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CS Unplugged activity

Clocks That Wrap Around: the modulo idea

About 20 minutesSoloPencil

Some numbers wrap around instead of growing forever, like the hands on a clock. This wraparound trick is called modulo, and computers use it all the time. You’ll count around a few clocks, then predict a computer’s own version of it.

Learn more: clocks are one example of modular arithmetic, a whole branch of math built on wraparound counting.

1. Look at the clock below. It has 10 spots, numbered 0 to 9.
2. Start at the number you’re given.
3. Count forward that many steps around the circle, going past 9 back to 0.
4. Write down the spot where you land.
5. Check yourself: it’s always the last digit of start + steps added normally.

A 10-clock: positions 0 through 9

For example: start at 7, count forward 4 steps: 8, 9, 0, 1. You land on 1. Quick check: 7 + 4 = 11, and 1 is the last digit of 11. That will always work on this clock, because it only has 10 spots.

Count forward

Start at	Count forward	Land on
2	3 steps
6	5 steps
3	47 steps

Counting backward

Counting backward works the same way, just going the other direction. Past 0, you wrap around to 9 instead.

For example: start at 2, count backward 5 steps: 1, 0, 9, 8, 7. You land on 7.

Start at	Count backward	Land on
3	5 steps
5	13 steps

Other clocks

The same wraparound idea works for any size clock, not just 10. Here are five you already use in real life.

12-hour clock	It is 9 o'clock. What time is it 6 hours later?
Days of the week	Today is Thursday (day 4). What day is it 10 days from now?
Even and odd	Is 17 even or odd?
24-hour & 360°	A 24-hour clock wraps from 23 back to 0; a full turn (360°) works the same way. A movie starts at 22:00 and runs for 3 hours. What time does it end? You spin a dial 370 degrees, starting at 0. Where does it point, as a number from 0 to 359?

Predicting a computer’s version

Python, a programming language, writes this exact same idea as %. 17 % 5 means “count forward 17 steps on a 5-spot clock (0, 1, 2, 3, 4), starting at 0. Where do you land?” Try it: 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 0, 1, 2. That’s 17 steps, landing on 2. So 17 % 5 is 2.

Expression	Predict the value
20 % 7
5 % 8
7 % 2

Count forward. 2+3→5. 6+5→1. 3+47→0.

Counting backward. 3−5→8. 5−13→2.

Other clocks. 12-hour clock: 3 o’clock. 24-hour clock: 01:00. Days of the week: Sunday. Degrees in a circle: 10 degrees. Even and odd: Odd (position 1).

Predicting %. 20 % 7 = 6. 5 % 8 = 5. 7 % 2 = 1.