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### Five Practices

Productive mathematical discussions lead to students who can think, reason and engage effectively in quantitative problem solving, characteristics which are much needed but may not routinely emerge from our classrooms. So just how can teachers create such discussions?

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# Grandi’s series

Consider 1 – 1 + 1 – 1 + ….

Is the answer 0, 1 or ½? Or all three?

Approaches:

[A]    (1 – 1) + (1 – 1) + (1 – 1) + … = 0 + 0 + 0 + … = 0

[B]     1 + (– 1 + 1) + (– 1 + 1 ) + (– 1 + 1) + … = 1 + 0 + 0 + 0 + … = 1

[C]     S = (1 – 1 + 1 – 1 + 1 – 1 +  …) , so

1 – S = 1 – (1 – 1 + 1 – 1 + 1 – 1 + …) = 1 – 1 + 1 – 1 + 1 – 1 + 1 … = S

1 – S = S

1 = 2S

S = ½