Patterns and Inductive Reasoning Geometry

Objectives/Assignment: Find and describe patterns. Use inductive reasoning to make real-life conjectures.

Finding & Describing Patterns Geometry, like much of mathematics and science, developed when people began recognizing and describing patterns. In this course, you will study many amazing patterns that were discovered by people throughout history and all around the world. You will also learn how to recognize and describe patterns of your own. Sometimes, patterns allow you to make accurate predictions. This is called inductive reasoning.

Ex. 1: Describing a Visual Pattern Describe the next figure in the pattern. 4 5 1 2 3

Ex. 1: Describing a Visual Pattern - Solution The sixth figure in the pattern has 6 squares in the bottom row. 5 6

Ex. 1: Describing a Number Pattern Describe a pattern in the sequence of numbers. Predict the next number. 1, 4, 16, 64 Many times in a number pattern, it is easiest listing the numbers vertically rather than horizontally.

Ex. 1: Describing a Number Pattern Describe a pattern in the sequence of numbers. Predict the next number. 1 4 16 64 How do you get to the next number? That’s right. Each number is 4 times the previous number. So, the next number is 256, right!!!

Ex. 1: Describing a Number Pattern Describe a pattern in the sequence of numbers. Predict the next number. -5 -2 4 13 How do you get to the next number? That’s right. You add 3 to get to the next number, then 6, then 9. To find the fifth number, you add another multiple of 3 which is +12 or 25, That’s right!!!

Ex. 1: Describing a Number Pattern Describe a pattern in the sequence of numbers. Predict the next number. 5,10,20,40… 80

Ex. 1: Describing a Number Pattern Describe a pattern in the sequence of numbers. Predict the next number. 1, -1, 2, -2, 3… -3

Ex. 1: Describing a Number Pattern Describe a pattern in the sequence of numbers. Predict the next number. 15,12,9,6,… 3

Ex. 1: Describing a Number Pattern Describe a pattern in the sequence of numbers. Predict the next number. 1,2,6,24,120… 720

Ex. 1: Describing a Number Pattern Describe a pattern in the sequence. Predict the next letter. O,T,T,F,F,S,S,E,… 1,2,3,4,5,6,7,8, N 9

Goal 2: Using Inductive Reasoning Much of the reasoning you need in geometry consists of 3 stages: Look for a Pattern: Look at several examples. Use diagrams and tables to help discover a pattern. Make a Conjecture. Use the example to make a general conjecture. Okay, what is that? A conjecture is an unproven statement that is based on observations. Discuss the conjecture with others. Modify the conjecture, if necessary. 3. Verify the conjecture. Use logical reasoning to verify the conjecture is true IN ALL CASES. (You will do this in Chapter 2 and throughout the book).

Ex. 2: Making a Conjecture Complete the conjecture. Conjecture: The sum of the first n odd positive integers is ________?_________. How to proceed: List some specific examples and look for a pattern.

Ex. 2: Making a Conjecture =12 =22 =32 =42 First odd positive integer: 1 = 1 1 + 3 = 4 1 + 3 + 5 = 9 1 + 3 + 5 + 7 = 16 The sum of the first n odd positive integers is n2.

Ex. 2: Making a Conjecture First even positive integer: 2 = 2 2 + 4 = 6 2 + 4 + 6= 12 2 + 4 + 6 + 8 = 20 2 + 4 + 6 + 8 + 10 = 30 The sum of the first 6 even positive integers is what? = 1x2 = 2x3 = 3x4 = 4x5 = 5x6 6x7 = 42

Note: To prove that a conjecture is true, you need to prove it is true in all cases. To prove that a conjecture is false, you need to provide a single counter example. A counterexample is an example that shows a conjecture is false.

Ex. 3: Finding a counterexample Show the conjecture is false by finding a counterexample. Conjecture: For all real numbers x, the expressions x2 is greater than or equal to x.

Ex. 3: Finding a counterexample- Solution Conjecture: For all real numbers x, the expressions x2 is greater than or equal to x. The conjecture is false. Here is a counterexample: (0.5)2 = 0.25, and 0.25 is NOT greater than or equal to 0.5. In fact, any number between 0 and 1 is a counterexample.

Ex. 3: Finding a counterexample Conjecture: The sum of two numbers is greater than either number. The conjecture is false. Here is a counterexample: -2 + -5 = -7 -7 is NOT greater than -2 or -5. In fact, the sum of any two negative numbers is a counterexample.

Ex. 3: Finding a counterexample Conjecture: The difference of two integers is less than either integer. The conjecture is false. Here is a counterexample: -2 - -5 = 3 3 is NOT less than -2 or -5. In fact, the difference of any two negative numbers is a counterexample.

Ex. 4: Using Inductive Reasoning in Real-Life Moon cycles. A full moon occurs when the moon is on the opposite side of Earth from the sun. During a full moon, the moon appears as a complete circle.

Ex. 4: Using Inductive Reasoning in Real-Life Use inductive reasoning and the information below to make a conjecture about how often a full moon occurs. Specific cases: In 2005, the first six full moons occur on January 25, February 24, March 25, April 24, May 23 and June 22.

Ex. 4: Using Inductive Reasoning in Real-Life - Solution A full moon occurs every 29 or 30 days. This conjecture is true. The moon revolves around the Earth approximately every 29.5 days.

Ex. 4: Using Inductive Reasoning in Real-Life - NOTE Inductive reasoning is very important to the study of mathematics. You look for a pattern in specific cases and then you write a conjecture that you think describes the general case. Remember, though, that just because something is true for several specific cases does not prove that it is true in general.

Ex. 4: Using Inductive Reasoning in Real-Life Use inductive reasoning and the information below to make a conjecture about the temperature. The speed with which a cricket chirps is affected by the temperature. If a cricket chirps 20 times in 14 seconds, what is the temperature? 5 chirps 45 degrees 10 chirps 55 degrees 15 chirps 65 degrees The temperature would be 75 degrees.

Find a pattern for each sequence. Use the pattern to show the next two terms or figures. 1. 3, –6, 18, –72, 360 2. Use the table and inductive reasoning. 3. Find the sum of the first 10 counting numbers. 4. Find the sum of the first 1000 counting numbers. Show that the conjecture is false by finding one counterexample. 5. The sum of two prime numbers is an even number. –2160; 15,120 55 500,500 Sample: 2 + 3 = 5, and 5 is not even -1

Conditional Statements If…then…

Conditional Statements *A conditional statement is also known as an if-then statement. If you are not completely satisfied then your money will be refunded. Every conditional has two parts, the part following the if, called the hypothesis, and the part following then, which is the conclusion. If Texas won the 2006 Rose Bowl game, then Texas was college football’s 2005 national champion. Hypothesis – Texas won the 2006 Rose Bowl game. Conclusion – Texas was college football’s 2005 national champion.

Conditional Statements This also works with mathematical statements. If T – 38 = 3, then T = 41 Hypothesis: T – 38 = 3 Conclusion: T = 41

Conditional Statements Sometimes the “then” is not written, but it is still understood: If someone throws a brick at me, I can catch it and throw it back. Hypothesis: Someone throws a brick at me Conclusion: I can catch it and throw it back.

Name the hypothesis: If you want to be fit, then get plenty of exercise. If You want to be fit Then Get plenty of exercise If you can see the magic in a fairy tale, you can face the future. You can see the magic in a fairy tale You can face the future

Name the conclusion: “If you can accept a deal and open your pay envelope without feeling guilty, you’re stealing.” – George Allen, NFL Coach If You can accept a deal and open your pay envelope without feeling guilty Then You’re stealing

Name the conclusion: “If my fans think I can do everything I say I can do, then they are crazier than I am.” – Muhammed Ali If My fans think I can do everything I say I can Then They are crazier than I am

Writing a Conditional You can write many sentences as conditionals. A rectangle has four sides. If a figure is a rectangle, then it has four sides. A tiger is an animal. If something is a tiger, then it is an animal. An integer that ends with 0 is divisible by 5. A square has four congruent sides.

Truth Value A conditional has a truth value of either true or false. To show a conditional is true, show that every time the hypothesis is true, then the conclusion is true. If you can find one counterexample where the hypothesis is true, but the conclusion is false, then the conditional is considered false.

Counterexamples To show that a conditional is false, find one counterexample. If it is February, then there are only 28 days in the month. This is typically true, however, because during leap years February has 29 days, this makes the conditional false. If a state begins with New, then the state borders an ocean. This is true of New York, and New Hampshire, but New Mexico has no borders on the water, therefore this is a false conditional.

Using a Venn Diagram If you live in Chicago, then you live in Illinois. Residents of Chicago Residents of Illinois

Converse The converse of a conditional switches the hypothesis and the conclusion. Conditional: If two lines intersect to form right angles, then they are perpendicular. Converse: If two lines are perpendicular, then they intersect to form right angles If two lines are not parallel and do not intersect, then they are skew.

Finding the Truth of a Converse It is possible for a conditional and its converse to have two different truth values. If an animal is a chicken, then it is covered in feathers. If an animal is covered in feathers, then it is a chicken. Although the first conditional is true, as all chickens have feathers, the second is false, as chickens are not the only animal with feathers.

Finding the Truth of a Converse

Finding the Truth of a Converse