Center of rotation stays fixed

4.MD.C.5 · take · grade 4

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

Representative Problem

Each green figure was made by rotating the orange figure clockwise about the red point. Write the label of the figure that was rotated through the largest angle. (The angle of rotation is greater than $0\degree$ and less than $360\degree$ .)

There are three cases, (A), (B), and (C). In each case the orange figure and the green figure meet at a single red point, which is the shared vertex. The green figure is obtained by rotating the orange figure clockwise about that red point.

In (A), the green figure was rotated clockwise by [ ].
In (B), the green figure was rotated clockwise by [ ].
In (C), the green figure was rotated clockwise by [ ].

Therefore, the figure rotated clockwise through the largest angle is [ ].

Show solution

Understand

In three cases (A), (B), (C), a green figure is made by rotating an orange figure clockwise about a shared red pivot point. We must find each rotation angle (between 0 and 360 degrees) and name the case rotated through the largest angle.

Givens

Each case has an orange figure and a green copy meeting at a red pivot point (the center of rotation, which stays fixed).
The green figure is the orange figure rotated clockwise about the red point.
In (A), the orange figure points up from the pivot and the green points to the right - a quarter turn.
In (B), the orange points up-and-to-one-side and the green points the opposite way (down) - a half turn.
In (C), the orange points to the right and the green is tilted on a slant well past straight-down - the biggest turn.

Unknowns

The clockwise rotation angle in each of (A), (B), (C).
Which case used the largest rotation angle.

Constraints

Each rotation angle is greater than 0 degrees and less than 360 degrees.
The red pivot point does not move.

Plan

#1 Draw a Diagram · also uses: #17 Visualize Spatial Relationships#3 Eliminate Possibilities

Each angle is found by drawing the ray from the pivot along a matching feature of the orange figure and of the green figure, then measuring the clockwise turn between them on the grid. Comparing the three measured angles identifies the largest. (Tool 3 is used on the genuinely finite set of three labeled cases, not on answer choices.)

Execute

#1 Draw a Diagram 4.MD.C.5

Pick a matching arm on the orange and green figures. In (A) the orange arm points straight up from the red pivot and the corresponding green arm points straight to the right. Turning straight-up to straight-right clockwise is a quarter turn.

\text{(A)} = 90^\circ

Up to right, going clockwise, is exactly one corner of a square - 90 degrees.

#1 Draw a Diagram 4.MD.C.5

In (B) the orange arm and the matching green arm point in exactly opposite directions across the pivot (orange up, green straight down). Opposite directions are a half turn.

\text{(B)} = 180^\circ

Pointing the exact opposite way is half of a full circle.

#17 Visualize Spatial Relationships 4.MD.C.5

In (C) the orange arm points to the right, and the green figure is tilted on a slant past straight-down (between a half turn and a three-quarter turn going clockwise). This slanted position is a larger clockwise turn than both 90 and 180 degrees - roughly 225 degrees.

\text{(C)} \approx 225^\circ

The slant shows the green figure has swung past the halfway point, so its turn is the biggest.

#3 Eliminate Possibilities 4.MD.C.5

Compare the three clockwise angles: (A) 90 degrees, (B) 180 degrees, (C) about 225 degrees. The largest is (C).

225^\circ > 180^\circ > 90^\circ

The figure swung furthest around the pivot was turned through the largest angle.

Answer: (C) - rotated clockwise through the largest angle (about 225 degrees, versus 90 degrees in (A) and 180 degrees in (B)).

Review

All three angles lie strictly between 0 and 360 degrees as required, and they increase from the smallest visible turn (A) to the half turn (B) to the slanted, furthest-swung (C), so naming (C) as the largest is consistent with the picture.

Create a physical representation (tool 10): pin a tracing of each orange figure at the red point and rotate it clockwise until it lands on the green figure; the case needing the most turning is the largest angle.

Standards · min grade 4

4.MD.C.5 Recognize angles as geometric shapes formed when two rays share an endpoint — Reading and comparing rotation angles formed at the shared pivot point in each case.

💡 Match an arm on each figure and measure the clockwise turn at the pivot - the figure that swings furthest around (C) used the biggest angle!