Practice · A clock face is twelve equal 30-degree parts · Sensim Math

Variant 1 easy

A clock shows 3:00.

Find the measure of the smaller angle formed by the hour hand and the minute hand.

Your answer

Show solution

1 · Understandwhat's really being asked

A clock reads 3:00; we want the smaller angle between the two hands.

Givens

The time is 3:00.
A full clock face is 360 degrees split into 12 equal 30-degree parts.

Unknowns

The smaller angle (in degrees) between the hour and minute hands.

Constraints

The hour hand also drifts 0.5 degrees for each passing minute.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #13 Convert to Algebra

Find each hand's angle from the 12 mark, then subtract; the smaller of the two arcs is the answer.

3 · Execute4 carry out the plan

1Minute hand

#13 Convert to Algebra 4.MD.C.5

The minute hand sweeps 6 degrees per minute.

6 \times 0 = 0^\circ

60 minutes make a full 360-degree turn, so each minute is 6 degrees.

2Hour hand

#13 Convert to Algebra 4.MD.C.6

The hour hand is 30 degrees per hour plus a half-degree drift each minute.

30 \times 3 + 0.5 \times 0 = 90^\circ

Each hour mark is 30 degrees, and the hour hand creeps forward as minutes pass.

3Subtract

#13 Convert to Algebra 4.MD.C.7

The gap between the hands is the difference of the two angles.

|90 - 0| = 90^\circ

Angles add and subtract along the circle.

4Smaller arc

#13 Convert to Algebra 4.MD.C.7

Keep the smaller of the gap and its 360-degree complement.

\min(90,\ 360 - 90) = 90^\circ

Two hands cut the circle into two arcs; we want the smaller one.

Answer: 90 degrees

4 · Reviewdoes it hold up?

90 degrees is between 0 and 180, as any smaller clock angle must be.

Another way: Count whole 30-degree hour steps between the hands, then adjust for the minute drift.

Standardsmin grade 4

4.MD.C.5 Recognize angles as geometric figures formed by two rays, and understand angle measure as a fraction of a 360-degree circle. — Seeing the clock as a 360-degree circle in 30-degree parts.
4.MD.C.6 Measure angles in whole-number degrees using a protractor. — Measuring each hand's angle in degrees.
4.MD.C.7 Recognize angle measure as additive; find an unknown angle by adding or subtracting parts. — Adding/subtracting angles to get the gap.

💡Takeaway. A clock is a circle cut into twelve 30-degree slices — measure each hand from 12, then take the difference.

Variant 2 easy

A clock shows 4:00.

Find the measure of the smaller angle formed by the hour hand and the minute hand.

Your answer

Show solution

1 · Understandwhat's really being asked

A clock reads 4:00; we want the smaller angle between the two hands.

Givens

The time is 4:00.
A full clock face is 360 degrees split into 12 equal 30-degree parts.

Unknowns

The smaller angle (in degrees) between the hour and minute hands.

Constraints

The hour hand also drifts 0.5 degrees for each passing minute.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #13 Convert to Algebra

Find each hand's angle from the 12 mark, then subtract; the smaller of the two arcs is the answer.

3 · Execute4 carry out the plan

1Minute hand

#13 Convert to Algebra 4.MD.C.5

The minute hand sweeps 6 degrees per minute.

6 \times 0 = 0^\circ

60 minutes make a full 360-degree turn, so each minute is 6 degrees.

2Hour hand

#13 Convert to Algebra 4.MD.C.6

The hour hand is 30 degrees per hour plus a half-degree drift each minute.

30 \times 4 + 0.5 \times 0 = 120^\circ

Each hour mark is 30 degrees, and the hour hand creeps forward as minutes pass.

3Subtract

#13 Convert to Algebra 4.MD.C.7

The gap between the hands is the difference of the two angles.

|120 - 0| = 120^\circ

Angles add and subtract along the circle.

4Smaller arc

#13 Convert to Algebra 4.MD.C.7

Keep the smaller of the gap and its 360-degree complement.

\min(120,\ 360 - 120) = 120^\circ

Two hands cut the circle into two arcs; we want the smaller one.

Answer: 120 degrees

4 · Reviewdoes it hold up?

120 degrees is between 0 and 180, as any smaller clock angle must be.

Another way: Count whole 30-degree hour steps between the hands, then adjust for the minute drift.

Standardsmin grade 4

4.MD.C.5 Recognize angles as geometric figures formed by two rays, and understand angle measure as a fraction of a 360-degree circle. — Seeing the clock as a 360-degree circle in 30-degree parts.
4.MD.C.6 Measure angles in whole-number degrees using a protractor. — Measuring each hand's angle in degrees.
4.MD.C.7 Recognize angle measure as additive; find an unknown angle by adding or subtracting parts. — Adding/subtracting angles to get the gap.

💡Takeaway. A clock is a circle cut into twelve 30-degree slices — measure each hand from 12, then take the difference.

Variant 3 easy

A clock shows 1:05.

Find the measure of the smaller angle formed by the hour hand and the minute hand.

Your answer

Show solution

1 · Understandwhat's really being asked

A clock reads 1:05; we want the smaller angle between the two hands.

Givens

The time is 1:05.
A full clock face is 360 degrees split into 12 equal 30-degree parts.

Unknowns

The smaller angle (in degrees) between the hour and minute hands.

Constraints

The hour hand also drifts 0.5 degrees for each passing minute.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #13 Convert to Algebra

Find each hand's angle from the 12 mark, then subtract; the smaller of the two arcs is the answer.

3 · Execute4 carry out the plan

1Minute hand

#13 Convert to Algebra 4.MD.C.5

The minute hand sweeps 6 degrees per minute.

6 \times 5 = 30^\circ

60 minutes make a full 360-degree turn, so each minute is 6 degrees.

2Hour hand

#13 Convert to Algebra 4.MD.C.6

The hour hand is 30 degrees per hour plus a half-degree drift each minute.

30 \times 1 + 0.5 \times 5 = 32.5^\circ

Each hour mark is 30 degrees, and the hour hand creeps forward as minutes pass.

3Subtract

#13 Convert to Algebra 4.MD.C.7

The gap between the hands is the difference of the two angles.

|32.5 - 30| = 2.5^\circ

Angles add and subtract along the circle.

4Smaller arc

#13 Convert to Algebra 4.MD.C.7

Keep the smaller of the gap and its 360-degree complement.

\min(2.5,\ 360 - 2.5) = 2.5^\circ

Two hands cut the circle into two arcs; we want the smaller one.

Answer: 2.5 degrees

4 · Reviewdoes it hold up?

2.5 degrees is between 0 and 180, as any smaller clock angle must be.

Another way: Count whole 30-degree hour steps between the hands, then adjust for the minute drift.

Standardsmin grade 4

4.MD.C.5 Recognize angles as geometric figures formed by two rays, and understand angle measure as a fraction of a 360-degree circle. — Seeing the clock as a 360-degree circle in 30-degree parts.
4.MD.C.6 Measure angles in whole-number degrees using a protractor. — Measuring each hand's angle in degrees.
4.MD.C.7 Recognize angle measure as additive; find an unknown angle by adding or subtracting parts. — Adding/subtracting angles to get the gap.

💡Takeaway. A clock is a circle cut into twelve 30-degree slices — measure each hand from 12, then take the difference.

Variant 4 easy

A clock shows 6:00.

Find the measure of the smaller angle formed by the hour hand and the minute hand.

Your answer

Show solution

1 · Understandwhat's really being asked

A clock reads 6:00; we want the smaller angle between the two hands.

Givens

The time is 6:00.
A full clock face is 360 degrees split into 12 equal 30-degree parts.

Unknowns

The smaller angle (in degrees) between the hour and minute hands.

Constraints

The hour hand also drifts 0.5 degrees for each passing minute.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #13 Convert to Algebra

Find each hand's angle from the 12 mark, then subtract; the smaller of the two arcs is the answer.

3 · Execute4 carry out the plan

1Minute hand

#13 Convert to Algebra 4.MD.C.5

The minute hand sweeps 6 degrees per minute.

6 \times 0 = 0^\circ

60 minutes make a full 360-degree turn, so each minute is 6 degrees.

2Hour hand

#13 Convert to Algebra 4.MD.C.6

The hour hand is 30 degrees per hour plus a half-degree drift each minute.

30 \times 6 + 0.5 \times 0 = 180^\circ

Each hour mark is 30 degrees, and the hour hand creeps forward as minutes pass.

3Subtract

#13 Convert to Algebra 4.MD.C.7

The gap between the hands is the difference of the two angles.

|180 - 0| = 180^\circ

Angles add and subtract along the circle.

4Smaller arc

#13 Convert to Algebra 4.MD.C.7

Keep the smaller of the gap and its 360-degree complement.

\min(180,\ 360 - 180) = 180^\circ

Two hands cut the circle into two arcs; we want the smaller one.

Answer: 180 degrees

4 · Reviewdoes it hold up?

180 degrees is between 0 and 180, as any smaller clock angle must be.

Another way: Count whole 30-degree hour steps between the hands, then adjust for the minute drift.

Standardsmin grade 4

4.MD.C.5 Recognize angles as geometric figures formed by two rays, and understand angle measure as a fraction of a 360-degree circle. — Seeing the clock as a 360-degree circle in 30-degree parts.
4.MD.C.6 Measure angles in whole-number degrees using a protractor. — Measuring each hand's angle in degrees.
4.MD.C.7 Recognize angle measure as additive; find an unknown angle by adding or subtracting parts. — Adding/subtracting angles to get the gap.

💡Takeaway. A clock is a circle cut into twelve 30-degree slices — measure each hand from 12, then take the difference.

Variant 5 medium

A clock shows 9:00.

Find the measure of the smaller angle formed by the hour hand and the minute hand.

Your answer

Show solution

1 · Understandwhat's really being asked

A clock reads 9:00; we want the smaller angle between the two hands.

Givens

The time is 9:00.
A full clock face is 360 degrees split into 12 equal 30-degree parts.

Unknowns

The smaller angle (in degrees) between the hour and minute hands.

Constraints

The hour hand also drifts 0.5 degrees for each passing minute.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #13 Convert to Algebra

Find each hand's angle from the 12 mark, then subtract; the smaller of the two arcs is the answer.

3 · Execute4 carry out the plan

1Minute hand

#13 Convert to Algebra 4.MD.C.5

The minute hand sweeps 6 degrees per minute.

6 \times 0 = 0^\circ

60 minutes make a full 360-degree turn, so each minute is 6 degrees.

2Hour hand

#13 Convert to Algebra 4.MD.C.6

The hour hand is 30 degrees per hour plus a half-degree drift each minute.

30 \times 9 + 0.5 \times 0 = 270^\circ

Each hour mark is 30 degrees, and the hour hand creeps forward as minutes pass.

3Subtract

#13 Convert to Algebra 4.MD.C.7

The gap between the hands is the difference of the two angles.

|270 - 0| = 270^\circ

Angles add and subtract along the circle.

4Smaller arc

#13 Convert to Algebra 4.MD.C.7

Keep the smaller of the gap and its 360-degree complement.

\min(270,\ 360 - 270) = 90^\circ

Two hands cut the circle into two arcs; we want the smaller one.

Answer: 90 degrees

4 · Reviewdoes it hold up?

90 degrees is between 0 and 180, as any smaller clock angle must be.

Another way: Count whole 30-degree hour steps between the hands, then adjust for the minute drift.

Standardsmin grade 4

4.MD.C.5 Recognize angles as geometric figures formed by two rays, and understand angle measure as a fraction of a 360-degree circle. — Seeing the clock as a 360-degree circle in 30-degree parts.
4.MD.C.6 Measure angles in whole-number degrees using a protractor. — Measuring each hand's angle in degrees.
4.MD.C.7 Recognize angle measure as additive; find an unknown angle by adding or subtracting parts. — Adding/subtracting angles to get the gap.

💡Takeaway. A clock is a circle cut into twelve 30-degree slices — measure each hand from 12, then take the difference.

Variant 6 medium

A clock shows 8:10.

Find the measure of the smaller angle formed by the hour hand and the minute hand.

Your answer

Show solution

1 · Understandwhat's really being asked

A clock reads 8:10; we want the smaller angle between the two hands.

Givens

The time is 8:10.
A full clock face is 360 degrees split into 12 equal 30-degree parts.

Unknowns

The smaller angle (in degrees) between the hour and minute hands.

Constraints

The hour hand also drifts 0.5 degrees for each passing minute.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #13 Convert to Algebra

Find each hand's angle from the 12 mark, then subtract; the smaller of the two arcs is the answer.

3 · Execute4 carry out the plan

1Minute hand

#13 Convert to Algebra 4.MD.C.5

The minute hand sweeps 6 degrees per minute.

6 \times 10 = 60^\circ

60 minutes make a full 360-degree turn, so each minute is 6 degrees.

2Hour hand

#13 Convert to Algebra 4.MD.C.6

The hour hand is 30 degrees per hour plus a half-degree drift each minute.

30 \times 8 + 0.5 \times 10 = 245^\circ

Each hour mark is 30 degrees, and the hour hand creeps forward as minutes pass.

3Subtract

#13 Convert to Algebra 4.MD.C.7

The gap between the hands is the difference of the two angles.

|245 - 60| = 185^\circ

Angles add and subtract along the circle.

4Smaller arc

#13 Convert to Algebra 4.MD.C.7

Keep the smaller of the gap and its 360-degree complement.

\min(185,\ 360 - 185) = 175^\circ

Two hands cut the circle into two arcs; we want the smaller one.

Answer: 175 degrees

4 · Reviewdoes it hold up?

175 degrees is between 0 and 180, as any smaller clock angle must be.

Another way: Count whole 30-degree hour steps between the hands, then adjust for the minute drift.

Standardsmin grade 4

4.MD.C.5 Recognize angles as geometric figures formed by two rays, and understand angle measure as a fraction of a 360-degree circle. — Seeing the clock as a 360-degree circle in 30-degree parts.
4.MD.C.6 Measure angles in whole-number degrees using a protractor. — Measuring each hand's angle in degrees.
4.MD.C.7 Recognize angle measure as additive; find an unknown angle by adding or subtracting parts. — Adding/subtracting angles to get the gap.

💡Takeaway. A clock is a circle cut into twelve 30-degree slices — measure each hand from 12, then take the difference.

Variant 7 medium

A clock shows 2:15.

Find the measure of the smaller angle formed by the hour hand and the minute hand.

Your answer

Show solution

1 · Understandwhat's really being asked

A clock reads 2:15; we want the smaller angle between the two hands.

Givens

The time is 2:15.
A full clock face is 360 degrees split into 12 equal 30-degree parts.

Unknowns

The smaller angle (in degrees) between the hour and minute hands.

Constraints

The hour hand also drifts 0.5 degrees for each passing minute.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #13 Convert to Algebra

Find each hand's angle from the 12 mark, then subtract; the smaller of the two arcs is the answer.

3 · Execute4 carry out the plan

1Minute hand

#13 Convert to Algebra 4.MD.C.5

The minute hand sweeps 6 degrees per minute.

6 \times 15 = 90^\circ

60 minutes make a full 360-degree turn, so each minute is 6 degrees.

2Hour hand

#13 Convert to Algebra 4.MD.C.6

The hour hand is 30 degrees per hour plus a half-degree drift each minute.

30 \times 2 + 0.5 \times 15 = 67.5^\circ

Each hour mark is 30 degrees, and the hour hand creeps forward as minutes pass.

3Subtract

#13 Convert to Algebra 4.MD.C.7

The gap between the hands is the difference of the two angles.

|67.5 - 90| = 22.5^\circ

Angles add and subtract along the circle.

4Smaller arc

#13 Convert to Algebra 4.MD.C.7

Keep the smaller of the gap and its 360-degree complement.

\min(22.5,\ 360 - 22.5) = 22.5^\circ

Two hands cut the circle into two arcs; we want the smaller one.

Answer: 22.5 degrees

4 · Reviewdoes it hold up?

22.5 degrees is between 0 and 180, as any smaller clock angle must be.

Another way: Count whole 30-degree hour steps between the hands, then adjust for the minute drift.

Standardsmin grade 4

4.MD.C.5 Recognize angles as geometric figures formed by two rays, and understand angle measure as a fraction of a 360-degree circle. — Seeing the clock as a 360-degree circle in 30-degree parts.
4.MD.C.6 Measure angles in whole-number degrees using a protractor. — Measuring each hand's angle in degrees.
4.MD.C.7 Recognize angle measure as additive; find an unknown angle by adding or subtracting parts. — Adding/subtracting angles to get the gap.

💡Takeaway. A clock is a circle cut into twelve 30-degree slices — measure each hand from 12, then take the difference.

Variant 8 medium

A clock shows 7:20.

Find the measure of the smaller angle formed by the hour hand and the minute hand.

Your answer

Show solution

1 · Understandwhat's really being asked

A clock reads 7:20; we want the smaller angle between the two hands.

Givens

The time is 7:20.
A full clock face is 360 degrees split into 12 equal 30-degree parts.

Unknowns

The smaller angle (in degrees) between the hour and minute hands.

Constraints

The hour hand also drifts 0.5 degrees for each passing minute.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #13 Convert to Algebra

Find each hand's angle from the 12 mark, then subtract; the smaller of the two arcs is the answer.

3 · Execute4 carry out the plan

1Minute hand

#13 Convert to Algebra 4.MD.C.5

The minute hand sweeps 6 degrees per minute.

6 \times 20 = 120^\circ

60 minutes make a full 360-degree turn, so each minute is 6 degrees.

2Hour hand

#13 Convert to Algebra 4.MD.C.6

The hour hand is 30 degrees per hour plus a half-degree drift each minute.

30 \times 7 + 0.5 \times 20 = 220^\circ

Each hour mark is 30 degrees, and the hour hand creeps forward as minutes pass.

3Subtract

#13 Convert to Algebra 4.MD.C.7

The gap between the hands is the difference of the two angles.

|220 - 120| = 100^\circ

Angles add and subtract along the circle.

4Smaller arc

#13 Convert to Algebra 4.MD.C.7

Keep the smaller of the gap and its 360-degree complement.

\min(100,\ 360 - 100) = 100^\circ

Two hands cut the circle into two arcs; we want the smaller one.

Answer: 100 degrees

4 · Reviewdoes it hold up?

100 degrees is between 0 and 180, as any smaller clock angle must be.

Another way: Count whole 30-degree hour steps between the hands, then adjust for the minute drift.

Standardsmin grade 4

4.MD.C.5 Recognize angles as geometric figures formed by two rays, and understand angle measure as a fraction of a 360-degree circle. — Seeing the clock as a 360-degree circle in 30-degree parts.
4.MD.C.6 Measure angles in whole-number degrees using a protractor. — Measuring each hand's angle in degrees.
4.MD.C.7 Recognize angle measure as additive; find an unknown angle by adding or subtracting parts. — Adding/subtracting angles to get the gap.

💡Takeaway. A clock is a circle cut into twelve 30-degree slices — measure each hand from 12, then take the difference.

Variant 9 hard

A clock shows 12:30.

Find the measure of the smaller angle formed by the hour hand and the minute hand.

Your answer

Show solution

1 · Understandwhat's really being asked

A clock reads 12:30; we want the smaller angle between the two hands.

Givens

The time is 12:30.
A full clock face is 360 degrees split into 12 equal 30-degree parts.

Unknowns

The smaller angle (in degrees) between the hour and minute hands.

Constraints

The hour hand also drifts 0.5 degrees for each passing minute.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #13 Convert to Algebra

Find each hand's angle from the 12 mark, then subtract; the smaller of the two arcs is the answer.

3 · Execute4 carry out the plan

1Minute hand

#13 Convert to Algebra 4.MD.C.5

The minute hand sweeps 6 degrees per minute.

6 \times 30 = 180^\circ

60 minutes make a full 360-degree turn, so each minute is 6 degrees.

2Hour hand

#13 Convert to Algebra 4.MD.C.6

The hour hand is 30 degrees per hour plus a half-degree drift each minute.

30 \times 0 + 0.5 \times 30 = 15^\circ

Each hour mark is 30 degrees, and the hour hand creeps forward as minutes pass.

3Subtract

#13 Convert to Algebra 4.MD.C.7

The gap between the hands is the difference of the two angles.

|15 - 180| = 165^\circ

Angles add and subtract along the circle.

4Smaller arc

#13 Convert to Algebra 4.MD.C.7

Keep the smaller of the gap and its 360-degree complement.

\min(165,\ 360 - 165) = 165^\circ

Two hands cut the circle into two arcs; we want the smaller one.

Answer: 165 degrees

4 · Reviewdoes it hold up?

165 degrees is between 0 and 180, as any smaller clock angle must be.

Another way: Count whole 30-degree hour steps between the hands, then adjust for the minute drift.

Standardsmin grade 4

4.MD.C.5 Recognize angles as geometric figures formed by two rays, and understand angle measure as a fraction of a 360-degree circle. — Seeing the clock as a 360-degree circle in 30-degree parts.
4.MD.C.6 Measure angles in whole-number degrees using a protractor. — Measuring each hand's angle in degrees.
4.MD.C.7 Recognize angle measure as additive; find an unknown angle by adding or subtracting parts. — Adding/subtracting angles to get the gap.

💡Takeaway. A clock is a circle cut into twelve 30-degree slices — measure each hand from 12, then take the difference.

Variant 10 hard

A clock shows 10:40.

Find the measure of the smaller angle formed by the hour hand and the minute hand.

Your answer

Show solution

1 · Understandwhat's really being asked

A clock reads 10:40; we want the smaller angle between the two hands.

Givens

The time is 10:40.
A full clock face is 360 degrees split into 12 equal 30-degree parts.

Unknowns

The smaller angle (in degrees) between the hour and minute hands.

Constraints

The hour hand also drifts 0.5 degrees for each passing minute.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #13 Convert to Algebra

Find each hand's angle from the 12 mark, then subtract; the smaller of the two arcs is the answer.

3 · Execute4 carry out the plan

1Minute hand

#13 Convert to Algebra 4.MD.C.5

The minute hand sweeps 6 degrees per minute.

6 \times 40 = 240^\circ

60 minutes make a full 360-degree turn, so each minute is 6 degrees.

2Hour hand

#13 Convert to Algebra 4.MD.C.6

The hour hand is 30 degrees per hour plus a half-degree drift each minute.

30 \times 10 + 0.5 \times 40 = 320^\circ

Each hour mark is 30 degrees, and the hour hand creeps forward as minutes pass.

3Subtract

#13 Convert to Algebra 4.MD.C.7

The gap between the hands is the difference of the two angles.

|320 - 240| = 80^\circ

Angles add and subtract along the circle.

4Smaller arc

#13 Convert to Algebra 4.MD.C.7

Keep the smaller of the gap and its 360-degree complement.

\min(80,\ 360 - 80) = 80^\circ

Two hands cut the circle into two arcs; we want the smaller one.

Answer: 80 degrees

4 · Reviewdoes it hold up?

80 degrees is between 0 and 180, as any smaller clock angle must be.

Another way: Count whole 30-degree hour steps between the hands, then adjust for the minute drift.

Standardsmin grade 4

4.MD.C.5 Recognize angles as geometric figures formed by two rays, and understand angle measure as a fraction of a 360-degree circle. — Seeing the clock as a 360-degree circle in 30-degree parts.
4.MD.C.6 Measure angles in whole-number degrees using a protractor. — Measuring each hand's angle in degrees.
4.MD.C.7 Recognize angle measure as additive; find an unknown angle by adding or subtracting parts. — Adding/subtracting angles to get the gap.

💡Takeaway. A clock is a circle cut into twelve 30-degree slices — measure each hand from 12, then take the difference.

Variant 11 hard

A clock shows 5:45.

Find the measure of the smaller angle formed by the hour hand and the minute hand.

Your answer

Show solution

1 · Understandwhat's really being asked

A clock reads 5:45; we want the smaller angle between the two hands.

Givens

The time is 5:45.
A full clock face is 360 degrees split into 12 equal 30-degree parts.

Unknowns

The smaller angle (in degrees) between the hour and minute hands.

Constraints

The hour hand also drifts 0.5 degrees for each passing minute.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #13 Convert to Algebra

Find each hand's angle from the 12 mark, then subtract; the smaller of the two arcs is the answer.

3 · Execute4 carry out the plan

1Minute hand

#13 Convert to Algebra 4.MD.C.5

The minute hand sweeps 6 degrees per minute.

6 \times 45 = 270^\circ

60 minutes make a full 360-degree turn, so each minute is 6 degrees.

2Hour hand

#13 Convert to Algebra 4.MD.C.6

The hour hand is 30 degrees per hour plus a half-degree drift each minute.

30 \times 5 + 0.5 \times 45 = 172.5^\circ

Each hour mark is 30 degrees, and the hour hand creeps forward as minutes pass.

3Subtract

#13 Convert to Algebra 4.MD.C.7

The gap between the hands is the difference of the two angles.

|172.5 - 270| = 97.5^\circ

Angles add and subtract along the circle.

4Smaller arc

#13 Convert to Algebra 4.MD.C.7

Keep the smaller of the gap and its 360-degree complement.

\min(97.5,\ 360 - 97.5) = 97.5^\circ

Two hands cut the circle into two arcs; we want the smaller one.

Answer: 97.5 degrees

4 · Reviewdoes it hold up?

97.5 degrees is between 0 and 180, as any smaller clock angle must be.

Another way: Count whole 30-degree hour steps between the hands, then adjust for the minute drift.

Standardsmin grade 4

4.MD.C.5 Recognize angles as geometric figures formed by two rays, and understand angle measure as a fraction of a 360-degree circle. — Seeing the clock as a 360-degree circle in 30-degree parts.
4.MD.C.6 Measure angles in whole-number degrees using a protractor. — Measuring each hand's angle in degrees.
4.MD.C.7 Recognize angle measure as additive; find an unknown angle by adding or subtracting parts. — Adding/subtracting angles to get the gap.

💡Takeaway. A clock is a circle cut into twelve 30-degree slices — measure each hand from 12, then take the difference.

Variant 12 hard

A clock shows 11:55.

Find the measure of the smaller angle formed by the hour hand and the minute hand.

Your answer

Show solution

1 · Understandwhat's really being asked

A clock reads 11:55; we want the smaller angle between the two hands.

Givens

The time is 11:55.
A full clock face is 360 degrees split into 12 equal 30-degree parts.

Unknowns

The smaller angle (in degrees) between the hour and minute hands.

Constraints

The hour hand also drifts 0.5 degrees for each passing minute.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #13 Convert to Algebra

Find each hand's angle from the 12 mark, then subtract; the smaller of the two arcs is the answer.

3 · Execute4 carry out the plan

1Minute hand

#13 Convert to Algebra 4.MD.C.5

The minute hand sweeps 6 degrees per minute.

6 \times 55 = 330^\circ

60 minutes make a full 360-degree turn, so each minute is 6 degrees.

2Hour hand

#13 Convert to Algebra 4.MD.C.6

The hour hand is 30 degrees per hour plus a half-degree drift each minute.

30 \times 11 + 0.5 \times 55 = 357.5^\circ

Each hour mark is 30 degrees, and the hour hand creeps forward as minutes pass.

3Subtract

#13 Convert to Algebra 4.MD.C.7

The gap between the hands is the difference of the two angles.

|357.5 - 330| = 27.5^\circ

Angles add and subtract along the circle.

4Smaller arc

#13 Convert to Algebra 4.MD.C.7

Keep the smaller of the gap and its 360-degree complement.

\min(27.5,\ 360 - 27.5) = 27.5^\circ

Two hands cut the circle into two arcs; we want the smaller one.

Answer: 27.5 degrees

4 · Reviewdoes it hold up?

27.5 degrees is between 0 and 180, as any smaller clock angle must be.

Another way: Count whole 30-degree hour steps between the hands, then adjust for the minute drift.

Standardsmin grade 4

4.MD.C.5 Recognize angles as geometric figures formed by two rays, and understand angle measure as a fraction of a 360-degree circle. — Seeing the clock as a 360-degree circle in 30-degree parts.
4.MD.C.6 Measure angles in whole-number degrees using a protractor. — Measuring each hand's angle in degrees.
4.MD.C.7 Recognize angle measure as additive; find an unknown angle by adding or subtracting parts. — Adding/subtracting angles to get the gap.

💡Takeaway. A clock is a circle cut into twelve 30-degree slices — measure each hand from 12, then take the difference.

A clock face is twelve equal 30-degree parts · 12 practice problems

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