Lesson 2.3.pptx jahahajjaammahdvevhsbebshjanama

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1. Inductive and Deductive Reasoning Math11n: Mathematics in the Modern World

2. Learning outcomes: At the end of this lesson, the students will be able to: Differentiate inductive reasoning and deductive reasoning. Give examples of inductive and deductive processes of inference; and Solve practical problems using either inductive or deductive reasoning.

3. There were two detectives investigating a case. Detective A walked into a house and saw muddy shoes, wet umbrellas , and puddles of water . He thought, “It must have rained today.” Detective B, on the other hand, remembered the weather forecast : “Whenever the forecast says there is a storm, heavy rain always follows.” He said: “Since the forecast predicted a storm, the rain must have caused the wet floor.”

4. Are you more like Detective A, who observes patterns and makes predictions? Or Like Detective B, who relies on rules, laws, or principles to ensure our conclusions? Lessons: Both detectives were correct, but they used different paths of reasoning . Learning how to use both makes us better problem-solvers in school, at work, and in everyday life.

5. What is Reasoning?

6. What is Reasoning? Reasoning is a process of drawing conclusions based on evidence or principles. Two main types: Inductive reasoning Deductive reasoning Both are essential in problem-solving, research, and decision-making.

7. Inductive vs. Deductive Inductive Reasoning is used to describe reasoning that involves using specific observations, such as observed patterns, to make a general conclusion. Moves from specific observations → general conclusion Pattern-based reasoning Conclusions are probable, not certain

8. You go to a cafe every day for a month, and every day, the same person comes at exactly 11 am and orders a cappuccino. Conclusion: This person always comes to the cafe at the same time and orders the same thing. E xample:

9. Guess a formula for the sum of the first n odd numbers. Conclusion: By inductive reasoning, we infer that the sum of the first n odd numbers is always .   E xample:

10. Inductive vs. Deductive Deductive reasoning (also called deduction) involves starting from a set of general premises and then drawing a specific conclusion that contains no more information than the premises themselves. Moves from general principle/law → specific conclusion Logic-based reasoning Conclusions are certain if the premises are true

11. E xample: Chickens are birds; all birds lay eggs. Conclusion: Therefore, chickens lay eggs. All mammals are warm-blooded. A dolphin is a mammal. Conclusion: Therefore, a dolphin is warm-blooded.

12. E xample: Given: Rule (general statement): If a number is divisible by 4, then it is also divisible by 2. Case (specific fact): 24 is divisible by 4. Since 24 is divisible by 4, it must also be divisible by 2. Conclusion: Therefore, 24 is divisible by 2.

13. Logic Puzzle A Logic Puzzle is derived from the mathematics field of deduction. It is basically a description of an event or any situation. Using the clues provided, one has to piece together what actually happened. This involves clear and logical thinking, hence the term “Logic” puzzles.

14. Example: Three musicians appeared at a concert. Their last names were Benton, Lanier, and Rosario. Each plays only one of the following instruments: guitar, piano, or saxophone. 1. Benton and the guitar player arrived at the concert together. 2. The saxophone player performed before Benton. 3. Rosario wished the Guitar player good luck. Who played each instrument?

15. Solution: The solution can be summarized using a chart. From Clue 1, Benton is not the guitarist. We mark X1 (this means “ruled out by clue 1”), in the guitar column of Benton’s row. From clue 2, Benton does not play saxophone, hence he must be the pianist. From Clue 3, Rosario is not the guitar play. Hence, Rosario plays saxophone. This leaves Lanier as the guitar player.

16. Guitar Piano Saxophone Benton X1 Yes X2 Lanier Yes No No Rosario X3 No Yes

17. Inductive vs. Deductive Key Differences Feature Inductive Deductive Direction Specific → General General → Specific Conclusion Probable Certain Use Prediction, discovery Proof, application

18. Every math test so far has been difficult → Next test will also be difficult. The sun has risen every day → It will rise tomorrow. Maria always gets sick after eating shrimp → Shrimp might cause her an allergy. E xample: (Inductive)

19. All humans need water to survive → Juan is a human → Juan needs water. If it rains, the ground gets wet → It rained this morning → Ground is wet. All even numbers are divisible by 2 → 28 is even → 28 is divisible by 2. E xample: (Deductive)

20. THANK YOU!