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

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
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 = ?