The minute hand sweeps 6 degrees a minute and the hour hand 0.5 degrees a minute, so the angle between them changes steadily and repeats. Nearly every clock problem is either an angle at a given moment, or a moment when the angle takes a given value.
●○○3 points, difficulty 1 of 33 ptLevel 5–6about 45s
What is the angle between the hands of a clock at exactly 3 o'clock?
Text description of the figure
A clock face with the hour hand pointing at 3 and the minute hand pointing at 12, with the angle between them marked.
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Twelve hour marks fill 360 degrees.
2The strategy
Work out how many degrees one hour mark is worth, then count how many hour marks lie between the two hands at this exact time.
3The full solution
One hour mark is 360 divided by 12 = 30 degrees. At 3 o'clock the hands are 3 marks apart, so the angle is 3 x 30 = 90 degrees.
Solution
The reliable way
The 12 hour marks divide 360 degrees, so each mark is 30 degrees. At exactly 3 o'clock the minute hand is at 12 and the hour hand is at 3, which is 3 marks apart. The angle is 3 x 30 = 90 degrees. The transferable idea: convert hour marks to degrees once and reuse the 30 for every clock question.
The elegant way
Three o'clock is exactly a quarter turn round the face, and a quarter of 360 is 90.
Why this is on the test: It fixes the 30-degrees-per-hour-mark fact that every harder clock question is built on.
Where it goes wrong
Each of these is written the way a student would say it to themselves.
“I forgot that the hour hand keeps moving between the numbers.”
“I found one time when the hands line up and forgot it happens repeatedly.”
“I measured the angle the wrong way round the clock face.”
Every problem in this skill
●○○3 points, difficulty 1 of 33 ptOne step from a skill you know.