Shade the grids to represent the fractions 1/2, 1/4, 1/8, 1/16, 1/32
You will end up with something like this:
Imagine doing this forever (you need a good imagination!)
Now imagine cutting out all of the infinite shaded pieces and joining them together to form a single shape. How big would the new shape be?
Would it fit in your school?
Would it fit in your classroom?
Can you hold it in your hands?
The sum of fractions we’ve been playing with is a series. A series is the sum of numbers in a pattern. Our pattern is that we get the next term by multiplying the previous term by 1/2.
Some series are arithmetic, like 2 + 4 + 6 + 8, where we add (or subtract) the same number. Some series, like ours, are geometric, where we multiply (or divide) by the same number. Also, some series are finite and some infinite. Our series is infinite.
Can an infinite series have a finite sum? Is there a sum for our series?
Watch Western University mathematician Graham Denham talk about this puzzle in the following videos:
What did you learn?
What did you learn about infinity?
What surprised you?
What math insights did you experience?
Share your learning with others, so they can experience math surprise and insight as well.