I. Calculating the squares of 2-digit numbers ending in 5:

examples: a) what is the square of 45?
          b) what is 75 x 75?

the "shortcut" solution:

  Step 1: take the "tens" place of the number we're calculating
  Step 2: multiply that number with that number + 1
  Step 3: append the "25" to the answer arrived from step 2

It's easier to illustrate:

a) what is the square of 45?
   i) the tens place of "45" is the number "4"
  ii) multiply "4" by "4+1" ==> 4 x 5 = "20"
 iii) append "25" to "20" ==> 2025
      the answer is 2025!

b) what is 75 x 75?
   i) "7"
  ii) 7 x 8 = 56
 iii) 5625
 

exercises (mentally, please):
c) what is the square of 55?
d) 95 x 95 = ?
 
 
 

II. Multiplying two numbers which are between 10 and 19:

examples: e) what is 12 x 14?
          f) 18 x 13 = ?

the "shortcut" solution:

  Step 1: take one of the numbers and add it to the "ones" place digit
          of the other number (remember the answer: this number is
          actually multiplied by 10, but you don't really need to
          do that since multiplying by 10 is a trivial operation)
  Step 2: multiply the "ones" place digits (remember the answer)
  Step 3: "append" the result from step 2 to step 1  (see illustration below)
 

illustration:

e) what is 12 x 14?
   i) we take "12" and add it to "4":  12 + 4 = 16
     (we could have taken the other number "14" and added it to "2",
      and we get the same answer of "16")
  ii) we multiply "2" by "4":  2 x 4 = 8
 iii) append the number "8" to "16":  168
                                            16
  (or you could think of it this way)     +   8
                                         =======
                                            168
      the answer is 168!

   note: in (i), the answer of 16 actually implies 160
         in (ii), it's probably safer to think of the answer as "08"

f) 18 x 13 = ?
   i) 18 + 3 = 21            (or: 13 + 8 = 21)
  ii) 8 x 3 = 24
 iii)      21
       +    24
         ======
           234
      the answer is 234!

exercises (do these mentally!):
g) 16 x 12 = ?
h) 17 x 19 = ?
 

III. Multiplying two numbers whose tens place are the same number:

examples: i) what is 23 x 24?
          j) 79 x 74 = ?

the "shortcut" solution: (well, it's not really a short "shortcut",
    but this method should be easier for mental calculation)

  Step 1: take one of the numbers and add it to the other number's
          "ones"-digit (remember the answer)
  Step 2: multiply the answer from step 1 by the "tens"-place of the
          original numbers (remember the answer)
  Step 3: multiply the "ones" place of each of the two numbers
          (remember the answer)
  Step 4: add (append) the answer from step 3 to the answer from step 2
          (see illustration below)

i) what is 23 x 24?
   i) "23" + "4" = "27"       (or: 24 + 3 = 27)
  ii) the "tens" place of the numbers "23" or "24" is the number "2",
      multiply "27" (answer from step 1) by "2":  27 x 2 = 54
 iii) multiply "3" x "4" (the "ones" digits of the two numbers):
       3 x 4 = 12
  iv) append/add the answers from step 2 and step 3:
                    54
                 +   12
                 =======
                    552
     the answer is 552!

j) 79 x 74 = ?
   i) "79" + "4" = 83         (or: 74 + 9 = 83)
  ii) 83 x "7" = 581       [I presume that you can do multiplications
                            of a 1-digit number by a 2-digit number
                            mentally with ease; if not, you SHOULD be
                            practicing mental multiplications of 1-digit
                            by 2-digit numbers before you proceed!!]
 iii) "9" x "4" = 36
  iv)     581
        +   36
        =======
          5846
    the answer is 5846!

exercises (do these MENTALLY!):
k) 21 x 22 = ?
l) 46 x 45 = ?
m) 86 x 89 = ?