Slides for class
To project in class and work in groups. They download as PowerPoint (.pptx).
Clock puzzles · Part 1
Slides to project (39 slides): recognising and drawing angles with the hands, five puzzles with minutes, the chimes, the clock face split in three and the clock without numbers. Each puzzle is followed by its solution and the answer is in the notes.
Clock puzzles · Part 2
Slides with hard puzzles (17 slides): when the hands meet, when the same angle comes back, the straight-line times, the 44 right angles a day, 84° between 7 and 8, the clock that runs fast and the second hand.
Teaching objectives
A lower-secondary lab (ages 12-14). A clock with hands is a protractor everyone carries in their head: a full turn, half a turn and a quarter turn are seen in the minute hand, and the hours give the multiples of 30°. The whole lab is built around a large clock that can be handled: the hands can be dragged (geared, like a real clock, or loose, like a toy clock), the angle between them is shaded amber, and time-lapse animations show the hour hand moving.
Rooms 1-14 are for every student; rooms 15-19 are competition problems (AMC, AHSME, Lewis Carroll).
What is learnt
- Full turn (360°), half turn (180°) and quarter turn (90°) as turns of the minute hand; right and straight angles.
- There are two angles between the hands, adding up to 360°; «the angle between the hands» is the smaller one.
- 30° between two numbers on the clock, 6° between two marks. Converting 24-hour times to 12-hour times.
- The key idea: the hour hand moves too, 0.5° per minute. That is why the angle at 3:30 is 75°, not 90°.
- Working out the angle at any time: position of the minute hand (6° per minute) minus position of the hour hand (30° per hour and 0.5° per minute), and 360° minus the difference if it is more than 180°.
- Making an angle with the hands: with the hour hand fixed (toy clock) and with the minute hand fixed (real clock, where the hour hand can only be in certain positions).
- How the angle changes with time: the minute hand gains 5.5° per minute on the hour hand.
- Competition problems: the angle alone does not give the time; when the same angle comes back (720/11 minutes between overlaps); how many times a day the hands meet (22) or make a right angle (44); from the angle to the time with an equation; the clock in the mirror (always a possible time), the rotated clock (never) and the clock without numbers; identical hands (132 ambiguous times in 12 hours).
Key mathematical ideas, by block
A · For everyone (rooms 1-14). Measuring angles as fractions of a turn; proportional turning (30° per hour, 6° per minute, 0.5° per minute); the angle between two rays as the absolute value of a difference reduced to [0°, 180°]; relative speed.
B · Competition problems (rooms 15-19). Linear equations with absolute value (|210 − 5.5·m| = 84); relative motion and number of overtakes; symmetry (the mirror shows 12:00 − t) versus rotation (which breaks the 12 : 1 ratio between the hands); congruences modulo 360° (143·H ≡ 0).
Material and sources
AMC 8 2003 #20 (4:20 → 10°), AHSME 1954 #50 (84° between 7 and 8), Lewis Carroll's and Dudeney's interchangeable hands
problem. Numbers checked with scripts/verify_angulos_reloj.mjs.
Room-by-room contents
Room 1 · Right, straight and the full turn
RECOGNISE multiples of 90° between the two hands. In the book, an animation: the short hand still on 12 and the minute hand sweeping 90°, 180°, 270° and 360°, marking 3, 6, 9 and 12.
Student tasks
- Gallery of 8 clocks with the hands anywhere: tap the right angles (4).
- Gallery: tap the straight angles (3).
- How big is the shaded angle from 12 to 9? (270°).
- How many right angles fit in a turn? (4).
Room 2 · With the hour hand broken
CREATE multiples of 90°: the hour hand is broken and does not move; the minute hand is dragged.
Student tasks
- Hour hand on 12, straight (6).
- Hour hand on 3, right: both positions (12 and 6).
- Hour hand on 8, straight (2).
- Hour hand on 1, right: both positions (4 and 10).
Room 3 · 30° between two numbers
Multiples of 30° that are not multiples of 90°. In the book, an animation marking each number with its degrees (30°, 60°…; those of 90° in grey) and the smaller angle between the hands.
Student tasks
- O'clock times with 60° (2 and 10), 150° (5 and 7) and 120°.
- The big angle at 10:00 (300°).
Room 4 · On the hour
From 24-hour to 12-hour times; 30° per gap; the big angle is a trap.
Student tasks
- 16:00 (120°), 20:00 (120°), 23:00 (30°), 19:00 (150°).
Room 5 · Make the angle: toy clock
Loose hands: the clock as a protractor; there are usually two positions.
Student tasks
- Hour hand on 12, 90° (15 or 45 min); on 6, 120°; on 9, 150°; on 12, 42° (7 or 53 min). Each case, both positions.
Room 6 · Make the angle: real clock
Minute hand fixed: the hour hand can only go to certain positions (h:30 or h:20).
Student tasks
- Half past: 75° (3:30 and 8:30), 45°, 15° (5:30 and 6:30).
- Twenty past: 50° (2:20) and 40° (5:20).
Room 7 · The clock in the mirror
In the mirror the real time and the reflected time add up to 12:00; the reflection is always a possible time and the angle does not change. Only 12 and 6 are shown; the hint (💡) shows the other numbers.
Student tasks
- Real time of three mirrored clocks (9:20, 7:10, 12:55), with «See without mirror».
- A time that looks the same (12:00 or 6:00).
- Does the angle change? (no).
Room 8 · Identical hands
With identical hands the time is ambiguous if swapping them gives another possible time: 143·H a multiple of 360°.
Student tasks
- Can we tell the time at 6:00? (yes: 12:30 is impossible).
- 1:00 and 25 s (we cannot tell: 12:05 also works).
Room 9 · The hour hand moves too
Key idea: the hour hand moves 30° per hour, 0.5° per minute. Animation from 3:00 to 3:30 with the trail of the hour hand.
Student tasks
- Bet: do the hands make 90° at 3:30? (less).
- How far the hour hand moves in 30, 20, 12 and 1 minutes (15°, 10°, 6°, 0.5°).
Room 10 · The clumsy clockmaker
Impossible times: the hour hand has to match the minute hand. First recognise them, then fix them.
Student tasks
- Gallery with o'clock, half past, quarter past and quarter to: tap the impossible ones (3).
- Gallery with ten past, twenty past, twenty to and ten to: tap the impossible ones (4).
- Fix the hour hand with the minute hand fixed: 3:30, 7:15, 10:20 and 4:45.
Room 11 · The rotated clock
Rotating adds the same to both hands and breaks the 12 : 1 relation: a rotated clock never shows a possible time. It is turned with 30° buttons.
Student tasks
- Upside-down clock (no possible time; it is not 8:10).
- Straighten it with the buttons until it shows a possible time (2:40).
- 12:00 fallen on its side (none).
- Does any time survive a half turn? (no).
- Clock without numbers hanging crooked: what time is it? (4:00; not 8:20: the hour hand is exactly on a mark, so it is on the hour).
Room 12 · Half past
The method: position of the minute hand minus position of the hour hand (book example: 3:30 → 75°). Only half-past times, which give whole degrees.
Student tasks
- 21:30 (105°), 12:30 (165°), 22:30 (135°), 18:30 (15°).
Room 13 · The angle at any time
Practice of the method with any number of minutes.
Student tasks
- 14:20 (50°), 18:50 (95°), 19:12 (144°) and 16:20 (10°, AMC 8 2003).
Room 14 · A little later
The minute hand gains 5.5° per minute on the hour hand: 165° every half hour.
Student tasks
- Degrees gained in 10 min (55).
- From 3:00 (90°) to 3:30 (75°) and to 3:20 (20°); from 4:00 to 4:10 (65°) and 4:30 (45°).
Room 15 · We only know the angle
The angle alone does not tell what it will be in half an hour.
Student tasks
- 90° now, and in half an hour? (it depends).
- O'clock time with 90° and 105° half an hour later (9:00); with 75° (3:00).
- Without the clock: on the hour with a right angle, what angle in 20 minutes? Two answers: 20° (3:00) and 160° (9:00).
Room 16 · The angle has not changed
The angle repeats when the minute hand gains a full turn or twice the angle; overlaps every 720/11 min.
Student tasks
- From 3:00 back to 90° (360/11 ≈ 32.7 min).
- First overlap after 12:00 (720/11 ≈ 65.5 min).
- Overlap between 2 and 3 (2:11).
Room 17 · How many times a day?
In 12 h the minute hand overtakes the hour hand 11 times; 12-hour time-lapse with a counter.
Student tasks
- Bet: overlaps per day (22).
- Right angles per day (44) and straight lines (22).
Room 18 · From the angle to the time
Equation |210 − 5.5·m| = angle (book example: 90° between 4 and 5).
Student tasks
- 84° between 7 and 8: 22.9 and 53.45 min (7:23 and 7:53; AHSME 1954).
- Straight line between 2 and 3 (43 7/11 min).
Room 19 · Identical hands: how many?
How many times cannot be read with identical hands, step by step: minute hand = 12 · hour hand; for the swap 144 · H = H; 143 positions; minus the 11 overlaps.
Student tasks
- Hour hand at 5° → minute hand at 60°.
- Hour hand at 20° → minute hand at 240°.
- Swap the hands of 12:10 (not a possible time).
- 12 · (12 · H) = 144 · H.
- Positions in one turn: 143.
- Ambiguous times in 12 h: 132 (Lewis Carroll and Dudeney).
Rooms to project
The most striking ones to show and discuss in class.