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# TCM Blog

### Reflecting on The Handshake Problem

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 the solution:

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 solving”?

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!) hypothesis:

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 take.

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.

 When the initial blog post came out, EllenM commented on the fact that the handshake problem could also be thought of as the "staircase problem". If you click on the Youtube video referenced in this posting, you can clearly see the "staircase" being used to solve the problem--a great connection! Hope more of you will share your comments on this post! Thanks!!Posted by: RalphC_79522 at 3/6/2014 1:33 PM

 Ralph, what is you opinion about following the exploration of The Handshake problem with the Locker problem? Would that be an appropriate next step for students? I really like the Locker Problem, but I am concerned that the variations of opening/closing the locker doors might needlessly introduce confusion. (The Locker problem can be found at http://www.nctm.org/publications/article.aspx?id=31567).Posted by: ElizabethS_05318 at 3/12/2014 10:31 AM

 Elizabeth- The locker problem is very effectively used with 6th grade students as a part of the "Connected Mathematics Project". If you have younger students you may need to scaffold them through the problem. :)Posted by: M LynnB_55212 at 3/13/2014 7:59 AM

 Great post.Posted by: LaikenJ_62015 at 3/13/2014 11:04 AM

 It's terrific to see comments being posted--thank you! Elizabeth, as M LynnB pointed out (and thank you, M LynnB for doing so!), the Locker Problem is great, but the pattern is more difficult to see, so younger students may require more scaffolding to help them with the problem, but it could certainly be a followup. And now, to be "really sneaky", I'll mention that the next "Math Task to Talk About", which will be posted next week I think (but I'm not certain) is perhaps a better followup to the Handshake Problem--the pattern is more obvious and would require less scaffolding for young students. You'll just have to visit the site again to see what it is [I know, I'm shameless, but I want to do SOMETHING to make sure people are reading the blog! :) ] Another possible followup was mentioned by DianaP in the comments to the original problem--the question of how many diagonals are there in a [take your pick of polygon here]. For example, in a quadrilateral there are 2, in a pentagon there are 5, in a hexagon there are 9, in a heptagon (7 sides) there are 14, octagon 20, etc. The "generalization" is something like what happens with the Handshake problem, but of course the "formula" is different. Like with the Handshake problem, a good variety of problem-solving strategies can be used in solving the problem.Posted by: RalphC_79522 at 3/13/2014 3:52 PM

 Hi Ralph,A little late with some comments but since this is one of my favorite problems, thought I would add some of my experiences. I first used this problem with my fifth graders. I wanted kids to use a variety of strategies and they sure did! Some acted it out during recess (yep -- during recess!). They lined up and the first kid went through the line and shook everyone else's hand and then got out of line -- the second kids went though and did the same. Oh yes, one kid was designated the recorder so he kept track of the handshakes. It didn't take them long to see the pattern of adding consecutive numbers starting with one less than the number of kids in the class and working their way down to 1 (alas the last kid did not shake any hands -- an important part of the discussion the next day!) Other kids made a table and solved a simpler problem starting with 2 kids (1 handshake) 3 kids (3 handshakes) 4 kids (6 handshakes) and saw the pattern of adding the next consecutive number. Others drew a picture -- similar to finding the diagonals of various polygons with x vertices (x being the number of students). They liked this one because they thought their drawings were awesome! When we discussed their strategies and their representations, the focus was soon on the patterns. While only one or two of the kids found the generalized expression x(x-1)/2 -- on their own...when they shared it with the class-- the others appreciated they could now find how many handshakes for any number of people (x). By the way, the pattern comes up in many problems and the kids would immediately recognize it as the "handshake pattern."Posted by: LindaG_42510 at 4/19/2014 9:04 AM

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