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Why do this problem?
gives an insight into modular arithmetic without
worrying too much about notation, by looking at the concept of
remainders. It gives students the opportunity to share ideas,
listen to each other justify their assertions, and come up with
convincing arguments and proofs using simple algebra.
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
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?
If the first day of this month was ... what can we say about the
first day of next month, and why?
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.
Students who are struggling with finding the remainder when
dividing by 7 could investigate division by 5 instead; this could
be a model for the working week ignoring Saturday and