Lesson 2
1. Lesson 2
1.5. Explore 5
Module 2: Logic and Geometry
In Lesson 1 you may have found that you had to adapt or revise your personal strategy for winning a game if your original strategy was not successful. In a similar manner, counterexamples tell us that a conjecture or strategy was not successful. When a counterexample is found, a conjecture may be revised or changed to limit it.
For instance, Vicky had made the following conjecture based on her pillow sketch.
Vicky’s conjecture is valid for 5 or less points. A counterexample was found for 6 points. This showed that the conjecture was false once there were 6 or more points on the circle. Vicky could revise her conjecture by limiting it to 5 or less points on the circumference of the circle.
If you want to see another example of using reasoning to find a counterexample to a conjecture, read “Example 3: Using reasoning to find a counterexample to a conjecture” on page 21 of your textbook.
In 1959, Arthur L. Samuel of the IBM research department programmed an IBM 704 computer to play a fair game of checkers with an opponent. The computer was capable of looking over its past games and modifying its strategy based on its experiences. It was one of the first learning machines that could steadily improve its ability to play games.
— Martin Gardner. The Colossal Book of Mathematics. (New York: W. W. Norton, 2001), pp. 471-472.
The process of making a conjecture and then revising it based on new information is characteristic of inductive reasoning. The following Try This activity provides you with an opportunity to play a game and make conjectures. Your conjectures can then be adapted based on the evidence you collect as the game progresses.
If you do not have the opportunity for group activities, you can use the Self-Check questions presented after Try This 2 to verify your understanding of inductive reasoning. Check with your teacher about what piece or pieces you should complete—the Try This 2 and/or the Self-Check 2 questions.
Try This 2
Play the I Like Game. Read over the instructions before playing this game with a partner or with a group.
After you have played a few rounds of the game, answer the following: How does the game model inductive reasoning?
In your response, identify how conjectures were used. Also, describe how finding a counterexample changed your strategy or revised your original conjecture.
Self-Check 2
Remember to check with your teacher about whether you are to complete Try This 2 and/or the Self-Check 2 questions for this portion of the lesson.
Complete “Further Your Understanding” question 2 on page 17 of your textbook. Answer