I first came across this problem in Marilyn Burns’ book, *About Teaching Mathematics*. That book was really instrumental for me as I started to realize I wanted my math class to center around rich problems, that valued student sense-making.

I like this problem because it seems incredibly easy, but isn’t. I also really like that there are many different approaches/problem-solving strategies that students can use here. I often use it at the beginning of a semester to engage student in a conversation about how multiple approaches that come from them are both valuable *and* mathematical. This problem can also be connected with class work on functions and visual patterns as a way to get students to make generalizations from observations. (See also, The Border Problem)

This problem involves figuring out how many handshakes will result if 9 people each shake hands with each other. I have used another version of the problem as well, set in a specific context. It is the same mathematically speaking – 9 people all shaking hands with each other – but instead of it being just any 9 people, it is the nine justices of the US Supreme Court.

I have included both the simplified version and the Supreme Court version in the attachments above.

**Describe how you solved the problem.**

I have known this problem for a while. The first time I solved it I used a 10 by 10 square. I created a matrix by writing the initials of the justices of the US Supreme Court along the left side and again along the bottom. Then I put an “X” in every square that represented a justice shaking their own hand, because you can’t shake your own hand. Then I counted the squares under the diagonal of “X”s – each square represents a handshake – and came up with 36 handshakes.

I was interested in the blank squares and so I counted them (in blue) and realized they were a mirror reflection of the number of handshakes I counted (in red).

Nine US justices are each going to shake hands with 8 other people which is 72 handshakes. But the key is not counting any handshakes more than once – Ruth Bader Ginsburg shaking hands with Sonia Sotomayor is the same as Sonia Sotomayor shaking hands with Ruth Bader Ginsburg – so we need to divide that total in half.

My visual helped me create a generalization that will give you the number of handshakes for any sized group of people.** n(n-1)/2** will give you the number of handshakes, where

*n*is the number of people in the group.

So in the case of the US Supreme Court, 9(8) is 72 and 72/2 is 36.

** ****Identify and describe a few specific challenges that your students will have in solving the problem? Describe how you might support the problem-solving efforts of struggling students without giving too much away.**

Some students will have a problem getting started. If that happens, as I walk around the class, I’ll ask them to explain the situation to me. Once they can do that to my satisfaction and ask if they can draw a visual representation of the situation. I might also either give out a list of problem-solving strategies and ask them to star any strategy they think might help. Then I ask them to try one.

If enough people are stuck, I sometimes stop everyone after 1-2 minutes and have a few volunteers talk about how they got started, being careful to cut them off before they get too far in their explanations.

I have noticed that some students get tripped up by the wording of the problem and be confused about the situation. Especially students who approach the problem through pure calculation. They either multiply 9 times 9 or 9 times 8 and get 81 or 72 respectively, and think that they are done because they don’t realize they are counting some of the handshakes twice.

If that happens, I’ll try to get them to generalize their procedure and get them to apply it to a smaller group of people. They’ll say something like, “I multiplied the total number of people, by one less person, since you can’t shake your own hand”. Then I’ll ask them to explain how many handshakes , using the same procedure, would result if 4 people shook hands. I would expect them to say either 16 (4×4) or 12 (4×3). I’ll then have 4 students stand up and act out the handshakes to test it. When they see it actually results in 6 handshakes, they’ll see the procedure doesn’t work and I’ll ask them to keep working. This is also an opportunity to connect them back to a closer reading of the text. If they act it out and actually get 12 handshakes, you can ask, “What does it mean in the problem when it says, ‘each person shakes hands exactly once with all the other people”?

I also think some students will use visual representations and get confused about what is being represented and count either too many or too few handshakes. If it is a visual representation I think is interesting, one thing I might do is have that student present it and then try to get the class to ask clarifying questions.

**What methods do you think students might use?**

**MAKE AN ORGANIZED LIST**Students might try to solve the problem by writing out all the combinations and counting the handshakes.- Person A shakes hands with person B, C, D, E, F, G, H, I – 8
- Person B shakes hands with C, D, E, F, G, H, I – 7
- Person C shakes hands with D, E, F, G, H, I – 6
- Person D shakes hands with E, F, G, H, I – 5
- Person E shakes hands with F, G, H, I – 4
- Person F shakes hands with G, H, I – 3
- Person G shakes hands with H, I – 2
- Person H shakes hands with I – 1
- Person I has already shaken hands with everyone
8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 + 0 = 36

or put in terms of the US Supreme Court (you might see if they can come up with any names and/or provide them with the names of the justices)

- Neil Gorsuch (NG) shakes hands with EK, SA, SB, SS, JR, AK, RBG, CT – 8
- Elena Kagan (EK) shakes hands with SA, SB, SS, JR, AK, RBG, CT – 7
- Samuel Alito (SA) shakes hands with SB, SS, JR, AK, RBG, CT – 6
- Steven Breyer (SB) shakes hands with SS, JR, AK, RBG, CT – 5
- Sonia Sotomayor (SS) shakes hands with JR, AK, RBG, CT – 4
- John Roberts (JR) shakes hands with AK, RBG, CT – 3
- Anthony Kennedy (AK) shakes hands with RBG, CT – 2
- Ruth Bader Ginsburg (RBG) shakes hands with CT – 1
- Clarence Thomas (CT) has already shaken hands with everyone

8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 36

2. **SOLVE A SIMILAR, SIMPLER PROBLEM **Students might also try to figure out the number of handshakes in smaller groups of people and see if there is a pattern. Two people – one handshake, three people – two handshakes, four people – six handshakes.

3. **ACT IT OUT **Students might try to get together in groups of nine and act it out.

4. **DRAW A PICTURE **Students might try to “act it out” using pictures and arrows.

**5. CONSTRUCT A TABLE AND LOOK FOR PATTERNS** Depending on their experience with functions, students might make a table with the number of and look for patterns

**What extension questions could you ask if students finish early?**

- Ask students to find the number of handshakes for different sized groups:

- How many handshakes would result if everyone in our class shook hands?
- There are 100 members of the US Senate. How many handshakes would result if they all shook hands?
- There are 435 representatives in the US House of Representatives. How many handshakes would result if they all shook hands.

2. Generalization for finding the number of handshakes for any sized group of people

- How could you figure out how many handshakes would result from any sized group?

**What do you want your students to get from working on this problem?**

There are two main takeaways I want students to get from working on this problem. The first is about problem-solving strategies and the second is about algebraic thinking.

I often give this problem in the beginning of the semester to talk about problem-solving strategies and to introduce a very important classroom norm:

Instead looking at the answers to many different (though similar) problems, we are going to look at multiple approaches to the same problem.

As I mentioned above, this problem can also be used to develop algebraic thinking, similar to our work with visual patterns, described in the CUNY HSE Math Curriculum Framework. If I am using it for that purpose, I will either save it for later in the semester, or even better I like to return to it after students have done some work with visual patterns and functions.

I have also used it in conjunction with a reading on working memory called *The Brain is Not for Thinking*, adapted from David Willingham’s book Why Students Don’t Like School?. With the reading, I try to connect their problem solving strategies with current neuroscience research as to why creating visual representations might help them.

#### Student Work

**Natasha**

Natasha shared a chart she drew to help us see her reasoning. In Natasha’s chart, each box represents a handshake between two justices. She had written the letters A through I up the side and the numbers 1 through 9 across the bottom. After Natasha explained how her picture worked, I asked her if we could put the letters A through I along the bottom of the

She went up to the board thinking her answer was 45 handshakes, but then she convinced herself it was 35 (or maybe 36!) handshakes. She realized it wasn’t 45 because as she was explaining how he chart works, she realized she was counting each person shaking their own hand.

Natasha wasn’t sure if it was 35 or 36 handshakes because she wasn’t sure if she needed to count that “last” box. The handshake in question is the box above that is circled with a question mark next to it. It represents what would be a handshakes between H and I.

Here’s a Natasha presenting her method (3:04-10:30).

I thought her decision of representing each handshake as a box was interesting and I wish I had made tiles available to students, (or square inch large graph paper). Using this visual model is an opportunity to see both the handshakes and the ones that already counted.

**Miyako & Ronlee**

Miyako and Ronlee teamed up to make this diagram.

Miyako wrote A through I across the top of her diagram representing the 9 justices. She also wrote A through I down the left side. She used a single “1” to represent each handshake. She also recorded the total number of handshakes under each column.

We can see Person A shakes 8 hands, Person B shakes 7 hands, Person C shakes 6 hands, Person D shakes 5 hands, Person E shakes 4 hands, Person F shakes 3 hands, Person G shakes 2 hands, Person H shakes 1 hand and Person I shakes no one’s hand (since they have already shaken everyone’s hand. Ron Lee add the diagonal line of “0”s, each showing us the handshakes we don’t count, because no one shakes their own hand.

Here’s Miyako and Ronlee presenting their collective method (10:30-13:16).

**Angel**

Angel made a different kind of diagram. He drew 9 stick figures, each representing a Supreme Court justice. He then moved left to right, imagining each person shaking hands with all the justices to the right of them. For example, the first justice (all the way on the left) shakes hands with the 8 justices standing to their right. The next justice already shook hands with the first person, and they don’t shake their own hand, so they only shake hands with the 7 people to their right. That pattern continues until the end.

Here’s Angel explaining his method, with my support (13:16-16:20)

###### Here are three samples of student work from another class

**Constance**

Constance’s approach was similar to Natasha. She labeled the parts of this to make it stand a little more on it’s own. Part of that was because she had an opportunity to draw it out on newsprint before she brought it to the board. I choose to look at this one first because I felt it was the most visual and concrete and accessible to the class. I also though it would allow most students to agree on the answer. It is interesting to see the impact of the different orientations (compare the shape/direction of the “triangle” that Natasha drew, to Miyako/Ronlee’s to the one drawn by Constance).

**Sofia**

Sofia made a chart to look for patterns in the size of a group and the number of handshakes. Unfortunately I only got a picture of the chart she put up on the board. What she wrote is in green – what I added during the class discussion of her strategy is in blue.

I didn’t want students to see the completed chart and think it just magically appeared to Sofia, so I asked her what challenges she had while creating her chart. I really wanted students to see that the chart was a tool that she used to find a pattern – she didn’t know beforehand what she’d find or even if she would find anything. After she presented when we went around saying what we appreciated about her strategy, I said I appreciated how she used what she had learned from our work with functions, that collecting data into charts can help us see different kinds of patterns in the way the numbers are changing.

Sofia said the hardest part for her was figuring out the number of handshakes for the smaller groups. She was able to draw pictures that were based off of our acting out with a group of 4. She got up to the 15 handshakes for the group of 6 before her pictures got too messy for her to keep track of. Then she noticed a pattern in the way the number of handshakes were changing and used that pattern to figure out how many handshakes there would be for a group of nine. After she explained the pattern she saw, I had the class use her pattern to figure out how many handshakes would result if there were ten people.

**Brandon**

*“Take the number of people in the room and multiply it by one less than the number of people (because you can’t shake your own hand). Then divide what you get by 2. That will give you the number of handshakes.”*

I saved this approach for last in part because Brandon came up with this generalization after the class agreed the answer was 36. He was one of the folks who initially thought it was 72. He noticed that when we acted out the group of 4, we expected 12, but got 6. Then for the group of 9, he expected 72, but there are only 36. Brandon was in Sofia’s class, so we were able to use the table she built to test out his generalization and we saw that it worked for each input and output. We were also able to consider his generalization with the visual provided by Constance. I asked students why Constance left certain spaces blank and they were able to tell each other that it was because those handshakes had already happened. That allowed them to understand that dividing by two keeps you from double counting (Once I shake your hand, when it’s your turn, we don’t need to shake again).

###### Final Thoughts

This is one of those problems that really do have a low entry and high ceiling. There will likely be a wide range of solution methods, but this problem engages adult basic education students and HSE-level students alike.

In the future, I will definitely make more manipulatives available for students, especially things like colored pencils, chess pieces, graph paper, and square inch tiles.

###### Additional Resources

- Here’s a lesson plan from the NCTM Illuminations website. It includes a student handout with a table similar to the one Sofia developed, asks students to generate some data and then to make a generalization about a group with
*n*people shaking hands. This lesson focuses on the algebra solution methods to this problem. - Metacognition in Math: Developing Problem-Solving Strategies – this video is about a different problem – The Bicycle Shop Problem, which is very similar to the Chickens and Goats Problem. It has more information about how I try to build student awareness of the strategies they use and the choices they make.