← Even number of flips returns original · Transformations Preserve Measures

Even number of flips returns original · 8 practice problems

4.G.A.3

Generated variants — 8

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

Variant 1 easy answer: After 5 right-flips the result is the mirror image of the original (a single right-flip).

Draw the shape that results after flipping the figure to the right 5 times.

The starting figure is an asymmetric shape drawn on a grid (shown on the left). Flipping to the right means reflecting the figure across a vertical line. After it has been flipped 5 times, draw the shape that results on the grid to the right.

Show solution
1 · Understandwhat's really being asked

An asymmetric grid shape is flipped to the right (reflected across a vertical line) 5 times in a row. We must draw the shape that results after all 5 flips.

Givens
  • A starting asymmetric grid shape (shown on the left).
  • The shape is flipped to the right (a reflection across a vertical line) 5 times.
Unknowns
  • The appearance of the shape after 5 right-flips.
Constraints
  • Each flip is the same right-flip (reflection across a vertical line).
  • Flips are applied repeatedly to the result of the previous flip.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

Flipping the same shape across the same line twice brings it back to the start, so the result repeats with period 2. I look at the small cases to find the pattern, then use whether the flip count is odd or even to pick the answer.

3 · Execute4 carry out the plan

1Try the easier case: what does one flip do?

#9 Solve an Easier Related Problem 4.G.A.3
One right-flip reflects the shape across a vertical line, producing its mirror image.
A single reflection across a vertical line is the 'flip to the right' practiced on grid paper.

2Flip a second time: back to start

#5 Look for a Pattern 4.G.A.3
Flipping the mirror image across the same vertical line again undoes the first flip, returning the original. So two flips = no change.
flip+flip=original\text{flip} + \text{flip} = \text{original}
Doing the same mirror move twice cancels out.

3Use odd/even to count 5 flips

#5 Look for a Pattern 4.G.A.3
Because every 2 flips return the original, only the leftover flip matters. 5 = 2 x 2 + 1, so 2 pairs cancel, leaving 1 extra flip(s).
5=2×2+15 = 2 \times 2 + 1
Pairing the flips off shows what an odd or even count behaves like.

4Draw the result

#1 Draw a Diagram 4.G.A.3
Since 5 is odd, the result is the mirror image of the original (a single right-flip). Draw it on the right grid.
An even number of identical flips returns the original; an odd number looks like a single flip.
Answer: After 5 right-flips the result is the mirror image of the original (a single right-flip).
4 · Reviewdoes it hold up?

The result must be either the original (even flips) or its mirror image (odd flips). 5 is odd, so the mirror image of the original (a single right-flip) is the correct outcome.

Another way: Cut out the shape, flip it over and over, and notice it alternates original, mirror, original, mirror.

Standardsmin grade 4
  • 4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Understanding that flipping a figure across a line is a reflection, and that two identical reflections return the original.
💡Takeaway. Two flips cancel out, so just check odd or even - 5 is odd, so it's the same as flipping once!
Variant 2 easy answer: After 6 right-flips the result is the original shape (unchanged).

Draw the shape that results after flipping the figure to the right 6 times.

The starting figure is an asymmetric shape drawn on a grid (shown on the left). Flipping to the right means reflecting the figure across a vertical line. After it has been flipped 6 times, draw the shape that results on the grid to the right.

Show solution
1 · Understandwhat's really being asked

An asymmetric grid shape is flipped to the right (reflected across a vertical line) 6 times in a row. We must draw the shape that results after all 6 flips.

Givens
  • A starting asymmetric grid shape (shown on the left).
  • The shape is flipped to the right (a reflection across a vertical line) 6 times.
Unknowns
  • The appearance of the shape after 6 right-flips.
Constraints
  • Each flip is the same right-flip (reflection across a vertical line).
  • Flips are applied repeatedly to the result of the previous flip.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

Flipping the same shape across the same line twice brings it back to the start, so the result repeats with period 2. I look at the small cases to find the pattern, then use whether the flip count is odd or even to pick the answer.

3 · Execute4 carry out the plan

1Try the easier case: what does one flip do?

#9 Solve an Easier Related Problem 4.G.A.3
One right-flip reflects the shape across a vertical line, producing its mirror image.
A single reflection across a vertical line is the 'flip to the right' practiced on grid paper.

2Flip a second time: back to start

#5 Look for a Pattern 4.G.A.3
Flipping the mirror image across the same vertical line again undoes the first flip, returning the original. So two flips = no change.
flip+flip=original\text{flip} + \text{flip} = \text{original}
Doing the same mirror move twice cancels out.

3Use odd/even to count 6 flips

#5 Look for a Pattern 4.G.A.3
Because every 2 flips return the original, only the leftover flip matters. 6 = 2 x 3 + 0, so 3 pairs cancel, leaving 0 extra flip(s).
6=2×3+06 = 2 \times 3 + 0
Pairing the flips off shows what an odd or even count behaves like.

4Draw the result

#1 Draw a Diagram 4.G.A.3
Since 6 is even, the result is the original shape (unchanged). Draw it on the right grid.
An even number of identical flips returns the original; an odd number looks like a single flip.
Answer: After 6 right-flips the result is the original shape (unchanged).
4 · Reviewdoes it hold up?

The result must be either the original (even flips) or its mirror image (odd flips). 6 is even, so the original shape (unchanged) is the correct outcome.

Another way: Cut out the shape, flip it over and over, and notice it alternates original, mirror, original, mirror.

Standardsmin grade 4
  • 4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Understanding that flipping a figure across a line is a reflection, and that two identical reflections return the original.
💡Takeaway. Two flips cancel out, so just check odd or even - 6 is even, so it's the same as flipping zero!
Variant 3 medium answer: After 8 right-flips the result is the original shape (unchanged).

Draw the shape that results after flipping the figure to the right 8 times.

The starting figure is an asymmetric shape drawn on a grid (shown on the left). Flipping to the right means reflecting the figure across a vertical line. After it has been flipped 8 times, draw the shape that results on the grid to the right.

Show solution
1 · Understandwhat's really being asked

An asymmetric grid shape is flipped to the right (reflected across a vertical line) 8 times in a row. We must draw the shape that results after all 8 flips.

Givens
  • A starting asymmetric grid shape (shown on the left).
  • The shape is flipped to the right (a reflection across a vertical line) 8 times.
Unknowns
  • The appearance of the shape after 8 right-flips.
Constraints
  • Each flip is the same right-flip (reflection across a vertical line).
  • Flips are applied repeatedly to the result of the previous flip.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

Flipping the same shape across the same line twice brings it back to the start, so the result repeats with period 2. I look at the small cases to find the pattern, then use whether the flip count is odd or even to pick the answer.

3 · Execute4 carry out the plan

1Try the easier case: what does one flip do?

#9 Solve an Easier Related Problem 4.G.A.3
One right-flip reflects the shape across a vertical line, producing its mirror image.
A single reflection across a vertical line is the 'flip to the right' practiced on grid paper.

2Flip a second time: back to start

#5 Look for a Pattern 4.G.A.3
Flipping the mirror image across the same vertical line again undoes the first flip, returning the original. So two flips = no change.
flip+flip=original\text{flip} + \text{flip} = \text{original}
Doing the same mirror move twice cancels out.

3Use odd/even to count 8 flips

#5 Look for a Pattern 4.G.A.3
Because every 2 flips return the original, only the leftover flip matters. 8 = 2 x 4 + 0, so 4 pairs cancel, leaving 0 extra flip(s).
8=2×4+08 = 2 \times 4 + 0
Pairing the flips off shows what an odd or even count behaves like.

4Draw the result

#1 Draw a Diagram 4.G.A.3
Since 8 is even, the result is the original shape (unchanged). Draw it on the right grid.
An even number of identical flips returns the original; an odd number looks like a single flip.
Answer: After 8 right-flips the result is the original shape (unchanged).
4 · Reviewdoes it hold up?

The result must be either the original (even flips) or its mirror image (odd flips). 8 is even, so the original shape (unchanged) is the correct outcome.

Another way: Cut out the shape, flip it over and over, and notice it alternates original, mirror, original, mirror.

Standardsmin grade 4
  • 4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Understanding that flipping a figure across a line is a reflection, and that two identical reflections return the original.
💡Takeaway. Two flips cancel out, so just check odd or even - 8 is even, so it's the same as flipping zero!
Variant 4 medium answer: After 9 right-flips the result is the mirror image of the original (a single right-flip).

Draw the shape that results after flipping the figure to the right 9 times.

The starting figure is an asymmetric shape drawn on a grid (shown on the left). Flipping to the right means reflecting the figure across a vertical line. After it has been flipped 9 times, draw the shape that results on the grid to the right.

Show solution
1 · Understandwhat's really being asked

An asymmetric grid shape is flipped to the right (reflected across a vertical line) 9 times in a row. We must draw the shape that results after all 9 flips.

Givens
  • A starting asymmetric grid shape (shown on the left).
  • The shape is flipped to the right (a reflection across a vertical line) 9 times.
Unknowns
  • The appearance of the shape after 9 right-flips.
Constraints
  • Each flip is the same right-flip (reflection across a vertical line).
  • Flips are applied repeatedly to the result of the previous flip.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

Flipping the same shape across the same line twice brings it back to the start, so the result repeats with period 2. I look at the small cases to find the pattern, then use whether the flip count is odd or even to pick the answer.

3 · Execute4 carry out the plan

1Try the easier case: what does one flip do?

#9 Solve an Easier Related Problem 4.G.A.3
One right-flip reflects the shape across a vertical line, producing its mirror image.
A single reflection across a vertical line is the 'flip to the right' practiced on grid paper.

2Flip a second time: back to start

#5 Look for a Pattern 4.G.A.3
Flipping the mirror image across the same vertical line again undoes the first flip, returning the original. So two flips = no change.
flip+flip=original\text{flip} + \text{flip} = \text{original}
Doing the same mirror move twice cancels out.

3Use odd/even to count 9 flips

#5 Look for a Pattern 4.G.A.3
Because every 2 flips return the original, only the leftover flip matters. 9 = 2 x 4 + 1, so 4 pairs cancel, leaving 1 extra flip(s).
9=2×4+19 = 2 \times 4 + 1
Pairing the flips off shows what an odd or even count behaves like.

4Draw the result

#1 Draw a Diagram 4.G.A.3
Since 9 is odd, the result is the mirror image of the original (a single right-flip). Draw it on the right grid.
An even number of identical flips returns the original; an odd number looks like a single flip.
Answer: After 9 right-flips the result is the mirror image of the original (a single right-flip).
4 · Reviewdoes it hold up?

The result must be either the original (even flips) or its mirror image (odd flips). 9 is odd, so the mirror image of the original (a single right-flip) is the correct outcome.

Another way: Cut out the shape, flip it over and over, and notice it alternates original, mirror, original, mirror.

Standardsmin grade 4
  • 4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Understanding that flipping a figure across a line is a reflection, and that two identical reflections return the original.
💡Takeaway. Two flips cancel out, so just check odd or even - 9 is odd, so it's the same as flipping once!
Variant 5 medium answer: After 12 right-flips the result is the original shape (unchanged).

Draw the shape that results after flipping the figure to the right 12 times.

The starting figure is an asymmetric shape drawn on a grid (shown on the left). Flipping to the right means reflecting the figure across a vertical line. After it has been flipped 12 times, draw the shape that results on the grid to the right.

Show solution
1 · Understandwhat's really being asked

An asymmetric grid shape is flipped to the right (reflected across a vertical line) 12 times in a row. We must draw the shape that results after all 12 flips.

Givens
  • A starting asymmetric grid shape (shown on the left).
  • The shape is flipped to the right (a reflection across a vertical line) 12 times.
Unknowns
  • The appearance of the shape after 12 right-flips.
Constraints
  • Each flip is the same right-flip (reflection across a vertical line).
  • Flips are applied repeatedly to the result of the previous flip.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

Flipping the same shape across the same line twice brings it back to the start, so the result repeats with period 2. I look at the small cases to find the pattern, then use whether the flip count is odd or even to pick the answer.

3 · Execute4 carry out the plan

1Try the easier case: what does one flip do?

#9 Solve an Easier Related Problem 4.G.A.3
One right-flip reflects the shape across a vertical line, producing its mirror image.
A single reflection across a vertical line is the 'flip to the right' practiced on grid paper.

2Flip a second time: back to start

#5 Look for a Pattern 4.G.A.3
Flipping the mirror image across the same vertical line again undoes the first flip, returning the original. So two flips = no change.
flip+flip=original\text{flip} + \text{flip} = \text{original}
Doing the same mirror move twice cancels out.

3Use odd/even to count 12 flips

#5 Look for a Pattern 4.G.A.3
Because every 2 flips return the original, only the leftover flip matters. 12 = 2 x 6 + 0, so 6 pairs cancel, leaving 0 extra flip(s).
12=2×6+012 = 2 \times 6 + 0
Pairing the flips off shows what an odd or even count behaves like.

4Draw the result

#1 Draw a Diagram 4.G.A.3
Since 12 is even, the result is the original shape (unchanged). Draw it on the right grid.
An even number of identical flips returns the original; an odd number looks like a single flip.
Answer: After 12 right-flips the result is the original shape (unchanged).
4 · Reviewdoes it hold up?

The result must be either the original (even flips) or its mirror image (odd flips). 12 is even, so the original shape (unchanged) is the correct outcome.

Another way: Cut out the shape, flip it over and over, and notice it alternates original, mirror, original, mirror.

Standardsmin grade 4
  • 4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Understanding that flipping a figure across a line is a reflection, and that two identical reflections return the original.
💡Takeaway. Two flips cancel out, so just check odd or even - 12 is even, so it's the same as flipping zero!
Variant 6 medium answer: After 13 right-flips the result is the mirror image of the original (a single right-flip).

Draw the shape that results after flipping the figure to the right 13 times.

The starting figure is an asymmetric shape drawn on a grid (shown on the left). Flipping to the right means reflecting the figure across a vertical line. After it has been flipped 13 times, draw the shape that results on the grid to the right.

Show solution
1 · Understandwhat's really being asked

An asymmetric grid shape is flipped to the right (reflected across a vertical line) 13 times in a row. We must draw the shape that results after all 13 flips.

Givens
  • A starting asymmetric grid shape (shown on the left).
  • The shape is flipped to the right (a reflection across a vertical line) 13 times.
Unknowns
  • The appearance of the shape after 13 right-flips.
Constraints
  • Each flip is the same right-flip (reflection across a vertical line).
  • Flips are applied repeatedly to the result of the previous flip.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

Flipping the same shape across the same line twice brings it back to the start, so the result repeats with period 2. I look at the small cases to find the pattern, then use whether the flip count is odd or even to pick the answer.

3 · Execute4 carry out the plan

1Try the easier case: what does one flip do?

#9 Solve an Easier Related Problem 4.G.A.3
One right-flip reflects the shape across a vertical line, producing its mirror image.
A single reflection across a vertical line is the 'flip to the right' practiced on grid paper.

2Flip a second time: back to start

#5 Look for a Pattern 4.G.A.3
Flipping the mirror image across the same vertical line again undoes the first flip, returning the original. So two flips = no change.
flip+flip=original\text{flip} + \text{flip} = \text{original}
Doing the same mirror move twice cancels out.

3Use odd/even to count 13 flips

#5 Look for a Pattern 4.G.A.3
Because every 2 flips return the original, only the leftover flip matters. 13 = 2 x 6 + 1, so 6 pairs cancel, leaving 1 extra flip(s).
13=2×6+113 = 2 \times 6 + 1
Pairing the flips off shows what an odd or even count behaves like.

4Draw the result

#1 Draw a Diagram 4.G.A.3
Since 13 is odd, the result is the mirror image of the original (a single right-flip). Draw it on the right grid.
An even number of identical flips returns the original; an odd number looks like a single flip.
Answer: After 13 right-flips the result is the mirror image of the original (a single right-flip).
4 · Reviewdoes it hold up?

The result must be either the original (even flips) or its mirror image (odd flips). 13 is odd, so the mirror image of the original (a single right-flip) is the correct outcome.

Another way: Cut out the shape, flip it over and over, and notice it alternates original, mirror, original, mirror.

Standardsmin grade 4
  • 4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Understanding that flipping a figure across a line is a reflection, and that two identical reflections return the original.
💡Takeaway. Two flips cancel out, so just check odd or even - 13 is odd, so it's the same as flipping once!
Variant 7 hard answer: After 17 right-flips the result is the mirror image of the original (a single right-flip).

Draw the shape that results after flipping the figure to the right 17 times.

The starting figure is an asymmetric shape drawn on a grid (shown on the left). Flipping to the right means reflecting the figure across a vertical line. After it has been flipped 17 times, draw the shape that results on the grid to the right.

Show solution
1 · Understandwhat's really being asked

An asymmetric grid shape is flipped to the right (reflected across a vertical line) 17 times in a row. We must draw the shape that results after all 17 flips.

Givens
  • A starting asymmetric grid shape (shown on the left).
  • The shape is flipped to the right (a reflection across a vertical line) 17 times.
Unknowns
  • The appearance of the shape after 17 right-flips.
Constraints
  • Each flip is the same right-flip (reflection across a vertical line).
  • Flips are applied repeatedly to the result of the previous flip.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

Flipping the same shape across the same line twice brings it back to the start, so the result repeats with period 2. I look at the small cases to find the pattern, then use whether the flip count is odd or even to pick the answer.

3 · Execute4 carry out the plan

1Try the easier case: what does one flip do?

#9 Solve an Easier Related Problem 4.G.A.3
One right-flip reflects the shape across a vertical line, producing its mirror image.
A single reflection across a vertical line is the 'flip to the right' practiced on grid paper.

2Flip a second time: back to start

#5 Look for a Pattern 4.G.A.3
Flipping the mirror image across the same vertical line again undoes the first flip, returning the original. So two flips = no change.
flip+flip=original\text{flip} + \text{flip} = \text{original}
Doing the same mirror move twice cancels out.

3Use odd/even to count 17 flips

#5 Look for a Pattern 4.G.A.3
Because every 2 flips return the original, only the leftover flip matters. 17 = 2 x 8 + 1, so 8 pairs cancel, leaving 1 extra flip(s).
17=2×8+117 = 2 \times 8 + 1
Pairing the flips off shows what an odd or even count behaves like.

4Draw the result

#1 Draw a Diagram 4.G.A.3
Since 17 is odd, the result is the mirror image of the original (a single right-flip). Draw it on the right grid.
An even number of identical flips returns the original; an odd number looks like a single flip.
Answer: After 17 right-flips the result is the mirror image of the original (a single right-flip).
4 · Reviewdoes it hold up?

The result must be either the original (even flips) or its mirror image (odd flips). 17 is odd, so the mirror image of the original (a single right-flip) is the correct outcome.

Another way: Cut out the shape, flip it over and over, and notice it alternates original, mirror, original, mirror.

Standardsmin grade 4
  • 4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Understanding that flipping a figure across a line is a reflection, and that two identical reflections return the original.
💡Takeaway. Two flips cancel out, so just check odd or even - 17 is odd, so it's the same as flipping once!
Variant 8 hard answer: After 20 right-flips the result is the original shape (unchanged).

Draw the shape that results after flipping the figure to the right 20 times.

The starting figure is an asymmetric shape drawn on a grid (shown on the left). Flipping to the right means reflecting the figure across a vertical line. After it has been flipped 20 times, draw the shape that results on the grid to the right.

Show solution
1 · Understandwhat's really being asked

An asymmetric grid shape is flipped to the right (reflected across a vertical line) 20 times in a row. We must draw the shape that results after all 20 flips.

Givens
  • A starting asymmetric grid shape (shown on the left).
  • The shape is flipped to the right (a reflection across a vertical line) 20 times.
Unknowns
  • The appearance of the shape after 20 right-flips.
Constraints
  • Each flip is the same right-flip (reflection across a vertical line).
  • Flips are applied repeatedly to the result of the previous flip.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

Flipping the same shape across the same line twice brings it back to the start, so the result repeats with period 2. I look at the small cases to find the pattern, then use whether the flip count is odd or even to pick the answer.

3 · Execute4 carry out the plan

1Try the easier case: what does one flip do?

#9 Solve an Easier Related Problem 4.G.A.3
One right-flip reflects the shape across a vertical line, producing its mirror image.
A single reflection across a vertical line is the 'flip to the right' practiced on grid paper.

2Flip a second time: back to start

#5 Look for a Pattern 4.G.A.3
Flipping the mirror image across the same vertical line again undoes the first flip, returning the original. So two flips = no change.
flip+flip=original\text{flip} + \text{flip} = \text{original}
Doing the same mirror move twice cancels out.

3Use odd/even to count 20 flips

#5 Look for a Pattern 4.G.A.3
Because every 2 flips return the original, only the leftover flip matters. 20 = 2 x 10 + 0, so 10 pairs cancel, leaving 0 extra flip(s).
20=2×10+020 = 2 \times 10 + 0
Pairing the flips off shows what an odd or even count behaves like.

4Draw the result

#1 Draw a Diagram 4.G.A.3
Since 20 is even, the result is the original shape (unchanged). Draw it on the right grid.
An even number of identical flips returns the original; an odd number looks like a single flip.
Answer: After 20 right-flips the result is the original shape (unchanged).
4 · Reviewdoes it hold up?

The result must be either the original (even flips) or its mirror image (odd flips). 20 is even, so the original shape (unchanged) is the correct outcome.

Another way: Cut out the shape, flip it over and over, and notice it alternates original, mirror, original, mirror.

Standardsmin grade 4
  • 4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Understanding that flipping a figure across a line is a reflection, and that two identical reflections return the original.
💡Takeaway. Two flips cancel out, so just check odd or even - 20 is even, so it's the same as flipping zero!