Write down two consecutive numbers.
Square each of them and find the difference.
Do the same for other pairs of consecutive numbers.
What do you notice?
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2^{2}  1^{2} = 3  3^{2}  2^{2} = 5  4^{2}  3^{2} = 7 
5^{2}  4^{2} = 9  6^{2}  5^{2} = 11  7^{2}  6^{2} = 13 
8^{2}  7^{2} = 15  9^{2}  8^{2} = 17  10^{2}  9^{2} = 19 
11^{2}  10^{2} = 21  12^{2}  11^{2} = 23  13^{2}  12^{2} = 25 
14^{2}  13^{2} = 27  15^{2}  14^{2} = 29  16^{2}  15^{2} = 31 
17^{2}  16^{2} = 33  18^{2}  17^{2} = 35  19^{2}  18^{2} = 37 
20^{2}  19^{2} = 39  21^{2}  20^{2} = 41  22^{2}  21^{2} = 43 
23^{2}  22^{2} = 45  24^{2}  23^{2} = 47  25^{2}  24^{2} = 49 
26^{2}  25^{2} = 51  27^{2}  26^{2} = 53  28^{2}  27^{2} = 55 
29^{2}  28^{2} = 57  30^{2}  29^{2} = 59  31^{2}  30^{2} = 61 
32^{2}  31^{2} = 63  33^{2}  32^{2} = 65  34^{2}  33^{2} = 67 
35^{2}  34^{2} = 69  36^{2}  35^{2} = 71  37^{2}  36^{2} = 73 
38^{2}  37^{2} = 75  39^{2}  38^{2} = 77  40^{2}  39^{2} = 79 
41^{2}  40^{2} = 81  42^{2}  41^{2} = 83  43^{2}  42^{2} = 85 
44^{2}  43^{2} = 87  45^{2}  44^{2} = 89  46^{2}  45^{2} = 91 
47^{2}  46^{2} = 93  48^{2}  47^{2} = 95  49^{2}  48^{2} = 97 
50^{2}  49^{2} = 99  51^{2}  50^{2} = 101  52^{2}  51^{2} = 103 
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Here a concise URL for a version of this page without the comments.
Transum.org/go/?Start=October10
Here is the URL which will take them to another activity involving square numbers.