Instructor's
Notes:
This demonstration came about several
years ago when I was trying to think of ways to help students sort out
the concepts and vocabulary of infinite series, and to understand the fundamental
idea of convergence of a series.
As mathematics teachers, a
"lets imagine that .. " scenario and formal definitions probably
worked just fine for us when we were students, but lots of my students
are in engineering and other applied fields, and many work better physically
and visually than they do abstractly. I wanted something that would get
their attention and would accurately introduce the abstract ideas of the
terms
of a series, the partial sums of a series,
and convergence or divergence of a series.
In fact, in one 50 minute lecture/demonstration/feast
we get all that plus alternating series and the divergence of the Harmonic
series.
Materials:
-
cardboard box
-
a dozen donuts and enough donut holes for the whole
class
-
napkins
-
a large knife
-
a small cutting board
Approximate script:
(A narration of using the donuts with the class.)
I come into class with everything inside the closed
box. The fact that I’m bringing a box to class automatically gets more
attention than usual.
I ask for 2 volunteers and pick them from near the
front of the class. Then I hold up the large knife, and ask for a third
volunteer. That usually gets more attention, and I pick the third volunteer
from near the front again. Finally, I give each volunteer a napkin and
I take out the cutting board and put a donut on it.
I tell the class that I am going to divide up a donut
among the volunteers in a strange mathematical way, and we are going to
keep track of how much each volunteer gets. (I also have them get out
their calculators.)
Step 1: I cut the donut into 4 (equal) pieces
and give each of the three volunteers
a piece. Then I go to the board and record
student A student
B
student C amount left
Step 1:
1/4
1/4
1/4
1/4
Step2: Then I cut the remaining 1/4 donut into
4 equal pieces and give each
student one piece. Now the recording is
student A student
B
student C amount left
Step 1:
1/4
1/4
1/4
1/4
Step 2:
1/16
1/16
1/16
1/16
so each student has a total of 1/4 + 1/16 = 5/16.
Step 3: Then I cut the remaining 1/16 donut into
4 equal pieces and give each
student one piece.
With this action on the board I add the line
Step 3: 1/64
1/64
1/64
1/64
so each student has 1/4 + 1/16 + 1/64 = 21/64 .
After one or two more steps the pieces are getting rather
small so I simply eat the last small piece and suggest that we analyze
what happens if I had continued the process.
The class is very willing to agree that the
"amount left" is quickly approaching zero, so in the "limiting case," the
whole donut is distributed equally among the three students and each student
gets 1/3 of the donut:
.
At this point I usually repeat the process with 4
different students as volunteers and cut the donut (and each remainder)
into 5 pieces. This gives us a series that gets closer and closer to 1/4
of the donut for each student:
.
Now I go back to the board and start attaching names
to the different parts of the information we have recorded:
-
the "pieces" 1/4, 1/16, 1/64, ... are called
the "terms" of the series
-
the "amounts after each step", 1/4, 5/16, 21/64,
.. are called the
"partial sums" of the series (abbreviated ps for use below)
-
and since the total amount each student is getting in
our first donut sharing example was approaching 1/3, we say that "the
limit of the series" is 1/3 (or "the
limit of the partial sums" is 1/3).
I usually ask for predictions about the following series:
and mention that we will show tomorrow that the predictions
are correct.
Onward
Next I ask for 1 volunteer and pick someone in the
middle or near the back of the room. This person gets
-
1/2 of a donut (partial
sum = 1/2),
-
then 1/3 (ps = 5/6),
-
then 1/4 (ps= 13/12
= 1.083333),
-
then 1/5 (ps = 77/60
= 1.283333 ),
-
then 1/6 (ps = 87/60
= 1.45 ).
About now I stop and ask for suggestions about how many
donuts the student will acquire if I keep handing out pieces of size 1/n.
After recording several of these student estimates
on the board, I suggest that perhaps a calculator isn't the answer to everything
and that some brain power will do very nicely.
Writing
I start by grouping the terms in the "usual" way
and justify that the sum in each group is greater
that or equal to 1/2 (so after 8 pieces
we have a total greater that 3/2.).
Next I include the terms
so now the partial sum is greater than 4/2=
2.
Now I include the terms
so the partial sum is greater than 5/2.
I mention that if we keep going, the partial sum
will eventually be larger than 3 donuts, 10 donuts, a billion donuts, etc.
and so we can declare the volunteer to be the "king
or queen of the universe of donuts".
I wrap things up here by mentioning that this series,
is called the Harmonic series,
and that we say it diverges since the
partial sums do not approach any finite limit. (Later in the course when
some student mistakenly concludes that since the terms of some series approach
zero then the series must converge, I refer them to the king or queen of
donuts.)
The rewards of mathematics
-- sugar for everyone
At this point I usually put the donut holes and the
napkins in a couple spots in the room and invite everyone to partake of
the holes; they have 2 minutes to get a couple and go back to their seats.
The give and take of
donut series
Finally, I usually have time for one last example.
I get out a large fancy frosted donut and ask for one last volunteer.
I give the student the whole
donut (which pleases them), but before they can take a bite,
I take back 1/2 of the donut (ps =
1/2). Then I give them 1/3 but take
back 1/4 (ps = 7/12). Usually I go to the board and ask how
to represent what has been happening as a series and hopefully someone
suggests
which I tell them is called the Alternating
Harmonic series.
Using the usual arguments, the students are willing
to accept that the partial sums of the Alternating Harmonic series are
less than 1 and greater than 1/2, and then between 1/2 and 5/6, and then
between 7/12 and 5/6, and usually someone with a calculator will tell me
that the partial sums are close to 0.62??? .
Class Time is up
By now we are always out of time, and I just mention
that at the next class we will formalize some of this reasoning and start
to examine some additional series.
Later classes
In future classes when the vocabulary is defined
and used, I refer back to the "donut meaning" of the words, and to the
results for the donut (geometric series) and the donut Harmonic series.
A few additional comments
A colleague in chemistry calculated that a "typical"
60g donut (see the 'Beginning' picture above) contains approximately 3
x 1023 atoms. So in the first demo (cutting each piece into
fourths), after about 39 steps we are subdividing a single atom.
With the Harmonic series, I sometimes point out that
even though we know that the "king or queen of the donut universe" eventually
has an unlimited number of donuts, it takes a lot of pieces to acquire
even a few entire donuts:
-
after 100 pieces they only have a bit more than 5 entire
donuts (about
5.187)
-
after 1,000,000 pieces they still have less than 15
entire donuts
(about 14.392)
-
and it takes more than 1043 pieces to get
to 100 entire donuts.
Certainly we could do this whole process as an "imagine
that ..." situation, but actually doing the physical cutting
and distributing (and adding) makes it much more real and memorable and
enjoyable for the students. (Many of my foreign students seem particularly
surprised that a mathematics teacher would actually give them sweets in
class.)
Bagels or apples or other foods work, but in morning
classes the donuts are very popular (and I have a sweet tooth).
For under $10 I definitely have their attention while
I introduce the main vocabulary and some early results of infinite series.
(Later, when they ask when the next donut lecture is, I mention that my
salary only affords one per term but that they are free to contribute to
my sweet tooth.)
The student comments at the end of the term always
include a few positive comments about the "donut lecture", and many students
seem to appreciate that I tried to make series real and physical and tasty
for them. The demo is very lo–tech, but I always have fun on donut day.
Credits:
This demo was submitted by
Dale
Hoffman
Department of Mathematics
Bellevue Community College
and is included in Demos
with Positive Impact with his permission.
