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

2. 85 = 80 + 5 Generalization: n is any number (e.g., 1, 2, 3, … ) Any number ending in 5 can be written in expanded form: n  10 + 5 For 85, n=8: 8  10+5

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

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

5. Graphical Approach : 5 n  10 25 A square of a length ending in 5 : n  10+5

6. Graphical Approach : 5 n  10 25 A square of a length ending in 5 : n  10+5

7. Graphical Approach : A square of length ending in 5 : n  10+5 5 n  10 25 (n  10)  (n  10) 5  (n  10)

8. Graphical Approach : 5 n  10 25 (n  10)  (n  10) 5  (n  10) A square of length ending in 5 : n  10+5

9. Graphical Approach : 5 n  10 25 (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

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

11. 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

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

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

14. Graphical Approach : Area: (n)  (n+1)  10  10 +25 10 n  10 25 = n  ( n+1 ) 25 100’s 10’s &1’s

15. Graphical Approach : 10 n  10 25 For 85, n=8 = n  ( n+1 ) 25 100’s 10’s &1’s 85  85 =

16. = n  ( n+1 ) 25 100’s 10’s &1’s 85  85 = Graphical Approach : 10 n  10 25 For 85, n=8 85  85 = 8  9 25 7 2 25

17. = n  ( n+1 ) 25 100’s 10’s &1’s 85  85 = Graphical Approach : 10 n  10 25 For 85, n=8 85  85 = 72 25

18. Examples 25  25 = 6 25 35  35 = 12 25 55  55 = 30 25 (n  10+5 )  (n  10+5 ) = n  (n+1) 25 100’s 115  115 = 495  495 = 995  995 = 132 25 2450 25 9900 25