- Students will solve problems involving handshakes between groups of people in various ways including using tables, diagrams, lists, or formulas. - A table shows that with 9 people there are 36 handshakes, following the pattern of adding the number of previous people to get the next number of handshakes. - Through exploration and questioning, students derive the formula for number of handshakes which is 1/2 * n * (n-1), where n is the number of people. This relationship is quadratic, not linear.

1. 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

2. 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.
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

3. 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:
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.
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

4. 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 n2, 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.