Even number of flips returns original

4.G.A.3 · take · grade 4

Archetype: Transformations Preserve Measures · step in a 8-type progression

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: a horizontal bar two squares wide across the top, followed by a one-square step down to the right, with two more squares added horizontally along the bottom. On the empty grid to the right, draw the shape after it has been flipped 17 times.

Show solution

Understand

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.

Givens

A starting asymmetric grid shape: a 2-wide horizontal bar on top, a one-square step down to the right, and two squares along the bottom.
The shape is flipped to the right (a horizontal flip / 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.

Plan

#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. 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.

Execute

#9 Solve an Easier Related Problem 4.G.A.3

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.

#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 shape. So two flips = no change.

\text{flip} + \text{flip} = \text{original}

Doing the same mirror move twice cancels out, just like turning a card over and back.

#5 Look for a Pattern 4.G.A.3

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 \times 8 + 1

Pairing the flips off shows an odd count behaves like a single flip.

#1 Draw a Diagram 4.G.A.3

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.

Answer: 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.

Review

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.

Make a physical representation (tool 10): cut out the shape, flip it over and over, and notice it alternates original, mirror, original, mirror - landing on 'mirror' for any odd number like 17.

Standards · min 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.

💡 Two flips cancel out, so just check odd or even - 17 is odd, so it's the same as flipping once!