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Transcript
Miss Stanley The Green School Geometry Fall 2011 Name _____________________________ Date ______________________________ Lesson 1.1 Patterns and Inductive Reasoning Finding and using a pattern. 3, 6, 12, 24… Triangle, Rectangle, Pentagon… What is the pattern? What is the pattern? What are the next three numbers in the sequence? What are the next three figures in the sequence? A conjecture is a conclusion you reach using inductive reasoning. Use the pattern to find the sum of consecutive odd numbers. 1 1+3 1+3+5 1+3+5+7 1+3+5+7+9 =1 =4 =9 = 16 = = 12 = 22 = 32 = 42 = Now use this same pattern to find the sum of the first 30 odd numbers. 1 + 3 + 5 + 7 + 9… First 30 odd numbers. = = A counterexample is an example where the conjecture is incorrect. Find a counterexample: The square of any number is greater than the original number. You can connect any three points to form a triangle. Any number and its absolute value are opposites. Classwork: Complete the following problems by the end of the class period. Find the next three numbers in the sequence. 3, 33, 333, 3333, … 81, 27, 9, 3, … 1 1 1 1 1, , , , ... 4 9 16 25 1, 1, 2, 3, 5, 8, 13, … Use the table to find the following sums. 2 2+4 2+4+6 2+4+6+8 2 + 4 + 6 + 8 + 10 =1•2 =2•3 =3•4 =4•5 =5•6 =2 =6 = 12 = 20 = 30 Find the sum of the first 6 positive even numbers. Find the sum of the first 30 positive even numbers. What conjecture can you make about the sum of the first 100 even numbers? In other words, what rule are you following? Find one counterexample to show that each conjecture is false? The sum of two numbers is greater than either number. The product of two positive numbers is greater than either number. The difference of two integers is less than either integer. The quotient of two proper fractions is a proper fraction.