Download to read offline
More Related Content
Similar to Inductive and Deductive Reasoning .pptx
Inductive and Deductive Reasoning .pptx
- 1. INDUCTIVE AND DEDUCTIVE REASONING Mathematics inthe Modern World
- 2. What is Reasoning? Reasoningis the process of using existing knowledge to draw conclusions, make predictions, or construct explanations. In mathematics, we use two primary types of logical reasoning: 1. Inductive Reasoning (Reasoning from specific examples to a general rule) 2. Deductive Reasoning (Reasoning from general rules to a specific conclusion)
- 3. Inductive Reasoning Making generalizationsbased on specific observations, patterns, and examples. • Process: Specific Examples Pattern → Recognition General Conclusion → • Nature: Probabilistic. The conclusion is likely but not certainly true. It's an educated guess.
- 4. USE INDUCTIVE REASONINGTO PREDICT A NUMBER 3, 6, 9, 12, 15, ? Each successive number is 3 larger than the preceding number. Thus, we predict that the next number in the list is 3 larger that 15, which is 18. Example:
- 5. USE INDUCTIVE REASONINGTO MAKE A CONJECTURE • Consider the following procedure: Pick a number. Multiply the number by 8, add 6 to the product, divide the sum by 2, and subtract 3. • Use inductive reasoning to make a conjecture about the relationship between the size of the resulting number and the size of the original number. Solution: Original Number: 5 Multiply by 8: 8 x 5 = 40 Add 6: 40 + 6 = 46 Divide by 2: 46 ÷ 2 = 23 Subtract 3: 23 – 3 = 20 We conjecture that the give procedures a number is four times the original number Example:
- 6. Predict the NextShape • Observe the pattern: • ▲, , , , , , ? ▼ ▲ ▼ ▲ ▼ • Each shape alternates between a triangle pointing up and a triangle pointing down. Thus, the next shape after is . ▼ ▲ Example:
- 7. Real-World Observation • Observation:Every time you see dark clouds forming in the sky, it rains later that day. This has happened 5 days in a row. • Conjecture: Dark clouds mean it will rain. (This is likely but not certain—it could be overcast without rain). Example:
- 8. Predict the NextNumber in a Sequence • Observe the pattern: 1, 4, 9, 16, 25, ? • *These are the perfect squares: 1², 2², 3², 4², 5². Thus, the next number is 6² = 36.* Example:
- 9. Real-World Pattern Recognition •Observation: For the past three years, the sales of umbrellas at your store have peaked in the month of July. • Conjecture: This year, umbrella sales will also peak in July. Example:
- 10. COUNTEREXAMPLES A statement isa true statement provided that is true in all cases. If you can find one case for which a statement is not true, called a counterexample, then the statement is a false statement.
- 11. Example • Every numberthat is multiple of 10 is divisible by 4. 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200 100 ÷ 4 = 25 120 ÷ 4 = 30 110 ÷ 4 = 27.5 With this example, we have shown that not all multiples of 10 are divisible by 4. So, we call 110 as a counter example.
- 12. Deductive Reasoning Applying generalprinciples, laws, or rules to reach a specific, certain conclusion. • Process: General Rule Specific Case → → Certain Conclusion • Nature: Certain. If the premises are true and the logic is valid, the conclusion must be true.
- 13. Examples: If a numberis divisible by 2, then it must be even. 12 is divisible by 2 Therefore, 12 is an even number.
- 14. Applying a MathematicalRule The sum of the angles in any triangle is 180°. Triangle ABC has angles measuring 90° and 30°. Conclusion: The third angle must be 60°. Example:
- 15. Applying a Definition Allsquares are rectangles (by definition, as they have four right angles). Shape XYZ is a square. Conclusion: Therefore, shape XYZ is a rectangle. Example:
- 16. Logical Syllogism All mammalsare warm-blooded. A dolphin is a mammal. Conclusion: Therefore, a dolphin is warm-blooded. Example:
- 17. Applying a GeometricProperty All radii of a given circle are congruent. In circle O, segments OA and OB are both radii. Conclusion: Therefore, OA is congruent to OB. Example:
- 18. Logical Argument If itis a public holiday, then the post office is closed. Today is a public holiday. Conclusion: Therefore, the post office is closed today. Example:
Editor's Notes
- #2 Let's start at the very beginning. In your own words, what does it mean to "reason" something out? (Pause for a few student answers. Likely responses: "to think logically," "to figure something out based on what you know," "to make a conclusion.") Facilitator: Excellent. So, reasoning is the process of using what we already know to draw new conclusions or make predictions. In math, we formalize this into two main types. Let's meet them.
- #3 Facilitator: The first type is Inductive Reasoning. The key is in the name: it's about inducing a pattern. You take specific observations and use them to make a broad, general rule. It's like being a detective, looking at clues and making your best guess about what happened. It's important to remember that inductive reasoning gives us a conclusion that is probable, but not guaranteed. It's an educated guess.
- #4 Facilitator: Okay, let's try one. Look at this sequence: 3, 6, 9, 12, 15, ? What do you notice? What pattern do you see? And what number do you think comes next? (Pause for answers. Students will likely say "it's adding 3 each time.") Facilitator: Exactly! You observed the specific examples, recognized the pattern of +3, and used that to make a general prediction. The next number is 18. This is inductive reasoning in action.
- #5 Facilitator: This one's a bit more involved. The procedure is: Pick a number. Multiply by 8, add 6, divide by 2, and subtract 3. Let's all pick a number. It can be any number. Don't shout it out, just do the steps in your head or on paper. (Give students a moment to calculate. Let's assume the original number is 5, as in the slide.) Facilitator: Okay, who got a final answer? (Get a few answers). Interesting. Now, what is the relationship between your original number and your final number? Look at the original and the final result. What do you see? (Pause. Students might say "it's double," or "it's four times." Guide them to see it's 4x.) Facilitator: That's the conjecture! No matter what number you started with, the result seems to be four times the original number. We've used specific examples to form a general rule. That's induction.
- #6 Facilitator: Patterns aren't just numbers. Look at these shapes: ▲, ▼, ▲, ▼, ▲, ▼, ? What's happening here? What comes next? (Pause. Students will almost certainly say "an up-pointing triangle.") Facilitator: Perfect. You saw the specific sequence, recognized the alternating pattern, and predicted the next specific instance. That's inductive reasoning.
- #7 Facilitator: Let's bring this into the real world. If you see dark clouds in the sky five days in a row, and it rains each time, what would you conclude on the sixth day when you see dark clouds? (Pause for answers: "It's going to rain.") Facilitator: Right! You're making a prediction based on a repeated observation. But is it a certainty? Could there be a day with dark clouds that doesn't rain? (Students will agree: yes, it's not certain.) Facilitator: Exactly. That's the probabilistic nature of induction. It's likely, but not guaranteed.
- #8 Facilitator: Back to numbers. Look at this: 1, 4, 9, 16, 25, ? What's the pattern this time? It's a bit different from just adding. (Pause. A student will likely identify them as "squares" or "1x1, 2x2, etc.") Facilitator: Brilliant. You recognized the general rule of perfect squares. So, the next specific number would be 6², which is 36.
- #9 Facilitator: One more real-world example. If a store's umbrella sales have peaked in July for three years in a row, what would the store owner logically expect to happen this July? (Pause for answer: "Sales will peak again.") Facilitator: Yes. They are using past specific data to predict a future specific outcome. This is how businesses use inductive reasoning for planning.
- #10 Facilitator: Now, here's the danger of inductive reasoning. Since it's not 100% certain, how can we prove a general rule is false? (Pause. Someone might say "find an example where it's not true.") Facilitator: Precisely. You only need one counterexample to break a rule. If I say "all birds can fly," what's a counterexample? (Answers: "Penguin!" or "Ostrich!") Excellent. Those are counterexamples.
- #11 Facilitator: Look at this statement: "Every number that is a multiple of 10 is divisible by 4." Is this true? Let's test some specific cases. 100 ÷ 4 = 25. True. 120 ÷ 4 = 30. True. What about 110? 110 ÷ 4 = 27.5. Not a whole number. So, what is 110? (Everyone: "A counterexample!") Facilitator: Correct. The statement is false because we found this one counterexample.
- #12 Facilitator: Now, let's meet the other type: Deductive Reasoning. This works in the opposite direction. Here, we start with a general rule that we know to be true, and we apply it to a specific situation to reach a certain conclusion. If the general rule is true and the logic is sound, the conclusion must be true. It's a watertight argument.
- #13 Facilitator: Simple one. Our general rule: "If a number is divisible by 2, then it is even." Now, take the specific number 12. We know 12 is divisible by 2. So, what must be true about 12? (Everyone: "It is even.") Facilitator: Yes. We used a general truth to deduce a specific, certain fact.
- #14 Facilitator: Geometry is full of deduction. The general rule: "The sum of angles in any triangle is 180°." Now, look at this specific triangle, ABC. We know two angles: 90° and 30°. So, what must the third angle be? How do we find it? *(Pause. Let someone explain the calculation: 180 - (90+30) = 60.)* Facilitator: Exactly. We deduced that the missing angle must be 60°. There is no other possibility.
- #15 Facilitator: We can also deduce things from definitions. The general definition: "All squares are rectangles" (because they have four right angles). Now, if I tell you that Shape XYZ is a square, what can you deduce about it? (Everyone: "It is a rectangle.") Facilitator: Perfect. This is a certain conclusion based on definitions.
- #16 Facilitator: This is a classic logical structure called a syllogism. General statement 1: "All mammals are warm-blooded." General statement 2 applied to a specific case: "A dolphin is a mammal." Therefore, what can we conclude about this specific dolphin? (Everyone: "It is warm-blooded.") Facilitator: This is a rock-solid deductive argument.
- #17 Facilitator: Another geometry example. The general property of a circle: "All radii of a given circle are congruent." In this specific circle, O, I have two segments: OA and OB. And we know they are both radii. So, what must be true about the lengths OA and OB? (Everyone: "They are congruent.") Facilitator: We deduced a specific fact from a general rule.
- #18 Facilitator: This is how we reason every day. General rule: "If it is a public holiday, then the post office is closed." Specific fact: "Today is a public holiday." Therefore, what is the certain conclusion about the post office today? (Everyone: "It is closed.") Facilitator: This is deduction. If the rule is true and the fact is true, the conclusion is inescapable.