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Transcript
GEOMETRY: CHAPTER 1 Ch. 1.1 Use Inductive Reasoning Ex. 1 Describe the pattern in the numbers -4, -12, -36, -108, …and write the next three numbers in the pattern. Ex. 1 Describe the pattern in the numbers -4, -12, -36, -108, …and write the next three numbers in the pattern. Notice that each number in the pattern is three times the previous number -4, -12, -36, -108, -324, -972, -2916, … So, the next three numbers in the pattern are -324, -972, and -2916. Ex. 2 Describe the pattern in the numbers 4.01, 4.03, 4.05, 4.07, …Write the next three numbers in the pattern. Ex. 2 (cont.) Describe the pattern in the numbers 4.01, 4.03, 4.05, 4.07, …Write the next three numbers in the pattern. The numbers are increasing by 0.02. The next 3 numbers are: 4.09, 4.11, 4.13. Go to: http://www.classzone.com/cz/books/geometry_2007_ na/resources/applications/animations/g7_1_1.html for more questions about number patterns. Inductive Reasoning A conjecture is an unproven statement that is based on observations. You use inductive reasoning when you find a pattern in specific cases and then write a conjecture for the general case. Ex. 3: Make and Test a Conjecture Use the following sums of odd integers: 5 + 7 = 12, 1 + 3 = 4, 19 + 23 = 42 CONJECTURE: The sum of any two odd integers is . . . To show that this is true, I need to test the conjecture with other examples. 43 + 33 = 76, 67 + 1=68, 25 + 7 = 32 Disproving Conjectures To show that a conjecture is true, you must show that it is true for all cases. You can show that a conjecture is false, however, by finding one counterexample. A counterexample is a specific case for which the conjecture is false. Ex. 4. Find a counterexample. A student makes the following conjecture that the difference of two numbers is always less than the larger number. Conjecture: the difference of two numbers is always less than the larger number. Ex. 4 (Cont.) Solution To find a counterexample, you need to find a difference that is greater than the larger number. 7- (-5) = 12 12 > 7 The difference, 12, is greater than the larger number, 7. Thus, the conjecture stating that “the difference of two numbers is always less than the larger number” is false. Images used for this presentation came from the following websites: http://z.about.com/d/math/1/0/M/2/perpendic ular.jpg http://education.yahoo.com/homework_help/ math_help/solutionimages/minigeogt/2/1/1 /minigeogt_2_1_1_25_10/f-10-24-1.gif