This past Monday was my 37th opening day as a classroom teacher and I forget every year how tiring the first week can be. I suppose that it is also possible that it is more tiring as I get a bit older!
For the first time in over a decade I am teaching an Algebra I class. This year on opening day I presented a handout on the handshake problem. You can find it here
We had a pretty terrific conversation about this problem. I have eight students in this class and I made sure that they understood that I count as a person in figuring out this problem. The question at hand, at least the initial question, is how many handshakes will happen if each person introduces themself to everybody else with a handshake. I asked one of my students to tell me how many handshakes he would need to make and he easily answered that he would have eight handshakes to make. Immediately, a student to my right then guessed that there would be a total of 64 handshakes. Her classmates did not see right away where her guess came from, but I could tell that they sort of stopped thinking much about their own guess since one was on the table. I asked her to explain her guess and she was a bit reluctant. Totally natural, it’s day one and she is a new student in our school. I asked if it was okay for me to try to read her mind. She granted me this right. I guessed that she thought if there were eight handshakes for one person, then eight squared felt like a natural conclusion. She agreed that this sounded like her thinking. Another student jumped right in then and said 72 would be a better guess. Since there are nine people total, if each shook eight hands, then there would be 72 handshakes. Everybody seemed pretty happy with this guess and I have to admit that I, too, was pretty happy with this even though I knew it was not right. I already had two students think out loud and share their guess. A good start to the year.
I pivoted and asked if it was okay to try and check this guess. I asked how many handshakes would happen with only one person in the room. Confused faces tried to process this odd question and we all agreed that no handshakes happen here. How about two people? One handshake. We were confident about this. Three people? I turned to the person on my right to imitate a handshake, then the person to my left. We all agreed on three handshakes for three people. Four people? Someone immediately said six handshakes. Much of the class quietly took in this guess without saying much. I was happy that a different student asked why that would be true. I backed up and asked if we all agreed that three people have three handshakes. We all did agree. Then, I asked if a new person enters the three person room, how many new handshakes would be needed now? A – ha, three new handshakes! Then, the student who guessed six immediately says ten for five people. Now, we are picking up steam. We have gone from 0 to 0+1, to 0+1+2, to 0+1+2+3, to 0+1+2+3+4. This piece, by the way, was my addition to the conversation. Looking back, I wish I had helpd back because I think it would not have taken much to get a student to say this. Sigh…
After this, I took over a bit and led them down a more algebraic thinking path to arrive at a more general formula. We have thirty minute classes on the first day and I feel like we packed in some pretty great math talk on this first day. Is this a problem that you’ve used in your classes? If so, I’d love to hear a little bit about what class and how the conversation unfolded. You can share here or reach out to me over on twitter @mrdardy
Happy 2024 – 2025 academic year!