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# Clock Arithmetic

### Why do this problem?

### Possible approach

### Key questions

See the questions in "Possible approach".

If the first day of this month was a Monday, what can we say about the first day of next month, and why?
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### Possible support

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Age 11 to 18

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

This problem gives an insight into modular arithmetic without worrying too much about notation, by using the image of a clock face and looking at the concept of remainders. It is a good preparation for More Adventures with Modular Arithmetic

*This problem featured in an NRICH Secondary webinar in April 2021.*

Task 1 to 5 are ideal for students to work on in pairs and justify their results to each other as they go along.

Before students start on the tasks, ask them some questions about time, such as:

- If the time is 10am, what time will it be in 3 hours time?
- If the time is 11pm, what time will it be in 15 hours time?
- If the time is 2pm, what time will it be in 20 hours time?
- If the time is 2pm, what time was it 5 hours ago?

Bring the class together to discuss their results (in particular their results for Task 5) before they tackle Task 6.

Before students start on Task 6, discuss why 702 days after a Monday will be a Wednesday, possibly by thinking about whole numbers of weeks and days left over. Students can then work out what day it will be in 15 days, 26 days, 234 days time.

Pose the question "If today is Monday, how many days from now is Wednesday?" Ask the students to give you as many answers as they can. (Does anyone suggest a negative number of days?).

As students work on Task 6 in pairs, circulate around the class listening out for good explanations that can be shared with the whole class at the end of the session.

See the questions in "Possible approach".

If the first day of this month was a Monday, what can we say about the first day of next month, and why?

Focussing on Tasks 1 to 5 and then moving on to Level 1 of Shifting Times Tables might be a good preparation for Task 6.

More Adventures with Modular Arithmetic and Clock Squares provide suitable follow-up questions.

Further reading on modular arithmetic can be found here

Further reading on modular arithmetic can be found here