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Why do this problem?
On first inspection this appears to be an opportunity to practice
mental calculation strategies, but it soon becomes apparent that
this context offers an opportunity to think about the structure of
numbers, and multiples in particular.
Give out the grid
and allow a little time for the students to find a couple of pairs
that add to a multiple of 11. Collect suggestions and display on
Set the challenge - how many can they find? Can anyone find them
When they are well into the problem, stop them and ask "What have
you noticed about the pairings?" Collect ideas and note them on the
board. If no one has suggested it, draw attention to the pairings
involving 9, 20 and 31.
"What is special about these numbers? What is special about their
Suggest that they return to the problem and use this insight to
find out how many pairings are possible.
Can this be done without listing them?
When appropriate, bring the class together and draw out ideas that
lead to an efficient strategy.
Offer the follow up
to consolidate the strategy.
What is special about the numbers in each pairing?
Are there some numbers that can only be used once? Why?
Are there some numbers that can be used many times? Why?
Both grids contain less than 30 possible pairings. Can you produce
a grid of numbers that has more than 30? What is the maximum number
of possible pairings in a 4x4 grid?
may provide an interesting follow-up challenge.
The grid below could be used to ask students to find pairs that add
to a multiple of 10.
The key questions are useful prompts to focus students on the
structure of the numbers rather than multiple calculations. This
could be useful preparation before going on to the main
useful to show the students that the sum of two multiples of 11 is
a multiple of 11, and the sum of two numbers in the form 11n+2,
11n-2 is a multiple of 11 as well.