Represent. 
0th:  Attempt to mentally add the fractions using
equivalent fractions with a common denominator. 
1st:  Use two display area. Let the one on the top
(further from the user) be for the numerator of the sum. Let the one on the bottom (closer to the user) be for
the denominator of the sum. 
2nd:  Represent the first numerator in storage on
the horizontal on the top left corner. 
3rd:  Represent the first denominator in storage on
the vertical side on the top left corner. 
4th:  Represent the second numerator in storage
on the horizontal in the bottom right corner. 
5th:  Represent the second denominator in storage on
the vertical of the bottom right corner. 
6th:  Represent the product of the two denominators in
the lower display area in a rectangular array of your choice. 
"Cross Multiply." 
7th:  "Cross Multiply" each numerator with the other
denominator placing the results in the numerator display area. This means,
multiply the (numerator) horizontal from the 1st with the (denominator) vertical from the 2nd, placing the
product in the lower left corner. Then, multiply the (numerator) horizontal from
the 2nd with the (denominator) vertical from the 1st, placing the product in the upper right corner.

Simplify and Reduce. 
8th:  Simplify the tiles in the numerator display area.

9th:  Remove tiles for both the numerator and denominators VERTICAL AND HORIZONTAL storage areas. 
10th:  Factor the numerator, if possible. 
11th:  Factor the denominator, if possible. 
12th:  Use the factors they
don't have in common as the new numerator and the new denominator.
Whenever possible, remove a top factor with an identical bottom factor.
TO REDUCE, USE THE FACTORS
THEY DON'T HAVE IN COMMON AS THE NEW NUMERATOR AND THE NEW DENOMINATOR. 

13th:  Repeat steps 9 and 10 and 11 and 12 as often as
needed. 