ACTIVITY 1.2.2 OBJECTIVE: Combine problem-solving strategies to discover a formula for triangular numbers The "Handshake Problem" is a famous problem often used to introduce students to the concepts involved in the sequence of triangular numbers. In the next activity, you will investigate these numbers by combining several of the problem-solving strategies introduced in this chapter. The Handshake Problem: There are 31 people at a party. If everyone shakes hands with each of the other party-goers exactly once, how many handshakes will there be? 1. To gain a better understanding of this problem, first consider a simpler problem. That is. what if there were 4 people at the party? a. Draw a picture to represent the handshakes between 4 people, if each person shakes hands with each of the other people exactly once. How many handshakes are there Explain why the total is not 4 x 3 (number of people x How many handshakes are there per person? altogether for 4 people? number of handshakes per person). Introduction to Problem Solving b. Now, make drawings to represent the handshakes between 2 people, 3 people and 5 people. Then, complete the following table with the information you discover in your drawings. ⚫ of people 2 3 4 5 # of shakes Total # per person of shakes c. Examine the table in part (b) and describe any patterns you notice. Can you predict the number of handshakes for 6 people? Check your result by drawing a picture. d. Based on your work in parts (a) - (c), are you able to write a closed formula that could be used to solve the handshake problem for 31 people, without the need to find the number of handshakes for 30 people? If so, state it here and then answer the question. In your answer, the leading digit should be 4 and your last digit should be 5. Does your formula work? 2. When counting the handshakes for 5 people, you may have drawn a picture such as the one at the right. You could first count the number of handshake for the first person; there are 4 handshakes. When you count the handshakes for the second person, how many new handshakes must be recorded?