Number properties, mental calculations, and estimation

          Number Properties

Closure for addition of fractions
The sum of any two fractions is another unique fraction.
EX:  -2/3 + 4/5 = -2x5 + 4x3 = -10 + 12 = 2
                              3x5     5x3     15     15    15
Closure for multiplication of fractions
The product of any two fractions is another unique fraction
EX:  -2/3 x 4/5 = -2x4 = -8
                              3x5    15        
Identity for addition
The sum of any fraction and 0 is the given fraction.  Zero + any integer = the given integer
EX:5/6 + 0 =5/6 + 0/6 = 5+0 = 5
                                          6       6
Identity for multiplication
http://intermath.coe.uga.edu/dictnary/descript.asp?termID=172
EX:  1 x (-4/5) = 1x-4 = -4
                               5        5
Addition is commutative and associative
http://www.learningwave.com/lwonline/numbers/com_assoc_add.html
commutative:  two fractions that are being added can be interchanged without changing the sum
associative:  in a sum of three fractions the middle numer may be grouped with either of the other two numbers.

Multiplication is commutative and associative
http://www.learningwave.com/lwonline/numbers/com_assoc_mult.html
commutative: two fractions that are being multiplied can be interchanged without changing the product
associative:  In a product of three fractions the middle number may be grouped with either of the other two numbers.

Inverses for addition and multiplication
Addition:  for every fraction there is another fraction called its opposite or inverse for addition such that the sum of the two fractions is 0. 
EX:  3/4 and -3/4
Multiplication:  for every fraction not equal to zero there is a nonzere fraction called its reciprocal or inverse for multiplication such that the product of the two numbers is 1.
EX:  3/8 and 8/3
          Mental Calculations

Compatible numbers
Numbers that can be conveniently combined in a given computation

Substitutions
EX:  2  7 + 1 = 2  7 + 1 + 1 =   3 1
             8    4        8    8     8         8

           Estimation

Rounding
The sum or difference of mixed numbers and fractions can be estimated by rounding each number to the nearest whole number.

Practice:  http://www.ixl.com/math/practice/grade-5-round-mixed-numbers

Compatible numbers
Replacing a fraction with a reasonably close and compatible fraction can be useful when estimating