CHAPTER 1TOOLS OF GEOMETRYSection 1.1 Patterns and Inductive Reasoning

INDUCTIVE REASONING: REASONING BASED ON PATTERNS YOU OBSERVE

Finding and Using a Pattern Examples: Find the next two terms of the sequence

• 3, 6, 12, 24,… Each term is being multiplied by 2 Next two terms: 48, 96

• 1, 2, 4, 7, 11, 16, 22, … Adding another 1 to the previous Next two terms: 29, 37

EXAMPLES (CONT.)

Monday, Tuesday, Wednesday… Thursday, Friday

CONJECTURE: A CONCLUSION YOU REACH USING INDUCTIVE REASONING

Make a conjecture about the sum of the first 30 odd numbers• 1 = 1• 1 + 3 = 4• 1 + 3 + 5 = 9• 1 + 3 + 5 + 7 = 16• 1 + 3 + 5 + 7 + 9 = 25• 1 + 3 + 5 + 7 + 9 + 11 = 36

Notice the pattern created (number of odd numbers, squared) The sum of the first 30 odd numbers would

be 302

= 12

.

= 22

.

= 32

= 42

.

= 52

.

= 62

EXAMPLES (CONT.)

Make a conjecture about the sum of the first 35 odd numbers• Since the sum of the first 30 odd numbers is 302,

then the sum of the first 35 odd numbers would be: 352

352 = 35 * 35, so:

TRUE OR FALSE?

If a statement is true, we can prove it is true for all cases

If a statement is false, we need to provide ONE counterexample.

A counterexample is a specific example for which the conjecture is false

EXAMPLES: FIND A COUNTEREXAMPLE FOR EACH CONJECTURE

The square of any number is greater than the original number.

Counterexample: Is

You can connect any three points to form a triangle

Counterexample: a straight line

EXAMPLES (CONT)

ANY number and its absolute value are opposites.

Counterexample: A positive number and its absolute value

ie: and

Alana makes a conjecture about slicing pizza. She says that if you use only straight cuts, the number of pieces will be twice the number of cuts. Draw an example that shows you can make 7 pieces using 3 cuts.

Classwork: pg. 6 # 1 – 6, 8, 11, 12, 19 – 22, 25 - 29

Homework: WS 1-1 #1-6, 13, 14