MMW Lesson 3_ Problem Solving

Mathematics in the Modern World Lesson 3: Problem Solving

Reasoning

  • Human beings use reasoning to make sound decisions

  • Two major types of reasoning: inductive and deductive

Inductive Reasoning

  • Used to come up with general conclusions (conjectures) based on observations

  • Conjectures can be wrong and can be negated by counterexamples

Examples of Inductive Reasoning

  • Sun rising in the east

  • Squares of odd numbers being odd

  • Number of regions in a circle made by connecting points

Deductive Reasoning

  • Does not rely on examples, but on general statements and assumptions

  • Used in formal mathematics to derive true statements from axioms

Examples of Deductive Reasoning

  • Starfish being invertebrates

  • Rational numbers being real numbers

Polya's Four Steps for Problem Solving

  1. Understand the problem

  2. Devise a plan

  3. Carry out the plan

  4. Look back and check the solution

Recommended Strategies

  • Draw a diagram

  • Solve a simpler problem

  • Make a table

  • Work backwards

  • Guess and check

  • Find a pattern

  • Use a formula or equation

  • Use logical reasoning

Example of Polya's Four Steps

  • Determining the number of magic cards Andrew has to trade

Handshakes Example

  • 30 attendees in a seminar

  • Each attendee shakes hands with everyone else

  • Determine the number of handshakes that took place

Polya's Four Steps Example

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  • 30 attendees were present in a seminar

  • Each attendee had to have a handshake with everyone in the room

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  • Step 2: Devise a plan

  • Start with solving simpler cases (3, 4, and 5 persons)

  • Draw a diagram representing persons as nodes and handshakes as arcs

  • Look for a pattern

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  • Step 3: Carry out the plan

  • Figures representing handshakes among 3, 4, and 5 persons

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  • A group of 3 persons makes 3 handshakes

  • A group of 4 persons makes 6 handshakes

  • A group of 5 persons makes 10 handshakes

  • For each case with k persons, each person has to have a handshake with the other k-1 persons

  • The product k(k-1) represents the number of handshakes from an individual perspective

  • Only half of k(k-1) represents the total number of handshakes

  • Generalized pattern: k(k-1)/2 represents the number of handshakes in a group of k persons

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  • Total number of handshakes in a group of 30 persons: 30(30-1)/2 = 870

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  • Step 4: Look back

  • Each person shakes hands with 29 others, resulting in 870 handshakes from an individual perspective

  • Half of 870 is 435

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  • Activity 1: Inductive reasoning may not always lead to a true conclusion

  • Activity 2: Conjecture about the final result of a mathematical operation using deductive reasoning

  • Activity 3: Solve problems using Polya's Four Steps

    • Susie's age is a multiple of 5 this year and a multiple of 7 next year. Find her present age.

    • Determine the number of perfect squares between 1,000,000 and