1.2 Inductive Reasoning

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1 Page 1 of Inductive Reasoning Goal Use inductive reasoning to make conjectures. Key Words conjecture inductive reasoning counterexample Scientists and mathematicians look for patterns and try to draw conclusions from them. A conjecture is an unproven statement that is based on a pattern or observation. Looking for patterns and making conjectures is part of a process called inductive reasoning. Geo-Activity Making a Conjecture Work in a group of five. You will count how many ways various numbers of people can shake hands. 2 people can shake 3 people can shake hands 1 way. hands 3 ways. Student Help STUDY TIP Copy this table in your notebook and complete it. Do not write in your textbook. 1 Count the number of ways that 4 people can shake hands. 2 Count the number of ways that 5 people can shake hands. 3 Organize your results in a table like the one below. People Handshakes 1 3?? 4 Look for a pattern in the table. Write a conjecture about the number of ways that 6 people can shake hands. Much of the reasoning in geometry consists of three stages. 1 Look for a Pattern Look at several examples. Use diagrams and tables to help discover a pattern. 2 Make a Conjecture Use the examples to make a general conjecture. Modify it, if necessary. 3 Verify the Conjecture Use logical reasoning to verify that the conjecture is true in all cases. (You will do this in later chapters.) 8 Chapter 1 Basics of Geometry

2 Page 2 of 6 Science 1 Make a Conjecture Complete the conjecture. Conjecture: The sum of any two odd numbers is?. Begin by writing several examples Each sum is even. You can make the following conjecture. ANSWER The sum of any two odd numbers is even. SCIENTISTS use inductive reasoning as part of the scientific method. They make observations, look for patterns, and develop conjectures (hypotheses) that can be tested. 2 Make a Conjecture Complete the conjecture. Conjecture: The sum of the first n odd positive integers is?. List some examples and look for a pattern ANSWER The sum of the first n odd positive integers is n 2. Make a Conjecture Student Help READING TIP Recall that the symbols and p are two ways of expressing multiplication. Complete the conjecture based on the pattern in the examples. 1. Conjecture: The product of any two odd numbers is?. S Conjecture: The product of the numbers (n 1) and (n 1) is?. S 1 p p p p p p Inductive Reasoning 9

3 Page 3 of 6 IStudent Help I C L A S S Z O N E. C O M MORE S More examples at classzone.com Counterexamples Just because something is true for several examples does not prove that it is true in general. To prove a conjecture is true, you need to prove it is true in all cases. A conjecture is considered false if it is not always true. To prove a conjecture is false, you need to find only one counterexample. A counterexample is an example that shows a conjecture is false. 3 Find a Counterexample Conjecture: The sum of two numbers is always greater than the larger of the two numbers. Here is a counterexample. Let the two numbers be 0 and 3. The sum is 0 3 3, but 3 is not greater than 3. ANSWER The conjecture is false. 4 Find a Counterexample Conjecture: All shapes with four sides the same length are squares. Here are some counterexamples. These shapes have four sides the same length, but they are not squares. ANSWER The conjecture is false. Find a Counterexample 3. If the product of two numbers is even, the numbers must be even. 4. If a shape has two sides the same length, it must be a rectangle. 10 Chapter 1 Basics of Geometry

4 Page 4 of Exercises Guided Practice Vocabulary Check 1. Explain what a conjecture is. 2. How can you prove that a conjecture is false? Skill Check Complete the conjecture with odd or even. 3. Conjecture: The difference of any two odd numbers is?. 4. Conjecture: The sum of an odd number and an even number is?. 5. Any number divisible by 2 is divisible by The difference of two numbers is less than the greater number. 7. A circle can always be drawn around a four-sided shape so that it touches all four corners of the shape. Practice and Applications Extra Practice See p Rectangular Numbers The dot patterns form rectangles with a length that is one more than the width. Draw the next two figures to find the next two rectangular numbers Triangular Numbers The dot patterns form triangles. Draw the next two figures to find the next two triangular numbers Homework Help Example 1: Exs Example 2: Exs Example 3: Exs Example 4: Exs Technology Use a calculator to explore the pattern. Write a conjecture based on what you observe ? ? ? ? ? 33 34? ? ? ? 1.2 Inductive Reasoning 11

5 Page 5 of 6 IStudent Help I C L A S S Z O N E. C O M HOMEWORK HELP Extra help with problem solving in Exs is at classzone.com Making Conjectures Complete the conjecture based on the pattern you observe. 13. Conjecture: The product of an odd number and an even number is?. 3 p p p p p p 63 16, p p ( 4) Conjecture: The sum of three consecutive integers is always three times? p p p p p p p p p Counting Diagonals In the shapes below, the diagonals are shown in blue. Write a conjecture about the number of diagonals of the next two shapes. Science sun Moon Phases 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. Suppose that one year, full moons occur on these dates: 5 9 Earth s orbit Earth moon moon s orbit T F S S M T 1 2 W July 21 Thursday August 19 Friday S M T W T F S September 18 Sunday T F S S M T W October 17 Monday S M T W T F S November 16 Wednesday F S S M T 1 W T December 15 Thursday S M T W T F S Not drawn to scale FULL MOONS happen when Earth is between the moon and the sun. Application Links C L A S S Z O N E. C O M Determine how many days are between these full moons and predict when the next full moon occurs. Error Analysis Show the conjecture is false by finding a counterexample. 17. Conjecture: If the product of two numbers is positive, then the two numbers must both be positive. 18. Conjecture: All rectangles with a perimeter of 20 feet have the same area. Note: Perimeter 2(length width). 19. Conjecture: If two sides of a triangle are the same length, then the third side must be shorter than either of those sides. 12 Chapter 1 Basics of Geometry

6 Page 6 of Telephone Keypad Write a conjecture about the letters you expect on the next telephone key. Look at a telephone to see whether your conjecture is correct. 21. Challenge Prove the conjecture below by writing a variable statement and using algebra. Conjecture: The sum of five consecutive integers is always divisible by five. x (x 1) (x 2) (x 3) (x 4)? Standardized Test Practice 22. Multiple Choice Which of the following is a counterexample of the conjecture below? Conjecture: The product of two positive numbers is always greater than either number. A 2, 2 B 1 2, 2 C 3, 10 D 2, Multiple Choice You fold a large piece of paper in half four times, then unfold it. If you cut along the fold lines, how many identical rectangles will you make? F 4 G 8 H 16 J 32 Mixed Review Describing Patterns Sketch the next figure you expect in the pattern. (Lesson 1.1) Y X Z 2 3 Algebra Skills Using Integers Evaluate. (Skills Review, p. 663) ( 3) ( 1) ( 5) 31. ( 2)( 7) ( 5) 33. ( 8) ( 2) Evaluating Expressions Evaluate the expression when x 3. (Skills Review, p. 670) 34. x x 36. x x x x 2 4x 40. 3x x Inductive Reasoning 13