1. Unit 1: The Art of Problem Solving Advanced General Mathematics

3. Ancient EgyptAncient Babylon The development of mathematics can be traced to the Egyptian and Babylonian cultures (3000 B.C. – A.D. 260) as a necessity to problem solving.

4. Ancient EgyptAncient Babylon To solve a problem or perform an operation, a cookbook-like recipe was given, and it was performed repeatedly to solve similar problems.

5. Ancient EgyptAncient Babylon By observing that a specific method worked for a certain type of problem, the Babylonians and Egyptians concluded that the same method would work for any similar type of problem.

6. Such a conclusion is called a conjecture. an educated guess based on repeated observations of a particular process or pattern Conjecture

7. This method of reasoning is called inductive reasoning. Inductive reasoning is characterized by drawing a general conclusion (making a conjecture) from repeated observations of specific examples. The conjecture may or may not be true. Inductive Reasoning

8. Ty went to Mr. Pizza on Tuesday to buy a slice of pizza for lunch. They had Ty’s favorite kind – Hawaiian – available in the pizza-by-the-slice rack. Ty decided to go to Mr. Pizza again on Wednesday, and once again, they had Hawaiian pizza available by-the-slice for lunch. Ty predicts that if he goes to Mr. Pizza again on Thursday, they will have Hawaiian available for pizza-by-the-slice.

9. In testing a conjecture obtained by inductive reasoning, it takes only one example that does not work to prove the conjecture is false. Such an example is called a counterexample. An example that proves a conjecture is false. Counterexample

10. Use inductive reasoning to predict the next term in the sequence of numbers. 1, 1, 2, 3, 5, Write a conjecture stating the pattern of the above sequence of numbers.

11. Use inductive reasoning to predict the next multiplication fact in the list of equations. 37 × 3 = 111 37 × 6 = 222 37 × 9 = 333 37 × 12 = 444

12. Use inductive reasoning to predict the next term in the sequence of numbers. 2, 9, 16, 23, 30, Inductive reasoning does not guarantee a true result, but it does provide means of making a conjecture.

13. Greek mathematicians such as Archimedes, Pythagoras, and Euclid developed deductive reasoning during the classical Greek period (600 B.C. – A.D. 450), where general concepts were applied to specific problems, resulting in a structured, logical development of mathematics. Euclid Pythagoras Inductive reasoning is useful, but since there is no assurance that any conjecture will always be true, mathematicians are reluctant to accept a conjecture as an absolute truth until it is formally proven using methods of deductive reasoning. Archimedes

14. Deductive reasoning is characterized by applying general principles to specific examples. Deductive Reasoning

15. To graduate from Chatfield High School, a student must have four math credits. Each of you needs four math credits in order to graduate from CHS. If you lift weights, you will get stronger. You lift weights. Therefore, you’ll get stronger. If the digits in a number add up to a multiple of 9, then that number is itself a multiple of 9. Therefore, 5,751 is a multiple of 9, since 5 + 7 + 5 + 1 = 18, and 18 is a multiple of 9.