Clock Arithmetic

Modular Arithmetic — Lesson 2 of 2

In this lesson:

• Explain why a clock wraps instead of counting up forever
• Add and subtract on a 12-hour clock
• Multiply on the clock, then wrap the total

What You Will Learn Today

By the end of this lesson you should be able to:

1. Explain why a clock wraps vs. the straight number line
2. Add and subtract on a clock with wrapping
3. Multiply on the clock — total first, then wrap
4. Solve elapsed-time word problems

What You Already Know from Lesson 1

From Lesson 1 — stepping on the number line:

• Forward steps = adding
• Backward steps = subtracting
• The number line goes on forever in both directions

Today: the line bends into a loop

9 o'Clock + 5 Hours = ?

Straight line → 14. Clock → 2. The clock loops back.

The Straight Line vs. the Clock

Number line: infinite — adding always gives a bigger number

Clock:

• Only 12 positions — loops back at 12
• Going all the way round returns you to the start
• 12 and 0 are the same position

(Timetable clocks wrap at 24 — same idea.)

A Number System That Wraps

The clock is a wrapping number system — also called clock arithmetic or modular arithmetic.

The two facts everything rests on:

• There are 12 positions — the clock loops back at 12
• 12 and 0 are the same position

Quick Check — Does the Clock Wrap?

Start at 10. Go forward 4 steps.

Count the positions: 11, 12, 1, 2

Where do you land?

Think before the next slide...

Adding on the Clock: Forward Past 12

When you step forward past 12, the clock loops back:

• Positions 13, 14, 15 don't exist on the clock face
• The hand re-enters at 1 and keeps counting

The forward wrap rule:

Example: 9 + 5 on the Clock

Start at 9. Step forward 5.

Total is above 12 — apply the wrap rule:

Answer: 2 o'clock

Subtracting on the Clock: Backward Below 1

When you step backward past 1, the clock loops back:

• The clock has no negative positions — no −1, −2, −4
• The hand re-enters at 12 and keeps going backward

The backward wrap rule:

Example: 11 + 6 on the Clock

Start at 11. Step forward 6.

Total is above 12 — wrap:

Answer: 5 o'clock

Predict Before You See the Answer

3 − 7 on the clock — what is the answer?

Think it through before the next slide:

• A learner who carried lesson 1's rules across might write −4
• A learner who applied the backward wrap might write ?

Write your answer before advancing.

3 − 7: The Backward Wrap in Action

Start at 3. Step backward 7.

Result is below 1 — apply the backward wrap rule:

Answer: 8 o'clock

Both Wrap Rules Side by Side

Extending What You Know to Multiplication

Multiplying on the clock = repeated forward steps, then one wrap:

• means stepping forward 5, four times
• Compute all steps on the straight line first
• Then wrap the total once

Don't wrap after each step — wrap the final total only.

Multiplying on the Clock: 4 × 5

Total is above 12 — wrap:

Answer: 8 o'clock

First compute the total. Then wrap once.

Different Problems, Same Clock Position

Both and land on 6 o'clock

Different problems — same clock position

Try It: Multiply and Then Wrap

Compute the total, then wrap:

Write your answers — total first, then clock position

When the Total Is Exactly 12

Subtracting 12 gives 0 — but there is no 0 o'clock.

Rule: A full lap lands on 12, not 0.

Word Problem: Bus Leaves Kampala at 10

Bus leaves at 10, travels 7 hours. Arrival time?

Translate:

Compute:

Wrap:

Read: 5 o'clock

Guided: It Is 8 o'Clock Now

What time will it be in 9 hours?

Translate:

Compute:

Wrap: above 12? → subtract 12

Read: as a clock time

On Your Own — No Scaffold

A school programme ended at 2 o'clock.
It had been running for 5 hours.
When did the programme start?

Translate → compute → wrap → read.

Watch Out — Three Common Errors

Key Takeaway: The Clock Wraps

✓ Finite wrapping system — 12 positions

✓ Past 12: subtract 12 | Below 1: add 12

✓ Multiply: total first, wrap once

✓ Full lap = 12, not 0

Answer as a clock time — not a bare number

Coming up: modular arithmetic.