Start by making sure everyone is convinced that 702 days after a Monday will be a Wednesday, 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. Make sure everyone understands that for the purposes of this problem we are always counting from Monday!

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?) Ask them to come up with a generalisation (possibly algebraic) for any Wednesday. When discussing their generalisations, focus on considering the number of days in a whole number of weeks with 2 extra, rather than simply extending the pattern in the sequence 2, 9, 16, 23...

Now they are ready to investigate the effects of adding or multiplying numbers on the remainder when we divide by 7. It may be worthwhile to do an example as a group:

$15 \div 7 = 2$ remainder $1$

$26 \div 7 = 3$ remainder $5$

$15 + 26 = 41$

$41 \div 7 = 5$ remainder $6$

Then give the students time to try a few examples of their own and write down what they notice. Make sure they can explain what happens when the remainders of each number add up to more than 7.

They can justify what they have noticed, possibly by using algebra or by giving a convincing argument based on whole numbers of weeks and days left over.

What will numbers have in common if they take us to a particular day of the week?

Investigate patterns when dividing by numbers other than 7. Does the same thing always happen? Students could be introduced to the language and notation of modular arithmetic; if the remainder is 2 when we divide 23 by 7, we write:

$23 \equiv 2$ mod $7$

and say "23 is congruent to 2 mod 7"

Further reading on modular arithmetic can be found here.