Practice · Four 90-degree turns return original · Sensim Math

Variant 1 medium

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Notice the full-turn pattern

#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 \times 90^\circ = 360^\circ

A whole turn lands the shape exactly where it started, like a clock hand going all the way around.

2Find the leftover turns in 4

#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 \times 1 + 0

Only the leftover turns past the full circles change the picture.

3Simplify the leftover turns

#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 \times 90^\circ\ \text{CCW} = 0^\circ\ \text{CCW}

Going partway around counterclockwise leaves only the leftover quarter-turns to draw.

4Draw the result

#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.

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. 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 medium

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Notice the full-turn pattern

#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 \times 90^\circ = 360^\circ

A whole turn lands the shape exactly where it started, like a clock hand going all the way around.

2Find the leftover turns in 13

#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 \times 3 + 1

Only the leftover turns past the full circles change the picture.

3Simplify the leftover turns

#5 Look for a Pattern 4.MD.C.5

After the full turns cancel, what remains is one 90-degree turn counterclockwise.

1 \times 90^\circ\ \text{CCW} = 90^\circ\ \text{CCW}

Going partway around counterclockwise leaves only the leftover quarter-turns to draw.

4Draw the result

#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).

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. 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 medium

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Notice the full-turn pattern

#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 \times 90^\circ = 360^\circ

A whole turn lands the shape exactly where it started, like a clock hand going all the way around.

2Find the leftover turns in 2

#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 \times 0 + 2

Only the leftover turns past the full circles change the picture.

3Simplify the leftover turns

#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 \times 90^\circ\ \text{CCW} = 180^\circ\ \text{CCW}

Going partway around counterclockwise leaves only the leftover quarter-turns to draw.

4Draw the result

#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).

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. 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 medium

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Notice the full-turn pattern

#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 \times 90^\circ = 360^\circ

A whole turn lands the shape exactly where it started, like a clock hand going all the way around.

2Find the leftover turns in 3

#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 \times 0 + 3

Only the leftover turns past the full circles change the picture.

3Simplify the leftover turns

#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 \times 90^\circ\ \text{CCW} = 270^\circ\ \text{CCW}

Going partway around counterclockwise leaves only the leftover quarter-turns to draw.

4Draw the result

#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.

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. 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 medium

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Notice the full-turn pattern

#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 \times 90^\circ = 360^\circ

A whole turn lands the shape exactly where it started, like a clock hand going all the way around.

2Find the leftover turns in 1

#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 \times 0 + 1

Only the leftover turns past the full circles change the picture.

3Simplify the leftover turns

#5 Look for a Pattern 4.MD.C.5

After the full turns cancel, what remains is one 90-degree turn counterclockwise.

1 \times 90^\circ\ \text{CCW} = 90^\circ\ \text{CCW}

Going partway around counterclockwise leaves only the leftover quarter-turns to draw.

4Draw the result

#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).

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. 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 medium

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Notice the full-turn pattern

#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 \times 90^\circ = 360^\circ

A whole turn lands the shape exactly where it started, like a clock hand going all the way around.

2Find the leftover turns in 9

#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 \times 2 + 1

Only the leftover turns past the full circles change the picture.

3Simplify the leftover turns

#5 Look for a Pattern 4.MD.C.5

After the full turns cancel, what remains is one 90-degree turn counterclockwise.

1 \times 90^\circ\ \text{CCW} = 90^\circ\ \text{CCW}

Going partway around counterclockwise leaves only the leftover quarter-turns to draw.

4Draw the result

#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).

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. 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 medium

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Notice the full-turn pattern

#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 \times 90^\circ = 360^\circ

A whole turn lands the shape exactly where it started, like a clock hand going all the way around.

2Find the leftover turns in 11

#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 \times 2 + 3

Only the leftover turns past the full circles change the picture.

3Simplify the leftover turns

#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 \times 90^\circ\ \text{CCW} = 270^\circ\ \text{CCW}

Going partway around counterclockwise leaves only the leftover quarter-turns to draw.

4Draw the result

#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.

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. 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 medium

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Notice the full-turn pattern

#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 \times 90^\circ = 360^\circ

A whole turn lands the shape exactly where it started, like a clock hand going all the way around.

2Find the leftover turns in 8

#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 \times 2 + 0

Only the leftover turns past the full circles change the picture.

3Simplify the leftover turns

#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 \times 90^\circ\ \text{CCW} = 0^\circ\ \text{CCW}

Going partway around counterclockwise leaves only the leftover quarter-turns to draw.

4Draw the result

#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.

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. 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 medium

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Notice the full-turn pattern

#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 \times 90^\circ = 360^\circ

A whole turn lands the shape exactly where it started, like a clock hand going all the way around.

2Find the leftover turns in 6

#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 \times 1 + 2

Only the leftover turns past the full circles change the picture.

3Simplify the leftover turns

#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 \times 90^\circ\ \text{CCW} = 180^\circ\ \text{CCW}

Going partway around counterclockwise leaves only the leftover quarter-turns to draw.

4Draw the result

#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).

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. 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 medium

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

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1Notice the full-turn pattern

#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 \times 90^\circ = 360^\circ

A whole turn lands the shape exactly where it started, like a clock hand going all the way around.

2Find the leftover turns in 20

#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 \times 5 + 0

Only the leftover turns past the full circles change the picture.

3Simplify the leftover turns

#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 \times 90^\circ\ \text{CCW} = 0^\circ\ \text{CCW}

Going partway around counterclockwise leaves only the leftover quarter-turns to draw.

4Draw the result

#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.

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. Four quarter-turns = a full circle back to start, so just find 20 divided by 4 - only the leftover 0 turns change the picture!

Four 90-degree turns return original · 10 practice problems

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