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You need a set of number cards 1-100, with the multiples of 5
removed. Shuffle the cards in front of the class and hand out one
to each student. Do not tell them
that the multiples of 5 are missing!

Ask each student to turn to their neighbour and work out how
to get from one number to the other and back again, using only
these two operations:

$\times 2$ and
$-5$

For example if the two numbers are 21 and 54 the chains could
be:

21, double, 42, take five, 37, take
five, 32, take five, 27, double, 54

and

54, take lots of fives, 9, double,
18, take five, 13, double, 26, take five, 21.

Pairs that are proving difficult to connect could be written
on the board and offered as a challenge for the whole class to
solve. Everyone should be
able to arrive at their partner's number!

Finally challenge the class to get to your number (which should be a
carefully chosen multiple of five). You may wish to offer a
prize......

Once the class give up, ask them to explain why it is
impossible.Display the 1-100 grid, choose 42 as your starting number and
explain that by using the operations above we are going to try to
visit all the numbers on the grid.

Demonstrate how the numbers are crossed out as they are
visited.

Ask the students to predict what will happen.

Will they be able to visit every number on the grid at least
once?

Hand out this 1-100 grid and allow some time for students to
work in pairs to check their predictions.

Bring the students together to link their ideas to the
findings from the earlier exercise.

What would have happened if they had started on a different
number?

Can they explain their results?

Ask if they think they will get the same sort of results with
other pairs of operations.

You may wish to suggest families of pairs of operations for
them to explore. Eg:

x3 and -5

x4 and -5

x5 and -5...

or

x5 and -2

x5 and -3

x5 and -4...

or they can try some families of their own choosing.

Hand out plenty of the 1-100 grids and
ask students to work in pairs or small groups and make a display of
their results.

Can they explain their findings and use these to begin to make
predictions about other pairs of operations? Encourage them to
justify their predictions.

You might find it useful to see if they can identify the pair
of operations that produced the patterns in the three grids below.
They can choose from either:

$\times 3$ and $-6$,

or

$\times 6$ and $-3$

See if they can spot which is which, and if the starting
number makes a difference.

What happens to multiples of 5 when they are doubled?

What about numbers that are 1 more, 2 more, 3 more and 4 more
than a multiple of 5?

What happens to multiples of 5 when 5 is subtracted from
them?

What about numbers that are 1 more, 2 more, 3 more and 4 more
than a multiple of 5?

Students may be interested in
this introductory reading on Modular Arithmetic.

Students could have a go at Take Three from Five which has a similar underlying structure. Can they use their insights from the previous problem to solve this challenge?