Supreme Court Handshake
Unit: Supreme Court Welcome
6th to 8th
In this investigation, students will solve the classic handshake problem, which can be stated as follows:
How many handshakes occur when a group of n people shake hands exactly once with every other person in the group?
Students will investigate this problem using multiple
representations: a concrete representation ("acting it out"), a
picture, a table of values, a graph, and an algebraic formula. The
problem is first presented within an interesting interdisciplinary
context; then, students explore the problem in various ways before
presenting their solutions to the class.
The handshake problem has an interesting context with the
Supreme Court. This lesson works well if used near the first Monday in
October, because that is the date that the Supreme Court convenes each
year. Open class by asking students what they already know about the
Supreme Court. You might ask them how many justices are currently on
the court; if they can name any of the current justices; what is the
ratio of males to females; who appoints justices to the court; and, who
was the most recent appointee.
After students share what they know, say, "Did you know that
the Supreme Court uses a lot of traditions? One of their traditions is
that every justice shakes hands with each of the other justices each
time they gather for a meeting. Chief Justice Melville W. Fuller
(1888‑1910) started this custom, saying that it shows ‘that the harmony
of aims, if not views, is the court’s guiding principle.’"
Then, present the problem to students as follows:
There are nine justices on the Supreme Court. How many
handshakes occur if each of them shakes hands with every other justice
You may wish to display the text of this problem on the chalkboard
or overhead projector for students to refer to. Allow students to work
in groups of four to find the number of handshakes that occur with a
group of nine people. Inform students that they will be presenting
their solutions to the entire class, so they should keep a record of
Help those who struggle by asking them to consider a simpler
problem. For instance, you may ask, "In your group, how many handshakes
are possible?" Help students model this by having them shake hands with
one another. Acting out the problem in this way may encourage groups to
combine to find the number of handshakes when there are more than
The specific problem of finding the number of handshakes for
the nine justices is a good way to introduce the problem, but the real
value of this problem is having students generalize their results.
Students who find the specific answer for nine people
[36 handshakes] should be asked to keep working and find a
general rule that will let them find the number of handshakes for a
group of n people. To help with this investigation, distribute the
Handshake Activity Sheet.
Handshake Activity Sheet
Students will solve this problem in a variety of ways. In
addition to acting it out, they may use pictures, tables, geometric (or
network) solutions, or organized lists. A table might be organized in
two columns, the first showing the number of people, and the second
showing the number of handshakes:
A pictorial or network solution could be drawn such that a dot
represents a person, and each line segment represents a handshake
between two people. (In the drawing below, this scheme has been used,
but color‑coding also shows that the first person (red) shakes hands
with eight people; then, the second person (blue) shakes hand with only
seven people, since he has already shaken hands with red; then, the
third person (yellow) shakes only six hands, because she has shaken
hands with red and blue; and so on.)
An organized list could also be used to show all the handshakes.
Note that every pair of numbers is included just once in the list
below; that is, if the pair 4‑6 is included, the pair 6‑4 is not also
included, because it represents the same handshake. Further, pairs with
the same number are not included, such as 7‑7, because they represent a
person shaking his or her own hand.
To allow varied approaches to be displayed, give each group a
transparency sheet and overhead marker so that they may create a visual
model to explain their solution to the class. Begin the discussion of
solution strategies with the physical model of the problem. Have
nine students stand in a line the front of the class. The first student
walks down the line, shaking hands with each person, while the class
counts the number of handshakes aloud (8). She then sits down. The next
student walks down the line, shaking hands with each person, while the
class counts aloud (7). The next student shakes 6 hands, then 5, 4, 3,
2, and 1. The last student has no hands to shake, since he has already
shaken the hands of all people in line before him, so he just sits
down. The total number of handshakes is
8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 36.
Now ask, "How many handshakes occur when there are 30 people?
How many handshakes occur with the whole class? Do we want everyone in
the class to stand up, and continue counting out loud?" Probe student
thinking to see if there is a different, or more efficient, way that
would make sense when considering larger groups.
Have each group use their transparency to explain their
various ways to get the solution. To engage students in examining
varied representations for the same problem, ask, "Does this make sense
to you? How is this group’s explanation similar to your explanation?
How is it different?"
Once all students are convinced that nine Supreme Court
Justices have a total of 36 handshakes, extend the problem. Ask, "How
many handshakes occur with 10 people?" Using the table, students may
see that one more is added in each row than was added in the previous
row; therefore, for 10 people, there would be 36 + 9 = 45 handshakes.
To allow students to investigate the relationship between number of people and number of handshakes, allow them to explore the
This interactive demonstration allows them to see a pictorial
representation of the situation as well as see the pattern of numbers
appear in a table. In particular, students can investigate the change
that occurs in the number of handshakes as the number of people
increases by 1, and noticing this change can be very powerful.
Handshake Online Activity
This is called a recursive relation, because the number of handshakes for n people can be described in terms of the number of handshakes that occurred for (n – 1) people.
Students may be comfortable adding on or computing manually for
groups up to 20 people. If that seems to be the case, and if students
are not looking for a generalized solution, pose the question, "What if
100 Senators greeted one another with a handshake when they met each
morning? How many handshakes would there be?" Distribute the
activity sheet, and allow time for students to complete the table and
discover relationships. (You might wish to display the activity sheet
as a transparency on the overhead projector and have the class work
together to fill in the first several rows. Many of the groups will
already have answers for the number of handshakes in groups of
Have various students explain the relationships they see. With
each suggestion, have the class decide if using that relationship will
allow them to determine the number of handshakes for 30, 100, or n people. Some possible relationships that students may see:
As a result of these discoveries, students should realize that the
number of handshakes for 30 people is 1 + 2 + 3 + … + 29 = 435. Value
all student suggestions, but keep probing to determine the number of
handshakes for 100 people.
To lead students to determine a closed‑form rule for the
relationship, have students look for a rule that uses multiplication,
and ask the following leading questions:
Students should see that the number of handshakes is equal to the
previous number of people multiplied by the current number of people,
divided by 2. In algebraic terms, the formula is:
Another way to attain the solution is to use an organized table.
If there are nine people, then we can list the individuals along the
top row and left column, as shown below. The entries within the table,
then, indicate handshakes. However, the handshakes in yellow cells
indicate that a person shakes his or her own hand, so they should not
be counted; and, the entries in red cells are the mirror images of the
entries in blue cells, so they represent the same handshakes and only
half of them should be counted. For nine people, there are 81 entries
in the table, but we do not count the nine entries along the diagonal,
and we only count half of those remaining. This gives ½(81 – 9) = 36.
In general, for n people, there are n2 entries in the table, and there are n entries along the diagonal. Therefore, the number of handshakes is ½(n2 – n), which is equivalent to the algebraic formula stated above.
When students arrive at the formula, ask, "Does it matter if you
multiply first and then divide by 2? Can you divide by 2 first and then
multiply?" [Because of the commutative property, order does not
matter.] This is an important point, because students can use mental
math to perform calculations with this formula in three different ways:
Students should decide which number to divide by 2, depending on whether n or (n – 1) is even. As an example, for 15 people, n = 15 and (n – 1) = = 14, so it makes sense to divide 14 by 2 and then multiply by 15: 7 × 15 = 105. On the other hand, for 20 people, n = 20 and (n – 1) = 19, so it makes sense to divide 20 by 2 and then multiply by 19: 10 × 19 = 190.
As a final step, students can plot the relationship between
number of people and number of handshakes. Students should describe the
shape of the graph and answer the following questions:
By the end of this lesson, students will have used (or at least
seen) a solution involving a table, a verbal description, a pictorial
representation, and a variable expression. It may be important to
highlight this to students, and it would be good to encourage students
to use all of these various types of representations. Each
representation provides different information and may offer insight
when solving problems.
Mr. and Mrs. Baker threw a party to which they invited five other
couples. When all the guests arrived, there were twelve people. Some of
them had met before, and some had not. All the people who had never met
shook hands. Then, Mr. Baker asked every guest (including his wife) how
many hands each of them had shaken. To his surprise, every person gave
a different answer. How many hands did Mrs. Baker shake?
The solution to this problem is found most easily by drawing the
situation. Students who attempt to solve this problem with a formula or
equation soon realize that it is not possible.
Questions for Students
1. If each justice shakes hands just once with everyone else, how many handshakes take place?
[There is a total of 36 handshakes with nine justices, as shown by the representations above.]
2. What is the number of handshakes in a group of n people?
[In a group of n people, there will be 1 + 2 + 3 + … + n handshakes. This can be shortened to the formula ½(n)(n – 1).]
3. How many handshakes will there be in a group of 40 people, and then
100 Senators? 40 x 39 / 2 = 780, 100 x 99/2 = 4950, certainly too many
to model one by one. The AppletforSupreme.doc may help students see
these relationships with the color.
[Using the formula, there are
½(40)(39) = 780 handshakes in a group of 40 people, and there are
½(100)(99) = 4950 handshakes in a group of 100 people.]
Supreme Court Welcome
Grade: 6th to 8th