Squaring of a number ending in 5 An approach to determine answers Quickly! Squaring of a number ending in 5

85 = Generalization: n is any number (e.g., 1, 2, 3, … ) Any number ending in 5 can be written in expanded form: n  For 85, n=8: 8  10+5

Graphical Approach : A square of a length ending in 5 : n  n  10

Graphical Approach : 55 n  10 A square of a length ending in 5 : n  10+5

Graphical Approach : 5 n  A square of a length ending in 5 : n  10+5

Graphical Approach : 5 n  A square of a length ending in 5 : n  10+5

Graphical Approach : A square of length ending in 5 : n  n  (n  10)  (n  10) 5  (n  10)

Graphical Approach : 5 n  (n  10)  (n  10) 5  (n  10) A square of length ending in 5 : n  10+5

Graphical Approach : 5 n  (n  10)  [(n  10)+5+5] n  10 Area: (n  10)  (n  10) 5  (n  10) A square of length ending in 5 : n  10+5

Graphical Approach : 10 n  (n  10)  [(n  10)+10] = (n  10)  [(n+1)  10] Area: A square of length ending in 5 : n  10+5

Graphical Approach : 10 n  10 (n  10)  [(n  10)+10] = (n  10)  [(n+1)  10] Area: + 25 A square of length ending in 5 : n  10+5

Graphical Approach : 10 n  10 Area: (n)  (n+1)  10 

Graphical Approach : Area: (n)  (n+1)  10  n  10 25

Graphical Approach : Area: (n)  (n+1)  10  n  = n  ( n+1 ) ’s 10’s &1’s

Graphical Approach : 10 n  For 85, n=8 = n  ( n+1 ) ’s 10’s &1’s 85  85 =

= n  ( n+1 ) ’s 10’s &1’s 85  85 = Graphical Approach : 10 n  For 85, n=8 85  85 = 8 

= n  ( n+1 ) ’s 10’s &1’s 85  85 = Graphical Approach : 10 n  For 85, n=8 85  85 = 72 25

Examples 25  25 =  35 =  55 = (n  10+5 )  (n  10+5 ) = n  (n+1) ’s 115  115 = 495  495 = 995  995 =