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Ch.1 The Art of Problem Solving

Ch.1 The Art of Problem Solving. How many outs are there in an inning of baseball? A farmer has 17 sheep, all but 9 die. How many are left? Is it legal for a man in Utah to marry his widow’s sister? How many went to St. Ives?. Current Event Think–Pair–Share & Essay.

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Ch.1 The Art of Problem Solving

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  1. Ch.1 The Art of Problem Solving How many outs are there in an inning of baseball? A farmer has 17 sheep, all but 9 die. How many are left? Is it legal for a man in Utah to marry his widow’s sister? How many went to St. Ives?

  2. Current Event Think–Pair–Share & Essay • “Today, independence starts later for adults” (6/13/10) • Read the Title – What does it imply about the article’s content? Discuss as a class. • Read the Article - As you read the paragraphs, note important statistics or statements and discuss with your partner. How do these relate back to and support the title? Do you have personal connections to the article? • After reading article, write a 3-paragraph essay responding to these questions: • What does the title of the article imply about its content? • What evidence in the article supports the title’s claim? • What future implications might exist after reading this article?

  3. 1.1 Solving Problems by Inductive Reasoning • McClane’s Water Jug Problem • Restate problem – • Plan – • Solve – • Check - • http://www.wikihow.com/Solve-the-Water-Jug-Riddle-from-Die-Hard-3

  4. Setting up your notes • Term or Concept • Explanation / Definition • EXAMPLES • Practice problems • Conjecture- • an educated guess based on repeated observations of a particular process or pattern • assuming that the same method would work for any similar type of problem • Similar to a scientific hypothesis that is to be tested

  5. Inductive reasoning – • Drawing a general conclusion (conjecture) from repeated observations of specific examples • The conjecture may or may not be true • Air Craft Investigation - documentary about ditched airplane on Hudson River • Sherlock Holmes – “The Band Saw” scene • Monsters Inc. – • They need screams to generate power for Monstropolis. However, their conjecture is false b/c… • “Don’t ever touch a child. Children are toxic to monsters.” Also a false conjecture… • Geometric proofs – All squares are rectangles, but not all rectangles are squares. Conjecture proven true.

  6. Example of inductive reasoning • SPECIFIC  GENERAL pattern (I: S  G) • What’s the next number in this pattern: • 2, 9, 16, 23, 30, ___ • Conjecture: Seems like 7 is added to each term, so the next number is 37. • Real answer: Next number is 7, as in July 7. The pattern were calendar dates in June.

  7. Counterexample - • When testing a conjecture, if one example does not work, it’s enough to prove the conjecture false • Conjecture: Children are toxic to monsters. Counter Ex: Sully is touched by a child, Boo, but does not die. Therefore, not all children are toxic.

  8. Pitfalls of inductive reasoning – • Conjecture is entirely false • All rectangles are squares. This conjecture can be proven false with one counterexample. • Conjecture is partially true, but fails after further investigation • Pluto is a planet in our solar system. • It doesn’t orbit the sun like other planets. • Therefore, Pluto is NOT a planet in our solar system.

  9. Deductive reasoning – • Method of proving a conjecture true by applying generally known principles to a specific example • GENERAL  SPECIFIC • Popularized by Greek mathematics as used by Euclid, Pythagoras, Archimedes, etc.

  10. Example of deductive reasoning • EX: People between 20 and 24 years old are taking longer to finish formal education. The median age for first-time marriages is 27. For example, my brother graduated college at age 25 and was married at 28. • Premise (generally held assumption or rule) PLUS Reason inductively or deductively to obtain conclusion  Logical argument

  11. 1.2 Applications of Inductive Reasoning – Number Patterns • Sequences – • Number sequence is a list of numbers having 1st, 2nd, 3rd, etc terms • Arithmetic or geometric sequences • Arithmetic sequences have a common difference between successive terms

  12. Arithmetic sequences – • Successive differences - method for finding sequential terms when a pattern is not obvious (this method does not work for Fibonacci sequence though) • EX: Find the next probable sequential term in this number pattern: • 5, 15, 37, 77, 141, _____

  13. Sum formulas • Use inductive reasoning to prove the pattern is true for that equation • Special sum formulas • For any counting number n, if you add successive numbers from 1 to n then square the sum, it equals the cube of each addend • (1 + 2 + 3 + … + n)2 = 13 + 23 + 33 + … + n3

  14. Gaussian Sum states if you add successive numbers from 1 to n, it equals n * (n+1) divided by 2. • 1 + 2 + 3 + … + n = [n(n+1)] / 2 • You show it works!

  15. The sum of the first n odd counting numbers equals n squared. • 1 + 3 + 5 + … + x = n2 • n numbers • You show it works!

  16. Figurate Numbers • Pythagoras (c. 540 BC) studied numbers having geometric arrangements of points • Use subscripts to represent which figurate number you want to calculate • T2 means “the second triangular number” • S4 means “the fourth square number” • P13 means “the thirteenth pentagonal number”

  17. Triangular numbers – 1, 3, 6, 10, 15, … • Drawings: • To calculate the Nth triangular number: • Tn = [n(n+1)] / 2 (the Gaussian sum) • EX: Find the 7th triangular number.

  18. Square numbers – 1, 4, 9, 16, 25, … • Drawings: • To calculate the Nth square number: • Sn = n2 • EX: Find the 12th square number.

  19. Pentagonal numbers – 1, 5, 12, 22, … • Drawings: • To calculate the Nth pentagonal number: • Pn = [n(3n – 1)] / 2 • EX: Find the 6th pentagonal number using the sum formula. • EX: Find the 6th pentagonal number using successive differences method.

  20. P. 17 #33 Complete the figurate number table Use Figurate number formulas and Successive Differences method to determine the missing values. (Do you notice any patterns?)

  21. 1.3 Strategies for Problem Solving • Logic Riddles - handout • General 4-step problem solving developed by George Polya (1888-1985) from Budapest, Hungary in his book “How to Solve It” • Step 1 – Understand the Problem • Read, re-read, ask “What must I find?”

  22. Step 2 – Devise a plan • Use any of these strategies….

  23. Step 3 – Carry out the plan • Using your strategy (Step 2), show your work and determine an answer. • Step 4 – Look back & check • Have you answered all parts of the original problem? • Do your answers make sense? • Write the complete answer in sentence form. • EX: The maximum height of the fireworks reaches 250 feet after 3 seconds.

  24. SAMPLE PROBLEMS • Using a Table or Chart – Solve Fibonacci’s Rabbit problem (p.21) • A pair of rabbits produce a pair of offspring after 1st month. Each offspring produce a pair of offspring in same manner. How many rabbit pairs will there be at end of 1 year?

  25. Working Backward – Determine a wager at the track (p.22) • Using Trial & Error – Find DeMorgan’s birth year (p.23)

  26. Set up equation / Guess & Check – Find the # of camels (Hindu math problem) (p.24)

  27. Draw a sketch – Straight 4 line segments puzzle (p.25) • Use common sense – Coin denominations (p.26)

  28. 1.4 Calculating, Estimating and Reading Graphs • Current Events • “Tornado Season” – Bar graph of Ohio’s tornadoes since 1950 • “Figures on retailing, jobs…” • Millbury, OH June 2010 http://www.myfoxatlanta.com/dpp/news/deadly-ohio-tornado-left-$100m-in-damage-060810 http://www.cnn.com/2010/US/06/06/midwest.storms/index.html?eref=rss_topstories&utm_source=feedburner&utm_medium=feed&utm_campaign=Feed%3A+rss%2Fcnn_topstories+%28RSS%3A+Top+Stories%29&utm_content=Google+Feedfetcher

  29. Tools of calculation – • Fingers, tally marks, handheld 4-function calculators, scientific calculators, graphing calculators • Estimation - • good to use when only a rough estimate, not an exact value is necessary

  30. Types of graphs - pictorial representations of data • Circle or pie chart • Sum of parts = 100% • Discrete data b/c data is categorical • EX: Favorite beverage survey …

  31. Your survey results show what is the favorite beverage of a group of teens. Lemonade 15; Cola 10; Cherry 5; Pepsi 20; Fanta 10. Construct a circle graph showing the different segments of the graph.

  32. Bar graph or Histogram (vertical or horizontal) • X-Y axes show comparisons • Discrete data b/c data is categorical • EX: Animal ages …

  33. Line graph • X-Y axes show changes or trends in data over time • Continuous data b/c data changes are always in flux • EX: Dolphin sightings …

  34. World Motor Vehicle Production Europe Japan U.S.A. Other Canada

  35. Chart Wizard Activity • Represent the Ohio Tornado Activity as a circle graph, bar graph and line graph.

  36. Review for Ch.1 Test • Practice questions • Bring personal calculator • Review notes & section problems

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