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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.
This printable worksheet may be useful: Elevenses.
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 the board.
Set the challenge  how many can they find? Can anyone find them all?
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 'partners'?"
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 grid 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?
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 problem.
8  42  72  12 
58  82  2  88 
23  28  18 
20

4  47  50  6 
This image may be 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, 11n2 is a multiple of 11 as well.
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?
Legs Eleven may provide an interesting followup challenge.