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Four 90-degree turns return original · 10 practice problems

4.MD.C.5

Generated variants — 10

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 answer: The starting shape unchanged: after a whole number of full turns it is back in its original orientation.

Draw the shape that results after turning the figure 90°90° counterclockwise 4 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 4 times.

Original Draw here
Show solution

Understand

An asymmetric grid shape is turned 90 degrees counterclockwise, and this same turn is repeated 4 times. We must draw the shape after all 4 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 4 times.
Unknowns
  • The orientation/appearance of the shape after 4 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 4 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 4 by 4: there are 1 full turns (4 turns that cancel out) with a remainder of 0. So 4 turns has the same effect as 0 turns of 90 degrees counterclockwise.
4=4×1+04 = 4 \times 1 + 0
Only the leftover turns past the full circles change the picture.
#5 Look for a Pattern 4.MD.C.5
After the full turns cancel, what remains is the original orientation (a whole number of full turns).
0×90∘ CCW=0∘ CCW0 \times 90^\circ\ \text{CCW} = 0^\circ\ \text{CCW}
Going partway around counterclockwise leaves only the leftover quarter-turns to draw.
#1 Draw a Diagram 4.MD.C.5
Draw the starting shape after the original orientation (a whole number of full turns), about its grid position.
After all the full turns cancel, only the leftover quarter-turns are left to draw.
Answer: The starting shape unchanged: after a whole number of full turns it is back in its original orientation.

Review

Only 4 distinct orientations are possible from 90-degree turns. Since 4 leaves a remainder of 0 when divided by 4, the answer is the 0-turns-CCW orientation - a valid one of the four possible pictures.

Create a physical representation (tool 10): cut out the shape and rotate it 90 degrees CCW 4 times, observing it returns to start every 4 turns.

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 4 divided by 4 - only the leftover 0 turns change the picture!
Variant 2 answer: The starting shape turned 90 degrees counterclockwise (one quarter-turn CCW).

Draw the shape that results after turning the figure 90°90° counterclockwise 13 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 13 times.

Original Draw here
Show solution

Understand

An asymmetric grid shape is turned 90 degrees counterclockwise, and this same turn is repeated 13 times. We must draw the shape after all 13 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 13 times.
Unknowns
  • The orientation/appearance of the shape after 13 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 13 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 13 by 4: there are 3 full turns (12 turns that cancel out) with a remainder of 1. So 13 turns has the same effect as 1 turns of 90 degrees counterclockwise.
13=4×3+113 = 4 \times 3 + 1
Only the leftover turns past the full circles change the picture.
#5 Look for a Pattern 4.MD.C.5
After the full turns cancel, what remains is one 90-degree turn counterclockwise.
1×90∘ CCW=90∘ CCW1 \times 90^\circ\ \text{CCW} = 90^\circ\ \text{CCW}
Going partway around counterclockwise leaves only the leftover quarter-turns to draw.
#1 Draw a Diagram 4.MD.C.5
Draw the starting shape after one 90-degree turn counterclockwise, about its grid position.
After all the full turns cancel, only the leftover quarter-turns are left to draw.
Answer: The starting shape turned 90 degrees counterclockwise (one quarter-turn CCW).

Review

Only 4 distinct orientations are possible from 90-degree turns. Since 13 leaves a remainder of 1 when divided by 4, the answer is the 1-turns-CCW orientation - a valid one of the four possible pictures.

Create a physical representation (tool 10): cut out the shape and rotate it 90 degrees CCW 13 times, observing it returns to start every 4 turns.

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 13 divided by 4 - only the leftover 1 turns change the picture!
Variant 3 answer: The starting shape turned 180 degrees (a half turn).

Draw the shape that results after turning the figure 90°90° counterclockwise 2 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 2 times.

Original Draw here
Show solution

Understand

An asymmetric grid shape is turned 90 degrees counterclockwise, and this same turn is repeated 2 times. We must draw the shape after all 2 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 2 times.
Unknowns
  • The orientation/appearance of the shape after 2 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 2 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 2 by 4: there are 0 full turns (0 turns that cancel out) with a remainder of 2. So 2 turns has the same effect as 2 turns of 90 degrees counterclockwise.
2=4×0+22 = 4 \times 0 + 2
Only the leftover turns past the full circles change the picture.
#5 Look for a Pattern 4.MD.C.5
After the full turns cancel, what remains is a 180-degree turn (a half turn).
2×90∘ CCW=180∘ CCW2 \times 90^\circ\ \text{CCW} = 180^\circ\ \text{CCW}
Going partway around counterclockwise leaves only the leftover quarter-turns to draw.
#1 Draw a Diagram 4.MD.C.5
Draw the starting shape after a 180-degree turn (a half turn), about its grid position.
After all the full turns cancel, only the leftover quarter-turns are left to draw.
Answer: The starting shape turned 180 degrees (a half turn).

Review

Only 4 distinct orientations are possible from 90-degree turns. Since 2 leaves a remainder of 2 when divided by 4, the answer is the 2-turns-CCW orientation - a valid one of the four possible pictures.

Create a physical representation (tool 10): cut out the shape and rotate it 90 degrees CCW 2 times, observing it returns to start every 4 turns.

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 2 divided by 4 - only the leftover 2 turns change the picture!
Variant 4 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.

Draw the shape that results after turning the figure 90°90° counterclockwise 3 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 3 times.

Original Draw here
Show solution

Understand

An asymmetric grid shape is turned 90 degrees counterclockwise, and this same turn is repeated 3 times. We must draw the shape after all 3 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 3 times.
Unknowns
  • The orientation/appearance of the shape after 3 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 3 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 3 by 4: there are 0 full turns (0 turns that cancel out) with a remainder of 3. So 3 turns has the same effect as 3 turns of 90 degrees counterclockwise.
3=4×0+33 = 4 \times 0 + 3
Only the leftover turns past the full circles change the picture.
#5 Look for a Pattern 4.MD.C.5
After the full turns cancel, what remains is three 90-degree turns counterclockwise, the same as one 90-degree turn clockwise.
3×90∘ CCW=270∘ CCW3 \times 90^\circ\ \text{CCW} = 270^\circ\ \text{CCW}
Going partway around counterclockwise leaves only the leftover quarter-turns to draw.
#1 Draw a Diagram 4.MD.C.5
Draw the starting shape after three 90-degree turns counterclockwise, the same as one 90-degree turn clockwise, about its grid position.
After all the full turns cancel, only the leftover quarter-turns are 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 3 leaves a remainder of 3 when divided by 4, the answer is the 3-turns-CCW orientation - a valid one of the four possible pictures.

Create a physical representation (tool 10): cut out the shape and rotate it 90 degrees CCW 3 times, observing it returns to start every 4 turns.

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 3 divided by 4 - only the leftover 3 turns change the picture!
Variant 5 answer: The starting shape turned 90 degrees counterclockwise (one quarter-turn CCW).

Draw the shape that results after turning the figure 90°90° counterclockwise 1 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 1 times.

Original Draw here
Show solution

Understand

An asymmetric grid shape is turned 90 degrees counterclockwise, and this same turn is repeated 1 times. We must draw the shape after all 1 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 1 times.
Unknowns
  • The orientation/appearance of the shape after 1 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 1 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 1 by 4: there are 0 full turns (0 turns that cancel out) with a remainder of 1. So 1 turns has the same effect as 1 turns of 90 degrees counterclockwise.
1=4×0+11 = 4 \times 0 + 1
Only the leftover turns past the full circles change the picture.
#5 Look for a Pattern 4.MD.C.5
After the full turns cancel, what remains is one 90-degree turn counterclockwise.
1×90∘ CCW=90∘ CCW1 \times 90^\circ\ \text{CCW} = 90^\circ\ \text{CCW}
Going partway around counterclockwise leaves only the leftover quarter-turns to draw.
#1 Draw a Diagram 4.MD.C.5
Draw the starting shape after one 90-degree turn counterclockwise, about its grid position.
After all the full turns cancel, only the leftover quarter-turns are left to draw.
Answer: The starting shape turned 90 degrees counterclockwise (one quarter-turn CCW).

Review

Only 4 distinct orientations are possible from 90-degree turns. Since 1 leaves a remainder of 1 when divided by 4, the answer is the 1-turns-CCW orientation - a valid one of the four possible pictures.

Create a physical representation (tool 10): cut out the shape and rotate it 90 degrees CCW 1 times, observing it returns to start every 4 turns.

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 1 divided by 4 - only the leftover 1 turns change the picture!
Variant 6 answer: The starting shape turned 90 degrees counterclockwise (one quarter-turn CCW).

Draw the shape that results after turning the figure 90°90° counterclockwise 9 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 9 times.

Original Draw here
Show solution

Understand

An asymmetric grid shape is turned 90 degrees counterclockwise, and this same turn is repeated 9 times. We must draw the shape after all 9 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 9 times.
Unknowns
  • The orientation/appearance of the shape after 9 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 9 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 9 by 4: there are 2 full turns (8 turns that cancel out) with a remainder of 1. So 9 turns has the same effect as 1 turns of 90 degrees counterclockwise.
9=4×2+19 = 4 \times 2 + 1
Only the leftover turns past the full circles change the picture.
#5 Look for a Pattern 4.MD.C.5
After the full turns cancel, what remains is one 90-degree turn counterclockwise.
1×90∘ CCW=90∘ CCW1 \times 90^\circ\ \text{CCW} = 90^\circ\ \text{CCW}
Going partway around counterclockwise leaves only the leftover quarter-turns to draw.
#1 Draw a Diagram 4.MD.C.5
Draw the starting shape after one 90-degree turn counterclockwise, about its grid position.
After all the full turns cancel, only the leftover quarter-turns are left to draw.
Answer: The starting shape turned 90 degrees counterclockwise (one quarter-turn CCW).

Review

Only 4 distinct orientations are possible from 90-degree turns. Since 9 leaves a remainder of 1 when divided by 4, the answer is the 1-turns-CCW orientation - a valid one of the four possible pictures.

Create a physical representation (tool 10): cut out the shape and rotate it 90 degrees CCW 9 times, observing it returns to start every 4 turns.

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 9 divided by 4 - only the leftover 1 turns change the picture!
Variant 7 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.

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
After the full turns cancel, what remains is three 90-degree turns counterclockwise, the same as one 90-degree turn clockwise.
3×90∘ CCW=270∘ CCW3 \times 90^\circ\ \text{CCW} = 270^\circ\ \text{CCW}
Going partway around counterclockwise leaves only the leftover quarter-turns to draw.
#1 Draw a Diagram 4.MD.C.5
Draw the starting shape after three 90-degree turns counterclockwise, the same as one 90-degree turn clockwise, about its grid position.
After all the full turns cancel, only the leftover quarter-turns are 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 - a valid one of the four possible pictures.

Create a physical representation (tool 10): cut out the shape and rotate it 90 degrees CCW 11 times, observing it returns to start every 4 turns.

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 - only the leftover 3 turns change the picture!
Variant 8 answer: The starting shape unchanged: after a whole number of full turns it is back in its original orientation.

Draw the shape that results after turning the figure 90°90° counterclockwise 8 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 8 times.

Original Draw here
Show solution

Understand

An asymmetric grid shape is turned 90 degrees counterclockwise, and this same turn is repeated 8 times. We must draw the shape after all 8 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 8 times.
Unknowns
  • The orientation/appearance of the shape after 8 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 8 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 8 by 4: there are 2 full turns (8 turns that cancel out) with a remainder of 0. So 8 turns has the same effect as 0 turns of 90 degrees counterclockwise.
8=4×2+08 = 4 \times 2 + 0
Only the leftover turns past the full circles change the picture.
#5 Look for a Pattern 4.MD.C.5
After the full turns cancel, what remains is the original orientation (a whole number of full turns).
0×90∘ CCW=0∘ CCW0 \times 90^\circ\ \text{CCW} = 0^\circ\ \text{CCW}
Going partway around counterclockwise leaves only the leftover quarter-turns to draw.
#1 Draw a Diagram 4.MD.C.5
Draw the starting shape after the original orientation (a whole number of full turns), about its grid position.
After all the full turns cancel, only the leftover quarter-turns are left to draw.
Answer: The starting shape unchanged: after a whole number of full turns it is back in its original orientation.

Review

Only 4 distinct orientations are possible from 90-degree turns. Since 8 leaves a remainder of 0 when divided by 4, the answer is the 0-turns-CCW orientation - a valid one of the four possible pictures.

Create a physical representation (tool 10): cut out the shape and rotate it 90 degrees CCW 8 times, observing it returns to start every 4 turns.

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 8 divided by 4 - only the leftover 0 turns change the picture!
Variant 9 answer: The starting shape turned 180 degrees (a half turn).

Draw the shape that results after turning the figure 90°90° counterclockwise 6 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 6 times.

Original Draw here
Show solution

Understand

An asymmetric grid shape is turned 90 degrees counterclockwise, and this same turn is repeated 6 times. We must draw the shape after all 6 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 6 times.
Unknowns
  • The orientation/appearance of the shape after 6 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 6 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 6 by 4: there are 1 full turns (4 turns that cancel out) with a remainder of 2. So 6 turns has the same effect as 2 turns of 90 degrees counterclockwise.
6=4×1+26 = 4 \times 1 + 2
Only the leftover turns past the full circles change the picture.
#5 Look for a Pattern 4.MD.C.5
After the full turns cancel, what remains is a 180-degree turn (a half turn).
2×90∘ CCW=180∘ CCW2 \times 90^\circ\ \text{CCW} = 180^\circ\ \text{CCW}
Going partway around counterclockwise leaves only the leftover quarter-turns to draw.
#1 Draw a Diagram 4.MD.C.5
Draw the starting shape after a 180-degree turn (a half turn), about its grid position.
After all the full turns cancel, only the leftover quarter-turns are left to draw.
Answer: The starting shape turned 180 degrees (a half turn).

Review

Only 4 distinct orientations are possible from 90-degree turns. Since 6 leaves a remainder of 2 when divided by 4, the answer is the 2-turns-CCW orientation - a valid one of the four possible pictures.

Create a physical representation (tool 10): cut out the shape and rotate it 90 degrees CCW 6 times, observing it returns to start every 4 turns.

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 6 divided by 4 - only the leftover 2 turns change the picture!
Variant 10 answer: The starting shape unchanged: after a whole number of full turns it is back in its original orientation.

Draw the shape that results after turning the figure 90°90° counterclockwise 20 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 20 times.

Original Draw here
Show solution

Understand

An asymmetric grid shape is turned 90 degrees counterclockwise, and this same turn is repeated 20 times. We must draw the shape after all 20 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 20 times.
Unknowns
  • The orientation/appearance of the shape after 20 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 20 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 20 by 4: there are 5 full turns (20 turns that cancel out) with a remainder of 0. So 20 turns has the same effect as 0 turns of 90 degrees counterclockwise.
20=4×5+020 = 4 \times 5 + 0
Only the leftover turns past the full circles change the picture.
#5 Look for a Pattern 4.MD.C.5
After the full turns cancel, what remains is the original orientation (a whole number of full turns).
0×90∘ CCW=0∘ CCW0 \times 90^\circ\ \text{CCW} = 0^\circ\ \text{CCW}
Going partway around counterclockwise leaves only the leftover quarter-turns to draw.
#1 Draw a Diagram 4.MD.C.5
Draw the starting shape after the original orientation (a whole number of full turns), about its grid position.
After all the full turns cancel, only the leftover quarter-turns are left to draw.
Answer: The starting shape unchanged: after a whole number of full turns it is back in its original orientation.

Review

Only 4 distinct orientations are possible from 90-degree turns. Since 20 leaves a remainder of 0 when divided by 4, the answer is the 0-turns-CCW orientation - a valid one of the four possible pictures.

Create a physical representation (tool 10): cut out the shape and rotate it 90 degrees CCW 20 times, observing it returns to start every 4 turns.

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 20 divided by 4 - only the leftover 0 turns change the picture!