Well, I’ve now been officially initiated
into the blogosphere (is that actually a word?)
I really appreciated those who took the time to comment on the first task, and I
am sincerely hoping that this blog entry, the discussion of the task,
encourages more discussion/comments.
So, how’d you do with the Handshake task?
If you missed it, here’s the link.
As you know, I love this problem! It’s overflowing with the variety of
problem-solving strategies that can be brought to bear in its solution—act it
out, draw a diagram, look for a pattern, solve a simpler problem, make an
organized list, make a table, use logical reasoning, . . . .
Young students (and math-anxious
teachers) can use (and combine) the
strategies of acting the problem out, solving a simpler problem, and looking
for a pattern, as they build up to the given problem. For example, have 2 students
act it out—1 handshake
3 students—3 handshakes
4 students—6 handshakes
5 students—10 handshakes
6 students—15 handshakes
I always find it interesting when students
act out the problem: They often go from random hand shaking, which they
discover is very hard to count, to
organized hand shaking (lining themselves up and going down the line), and
finally (what we hope for) to the realization that when an extra person joins
the group, they don’t have to repeat all the handshakes that came before, but
rather just add on how many handshakes the new person has to do. Therefore, the
7th person would have to do 6 handshakes, 15 + 6 = 21, so 7 students—21 handshakes.
Often at this point (if not before), the pattern-seeking students will see that
as the number of people goes up by 1, the number of handshakes goes up by first 2,
then 3, then 4, then 5, and so on. Continuing, they arrive at
8 students—21 + 7 = 28 handshakes
9 students—28 + 8 = 36 handshakes
10 students—36 + 9 = 45 handshakes
Older students (and teachers) will
tackle the whole problem with a combination of organized listing and looking
for a pattern. They mentally or physically line up 10 people and reason that
the 10th person will go down the line and shake hands with 9 people; the next
person will go down the line and shake hands with 8 people; the next one, 7; the
next one, 6, and so on. So the total number of handshakes is 9 + 8 + 7 + 6 + 5
+ 4 + 3 + 2 + 1 = 45 handshakes.
This usually leads to the discovery of a
generalization: for 20 people, add 19 + 18 + 17 + … + 1. The generalization
still takes a lot of computation to find the total number of handshakes, but
it’s certainly within the capabilities of students in elementary school. A wonderful
YouTube video shows two grade 3 girls solving the problem in this manner,
using linking cubes as an aid, for the number of handshakes for 35 people! How’s
that for “perseverance in problem
With teachers and older students, if no
one has yet suggested this problem-solving strategy, I like to demonstrate
using logical reasoning and throw in my incredibly well-reasoned (but incorrect!)
Well, if I’m one of the 10 people at the
party, I shake hands with 9 other people. So does every other person at the
party. Since there are 10 of us, and we each shake hands with 9 other people,
it’s obvious that the answer is 10 × 9 = 90 handshakes. But this doesn’t match
the answer arrived at by using the other methods. What’s wrong?
After some pondering, they realize that
I’ve broken one of the ground rules and have counted every handshake twice, so
the correct answer would be 90 ÷ 2 = 45.
This discovery will then lead to the algebraic generalization that for any
group of n people, the number of
handshakes will be
n(n – 1)/2,
which matches the formula for
finding the sum of all the numbers up to (but not including) a given number.
The process is a nice, concrete way of showing why that formula makes sense: n people would each shake (n – 1) hands, but that counts every
handshake twice, so we have to divide by 2. Thus,
n(n – 1)/2.
The handshake problem has many variations in presentation. A way
of incorporating the problem into a history context is effectively shown on
NCTM’s Illuminations website, which discusses the tradition of the Supreme
Court Justices all shaking hands with one another before each session. Then
follows the Handshake problem, which asks how many handshakes that scenario would
The Illuminations website also has an
applet that draws a diagram, along with creating a chart, for the number of
handshakes for 2 people up to 12 people.
And, for students in grade 6 and beyond, Illuminations has a nice extension/connection between the Handshake problem
and triangular numbers.
OK—so there you have it. I hope you’ll
agree with me that this is indeed a “Math Task To Talk About.” Maybe you have
some other interesting connections to the Handshake problem, great ways that your students thought about it, or thought-provoking
activities that build on it. Please share!
Ralph Connelly is Professor Emeritus in the Faculty of Education at Brock University in Ontario, where he taught elementary math methods courses for 30+ years. He is active in both NCTM, where he’s served on several committees, currently the Editorial Panel of TCM, and NCSM, where he’s served two terms as Canadian Director as well as on numerous committees.