Problem
Angle between the hands at a given time
Two clocks read 4:10 and 2:37. The hour hand drifts past the number as minutes pass. Take the smaller of the two angles. Find the angle on each clock.
Reasoning · Grade 6-1 Clocks and Angles
Your answer
Watch out The difference is tempting, but past 180 you must subtract from 360.
How to solve
Strategy Draw a Diagram — The dial is already a protractor if I read it that way: one big division is 30 degrees and one small tick is 6 degrees. So I draw a ray from the centre straight up to the 12 and use it as a starting line, then measure each hand separately from that line. That turns one hard picture question into two easy subproblems — 'where is the minute hand?' and 'where is the hour hand?' — and a subtraction. Watching the units (degrees per minute for each hand) is what stops the classic mistake of leaving the hour hand parked on the numeral.
1STEP 1
Turn the dial into a protractor
A big division is 30 degrees, a small tick 6.
360° ÷ 12 = 30° (one big division), 360° ÷ 60 = 6° (one small tick)
Grade 4 already says an angle is measured by what fraction of a full turn it cuts off. A clock face is that idea printed on cardboard: the tick marks are the degree marks, you just have to know what each one is worth.
4.MD.C.5Draw A Diagram2STEP 2
Find how fast each hand moves
The hour hand moves 0.5 degrees a minute.
minute hand: 360° ÷ 60 = 6° per minute; hour hand: 30° ÷ 60 = 0.5° per minute
This is a rate, exactly like 'kilometres per hour', but the thing that piles up is turning instead of distance. Once you know degrees per minute, any number of minutes just multiplies.
6.RP.A.3Analyze The UnitsEach hand turns at its own fixed rate, so any number of minutes can be turned straight into degrees.
Why?
One complete turn of the dial is 360 degrees, and the marks cut that turn into equal slices the hands step through.
🧱Angles around a point make 360 degreesAngles that fit side by side all the way around one point fill exactly one full turn, 360 degrees.
Why?
Because every minute is worth the same fixed amount of turning, the degrees pile up in step with the minutes.
🧱A bigger unit is a fixed count of smaller unitsA bigger unit always equals the same fixed number of smaller units (1 hour = 60 minutes, 1 km = 1000 m), so a measure can be renamed between them.
3STEP 3
Clock (1), 4:10: place the minute hand
At 4:10 the minute hand is at 60 degrees.
10 × 6° = 60°
The minute hand is the easy one, because minutes and ticks are the same count. Ten minutes means ten ticks, no drift to worry about.
4.MD.C.6Identify Subproblems4STEP 4
Clock (1), 4:10: place the hour hand, drift included
With the drift the hour hand is at 125 degrees.
4 × 30° + 10 × 0.5° = 120° + 5° = 125°
Angle measure adds: the turn from the 12 to the 4 and the extra turn made during ten minutes sit side by side with no gap and no overlap, so you can just add the two numbers.
4.MD.C.7Identify Subproblems5STEP 5
Clock (1): subtract to get the angle
Subtracting gives 65 degrees.
125° - 60° = 65°
When two angles share a starting ray, the gap between them is a subtraction — the same 'take the small one away from the big one' move you use with a protractor.
4.MD.C.7Identify Subproblems6STEP 6
Clock (2), 2:37: place the minute hand
At 2:37 the minute hand is at 222 degrees.
37 × 6° = 222°
Thirty-seven ticks is more than half the dial, so you should expect a number bigger than 180 — a quick way to catch a slip before it spreads.
4.MD.C.6Identify Subproblems7STEP 7
Clock (2), 2:37: place the hour hand, drift included
The hour hand is at 78.5 degrees.
2 × 30° + 37 × 0.5° = 60° + 18.5° = 78.5°
Multiplying by 0.5 is just halving, so 37 times 0.5 is half of 37, which is 18.5. No hard decimal work is needed, only the willingness to remember the drift at all.
5.NBT.B.7Identify Subproblems8STEP 8
Clock (2): subtract, then check which angle is the smaller one
The difference 143.5 degrees is already the smaller.
222° - 78.5° = 143.5°, 360° - 143.5° = 216.5°
The two angles at the centre always add to a full turn, so once you have one of them the other is a single subtraction — and you keep whichever one is at most 180 degrees.
4.MD.C.7Identify SubproblemsAnswer
65, 143.5 degrees
125 − 60 = 65, 222 − 78.5 = 143.5
Both answers are in degrees and both are at most 180 degrees, which is what 'the smaller of the two angles' demands. Check them against the pictures: at 4:10 the hands are a bit more than two big divisions apart, and two big divisions is 60 degrees, so an answer a little above 60 is right — 65 degrees fits, while an answer near 120 would mean the hands were four divisions apart, which the picture flatly denies. At 2:37 the hands are nearly five big divisions apart, and 5 times 30 is 150 degrees, so an answer just under 150 is right — 143.5 degrees fits. Notice also that forgetting the hour hand's drift would have given 120 minus 60 = 60 degrees for (1) and 222 minus 60 = 162 degrees for (2); both would put the hour hand exactly on a numeral, which neither picture shows.
Takeaway
Measure each hand from the 12 and subtract — and never forget that the hour hand has already crept past its numeral, half a degree for every minute.
- Turn the dial into a protractor
- Find how fast each hand moves
- Clock (1), 4:10: place the minute hand
- Clock (1), 4:10: place the hour hand, drift included
- Clock (1): subtract to get the angle
- Clock (2), 2:37: place the minute hand
- Clock (2), 2:37: place the hour hand, drift included
- Clock (2): subtract, then check which angle is the smaller one