Problem
Four 90-degree turns return original
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.
Measurement & data
Your answer
Watch out It's tempting to think 11 turns must land somewhere completely new, but the orientation only has 4 possible pictures - it cycles back every 4 turns, so only the remainder after dividing by 4 matters.
How to solve
Strategy Look for a Pattern — 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.
1STEP 1
Notice the full-turn pattern
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°
A whole turn lands the shape exactly where it started, like a clock hand going all the way around.
4.MD.C.5Solve An Easier Related ProblemTurning the shape 90 degrees four times brings it back to its exact starting orientation, so the possible orientations repeat every four turns.
Why?
Four turns of 90 degrees add up to exactly 360 degrees, which is one whole turn around the center.
Why?
Four equal turns of 90 degrees is four groups of 90, and four groups of 90 count up to 360.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Why?
Repeating the same 90-degree turn is a fixed cycle, and because one whole turn lands the shape back on its start, that cycle is exactly four turns long.
Why?
A fixed repeating cycle of length four comes back to the same position every four steps.
🧱A cycle returns after a full periodA fixed repeating cycle of length p comes back to the same position every p steps.
2STEP 2
Find the leftover turns in 11
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 + 3
Only the leftover turns past the full circles change the picture.
4.MD.C.5Look For A Pattern3STEP 3
Simplify three CCW turns
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° CW
Going three quarters of the way around counterclockwise is the same as one quarter clockwise.
4.MD.C.5Look For A Pattern4STEP 4
Draw the result
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.
4.MD.C.5Draw A DiagramAnswer
The starting shape turned 90 degrees clockwise (the same as turning it 270 degrees counterclockwise): the asymmetric spiral shape rotated one quarter-turn clockwise.
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.
Takeaway
Four quarter-turns = a full circle back to start, so just find 11 divided by 4 - the leftover 3 turns are the same as one turn clockwise!
- Notice the full-turn pattern
- Find the leftover turns in 11
- Simplify three CCW turns
- Draw the result
Where next?
Level upsuggested
Another one like thissuggested
Step back and solidifysuggested