These and other activities included in our book, , now available from our Math Dance with Dr. Schaffer and Mr. SternBook Shop(http://www.mathdance.org/store.html) |

How Many Ways are there to Shake Hands?

A new look at an everyday gesture

Grades |
K—12 |

Time |
25-45 minutes |

Concepts |
Counting, combinations, problem definition, sequence, dynamics |

Groups of |
2, 3 |

Space |
Regular classroom with desks moved to sides, or larger |

Related activities |
Circle Symmetry uses handshakes to explore symmetry instead of combinatorics |

We all shake hands. Some handshakes are ordinary, and some are more unusual. Kids are especially creative about handshakes, often inventing complex "secret" handshakes.

In this activity we ask the seemingly innocent question: how many ways are there to shake hands? This seemingly simple question will lead us deep into exercises that range from specific mathematical counting problems to open-ended creation of movement sequences. The activities are fun because they ask us to look at an everyday action in a new way.

[*In their "hand-shake" dance, Dr. Schaffer and Mr. Stern keep missing each other's hands. Then when they finally do shake, the hands are stuck! This leads to a discussion with the audience about counting.]*

7-1. Try this (5-10 minutes)

Have students pair up. Have each pair answer the following questions.

• What "secret handshakes" do you already know?

• What's the biggest handshake you can make up? What's the smallest?

• What's the most unusual handshake you can make up? Try do something different from what anyone else is doing.

[Illustration: kids making wild handshakes.]

7-2. Counting handshakes (5 minutes)

**"Let's say that we're only dealing with the simplest hand-shake. By this we mean that right hand to right hand counts as only one handshake, no matter how you do it (see illustration showing a normal right-to-right, and two right-to-right variations. How many ways are there for two people to shake hands, if they can only use one hand at a time."**

[Illustration: show three handshakes that count as the same even though they are different.]

**The answer** we originally had in mind is four handshakes:

Shake #1 |
first person's right hand |
to second person's right hand |

Shake #2 |
first person's right hand |
to second person's left hand |

Shake #3 |
first person's left hand |
to second person's right hand |

Shake #4 |
first person's left hand |
to second person's left hand |

[illustration showing all four handshakes]

**Other answers**. The above chart indicates there are four ways. But we find that students often come up with other answers. Rather than telling them the answer we show above, have them explain their answer, whether it agrees with ours or not. The answers we have heard have ranged from "two" to "infinite" to "it depends." Here are two other answers based on different interpretations of what counts as a handshake:

Six handshakes (Four handshakes between the two people, plus two more handshakes in which one person or the other shakes their own hand):

Shake #1 |
first person's right hand |
to second person's right hand |

Shake #2 |
first person's right hand |
to second person's left hand |

Shake #3 |
first person's left hand |
to second person's right hand |

Shake #4 |
first person's left hand |
to second person's left hand |

Shake #5 |
first person's left hand |
to first person's right hand |

Shake #6 |
second person's left hand |
to second person's right hand |

[illustration showing all six handshakes]

Three handshakes (Same as the four handshake solution, but we don't distinguish whose hands are doing the shaking, so right to left and left to right count as the same):

Shake #1 |
right |
to right |

Shake #2 |
right |
to left |

Shake #3 |
left |
to left |

**After the discussion**, show the students our method of counting which is diagrammed above. They might not all agree with our method.

**If you have time**, it is valuable to let students defend their answers based on their interpretations of the rules. The point is that it is important for students to be able to find answers that are consistent with their interpretation of the rules, whatever that interpretation may be. Discussing different interpretations of the rules will help the class come to a clearer understanding of just what the rules are..

7-3. Expanding the handshake problem (5-10 minutes)

Now have students get in groups of three. Since this question is a bit trickier, for these activities two people in the trio are the "Hand-shakers" and one is the "Counter."

• How many ways are there for two people to shake hands, if they can use ONE or TWO hands at a time?

• Have pairs demonstrate and count their ways to do it. If time allows, can the class make a chart on the board similar to chart we made above? Since students can use one or two hands, there are more choices.

• Note: The answer will undoubtedly vary from student to student. That’s okay. The point is to have students explain their rules and show that their counts are consistent with those rules, not for everyone to get the same answer.

[Illustration: photos of at least a couple different groups making handshakes]

Math note: Problem solving strategies

In mathematics as well as in dance, it is important to give students open-ended activities in which they have room to develop their own strategies. Counting is not just memorizing the sequence of numerals, 1,2,3, …, but figuring out how to organize what you are counting so that you do not miss anything, and do not count anything twice. In everyday life we often encounter situations in which we need to improvise new ways to count things. This is not as easy as it sounds!

7-4. Handshake dances: (10-20 minutes)

**Make up a dance**. Students are still in groups of three. For this activity, any type of movement is acceptable, as long as it's safe!

• Make up a sequence of 3 to 6 hand shakes (depending on the age).

If students want more direction, ask them these questions (or make up your own):

• Can you make it really high, or really low? Can it involve a jump or a turn?

• How can you shake hands without using your hands?

• What would it look like to shake hands under water?

• Can you think of a handshake that is completely different from any handshake you have seen before?

**Practice**. Once each trio has decided on their handshake sequence, have them practice the sequence so they can do it clearly without hesitating and without talking. Urge the students to think about performance aspects of their handshake sequences.

• How do the people come together to shake hands?

• How do they separate or finish the sequence? A strong beginning and ending can add much to a dance sequence.

• Try varying the dynamics, sometimes moving with exaggerated slowness, sometimes with great speed.

• Think about character - who are they and why are they shaking hands?

**Perform**. Then have each group demonstrate their hand-shake sequence.

• Have each group perform their sequence for the rest of the group. You may want to ask everyone who is not performing to sit down so people can see better.

• You might want to add music. Decide on an order for the groups of the class (e.g. Tommy’s group, then Tammy’s group, etc.). Have the groups go in order and perform to music. Ask each group to be ready so they can start as soon as the previous group is finished. We find that adding music makes this activity feel more like a performance.

**Variations**. Once groups have mastered their sequences, you can challenge them to:

• Reverse the order of the handshakes in your sequence.

• Reverse the movements themselves (more difficult).

• Switch the orientation so that a right to left shake becomes a left to right shake, thus performing the mirror image.

• Construct transitions from one group's performance to another.

[Illustration: sequence of at least 3 handshakes being performed.]

7-5. Further ideas

**More dance ideas**. Here are other movement activities that build on the handshake dances. See also the Threesies activities, which uses handshakes to explore symmetry instead of counting.

• **Remember**. Come back to the handshake exercise a day or a week or a month later. Who can remember their secret handshakes? The handshakes provide a good classroom fill-in activity.

• **Story**. Tell a nonverbal story with the handshakes.

• **Written description**. Have students write descriptions of their handshake sequences in words and pictures, and then trade descriptions. Can another group reconstruct the sequence from the written description? [illustration]

• **Begin and end**. Create a movement sequence that begins and ends with a handshake, but can do anything else inbetween.

Dance note: Everyday movement

Everyday movements like shaking hands are a great way to get non-dancers into doing and creating movement sequences. The gesture of shaking hands is familiar, so students don’t feel intimidated. And creating new handshakes is fun, so students don’t feel self-conscious. Yet as students practice and refine their handshake sequences, they quickly become engaged in the real challenges of choreography.

Although many types of dance like classical ballet take years of practice to master, others are built on everyday movements that anyone can do. For instance (cite a major dance company and what they did) the XXX dance company made a dance called XXX entirely out of XXX. Many hip hop moves started off as everyday gestures. All over the world folk dances are made out of the everyday movements of work and play. Of course it still takes concentration and focus to perform these dances well, but by using everyday movement they invite observers to put themselves in the dancers shoes.

**More math ideas**. Here are other counting questions involving handshakes and other everyday movements.

• **Handshakes generalized**. Generalize the problem of counting handshakes. How many ways are there for two people to shake hands if each person has three hands? Four hands? Five? Can you find a pattern in your answers? For more advanced students: Can you find an algebraic formula that gives the number of ways to shake hands if each person has n hands? Can you think of other ways to generalize the problem?

• **Pairs**. Get in groups of four. Have each pair of people shake hands once. How many handshakes are there all together? (For this problem we don't care what the handshakes look like, we only care about who has shaken hands with whom.) Can you perform all the handshakes in sequence without repeating any? Can you find an order that is easy to remember?

• **Pairs generalized. **How many different pairs of people can shake hands in a group of three people? Five people? Six? Write down your answers and see if you can find a pattern in the numbers. Can you predict what the answer will be for seven people without actually counting all the pairs? Can you draw picture that helps you count all the pairs of people?

• **Lots of pairs**. If everyone in your class wanted to shake hands with every other person, how many handshakes would there be altogether? How long would it take for every person in the world to shake hands with every other person in the world? Note: this question is deliberately ambiguous. How many people are there in the world? How long does a handshake take? Do you have to count travel time? Clarifying the question is part of the problem here.

• **Order**. In how many different orders can three people stand in a line? Can you make a movement sequence that includes every order just once? Try the same game with four people.

• **Stairs**. How many different ways are there to walk up a flight of six stairs, if each time you walk you can go up one or two steps? What about for a flight of seven stairs? Eight? What other rules can you make for going up a flight of stairs?

Math note: Combinatorics

The area of mathematics that deals with counting combinations of things is called combinatorics, which is part of discrete mathematics. The word "discrete" refers to the fact that we are counting discrete things, like handshakes or people, which only occur in whole units, instead of continuous things, like water or speed, which can be measured in decimal fractions. Discrete math has had a big resurgence in the recent years because it is so important in computer science. Combinatorics is one of the most accessible areas of modern mathematics, since many of the questions are easy for anyone to understand.

7-6. Assessment

**Assessing the dance**. Once students have demonstrated their handshake sequences, they can develop them further into dances. Here are some things to look for in critiquing their performances:

• **Clarity of movement**. Is the movement from one handshake to the next clear and concise, or are there distracting elements that look out of place?

• **Ensemble**. Do the member of a group move together in a unified way to create a cohesive effect?

• **Overall structure**. Does the sequence have a clear beginning and end? Does one movement flow logically into the next?

• **Inventiveness**. Are the handshakes mundane or unusual? Are the handshakes similar to each other or more varied?

• **Accuracy**. Some of the dance challenges, like Pairs under Math Challenges, ask a group to go through a series of combinations exactly once. Were all the combinations included? Did they miss any? Did they repeat any?

**Assessing the math**. In this exercise we ask students to invent systematic counting methods to handle unfamiliar situations. Here are some things to look for as you assess their mathematical work:

• **Explanation**. Did the student explain their interepretation of the rules clearly?

• **Follow the rules**. Was the interpretation consistent with the assigned problem?

• **Accuracy**. Count consistent with rules

• **Systematic**. Miss anything? Duplicates?

• **Clearly organized**? Draw pictures.

• **Formulas**. For more advanced students: generalize, formulas.

7-7. Resources (not finished; will mention particular publications)

Music. We prefer music that has a sense of continuous forward movement, without sudden changes. Music without words tends to be less distracting. Some of our favorites are:

• Deep Forest

• Drumming

• African

• New Age

Here are some resources for counting problems:

• Anno’s counting books

• Web sites about counting problems

• Books on discrete math

And other books about sign language and handshakes

• Books on gestures with the hands

• Books on sign language(?)

These and other activities included in our book, , now available from our Math Dance with Dr. Schaffer and Mr. SternBook Shop(http://www.mathdance.org/store.html) |