Sensim Math · Depth 한국어

4-1 · Plane Figure Movement

Four 90-degree turns return original

4.MD.C.5 · take · grade 4

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

▶ Practice — 10 problems

Draw the shape that results after turning the figure 90°90° counterclockwise 11 times.

The starting figure is an asymmetric shape on a grid, with an inward-bent, spiral-like outline. On the empty grid to the right, draw the shape after it has been turned 90°90° counterclockwise 11 times.

Original Draw here
Show solution

Understand

An asymmetric grid shape is turned 90 degrees counterclockwise, and this same turn is repeated 11 times. We must draw the shape after all 11 turns.

Givens
  • A starting asymmetric (inward-bent, spiral-like) shape on a grid.
  • Each move turns the shape 90 degrees counterclockwise.
  • The move is repeated 11 times.
Unknowns
  • The orientation/appearance of the shape after 11 turns of 90 degrees counterclockwise.
Constraints
  • Every turn is the same 90-degree counterclockwise rotation.
  • Four 90-degree turns make a full 360-degree turn, returning the original.

Plan

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

Four quarter-turns equal one full turn and bring the shape back to start, so the orientations repeat with period 4. I find the remainder of 11 divided by 4 to know how many effective turns remain, then draw that orientation.

Execute

#9 Solve an Easier Related Problem 4.MD.C.5
Turning 90 degrees four times equals 360 degrees, a full turn, which returns the original shape. So orientations cycle every 4 turns.
4×90∘=360∘4 \times 90^\circ = 360^\circ
A whole turn lands the shape exactly where it started, like a clock hand going all the way around.
#5 Look for a Pattern 4.MD.C.5
Divide 11 by 4: there are 2 full turns (8 turns that cancel out) with a remainder of 3. So 11 turns has the same effect as 3 turns of 90 degrees counterclockwise.
11=4×2+311 = 4 \times 2 + 3
Only the leftover turns past the full circles change the picture.
#5 Look for a Pattern 4.MD.C.5
Three 90-degree turns counterclockwise (270 degrees CCW) leave the shape in the same orientation as one 90-degree turn clockwise.
3×90∘ CCW=270∘ CCW=90∘ CW3 \times 90^\circ\ \text{CCW} = 270^\circ\ \text{CCW} = 90^\circ\ \text{CW}
Going three quarters of the way around counterclockwise is the same as one quarter clockwise.
#1 Draw a Diagram 4.MD.C.5
Draw the starting shape turned 90 degrees clockwise (equivalently 270 degrees counterclockwise) about its grid position: what pointed up now points right, and the spiral bend rotates a quarter turn clockwise.
After all the full turns cancel, one clockwise quarter-turn is all that is left to draw.
Answer: The starting shape turned 90 degrees clockwise (the same as turning it 270 degrees counterclockwise): the asymmetric spiral shape rotated one quarter-turn clockwise.

Review

Only 4 distinct orientations are possible from 90-degree turns. Since 11 leaves a remainder of 3 when divided by 4, the answer is the 3-turns-CCW orientation, equal to one clockwise quarter-turn - a valid one of the four possible pictures.

Create a physical representation (tool 10): cut out the shape and rotate it 90 degrees CCW eleven times, observing it returns to start every 4 turns and ends in the 3rd position (one clockwise turn).

Standards · min grade 4

  • 4.MD.C.5 Recognize angles as geometric shapes formed when two rays share an endpoint — Measuring turns as angles and adding 90-degree quarter-turns up to and past a full 360-degree rotation.
💡 Four quarter-turns = a full circle back to start, so just find 11 divided by 4 - the leftover 3 turns is the same as one turn clockwise!