And then once the students are seated, I want to generate some enthusiasm for mathematical problem solving by giving my students an opportunity to do some interesting math on the very first day of school by posing an intriguing and mathematically rich problem for the students to consider and solve. The handshake problem is a fun way to combine these two first day of school goals--it goes nicely with the idea of meeting and greeting new people, it is pretty easy to set up and explain the problem, it is accessible for most students, there are multiple ways to solve this problem and it contains some intriguing and elegant mathematical patterns.
After I shake every student's hand at the door, I ask the students to get up and shake the hand of every other person in the room. I play James Taylor's version of Getting Know You while students walk around the room shaking each others' hands. Thank You For Being a Friend would also be good background music to play. Then I ask "I wonder how many handshakes it took for every person in this room to shake hands with each person in the room."
Here are some ideas I would like to try this year to improve my presentation of the problem and to get students more engaged in the problem by asking them what they notice and wonder about this activity. I will have a slide with a photo of each child on the screen and I will get the discussion started by asking several students if they shook hands with all the students in the room. I could also ask a child if they shook hands with another particular child and I will start to draw lines on the slide connecting any two students that acknowledge that they did shake hands with each other. This would be a great time to discuss the question about how we count handshakes: if two people shake hands does this count as one handshake or two handshakes? After I have drawn several lines connecting various students in the classroom, I will ask some general questions to get the students thinking about this problem: What are you wondering about? What mathematical questions could we ask about this activity? What is the problem? What do you find interesting about this problem? What do you notice about this problem?, etc. Hopefully through this discussion, the students will come up with the mathematical question posed by the classic handshake problem: "if you have a room full of people and everyone shakes hands, how many total handshake are there?" and suggest some possible ways to think about and solve this problem. Then I will give the students some time to discuss this problem in small groups. Other possibilities for class discussion, either before or after the students work on the problem in small groups, could include showing a slide of all the students, but this time the photos would be shown in a circle instead of in rows. In addition, I could show a slide with one student and ask how many handshakes?, two students: how many handshakes?, three students: how many handshakes?, etc. and then discuss possible patterns that emerge.
I plan to use this problem for the first Problem of the Week (POTW) for the 2015-2016 school year:) And I am hoping that it will get the class off to a good start in terms of working on challenging math problems in small groups, discussing and sharing our mathematical thinking with others and finding multiple ways to solve problems.