What’s My Sum?
Teacher Notes
Objective
Students use paper cutting to understand how an infinite sum can result in a finite
number and derive the formula for an Infinite Geometric Series.
Standards
Alg II 22.0 Students find the general term and the sums of arithmetic series and of
both finite and infinite geometric series.
Alg II 23.0 Students derive the summation formulas for arithmetic series and for
both finite and infinite geometric series.
MA8.0
Students are familiar with the notion of the limit of a sequence and the
limit of a function as the independent variable approaches a number
or infinity. They determine whether certain sequences converge or
diverge.
Materials
Piece of Scratch Paper
Scissors
Copies, What’s My Sum? Part I, What’s My Sum? Part II, What’s My Sum? Part III
Teacher Directions
Pass out What’s My Sum? Part I. You may want to model the first cut for the
students. Although, in some classes you may find you don’t need to model it for
them.
Teacher Notes
 Problem #4- student teams may need help with the fact that this sum is really
never ending. They may use ellipsis (...) to show that the sum is infinite.
 Discuss problems 1-5, and then depending on when you choose to do this
activity, the students may or may not know how to write the sigma notation in
number 6.
 Stop after question 6, and complete What’s My Sum? Part II as a class. Then
complete #7 from What’s My Sum? Part I.
 What’s My Sum? Part III are also area problems, similar to what they just put
together.
 You may want to model problem (a) for the groups. If students struggle to see
it visually, cut a piece of square paper into fourths and this time make 3 piles.
Then take the fourth sheet and cut that into fourths and distribute the pieces
into the 3 piles with one left over, etc.
What’s My Sum? Part I
Follow the directions below:
Take a sheet of paper, cut it into 3 equal pieces
Take two of the pieces and put them in two separate piles (pile A and pile B)
Take the third piece, cut it into 3 equal pieces.
Take two of the pieces of paper you just cut and put one into pile A and one
into pile B
 Continue this process until you have “nothing” left to cut.




Suppose the area of the original paper was 1 square unit.
1. What is happening to the third piece as you continue cutting?
2. Look at pile A and at pile B, how much of the area of the original paper is just
in pile A? (Assume that you continue to cut the third piece so small that the
entire area of the initial paper is almost included).
3. Take Pile A and on each piece of paper, write the area of that piece directly on
it. Remember the area of the original paper was considered 1 square unit.
4. Write the area of pile A as a sum of all of the pieces (note you do not have to
add them, just write them as a sum which should be an infinite number of
pieces).
Irvine Math Project
1
5. What does the sum in #4 get closer and closer to the more you cut up the 3rd
piece of paper? (remember your answer to #1)
6. So even though the sum never ends, we say that the sum converges to a
number. This is an example of a convergent series, i.e. an infinite series with
a finite sum. Write this convergent series using sigma notation.
Stop Here! Complete Part II, and then come back to the following
questions.
7. In # 4, what is the value of “a”? (Hint: you can also look at #3 in Part II to help
you determine the value of “a”.)
What is the value of “r”?
Calculate the sum of this series by using the formula you just came up with on
page 3.
Irvine Math Project
2
What’s My Sum? Part II
You will now figure out the formula for the sum of a generic
geometric series,
a + ar + ar2 + ar3 + …
Let’s use some brain power…..
1. Suppose that S = a + ar + ar2 + ar3 + …
2. The sum of the geometric series above equals S. Take the above equation and
multiply both sides by r (be sure to distribute the r). Write your result below.
3. Subtract the right side expression of #2 from the right side expression in #1. The
left side is already done for you.
S – Sr = __________________________________
4. Solve the above equation you wrote for S.
Irvine Math Project
3
5. Write the formula below.
Sum of an infinite geometric series:
6. Use this formula to go back and complete #7 on page 2.
What’s My Sum? Part III
The following pictures illustrate an infinite series with a finite sum. Write the
series and the sum. Assume that the original area is 1 square unit.
a)
Irvine Math Project
b)
4
Series:
Series:
Sum:
Sum:
c)
Series:
Irvine Math Project
Sum:
5