A = 1, B = 2, C = 4. The largest is 4, which is on face C.
C=4>B=2>A=1C = 4 > B = 2 > A = 1
Face C it is.
Answer: C
4 · Reviewdoes it hold up?

All six faces together are 6, 5, 3, 1, 2 and 4 -- the numbers 1 to 6 with none missing and none repeated, which is what a die must show.

Another way: Because opposite faces add to 7, the largest of A, B and C is opposite the smallest of the three given faces. The order of A, B, C is C > B > A either way.

Standardsmin grade 6
  • 6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles — Folding the net to find which faces end up opposite.
💡Takeaway. Fold a strip of four squares and the ends meet, so every other square in the strip is opposite its neighbour's neighbour.
Variant 11 medium answer: C

On a die, the two faces that are parallel to each other have pip counts that add up to 77. In the net of the die shown at the right, among the faces labeled A, B, and C, write the label of the face that gets the smallest number of pips. (A die is a cube.)

The net is shaped like a cross/staircase. Four faces run in a single horizontal row; from left to right they are A, B, the face with 33 pips, and the face with 22 pips. The face with 66 pips is attached above B, and the face below A is C.

A B C
Show solution
1 · Understandwhat's really being asked

A net of a die has three faces already showing 3, 2 and 6 pips, and three faces labelled A, B and C. We need which of A, B, C ends up with the smallest number of pips.

Givens
  • Opposite faces of a die add to 7.
  • In the row of four, the third face shows 3 and the fourth shows 2.
  • The flap above B shows 6.
Unknowns
  • Which of A, B and C has the smallest pip count.
Constraints
  • Every face of a die shows between 1 and 6 pips, all different.
2 · Planchoose the strategy

#10 Create a Physical Representation · also uses: #17 Visualize Spatial Relationships#1 Draw a Diagram

Fold the net in your head -- or on paper -- to see which squares land opposite each other. Once the pairs are known, the add-to-7 rule fills in all three unknown faces and they can just be compared.

3 · Execute4 carry out the plan

1Fold the strip of four into a band

#10 Create a Physical Representation 6.G.A.4
The four faces A, B, 3, 2 are in one straight row. A strip of four squares wraps all the way around the cube and its two ends join, so the 1st and 3rd squares land opposite each other, and so do the 2nd and 4th.
A opposite the third square, B opposite the fourth.

2Find A and B using the sum-7 rule

#17 Visualize Spatial Relationships 6.G.A.4
A is opposite the face with 3 pips, so A and 3 must add to 7. That makes A = 4. B is opposite the face with 2 pips, so B = 7 - 2 = 5.
A=7−3=4,B=7−2=5A = 7 - 3 = 4, \quad B = 7 - 2 = 5
A shows 4, B shows 5.

3Find C from the last pair of faces

#1 Draw a Diagram 6.G.A.4
The two flaps that are not in the main strip are the 6-face (above B) and C (below A). Those two become the top and bottom of the cube, so they are the last opposite pair.
C=7−6=1C = 7 - 6 = 1
C shows 1.

4Compare A, B, and C

#17 Visualize Spatial Relationships 6.G.A.4
Now compare the three numbers: A = 4, B = 5, C = 1. The smallest is 1, which is on face C.
B=5>A=4>C=1B = 5 > A = 4 > C = 1
Face C it is.
Answer: C
4 · Reviewdoes it hold up?

All six faces together are 3, 2, 6, 4, 5 and 1 -- the numbers 1 to 6 with none missing and none repeated, which is what a die must show.

Another way: Because opposite faces add to 7, the smallest of A, B and C is opposite the largest of the three given faces. The order of A, B, C is B > A > C either way.

Standardsmin grade 6
  • 6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles — Folding the net to find which faces end up opposite.
💡Takeaway. Fold a strip of four squares and the ends meet, so every other square in the strip is opposite its neighbour's neighbour.
Variant 12 medium answer: A

On a die, the two faces that are parallel to each other have pip counts that add up to 77. In the net of the die shown at the right, among the faces labeled A, B, and C, write the label of the face that gets the largest number of pips. (A die is a cube.)

The net is shaped like a cross/staircase. Four faces run in a single horizontal row; from left to right they are A, B, the face with 22 pips, and the face with 66 pips. The face with 33 pips is attached above B, and the face below A is C.

A B C
Show solution
1 · Understandwhat's really being asked

A net of a die has three faces already showing 2, 6 and 3 pips, and three faces labelled A, B and C. We need which of A, B, C ends up with the largest number of pips.

Givens
  • Opposite faces of a die add to 7.
  • In the row of four, the third face shows 2 and the fourth shows 6.
  • The flap above B shows 3.
Unknowns
  • Which of A, B and C has the largest pip count.
Constraints
  • Every face of a die shows between 1 and 6 pips, all different.
2 · Planchoose the strategy

#10 Create a Physical Representation · also uses: #17 Visualize Spatial Relationships#1 Draw a Diagram

Fold the net in your head -- or on paper -- to see which squares land opposite each other. Once the pairs are known, the add-to-7 rule fills in all three unknown faces and they can just be compared.

3 · Execute4 carry out the plan

1Fold the strip of four into a band

#10 Create a Physical Representation 6.G.A.4
The four faces A, B, 2, 6 are in one straight row. A strip of four squares wraps all the way around the cube and its two ends join, so the 1st and 3rd squares land opposite each other, and so do the 2nd and 4th.
A opposite the third square, B opposite the fourth.

2Find A and B using the sum-7 rule

#17 Visualize Spatial Relationships 6.G.A.4
A is opposite the face with 2 pips, so A and 2 must add to 7. That makes A = 5. B is opposite the face with 6 pips, so B = 7 - 6 = 1.
A=7−2=5,B=7−6=1A = 7 - 2 = 5, \quad B = 7 - 6 = 1
A shows 5, B shows 1.

3Find C from the last pair of faces

#1 Draw a Diagram 6.G.A.4
The two flaps that are not in the main strip are the 3-face (above B) and C (below A). Those two become the top and bottom of the cube, so they are the last opposite pair.
C=7−3=4C = 7 - 3 = 4
C shows 4.

4Compare A, B, and C

#17 Visualize Spatial Relationships 6.G.A.4
Now compare the three numbers: A = 5, B = 1, C = 4. The largest is 5, which is on face A.
A=5>C=4>B=1A = 5 > C = 4 > B = 1
Face A it is.
Answer: A
4 · Reviewdoes it hold up?

All six faces together are 2, 6, 3, 5, 1 and 4 -- the numbers 1 to 6 with none missing and none repeated, which is what a die must show.

Another way: Because opposite faces add to 7, the largest of A, B and C is opposite the smallest of the three given faces. The order of A, B, C is A > C > B either way.

Standardsmin grade 6
  • 6.G.A.4 Represent three-dimensional figures using nets made up of rectangles and triangles — Folding the net to find which faces end up opposite.
💡Takeaway. Fold a strip of four squares and the ends meet, so every other square in the strip is opposite its neighbour's neighbour.