Beyond Handshakes

• Lesson
• 1
• 2
6-8
1

Using spreadsheets, students will explore another pattern, that of the triangular numbers. This exploration will enhance students’ ability to generalize a pattern with variables.

In this lesson, students will examine triangular numbers and see the relationship to the handshake problem, which was explored in the first lesson of this unit, Supreme Court Welcome. Students will first review the handshake problem and get comfortable with spreadsheets as a tool for representing patterns.

Open the class by displaying a spreadsheet program, and check for background knowledge of spreadsheet use.

Use the Handshake spreadsheet to review the handshake problem. The tabs at the bottom of each page can be used to access different sheets in the file. Handshake Graph shows a graph that compares the number of people to the number of handshakes. Shake Table shows a table for the same data. Review the formula that students discovered for the number of handshakes that occur with n people: n(n – 1) ÷ 2. The formula can also be thought of as the number of people (n) times the previous number of people (n – 1), all divided by two. Ask students to express this for any number of people. Help students to understand the spreadsheet by reviewing the following:

• Columns are labeled with letters, A, B, C; rows are labeled with numbers, 1, 2, 3.
• Cells are named according to the letter of the column and number of the row: A1, B2, C1, D4, etc.
• The cell in the fifth column and seventh row, for example, is E7.

On the Shake Table of the Handshake Spreadsheet, the number 1 is listed in cell A3. Ask students, "How can we write a formula to get the number 2 in A4, to get 3 in A5, and so on? Explain that the numbers could be written by hand, but using a formula allows us to generalize and extend this indefinitely. [The formula "=A3 + 1" should be placed in A4. Then, the fill down command can be used from the Edit menu to place the formula in subsequent cells.] Show students the formula that appears in some of the cells. They should notice that the spreadsheet changed the cell number appropriately so that 1 was always added to the previous cell.

Ask students how to use the numbers in the column for people to generate the numbers in the column for handshakes. [There are no handshakes with just 1 person, so write 0 in B3. In B4, insert the formula "=(A4*A3)/2." Again, copy the formula into subsequent cells using the fill down command.] As above, have students examine the formulas that appear in the cells. They should again notice that the formula was automatically updated in each cell to use the number of people from the correct row.

Next, use the spreadsheet to make the graph. Examine the graph, and have students compare it to the graphs they made during the previous lesson. Ask, "Is this graph linear?" [No, it is not linear.] Again, ask students to explain how they know.

Transition to manipulatives to explore triangular numbers. Pass out tiles. Distribute the Triangular Numbers Activity Sheet. Ask students to draw the fifth triangular number on their sheet; or, using algebra tiles or other manipulatives, ask them to build it. Then, working individually, students should complete the table on the activity sheet. Walk around the room as students work. When most students are done, have them share their work with a partner. Then, discuss the results as a class.

Students should recognize the pattern of 1, 3, 6, 10, as these are the same numbers from the handshake problem. Ask them to explain the pattern in the table. [The number of dots is equal to the figure number multiplied by the next figure number and divided by 2.] Ask, "How is this pattern different from the handshake problem?" [We are examining the next number, not the previous number!] Allow students to draw a graph on the activity sheet. Again, have students enter this information into a spreadsheet and create a graph on the computer.

Students have now had practice with a table, graph, and variable expression for another verbal problem. A picture can also be used to describe the triangular numbers, as shown below. The entire rectangle contains n(n + 1) dots. When the rectangle is divided in half, it forms one of the triangular numbers, and the number of dots is n(n + 1) ÷ 2.

To provide another example of the triangular number pattern, have students examine the Sum of Integers Spreadsheet.

This is the list of the first n positive integers. Ask students, "How can we apply what we just discovered to another situation?" Students will recognize the nth sum; that is, 1 + 2 + 3 + … + n = n(n + 1) ÷ 2. Students should predict whether this graph will be linear. [No, the rate of change is not constant; in addition, n multiplied by (n + 1) will yield a term of n2 in the equation.]

Assessment Options

1. Have students print their spreadsheet, their spreadsheet showing formulas, and their graph to demonstrate mastery of using the spreadsheet to model a problem.
2. Have students demonstrate the sum of the first 24 positive integers [300] in more than one way. [24 x 25 / 2 = 300]. Students may use the spreadsheet, Gauss’ method, n(n+1)/2, a calculator, diagram, etc.

Extensions

1. Examine the sum of the first n odd positive integers. For example, 1 + 3 + 5 = 9. Further, 1 + 3 + 5 + 7 = 16. What variable expression would represent the sum of the first n odd positive integers? [n2.]
2. Examine pyramidal numbers by making a model of the first 4 triangular numbers out of ping pong balls (attach them together). What happens when you stack them up (1, 3, 6, 10 are the triangular numbers; the first pyramid is 1, then 4, then 10, then 20). What are these numbers called? [Pyramidal numbers.] The third and fourth pages of the Triangular Numbers Activity Sheet can be used to explore the pyramidal numbers.
 Triangular Numbers Activity Sheet
3. Ask students to examine Pascal's triangle.
 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1

Where are the triangular numbers in Pascal's triangle? Where are the pyramidal numbers?

4. A story is often told that Carl Frederick Gauss astonished his primary teacher when he quickly found the sum of 1 + 2 + 3 + … + 100. As the story goes, he supposedly recognized that 1 + 100 = 101; 2 + 99 = 101; 3 + 98 = 101; and so on. There are 50 pairs that sum to 101, so the total sum is 50 × 101 = 5050. Even though the story may not be true, it provides an interesting setting for the patterns studied in this lesson. Have students research Gauss. What are his contributions to mathematics?

Questions for Students

1. How are triangular numbers similar related to the handshake problem? How are they different?

[The pattern is to multiply successive numbers and divide by 2; however, the handshake problem multiplies by the previous number, but the triangular numbers multiply by the next number.]

2. How do you know if a graph will be linear or not? How can you tell from a table of values? How can you tell from an equation?

[A curve will not have a constant rate of change from term to term in a table, but a straight line will. With an equation, if the highest power of the variable is greater than 1, a curve will result.]

3. What is the generalized form for the sum of the first n positive integers?

[n(n + 1) ÷ 2.]

Teacher Reflection

1. Were the students engaged when using the spreadsheets? How did the use of technology enhance this lesson?
2. What connections did students make between the various problems that were related? What connections did students make between the various representations?
3. How did this lesson address auditory, tactile and visual learning styles?

Supreme Court Welcome

6-8
Experience the triangular numbers in an interesting real‑world context of the Supreme Court.

Supreme Court Handshake

6-8
During this lesson, students will explore the handshake problem, a classic problem in mathematics that asks, "How many handshakes occur when n people shake hands with each other?" Groups work to determine how many handshakes take place among the nine Supreme Court justices, and then generalize to the number of handshakes in any size group. Students explore the problem using a verbal description, a table, a graph, a picture, and an algebraic formula.

Learning Objectives

Students will:

• Understand the connections among the handshake problem, the triangular numbers, and the sums of positive integers.
• Use formulas to extend patterns in a spreadsheet, and graph those relationships.

NCTM Standards and Expectations

• Represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules.
• Relate and compare different forms of representation for a relationship.
• Use symbolic algebra to represent situations and to solve problems, especially those that involve linear relationships.
• Model and solve contextualized problems using various representations, such as graphs, tables, and equations.