CLASSIC HANDSHAKE PROBLEM: A REAL 'HANDS-ON' MATH LESSON
GUIDED INQUIRY LESSON TO BUILD CRITICAL THINKING AND PROBLEM-SOLVING SKILLS; GRADES 6 -??
The Handshake Problem
With 3 people, only 3 different handshakes are possible. Why? How about 5 people? 10? n??
PREFACE
Of course the big question is why and how you would ‘take valuable time’ from the ‘required’ topics that will appear on standardized testing. I believe that integrating sequence-type problems into the math curriculum deepens understanding of algebraic concepts and naturally develops mathematical reasoning and problem-solving. Not to mention they are still part of Common Core Standards currently in place in 41 states and territories:
Integrates algebraic reasoning (CCSS.MATH.CONTENT.HSA.SSE.B.3).
Q: Are any of you still paying attention to Common Core?
Q: Does assigning more difficult problems from the text or from teacher-constructed worksheets replace guided inquiry?
So what are some of the typical approaches that we model for this type of question? The same thing mathematicians do -- they start with a smaller problem and look for a pattern! This problem begins with developing a sequence and can be taken through second year algebra and combinatorics or as far as the teacher chooses to go! Triangular numbers, anyone?
SUGGESTED INQUIRY LESSON
“If everyone in our class shook hands with each other exactly once, how many total handshakes would there be?”
[Of course this is not what we expect students to do first but we can ask, “How do you think mathematicians would have approached this at the beginning?” Naturally most math teachers would have students act this out in their small groups!]
“In your groups solve for small numbers of people (2–4) and find a way to list the possible outcomes.”
2 people: 1 handshake (A-B)
3 people: 3 handshakes (A-B, A-C, B-C).
4 people : 6 handshakes (A-B, A-C, A-D, B- C, B-D, C-D).
Create a table to represent data - it is very important for students to see two different sets of numbers and use variables to represent them. When they focus only on differences, they are ignoring the pairing which will become the basis for understanding functions.
Number of People | 2 | 3 | 4 | 5 |…
# of Handshakes: | 1 | 3 | 6 | 10|…
1st Observation: Differences Of Course!
- Differences between terms: 2, 3, 4, 5... (increasing by 1 each time).
NOTE: In the upper level classes, they will learn that, since the second differences 1,1,1,… are constant, the general formula needed must be a quadratic. You mean, there’s still one reader left at this point!
4. Generalize the Formula - going past differences for now
"How can we predict handshakes for *n* people?"
**Key Insight: Each person shakes hands with *n–1* others, but this counts each handshake twice.
FORMULA : Total handshakes = n(n−1)2
5. Applying the Formula
- Solve for the class size (e.g., 25 students):
(25×24)/2 = 300 handshakes.
6. OPTIONAL EXTENSION: Connect to Algebra and Combinations
- Explain that this formula is a combinatorial concept (C(n,2)).
- Show how it is directly related to TRIANGULAR #’s:1+2+3+…+n
7. Real-world Link: Relate to networking, sports tournaments, etc
Still no time for these kinds of lessons? Yes, Dave, you’re no longer in the classroom so you have no clue!😉😉
CONCLUSION
This lesson not only teaches a formula but also models how mathematicians think—by breaking down problems, testing ideas, refining conclusions. Finally, and perhaps most significantly, it promotes a deeper understanding of how algebra is the language of number patterns!
Final Note
I deeply appreciate, not only likes, restacks and subscriptions, but, more importantly, your feedback about whether these efforts are worthwhile.😊
Dave