1. Notes 1.1 – Representing Number Patterns

2. I. Problem
10 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?

3. Ideas?
Let’s start with a smaller number of people and work our way up to 10.
See a pattern developing?

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

5. II. Making a Conjecture
A ____________is a statement based on observation that is believed to be _______.
What would your conjecture be? Write it down now.
conjecture
TRUE
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.

6. 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?
Almost exactly like the handshake situation, right?

7. 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.
sum
arithmetic
sequence

8. IV. Sequences
A ____________ is any arrangement of numbers that shows a pattern
SEQUENCE
Example -
# term 1 2 3 4 5 6 …
Term 10 15
… means the sequence continues

9. V. Examples
Predict the next three terms of each sequence.

10. VI. Rectangular and Triangular Numbers
___________ __________are a sequence of numbers
that can be visualized as a ______________.
Rectangular Numbers
RECTANGLE
Determine how many dots would be in the next figure.

11. So…the next rectangle would be…2 6 12 20
So…the next rectangle would be…
5x6=30

12. Triangular Numbers: Numbers that can be visualized by a _____________.
TRIANGLE
2x3÷2
1x2 ÷2
3x4÷2
4x5÷2 1 3 6 10
Determine how many dots would be in the next figure.
5x6÷2=15

13. 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?

14. 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 number
divide 2

15. 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.

16. 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?