Try adding up consecutive (in a row) odd numbers starting from 1. Here are some examples:
1 + 3 + 5 + 7 = 16
1 + 3 + 5 = 9
1 + 3 + 5 + 7 + 9 + 11 = 36
Do you notice anything special? What's this have to do with the picture?
Further Adventure. If you add up consecutive odd numbers in a fraction, sometimes you get strange but wonderful results.
\[frac{1}{3} = frac{1 + 3}{5 + 7} = frac{1 + 3 + 5}{7 + 9 + 11}\]
\[frac{1}{8} = frac{1 + 3 + 5}{7 + 9 + 11 + 13 + 15 + 17}\]
Can you draw a picture that shows what's going on here?
1 + 3 + 5 + 7 = 16
1 + 3 + 5 = 9
1 + 3 + 5 + 7 + 9 + 11 = 36
Do you notice anything special? What's this have to do with the picture?
Further Adventure. If you add up consecutive odd numbers in a fraction, sometimes you get strange but wonderful results.
\[frac{1}{3} = frac{1 + 3}{5 + 7} = frac{1 + 3 + 5}{7 + 9 + 11}\]
\[frac{1}{8} = frac{1 + 3 + 5}{7 + 9 + 11 + 13 + 15 + 17}\]
Can you draw a picture that shows what's going on here?