Section 1.1 Patterns & Inductive Reasoning: Section 1.1 Patterns & Inductive Reasoning Mrs. Adams Geometry ~ Fall 2007 Notetaking Guide p. 1-3


Slide2: Vocabulary Conjecture: an unproven statement that is based on observations Inductive reasoning: a process that includes looking for patterns and making conjectures


Slide3: Vocabulary Counterexample: an example that shows a conjecture is false


Slide4: Example 1: Describing a Visual Pattern Solution: Sketch the next figure in the pattern.


Slide5: Checkpoint #1 Solution: Sketch the next figure in the pattern.


Slide6: Example 2: Describing a Number Pattern Solution: Describe a pattern in the sequence of numbers. Predict the next number. a. 128, 64, 32, 16,… Each number is ________ the previous number. The next number is ___. one half 8


Slide7: Solution: Describe a pattern in the sequence of numbers. Predict the next number. b. 5, 4, 2, -1,… Subtract __ to get the 2nd #, then subtract __ to get the 3rd #, then subtract __ to get the 4th #. To find the 5th #, subtract __ from the 4th #. 4 3 2 1 Example 2: Describing a Number Pattern


Slide8: Solution: Describe a pattern in the sequence of numbers. Predict the next number. b. 5, 4, 2, -1,… So, the next number is ___ – __, or ___. -1 -5 4 Example 2: Describing a Number Pattern


Slide9: Checkpoint #2 Solution: Describe a pattern in the sequence of numbers. Predict the next number. 4, -20, 100, -500,… Each number is -5 times the previous number; 2500.


Slide10: Checkpoint #3 Solution: Describe a pattern in the sequence of numbers. Predict the next number. 10, 20, 40, 70, 110, … Numbers after the first are found by adding consecutive multiples of 10; 160.


Slide11: Example 3: Complete the conjecture. Solution: The sum of the first n even positive integers is ___. List some specific examples & look for a pattern. ? first even integer: 2 = 1(___) sum of first 2 even integers: 2+4 = ___= 2(___) sum of first 3 even integers: 2+4+6 = ___= 3(___) 2 6 3 12 4


Slide12: Example 3: Complete the conjecture. Solution: The sum of the first n even positive integers is ___. List some specific examples & look for a pattern. ? sum of first 4 even integers: 2+4+6+8= ___=4(___) Conjecture: The sum of the first n even positive integers is _________. 20 5 n (n + 1)


Slide13: Example 4: Finding a counterexample. Conjecture: Show the conjecture is false by finding a counterexample. If the difference of two numbers is odd, then the greater of the two numbers must also be odd. So, the conjecture is _____. false Solution: Counterexample: ___ – ___ = __ 8 3 5


Slide14: Checkpoint #5 A Possible Solution: Show the conjecture is false by finding a counterexample. Conjecture: The difference of two negative numbers is always negative. – 2 – (–6) = 4


Slide15: Assignment Textbook pp. 6 - 8 (12-24 even, 29-31 all, 34-38 even, 60-71 all)