Reverse the moves to recover original

4.G.A.3 · take · grade 4

Archetype: Work Backwards to Recover a Start Value · step in a 9-type progression

The figure shown is the result of flipping some shape upward 7 times and then turning it $90°$ clockwise. Draw the original shape.

The grid on the right shows the shape after the moves. It looks like the numeral 4, made of a slanted line coming down from the top, a vertical right edge, and a part that extends downward. On the empty grid to the left, draw the original shape.

Show solution

Understand

Some original shape was flipped upward 7 times and then turned 90 degrees clockwise, giving the figure shown on the right (it looks like the numeral 4). We must work backward to draw the original shape.

Givens

The final figure on the right grid looks like a numeral 4 (slanted line from the top, vertical right edge, and a downward extension).
The moves applied to the original were: flip upward 7 times, then turn 90 degrees clockwise.

Unknowns

The appearance of the original shape (to be drawn on the left grid).

Constraints

Flipping upward is a reflection across a horizontal line; doing it twice returns the shape (so 7 flips = 1 flip).
To reverse a sequence of moves, undo them in the opposite order with each move's inverse.

Plan

#11 Work Backwards · also uses: #5 Look for a Pattern#1 Draw a Diagram

We are given the end state and must find the start, which is exactly the trigger for Work Backwards: undo the last move first. The flip-count pattern (7 flips behave like 1 flip) simplifies the undo, and a diagram lets us draw each reversed step on the grid.

Execute

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

Flipping upward twice returns the shape to its position, so flips cancel in pairs. 7 = 6 + 1 (three cancelling pairs plus one), so 7 upward flips have the same effect as a single upward flip.

7 = 2 \times 3 + 1

An odd number of identical flips acts like just one flip.

#11 Work Backwards 4.MD.C.5

The last move was a 90-degree clockwise turn, so to reverse it we turn the shown numeral-4 figure 90 degrees counterclockwise. This puts the figure back to its state right after the flips.

\text{undo } 90^\circ\,\text{CW} = 90^\circ\,\text{CCW}

To unwind moves you reverse the order and reverse each move - a clockwise turn is undone by a counterclockwise turn.

#11 Work Backwards 4.G.A.3

Next undo the flipping. Since 7 upward flips act like one upward flip, and a flip is its own undo, flip the figure upward one time (reflect it across a horizontal line). The result is the original shape.

A flip reverses a flip, so one upward flip undoes the odd number of upward flips.

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

Take the shown numeral-4 figure, turn it 90 degrees counterclockwise, then flip it upward once; draw the resulting shape on the empty left grid. That is the original shape before any moves.

Performing the reversed moves in reverse order reconstructs the start exactly.

Answer: The original shape is obtained by turning the shown figure 90 degrees counterclockwise and then flipping it upward once. Drawing those two reversed moves on the left grid gives the original shape.

Review

Applying the original moves (flip up 7 times = flip up once, then turn 90 degrees CW) to our reconstructed shape returns the shown numeral-4 figure, confirming the work-backwards steps are consistent.

Create a physical representation (tool 10): cut out the shown figure and literally perform the inverse moves in reverse order (turn CCW, then flip up) to see the original take shape.

Standards · min grade 4

4.G.A.3 Recognize a line of symmetry for a two-dimensional figure — Understanding flips as reflections and that a flip undoes a flip when reversing the moves.
4.MD.C.5 Recognize angles as geometric shapes formed when two rays share an endpoint — Undoing the 90-degree clockwise turn with a 90-degree counterclockwise turn.

💡 To go back to the start, undo the moves in reverse: turn it back the other way, then flip it once - reversing moves is just Grade 4 flip-and-turn thinking!