The Next Square

The Next Square


Here is a table of squares from 1 to 20. I have started a table of differences for you to (copy and) complete:

Number Square Difference ......... Number Square Difference
1 1 ......... 11 121
3 .........
2 4 ......... 12 144
5 .........
3 9 ......... 13
7 .........
4 16 ......... 14
.........
5 25 ......... 15
.........
6 36 ......... 16
.........
7 49 ......... 17
.........
8 64 ......... 18
.........
9 81 ......... 19
.........
10 100 ......... 20 400
21 .........
11 121 ......... 21


Notice that in the difference column 1+2 = 3, 2+3=5, etc.
Predict the squares of 20 to 30. The last digit of each number should make sense. 30
² = 900
Write down your version of the rule for finding the next square.I reckon I can write a rule for finding the previous square:

Suppose I know that 40 squared is 1600; then I think 39² is 1600 – (40+39) which is 1600 – 79, or 1521. 92 is 81 so the last digit looks right. 
Furthermore, I can use my rule again to find 38²….  1521 – (39+38)    
and 1521 – 77 =1521 + 23 –100 = 1444;       8x8=64 so this ends with the right digit too.
So what are the squares of 37, 36 and 35? Check that each of these make sense and that 1225 is one of your answers.


Exercise: Find the squares of each of these: you should not need a calculator at all.

1 a) 40  b) 50  c)  100  d) 700  e) 1000

2 a) 41  b) 51  c)  101  d) 701  e) 1001

3 a) 42  b) 52  c)  102  d) 702  e) 1002

4 a) 39  b) 49  c)   99  d) 699  e) 999

5 a) 61  b) 59  c) 105  d) 703  e) 1004


Check each answer is the right sort of size and that it finishes with the correct last digit. This is generally called checking for sense and only a fool doesn’t do such a check.



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