Problem
A clock face is twelve equal 30-degree parts
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.
Measurement & data
Your answer
°
Watch out It's easy to count the wrong way around the clock and get 240° (8 gaps) instead of the smaller angle the question actually asks for.
How to solve
Strategy Draw a Diagram — 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.
1STEP 1
Degrees in one hour-mark gap
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 ÷ 12 = 30^\circ
Twelve equal slices of a full circle each measure 30 degrees.
4.MD.C.6Identify SubproblemsOne gap between two neighboring clock numbers measures 30 degrees.
Why?
A full turn around the clock is 360 degrees, and the 12 numbers split that turn into 12 gaps of the same size, so each gap gets an equal share of the 360 degrees.
Why?
The 12 gaps sit next to each other with no overlap and no space left over, so all of them together make exactly the whole 360-degree turn.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
Because the 12 gaps are equal, finding one gap means splitting 360 into 12 equal shares, and splitting a total back into equal groups is the division that undoes the grouping.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
2STEP 2
Count the gaps from 12 to 4
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 gaps
From 12 to 4 the shorter side spans four hour marks.
4.MD.C.7Draw A Diagram3STEP 3
Multiply to get the angle
Four gaps of 30 degrees each give 4 × 30 = 120 degrees.
4 × 30^\circ = 120^\circ
Adding four 30-degree slices gives 120 degrees, the smaller of the two gaps.
4.MD.C.7Analyze The UnitsAnswer
120 °
4 × 30° = 120°
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.
Takeaway
This only needs Grade 4 angle sense: a clock is 12 equal 30-degree slices, so just count the slices between the hands!
- Degrees in one hour-mark gap
- Count the gaps from 12 to 4
- Multiply to get the angle
Where next?
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