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)

12

13

14

15

16

17

18

19

(7 handshakes)

23

24

25

26

27

28

29


(6 handshakes)

34

35

36

37

38

39


(5 handshakes)

45

46

47

48

49


(4 handshakes)

56

57

58

59


(3 handshakes)

67

68

69


(2 handshakes)

78

79


(1 handshake)

89

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:
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

11

12

13

14

15

16

17

18

19

2

21

22

23

24

25

26

27

28

29

3

31

32

33

34

35

36

37

38

39

4

41

42

43

44

45

46

47

48

49

5

51

52

53

54

55

56

57

58

59

6

61

62

63

64

65

66

67

68

69

7

71

72

73

74

75

76

77

78

79

8

81

82

83

84

85

86

87

88

89

9

91

92

93

94

95

96

97

98

99

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.