Notes 1.1 – Representing Number Patterns

I. Problem10 people in a room are instructed to shake hands

with each other. How many handshakes would be required for each person to shake hands with each person 1 time?

Ideas?

Let’s start with a smaller number of people and work our way up to 10.

People

Shakes

1 2 3 4 50 1 3 6 10

See a pattern developing?

+1 +2 +3 +4

Why?

Every time someone new enters the room, everyone else must shake their hand.

People

Shakes

6 7 8 9 1015 21 28 36 45

+6 +7 +8 +9+5

II. Making a ConjectureA ____________is a statement based on

observation that is believed to be _______.

What would your conjecture be? Write it down now.

conjectureTRUE

The total number of handshakes among any number of students is the sum of the numbers from 1 to one less than the number of people in the room.

Almost exactly like the handshake situation, right?

III. Your Turn…Example 1 – A football conference consists of 8

teams. How many games will be played in a season if each team is to play every other team in the conference exactly one time?

1+2+3+4+5+6+7= 28!!!

The shortcut…

Notice that this is a ________ of what we call an __________ _________– a pattern of numbers that increases by a constant sum. There is a mathematical formula for this.

first number + last numbersum= the number of terms

2

4 7 28 8

72

1 + 7

72

sumarithmetic sequence

sum=

… means the sequence continues

IV. Sequences

A ____________ is any arrangement of numbers that shows a pattern

Example -

# term 1 2 3 4 5 6 …

Term 0 1 3 6 10 15 …

SEQUENCE

V. Examples

A.) 13,16,19,22,... B.) 1,3,7,13,21,...

25,28,31 31,43,57

2, 4, 6, 8, 10, 12, 14 3, 3, 3,...

Pattern :

Predict the next three terms of each sequence.

Pattern :

VI. Rectangular and Triangular Numbers

Determine how many dots would be in the next figure.

___________ __________are a sequence of numbers that can be visualized as a ______________.Rectangular Numbers

RECTANGLE

So…the next rectangle would be…

1x2 2x3 3x4 4x5

Determine how many dots would be in the next figure.

1x2 ÷2 2x3÷2

3x4÷2 4x5÷2

Triangular Numbers: Numbers that can be visualized by a _____________.TRIANGLE

Notice how the triangular numbers are the same numbers from the handshake problem.

Remember when we added the numbers all the way up to one less than the number of people?

1,3,6,10,...

0+1+2+3+4+5+6+7+8+9=45

This gives us another way to find the sum of a set of consecutive numbers!!!

First, find the __________ _______ associated with the term, and then ________ it by ____.

rectangular numberdivide 2

For example, for a room full of 10 people or the sum of

Is the triangular number for 9.

Find the rectangular number for 9 and divide it by 2.

9 10 2=45

1+2+3+4+5+6+7+8+9

Your turn…

1.) Find the 15th triangular number.

2.) Find the sum of the numbers 1 through 100.

3.) How many handshakes would take place between 20 people?

20 19 2=190

15 16 2=120

100 101 2=5050