## Supreme Court Handshake

During this lesson, students will explore the handshake problem, a
classic problem in mathematics that asks, "How many handshakes occur
when *n* people shake hands with each other?" Groups work to
determine how many handshakes take place among the nine Supreme Court
justices, and then generalize to the number of handshakes in any size
group. Students explore the problem using a verbal description, a
table, a graph, a picture, and an algebraic formula.

In this investigation, students will solve the classic handshake problem, which can be stated as follows:

How many handshakes occur when a group ofnpeople 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 exactly once?

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 their work.

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 four people.

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&nsbp;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:

People | Handshakes |

1 | 0 |

2 | 1 |

3 | 3 |

4 | 6 |

5 | 10 |

6 | 15 |

7 | 21 |

8 | 28 |

9 | 36 |

10 | 45 |

11 | 55 |

12 | 66 |

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.

(8 handshakes) | 1-2 | 1-3 | 1-4 | 1-5 | 1-6 | 1-7 | 1-8 | 1-9 |

(7 handshakes) | 2-3 | 2-4 | 2-5 | 2-6 | 2-7 | 2-8 | 2-9 | |

(6 handshakes) | 3-4 | 3-5 | 3-6 | 3-7 | 3-8 | 3-9 | ||

(5 handshakes) | 4-5 | 4-6 | 4-7 | 4-8 | 4-9 | |||

(4 handshakes) | 5-6 | 5-7 | 5-8 | 5-9 | ||||

(3 handshakes) | 6-7 | 6-8 | 6-9 | |||||

(2 handshakes) | 7-8 | 7-9 | ||||||

(1 handshake) | 8-9 |

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 Handshake Activity. 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.

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 1‑10 people.

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:

- Add the number of previous people to their number of handshakes, and that will give the next number of handshakes; For instance, there were 6 handshakes with 4 people; therefore, there are 6 + 4 = 10 handshakes for a group of 5 people.
- The differences between the numbers in the second column form a linear patern, 1, 2, 3, 4, ….

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:

- For 7 people, there are 21 handshakes. How is 7 related to 21? [Multiply by 3.]
- For 9 people, there are 36 handshakes. How is 9 related to 36? [Multiply by 4.]
- What about for 8 people? There are 28 handshakes. How is 8 related to 28? [Multiply by 3.5.]

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:

[n(n-1)]/2

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 *n*^{2} entries in the table, and there are *n* entries along the diagonal. Therefore, the number of handshakes is ½(*n*^{2} – *n*), which is equivalent to the algebraic formula stated above.

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |

1 | 1-1 | 1-2 | 1-3 | 1-4 | 1-5 | 1-6 | 1-7 | 1-8 | 1-9 |

2 | 2-1 | 2-2 | 2-3 | 2-4 | 2-5 | 2-6 | 2-7 | 2-8 | 2-9 |

3 | 3-1 | 3-2 | 3-3 | 3-4 | 3-5 | 3-6 | 3-7 | 3-8 | 3-9 |

4 | 4-1 | 4-2 | 4-3 | 4-4 | 4-5 | 4-6 | 4-7 | 4-8 | 4-9 |

5 | 5-1 | 5-2 | 5-3 | 5-4 | 5-5 | 5-6 | 5-7 | 5-8 | 5-9 |

6 | 6-1 | 6-2 | 6-3 | 6-4 | 6-5 | 6-6 | 6-7 | 6-8 | 6-9 |

7 | 7-1 | 7-2 | 7-3 | 7-4 | 7-5 | 7-6 | 7-7 | 7-8 | 7-9 |

8 | 8-1 | 8-2 | 8-3 | 8-4 | 8-5 | 8-6 | 8-7 | 8-8 | 8-9 |

9 | 9-1 | 9-2 | 9-3 | 9-4 | 9-5 | 9-6 | 9-7 | 9-8 | 9-9 |

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:

- Multiply
*n*by (*n*– 1), and then divide by 2; - Divide
*n*by 2 , and then multiply by (*n*– 1); or, - Divide (
*n*– 1) by 2 , and then multiply by*n*.

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:

- Is the relationship linear? [No, it is nonlinear.]
- How would you know from the table that the relationship is not linear? [There is not a constant rate of change.]
- How would you know from the variable expression that the relationship is not linear? [The variable
*n*is multiplied by (*n*– 1), and the product contains*n*^{2}, which means the curve will be quadratic.] - How would you know from the graph that the relationship is not linear? [The graph is a curve, not a straight line.]

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.

- Handshake Activity Sheet
- Graph paper

**Assessments**

- Have the students write their solution to the original question, and demonstrate how they know. What is the number of handshakes in any size group?
- How many handshakes would take place if all twelve math teachers shook hands when they attended the math department meeting? Show two ways to explain your solution.
- At a party, everyone shook hands with everyone else. There were 45 handshakes. How many people were at the party? Show how you know.

**Extensions**

- Have students examine Gauss’s method for finding sums of numbers: 1 + 2 + 3 + … = ½(
*n*)(*n*+ 1). - Have students determine the number of diagonals in a polygon. The number of diagonals is ½(
*n*)(*n*– 1). They should consider how this problem is related to the handshake problem. - Students should attempt to solve the following problem.
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

npeople, there will be 1 + 2 + 3 + … +nhandshakes. 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.]

**Teacher Reflection**

- What were some of the ways that the students illustrated that they were actively engaged in the learning process about the number of handshakes?
- What content areas did you integrate within the lesson about the Supreme Court? Was this integration appropriate and successful?
- How did you challenge the achievers?
- Did you connect the handshake problem to the number of diagonals in a polygon? If so, how did this enhance the lesson?
- How did students demonstrate understanding of the pattern? How did they demonstrate understanding of the variable expression?

### Beyond Handshakes

### Learning Objectives

Students will:

- Explain the generalized solution to the handshake problem.
- Understand the handshake problem is non-linear, and be able to explain how you know.