2.1 Inductive Reasoning Inductive Reasoning: is the process of forming a conjecture based on a set of observations One of the most important things in mathematics…one of the most important things mathematics can teach us is that of searching for patterns. This is really the fundamental strength of mathematical thinking….Thinking about things in context, how things follow, how they fit in, and then once we have firm understanding of that, trying to then predict what is going to come next and looking for patterns. (In mathematics this is known as inductive reasoning) So the idea of looking at evidence and trying to use that evidence to figure out what a pattern is, what will come next, predicting what will happen in the future. This is such a powerful tool and really something that mathematics can offer us if we embrace it, even in our own daily lives. Forget about geometry, which is real easy to do. I will show you induction with some simple examples Conjecture is a statement that is believed to be true but not yet proved.

Ex. 1 Use inductive reasoning to form a conjecture 1, 2, 3, 4, 5, 6, 7, …. *Purple is your given *Green is your guess of what comes next Let’s look at an example…with a sequence of number. 1,2,3,4 what is going to come next? It could be anything 5, 90? Taking the previous number and adding one… these are successive numbers so you would guess 5,6,7,8,…. That is inductive reasoning …..that is inductive reasoning looking at the evidence to find a pattern and using that evidence to predict what is going to come next. Image the power of this in your everyday life…if you can really harness this….the reality actually is… the truth is math is much easier then life…it’s easier to predict what is going to come next here and much harder to predict if the person who your going out with is going to go out with you in a month….if you are really good at inductive reasoning you would know this. Conjecture The sequence increases by one each time Lesson 1-1 Point, Line, Plane

Ex. 2 Inductive Reasoning , 30, 35, 40, 45, …. 5, 10, 15, 20, 25 Conjecture 1: The sequence goes up by five each time. Conjecture 2: ends in a 5, then a 0, then a 5, and so on…and all the numbers in the tens place appear twice in the sequence. Here is another example…. 5, 10, 15, 20, 25 1st pattern: Consecutive multiples of 5… so 30,35,40 2nd pattern: 5, 0 , 5, 0 and all the counting numbers appear twice that is another pattern 1 and 1, 2 and 2, and then 3 and 3 (that is another pattern) if you would use this pattern you would again predict just like with the multiple of 5 …..30,35,40, 45 and so on Inductive reasoning draws on any and all patterns Lesson 1-1 Point, Line, Plane

Ex. 3 Inductive Reasoning Conjecture 1: every term in the sequence is rotated counterclockwise 90 degrees. Let’s look at a more abstract example…one that is a little more geometric I am going to create something here….. a sequence…remember…patterns don’t always have to be numbers…. in fact in our livess patterns are not usually numbers Blob is on the left….then…then….turning 90 degrees counter clockwise Inductive reasoning Conjecture 2: continuously repeats the four positions of right, bottom, left, top, right, bottom, left, top, … Lesson 1-1 Point, Line, Plane

Lesson 1-1 Point, Line, Plane Ex 4 Describe how to sketch the fourth figure in the pattern. Then sketch the fourth figure. Conjecture: ?? Lesson 1-1 Point, Line, Plane

Lesson 1-1 Point, Line, Plane EXAMPLE I Describe the pattern in the numbers –7, –21, –63, –189,… and write the next three numbers in the pattern. Lesson 1-1 Point, Line, Plane

Lesson 1-1 Point, Line, Plane EXAMPLE II Numbers such as 3, 4, and 5 are called consecutive integers. Make and test a conjecture about the sum of any three consecutive integers using inductive reasoning. Hint: We must first gather data before we make predictions Lesson 1-1 Point, Line, Plane

Lesson 1-1 Point, Line, Plane EXAMPLE IV Make and test a conjecture about the sign of the product of any three negative integers. Hint: We must first gather data before we make predictions Lesson 1-1 Point, Line, Plane

Counterexamples in Real Life All birds can fly. A basketball player must be tall in order to be good at dunking baskets. Students with low grade-point averages in high school do not contribute to the academic community. That is, they are devices used in debating or argumentation to offer rebuttal to proposed generalizations or definitions. Students with low grade-point averages in high school, like Albert Einstein, sometimes become famous scientists. Lesson 1-1 Point, Line, Plane

Counterexample in Math A counter example in math is an example for which the conjecture is false. ** It is one number or one picture or one set of numbers….it is not a written reason!!! * It is one number or one picture or one set of numbers….it is NOT a written reason!!! Lesson 1-1 Point, Line, Plane

EXAMPLE A – Counter Examples Conjecture: The sum of two numbers is always greater than the larger number. Are they any counterexamples that exist to disprove this conjecture? Lesson 1-1 Point, Line, Plane

EXAMPLE V – Counter Examples Find a counterexample to show that the following conjecture is false. Conjecture: The value of x is always greater than the value of x. 2 Lesson 1-1 Point, Line, Plane

2.3 Apply Deductive Reasoning Deductive Reasoning uses facts, definitions, and the laws of logic to form a logical argument. This is different from inductive reasoning, which uses specific examples and patterns to form a conjecture.

Lesson 1-1 Point, Line, Plane Laws of Logic Law of Detachment If the hypothesis of a conditional statement is true, then the conclusion is also true. Law of Syllogism If A, then B. If B, then C. If A, then C Ex 2. If x >25, then x >20 If x>5, then x >25. 2 Ex. 1 If Rick takes Chemistry this year, then Jesse will be Rick’s partner. If Jess is Rick’s lab partner, then Rick will get an A in chemistry. Ex. 1 If two segments have the same length, then they are congruent. You know that BC =XY Ex.2 Mary goes to the movies every Friday and Saturday night. Today is Friday Lesson 1-1 Point, Line, Plane

Logic Puzzle (Nicknames) Four friends: Dave, Mike, John, and Terry, are nicknamed Stick, Batman, Atomic Head, and Feaser, but not in that order. Which friend has which nickname? A. John is faster than Batman but not as strong as Atomic Head. B. Batman is stronger than Terry but slower than Feaser. C. Dave is faster than both Stick and John, but not as strong as Batman Strength and speed are independent qualities. Dave Mike John Terry Stick Batman Atomic Head Feaser x x x x x x x x x x x x Lesson 1-1 Point, Line, Plane