Geometry Inductive Reasoning and Conjecturing Section 2.1

Inductive Reasoning and Conjecturing 10-Aug-15…\GeoSec02_01.ppt2 Inductive Reasoning - Reasoning that uses a number of specific examples to arrive at a plausible generalization or prediction. You use inductive reasoning when you “see” or “notice” a pattern in a sequence of numbers, in a painted object, or in the behavior of someone or something. Once we recognize a pattern we can make a conjecture related to the pattern behavior and determine if a conclusion is possible. Conjecture - An educated guess. For example: given the sequence 1, 3, 5, 7, … we can make a conjecture that the next three numbers in the sequence is 9, 11, 13, …

Inductive Reasoning and Conjecturing 10-Aug-15…\GeoSec02_01.ppt3 Conclusions arrived at by inductive reasoning lack the logical certainty (which means it may not be true all the time) as those arrived at by deductive reasoning. Sherlock Holmes uses deductive reasoning when trying to determine “who done it?” Mathematicians use inductive reasoning when they notice patterns in numbers, nature, or something’s behavior. They set up specific examples that explore those patterns. Once they have a good idea how the pattern works, they then make a conjecture and try to generalize that conjecture. This generalization uses deductive reasoning to arrive at a mathematical proof of the patterns they explored.

Inductive Reasoning and Conjecturing 10-Aug-15…\GeoSec02_01.ppt4 Given points A, B, and C, AB AB = 10, BC BC = 8, AC AC = 5, is there a conjecture we can make? First, draw the figure and make a conjecture. Is it possible for C to be between A and B? AB10 CB 8 AC5 AB C 8 5 So what is your conjecture?

Inductive Reasoning and Conjecturing 10-Aug-15…\GeoSec02_01.ppt5 Given that points P, Q, and R are collinear. What kind of conjecture could you make about which point is between which points? How about Q is between P and R? Is this conjecture true or false? Suppose it is true, could the following also be true? Have I violated any of the given with either of the examples? PRQ PQR The answer is no. Our second figure, which contradicts the first figure, is called a counterexample. A counterexample is a false or contradictory example that disproves a conjecture,

Inductive Reasoning and Conjecturing 10-Aug-15…\GeoSec02_01.ppt6 Given: D O F E G (3p + 24) o (5p - 4) o (3p + 24) o + (5p - 4) o = 180 o 3p p - 4 = 180 8p + 20 = 180 8p = p = p = 20  DOF = (3p + 24) o = (3 x ) o = ( ) o = 84 o  DOG = (5p - 4) o = (5 x ) o = ( ) o = 96 o Find  DOF and  DOG.

Inductive Reasoning and Conjecturing 10-Aug-15…\GeoSec02_01.ppt7 Suppose  MON is a right angle and L is in the interior of  MON. If m  MOL is 5 times m  LON, find m  LON. M ON L 90 =  MON +  LON = 5(  LON) +  LON = 6 (  LON) =  LON 15 =  LON 90 6

Inductive Reasoning and Conjecturing 10-Aug-15…\GeoSec02_01.ppt8 Summary Through inductive reasoning, we take specific examples of some process and use it to find general patterns within the boundaries of the process. From these general patterns, we can declare a conjecture based on the patterns. Conjectures derived from inductive reasoning are not always true, but they can be the basis for a strategy to use in deductive reasoning. Then we can test the conjecture to determine if the process is true or false.

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