Presentation on theme: "Using Inductive Reasoning to Make Conjectures 2-1"— Presentation transcript:
1 Using Inductive Reasoning to Make Conjectures 2-1 Warm UpLesson PresentationLesson QuizHolt Geometry
2 Complete each sentence. Are you ready?Complete each sentence.1. ? points are points that lie on the same line.2. ? points are points that lie in the same plane.3. The sum of the measures of two ? angles is 90°.
3 ObjectivesTSW use inductive reasoning to identify patterns and make conjectures.TSW find counterexamples to disprove conjectures.
4 Biologists use inductive reasoning to develop theories about migration patterns.
6 Example 1: Identifying a Pattern Find the next item in the pattern.January, March, May, ...
7 Example 2: Identifying a Pattern Find the next item in the pattern.7, 14, 21, 28, …
8 Example 3: Identifying a Pattern Find the next item in the pattern.
9 Example 4Find the next item in the pattern 0.4, 0.04, 0.004, …
10 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.
11 Example 5: Making a Conjecture Complete the conjecture.The sum of two positive numbers is ? .
12 Example 6: Making a Conjecture Complete the conjecture.The number of lines formed by 4 points, no three of which are collinear, is ? .
13 Example 7Complete the conjecture.The product of two odd numbers is ? .
14 Example 8: 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 BlowsHeight of Blue-whale Blows25292724Height of Humpback-whale Blows879
15 Example 9Make a conjecture about the lengths of male and female whales based on the data.Average Whale LengthsLength of Female (ft)4951504847Length of Male (ft)454446
16 To show that a conjecture is always true, you must prove it. 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.A counterexample can be a drawing, a statement, or a number.
17 Inductive Reasoning1. Look for a pattern.2. Make a conjecture.3. Prove the conjecture or find a counterexample.
18 Example 10: Finding a Counterexample Show that the conjecture is false by finding a counterexample.For every integer n, n3 is positive.
19 Example 11: Finding a Counterexample Show that the conjecture is false by finding a counterexample.Two complementary angles are not congruent.
20 Monthly High Temperatures (ºF) in Abilene, Texas Example 12: Finding a CounterexampleShow that the conjecture is false by finding a counterexample.The monthly high temperature in Abilene is never below 90°F for two months in a row.Monthly High Temperatures (ºF) in Abilene, TexasJanFebMarAprMayJunJulAugSepOctNovDec8889979910710911010610392
21 Example 13Show that the conjecture is false by finding a counterexample.For any real number x, x2 ≥ x.
22 Example 14Show that the conjecture is false by finding a counterexample.Supplementary angles are adjacent.
23 Planets’ Diameters (km) Example 15Show that the conjecture is false by finding a counterexample.The radius of every planet in the solar system is less than 50,000 km.Planets’ Diameters (km)MercuryVenusEarthMarsJupiterSaturnUranusNeptunePluto488012,10012,8006790143,000121,00051,10049,5002300
25 Lesson QuizFind the next item in each pattern.1. 0.7, 0.07, 0.007, … 2.0.0007Determine if each conjecture is true. If false, give a counterexample.3. The quotient of two negative numbers is a positive number.4. Every prime number is odd.5. Two supplementary angles are not congruent.6. The square of an odd integer is odd.truefalse; 2false; 90° and 90°true