Problem
Read the time on a numberless clock
The dial has tick marks but no numbers at all. Which long tick is the 12 is unknown. The hour hand sits 4 degrees from a tick. Work out the time.
Reasoning · Grade 6-1 Clocks and Angles
Your answer
Watch out Reading the top tick as 12 is tempting, but nothing says it is.
How to solve
Strategy Work Backwards — Normally you are told the time and asked for an angle; here the angle is given and the time is missing, so the whole solution runs the usual chain in reverse — that is tool 11. Tool 8 supplies the hinge: the hour hand's speed is a rate in degrees per minute, so dividing an angle in degrees by that rate returns a number of minutes, and checking that the units come out as minutes is what makes the step trustworthy. Once the minutes are known the minute hand becomes a signpost: it must be pointing at that minute, which pins down where the missing 12 sits, and tool 1 finishes the job — mark the 12 on the dial, write the numbers round clockwise, and simply read off the tick the hour hand is sitting on.
1STEP 1
Work out how fast the hour hand creeps
The hour hand moves 0.5 degrees a minute.
360°/(12 hours) = 30° per hour, 30°/(60 min) = 0.5° per minute
Turning the hour hand's motion into a per-minute rate is the same kind of step as finding a price per item, and it is what lets a measurement in degrees be traded for a measurement in minutes.
6.RP.A.2Analyze The Units2STEP 2
Run the rate backwards to get the minutes
Four degrees means 8 minutes have passed.
4° ÷ 0.5° per minute = 8 minutes
Forwards you would multiply minutes by 0.5 to get degrees; backwards you divide degrees by 0.5 to get minutes, and 4 is eight halves, so 8 minutes have gone by.
6.RP.A.3Work BackwardsRunning the hour hand's rate backwards turns how far it has crept into how many minutes have passed.
Why?
The hour hand creeps the same fixed amount every minute, so its position and the minutes are locked together.
🧱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.
Why?
Dividing by that rate undoes multiplying by it, so the creep can be read backwards into a count of minutes.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
3STEP 3
Check which side of the tick the hour hand is on
Check which side of the tick it sits on.
An angle on a dial has a direction as well as a size, and reading it the wrong way round changes the answer by more than an hour, so it is worth naming out loud before going on.
4.MD.C.5Work Backwards4STEP 4
Use the minute hand to find where the 12 is
Winding the minute hand back 8 shows where the 12 is.
8 ticks × 6° = 48°
Counting small ticks is exactly how you read minutes off any clock; here the count is simply done in reverse, using a known minute to find the missing starting mark.
3.MD.A.1Work Backwards5STEP 5
Number the dial clockwise from that 12
Number the dial clockwise from that 12.
Once one number on a dial is known, every other number is forced, because the 12 long ticks are equally spaced 30 degrees apart all the way round.
4.MD.C.5Draw A Diagram6STEP 6
Read the hour hand against the numbered dial
Reading the hour hand gives 10:08.
12 - 2 = 10 → 10 o'clock + 8 minutes = 10{:}08
Reading a clock is easy once the numbers are on it, so all the real work went into finding the 12; after that it is an ordinary time-telling question.
3.MD.A.1Draw A DiagramAnswer
10:08
4 ÷ 0.5 = 8
Both hands are checked against 10:08 independently. The hour hand at 10:08 should sit 10 × 30° + 8 × 0.5° = 304° clockwise from the 12, which is 4° past the 10 mark at 300° — matching the marked angle exactly. The minute hand should sit at 8 × 6° = 48° from the 12. The gap between the hands is then 304° - 48° = 256° one way, so 360° - 256° = 104° the short way, and in the picture the hands really are a bit more than a right angle apart, with the minute hand ahead of the hour hand going clockwise. The size of the answer is sensible too: 4° is a tiny fraction of the 30° between long ticks, about one seventh, so the minutes had to be a small number, and 8 out of 60 is indeed about one seventh of the way through the hour.
Takeaway
The hour hand's tiny 4-degree lean is a clock in itself: at half a degree a minute it says 8 minutes past, and the minute hand then shows you where the missing 12 must be.
- Work out how fast the hour hand creeps
- Run the rate backwards to get the minutes
- Check which side of the tick the hour hand is on
- Use the minute hand to find where the 12 is
- Number the dial clockwise from that 12
- Read the hour hand against the numbered dial