How Many Handshakes

How Many Handshakes? Target audience: Algebra II Collect data in groups of two to four students.

Objective: To collect and analyze data that models a quadratic function. Materials Needed: Graphing Calculator and lab sheets Introduce the Experiment: Describe the exploration. To collect the data the students can simulate the handshakes in small groups or draw polygons where vertices represent people. Organize the Data Collection: Have the students work in groups of two, three or four. Distribute the lab sheets. Each group should do its own experiment collecting the data.

NAME________________________________________________________________ DATE________________________ How Many Handshakes?

You are attending a party with a large group of people that you do not know and you would like to meet everyone. You are going to investigate how many distinct handshakes will occur if each person shakes hands. In this exploration you will collect data and determine the mathematical model that best represents your data. Part 1 1.

If there are 28 people in a room, estimate how many distinct handshakes will occur if each person shakes hands?

2.

Let’s simplify this problem, beginning with no one in the room. If there is no one in the room, then no handshakes occur. Continue and complete the table. (Hint: You can either simulate the handshakes in small groups or draw polygons where vertices represent people.) # of People

0

1

2

3

4

5

6

7

8

9

10

# of Distinct Handshake s 3.

Explain how you got the values in the handshake row. Do you see any patterns?

4.

Plot the # of People versus the # of Distinct Handshakes. Assign the # of People as the independent variable. Set your window to

5.

Part 2

[ −1,10] 1 x [ −5,30] 5 .

Describe the shape of your graph.

6.

The data set seems to be quadratic in the form of y = ax 2 + bx + c . Select three points, set up a system of equations and solve to find the values of a, b, and c. Graph the data and this equation on the same set of axes. Is this a good fit? Explain.

7.

Use your calculator to find the quadratic regression QuadReg on the number of people versus the number of distinct handshakes. Graph this equation on your data. Is this a good fit? How does this compare to your equation in #6? y=

8.

Use your equation from #7 to find the number of distinct handshakes that will occur in a room with 28 people. How does this compare to your estimate?