Problem
Even number of flips returns original
An asymmetric shape on a grid 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.
Geometry
Your answer
Watch out It's tempting to assume that flipping many times (17) just cancels back to the original shape, without checking whether 17 is odd or even.
How to solve
Strategy Look for a Pattern — Flipping the same shape across the same line twice brings it back to the start, so the result repeats with period 2. Instead of drawing 17 separate flips, I look at the small cases (1 flip, 2 flips) to find the pattern, then use whether 17 is odd or even to pick the answer.
1STEP 1
Try the easier case: what does one flip do?
One right-flip reflects the shape across a vertical line, producing the mirror image: the step that went down-to-the-right now goes down-to-the-left.
A single reflection across a vertical line is exactly the 'flip to the right' a fourth grader practices on grid paper.
4.G.A.3Solve An Easier Related Problem2STEP 2
Flip a second time: back to start
Flipping the mirror image across the same vertical line again undoes the first flip, returning the original shape. So two flips = no change.
flip + flip = original
Doing the same mirror move twice cancels out, just like turning a card over and back.
4.G.A.3Look For A PatternFlipping the shape across the same vertical line a second time returns the exact original shape, so two flips make no change.
Why?
Two flips leave the shape unchanged because every square is carried away and then brought right back to its own place.
Why?
The first flip carries each square across the line to a matching spot the same distance on the other side.
Why?
Flipping across a line is the same as folding the paper on that line, and a fold lays each square exactly onto its matching spot on the other side.
🧱Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).
Why?
The second flip carries each square from that matching spot back across the same line to exactly where it started.
Why?
The matching spot sits the same distance from the line as the original, so folding across the line lays it back exactly onto the starting square.
🧱Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).
3STEP 3
Use odd/even to count 17 flips
Because every 2 flips return the original, only the leftover flip matters. 17 = 16 + 1, and 16 is even (eight pairs that cancel), leaving 1 extra flip.
17 = 2 × 8 + 1
Pairing the flips off shows an odd count behaves like a single flip.
4.G.A.3Look For A Pattern4STEP 4
Draw the result
Since 17 is odd, the result is the shape after exactly one right-flip: the mirror image of the original. Draw the same shape with the one-square step going down to the LEFT instead of to the right (the top 2-wide bar and bottom two squares are mirrored left-to-right).
An odd number of identical flips always looks like a single flip.
4.G.A.3Draw A DiagramAnswer
The mirror image of the starting shape (the same shape flipped once to the right): the figure reflected across a vertical line, with the one-square step now going down to the left.
The result must be either the original (even flips) or its mirror image (odd flips) - no other shape is possible from repeated identical flips. 17 is odd, so the mirror image is the correct and reasonable outcome.
Takeaway
Two flips cancel out, so just check odd or even - 17 is odd, so it's the same as flipping once!
- Try the easier case: what does one flip do?
- Flip a second time: back to start
- Use odd/even to count 17 flips
- Draw the result
Where next?
Another one like thissuggested
Step back and solidifysuggested