A clock face is twelve equal 30-degree parts

4.MD.C.64.MD.C.7 · adapt · grade 4

Archetype: Angle Facts in a Figure · step in a 13-type progression

When a clock shows 4 o'clock, find the smaller angle formed between the minute hand and the hour hand.

Show solution

Understand

At exactly 4 o'clock the minute hand points to 12 and the hour hand points to 4; find the smaller angle between the two hands.

Givens

The clock shows 4:00.
The minute hand points to 12 and the hour hand points to 4.
A full clock face is a circle of 360 degrees split into 12 equal number marks.

Unknowns

The smaller angle between the minute hand and the hour hand

Constraints

The angle is measured as the size of the gap between the hands; we want the smaller of the two possible gaps.

Plan

#1 Draw a Diagram · also uses: #7 Identify Subproblems#8 Analyze the Units

Find how many degrees one number-to-number gap is by dividing the full 360 degrees into 12 equal parts, then count the gaps between 12 and 4 and multiply.

Execute

#7 Identify Subproblems 4.MD.C.6

A whole turn is 360 degrees split evenly among the 12 numbers, so each gap from one number to the next is 360 ÷ 12 = 30 degrees.

360^\circ \div 12 = 30^\circ

Twelve equal slices of a full circle each measure 30 degrees.

#1 Draw a Diagram 4.MD.C.7

The minute hand is at 12 and the hour hand is at 4. Going from 12 to 4 the short way passes 4 number gaps (12→1→2→3→4).

4 \text{ gaps}

From 12 to 4 the shorter side spans four hour marks.

#8 Analyze the Units 4.MD.C.7

Four gaps of 30 degrees each give 4 × 30 = 120 degrees. The other way around would be 8 × 30 = 240 degrees, so the smaller angle is 120 degrees.

4 \times 30^\circ = 120^\circ

Adding four 30-degree slices gives 120 degrees, the smaller of the two gaps.

Answer: 120°

Review

120 degrees is more than a right angle (90°) but less than a straight angle (180°), which fits the wide-but-not-flat gap you see at 4:00. The two angles 120° and 240° add to 360°, confirming 120° is the smaller.

Look for a pattern (tool 5): each whole hour the hour hand is 30° farther from 12, so at 4:00 it is 4 × 30° = 120° from the minute hand at 12.

Standards · min grade 4

4.MD.C.6 Measure angles in whole-number degrees using a protractor — Establishing that each of the 12 equal clock gaps measures 30 degrees.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Adding four 30-degree gaps to get the 120-degree angle between the hands.

💡 This only needs Grade 4 angle sense: a clock is 12 equal 30-degree slices, so just count the slices between the hands!