Problem
Angles swept by the clock hands
Liam does homework from 5:24 p.m. to 6:04 p.m.. Both clock hands keep turning the whole time. Work out the angle each hand turns through. Find how much further the minute hand turned.
Reasoning · Grade 4-1 Clocks and Angles
Your answer
Watch out Subtracting the clock faces confuses; the stretch is 40 minutes.
How to solve
Strategy Analyze the Units — Both hands turn at a steady rate, so the whole problem becomes a units question: how many degrees per minute does each hand turn? Once I know the degrees-per-minute for each hand, multiplying by the elapsed minutes gives each angle. To find those rates I first solve an easier version — how far does each hand go in one single minute — using the facts that the minute hand needs 60 minutes for a whole circle and the hour hand needs 60 minutes for just one big-number step. Sketching the clock face keeps the elapsed time and the direction of turning straight.
1STEP 1
Find how long the homework took
The homework lasted 40 minutes.
(60 - 24) + 4 = 36 + 4 = 40 minutes
Hopping to the next whole hour first turns an awkward subtraction across an hour boundary into two easy counts that a fourth grader can do on a drawn time line.
3.MD.A.1Draw A Diagram2STEP 2
Find the minute hand's rate, then its angle
At 6 degrees a minute the long hand turns 240 degrees.
360° ÷ 60 = 6° per minute, 6° × 40 = 240°
Degrees are just equal slices of one full turn, so sharing 360 degrees fairly among the 60 minute marks is the same kind of division as sharing 360 marbles among 60 children.
4.MD.C.5Analyze The UnitsSharing 360 degrees fairly among the 60 minute marks gives what one minute of turning is worth.
Why?
One complete turn around the centre is 360 degrees, and the minute marks cut that turn into 60 equal slices.
🧱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, any number of minutes is that amount repeated.
🧱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
Find the hour hand's rate the easy way
The short hand turns 1 degree every 2 minutes.
360° ÷ 12 = 30° per hour, 30° ÷ 60 = 1° every 2 minutes
Asking the smaller question — how far does the short hand go in one hour — is much easier than staring at 40 minutes, and once the one-hour answer is known the rest is simple scaling.
4.MD.C.5Solve An Easier Related Problem4STEP 4
Find the hour hand's angle for 40 minutes
In 40 minutes that is 20 degrees.
40 ÷ 2 = 20°
Counting how many 2-minute chunks fit inside 40 minutes is plain division within the times tables, and each chunk is worth exactly one degree.
3.OA.C.7Analyze The Units5STEP 5
Subtract the two angles
The difference is 240 − 20 = 220 degrees.
240° - 20° = 220°
Angle measures add and subtract exactly like lengths do, so once both turns are written in degrees the question becomes ordinary subtraction.
4.MD.C.7Analyze The UnitsAnswer
220 degrees
240 − 20 = 220
The units work out: degrees per minute times minutes leaves degrees, and 220 degrees is the difference of two angle measures, so it is properly an angle. The size is sensible too. In 40 minutes the long hand covers two thirds of the dial, which is between a half turn (180 degrees) and a full turn (360 degrees), and 240 degrees sits right there. The short hand only shuffles from just past 5 to just past 6, less than one big step, so its 20 degrees must be under 30 degrees — and it is. A neat check: the long hand always turns 12 times as far as the short hand in the same time, and 20 times 12 is exactly 240.
Takeaway
Every minute the long hand turns 6 degrees while the short hand turns only half a degree — that is why one falls so far behind the other!
- Find how long the homework took
- Find the minute hand's rate, then its angle
- Find the hour hand's rate the easy way
- Find the hour hand's angle for 40 minutes
- Subtract the two angles