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Here are a selection of solutions that were sent in. They vary slightly in the degree of their explanations. 
First, Daniel from North Ferriby School wrote:


If the number is odd and I am making the biggest odd number I put 9 at the front, unless the number is nine in which case I put 7 after the number.
If the number is even and I am making the biggest odd number I put 9 after the number
If the number is odd and I am making the biggest even number I put 8 after the number.
If the number is even and I am making the biggest even number I put 9 in front.

 Tom and Jeevan from Devonshire Primary School said:

If you are trying to make the largest even number using an odd number you put the odd number in the tens column and in the units column you put 8.
But if you're doing it with an even number then you put 9 in the tens column and then put the even number in the units column.

Peter, Henry, Charlie, Lulu and Samuel at Weald C of E Primary School assumed that you could also get a zero.  They went on to think about how you would make the smallest even number:

Largest Even
1. If you get an odd number, put an eight in the units column.
2. If you get an even number, put a nine in the tens column.
3. If you get a zero, put a nine in the tens column.

Smallest even
1. If you get an odd number, add a zero in the units column.
2. If you get an even number, add a one in the tens column
3. If you get a zero, put a one in the tens column.

Homeroom 7 at Kororoit Creek Primary School, Australia looked at a strategy for making the largest odd number too:

The largest two-digit odd number we could make is 97. We could not make 99 because we only have one 9.
If you pulled out an odd number first this should always go in the 'ones' column. Unless you pull out a number 9. The 9 will always go first because 97 is the biggest number we can make. e.g. 97, 79, 59, 39, 19.
If you pull out an even number first it should always go in the 'tens' columns, otherwise your two-digit number will not be odd. You should still always choose the number 9 as your second card to make the largest number possible. e.g. 89, 69, 49, 29.

Year 2 from Northam St George's School worked in pairs to explore the largest even numbers. You can see three of the pupils' work here.pdf .

Finally, Nicole from New Jersey sent in the following;

The card is a 3.  I would choose the largest even number from 1 to 9, which is 8.  Since 3 is odd and 8 is even, then the 3 goes to the tens place and the 8 in the units place.  Therefore, the largest possible even number using 3 would be 38.
The card is a 6.  I would choose the largest possible number from 1 to 9, which is 9.  Since 6 is even and  9 is odd, then the 9 goes in the tens place, while the 6 goes in the units place.  Therefore, the largest possible even number using 6 would  be 96.

Method: Finding the largest two-digit even number
1. If the first card is odd, (1,3,5,7,9), then that odd number goes in the tens place.  Find the largest even number from 1 to 9, which will always be 8.  Then 8 will go in the units place.
2. If the first card is even, (2,4,6,8), then that even number will be in the units place.  Find the largest number from 1 to 9, which will always be 9.  Then the 9 will go in the tens place.

Method: Finding the largest two-digit odd number
3. If the first card is odd, (1,3,5,7,9), then that odd number goes in the units place.  Find the largest number from 1 to 9, which will always be 9.  Then 9 will go in the tens place.
4. If the first card is even, (2,4,6,8), then that even number will be in the tens place.  Find the largest odd number from 1 to 9, which will always be  9.  Then the 9 will go in the units place.

Thank you all for sending in solutions to show ways of getting solutions to this challenge.