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.
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
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 pattern, 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 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
|
| 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.