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Holt McDougal Geometry 2-1 Using Inductive Reasoning to Make Conjectures Toolbox Pg. 77 (11-15; 17-22; 24-27; 38 why 4 )
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Holt McDougal Geometry 2-1 Using Inductive Reasoning to Make Conjectures How do you use inductive reasoning to identify patterns and make conjectures? How do you find counterexamples to disprove conjectures? Essential Questions
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Holt McDougal Geometry 2-1 Using Inductive Reasoning to Make Conjectures Find the next item in the pattern. Example 1A: Identifying a Pattern January, March, May,... The next month is July. Alternating months of the year make up the pattern.
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Holt McDougal Geometry 2-1 Using Inductive Reasoning to Make Conjectures Find the next item in the pattern. Example 1B: Identifying a Pattern 7, 14, 21, 28, … The next multiple is 35. Multiples of 7 make up the pattern.
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Holt McDougal Geometry 2-1 Using Inductive Reasoning to Make Conjectures Find the next item in the pattern. Example 1C: Identifying a Pattern In this pattern, the figure rotates 90° counter- clockwise each time. The next figure is.
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Holt McDougal Geometry 2-1 Using Inductive Reasoning to Make Conjectures When several examples form a pattern and you assume the pattern will continue, you are applying inductive reasoning. Inductive reasoning is the process of reasoning that a rule or statement is true because specific cases are true. You may use inductive reasoning to draw a conclusion from a pattern. A statement you believe to be true based on inductive reasoning is called a conjecture.
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Holt McDougal Geometry 2-1 Using Inductive Reasoning to Make Conjectures Complete the conjecture. Example 2: Making a Conjecture The sum of two positive numbers is ?. The sum of two positive numbers is positive. List some examples and look for a pattern. 1 + 1 = 23.14 + 0.01 = 3.15 3,900 + 1,000,017 = 1,003,917
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Holt McDougal Geometry 2-1 Using Inductive Reasoning to Make Conjectures Complete the conjecture. Example 2B: Making a Conjecture The number of lines formed by 4 points, no three of which are collinear, is ?. Draw four points. Make sure no three points are collinear. Count the number of lines formed: The number of lines formed by four points, no three of which are collinear, is 6.
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Holt McDougal Geometry 2-1 Using Inductive Reasoning to Make Conjectures Example 3: Biology Application The cloud of water leaving a whale’s blowhole when it exhales is called its blow. A biologist observed blue-whale blows of 25 ft, 29 ft, 27 ft, and 24 ft. Another biologist recorded humpback- whale blows of 8 ft, 7 ft, 8 ft, and 9 ft. Make a conjecture based on the data. Heights of Whale Blows Height of Blue-whale Blows25292724 Height of Humpback-whale Blows 8789
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Holt McDougal Geometry 2-1 Using Inductive Reasoning to Make Conjectures Example 3: Biology Application Continued The smallest blue-whale blow (24 ft) is almost three times higher than the greatest humpback- whale blow (9 ft). Possible conjectures: The height of a blue whale’s blow is about three times greater than a humpback whale’s blow. The height of a blue-whale’s blow is greater than a humpback whale’s blow.
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Holt McDougal Geometry 2-1 Using Inductive Reasoning to Make Conjectures To show that a conjecture is false, you have to find only one example in which the conjecture is not true. This case is called a counterexample. To show that a conjecture is always true, you must prove it. A counterexample can be a drawing, a statement, or a number.
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Holt McDougal Geometry 2-1 Using Inductive Reasoning to Make Conjectures Inductive Reasoning 1. Look for a pattern. 2. Make a conjecture. 3. Prove the conjecture or find a counterexample.
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Holt McDougal Geometry 2-1 Using Inductive Reasoning to Make Conjectures Show that the conjecture is false by finding a counterexample. Example 4A: Finding a Counterexample For every integer n, n 3 is positive. Pick integers and substitute them into the expression to see if the conjecture holds. Let n = 1. Since n 3 = 1 and 1 > 0, the conjecture holds. Let n = –3. Since n 3 = –27 and –27 0, the conjecture is false. n = –3 is a counterexample.
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Holt McDougal Geometry 2-1 Using Inductive Reasoning to Make Conjectures Show that the conjecture is false by finding a counterexample. Example 4B: Finding a Counterexample Two complementary angles are not congruent. If the two congruent angles both measure 45°, the conjecture is false. 45° + 45° = 90°
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Holt McDougal Geometry 2-1 Using Inductive Reasoning to Make Conjectures Summary D-What did we do? L-What did I learn? I-What was interesting? Q-What questions do I have?
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