Ch.1 The Art of Problem Solving

1. How many outs are there in an inning of baseball?

2. A farmer has 17 sheep, all but 9 die. How many are left?

3. Is it legal for a man in Utah to marry his widow’s sister?

4. How many went to St. Ives?

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?

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

                                                                            

                

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

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.

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.

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.

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.

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.

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

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

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, _____

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

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!

The sum of the first n odd counting numbers equals n squared. 1 + 3 + 5 + … + x = n2

n numbers

You show it works!

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”

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.

Square numbers – 1, 4, 9, 16, 25, … Drawings:

To calculate the Nth square number: Sn = n2

EX: Find the 12th square number.

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.

P. 17 #33 Complete the figurate number table

Figurate Number 1st 2nd 3rd 4th 5th 6th 7th 8th

Triangular 1 3 6 10 15 21Square 1 4 9 16 25Pentagonal 1 5 12 22Hexagonal 1 6 15Heptagonal 1 7Octagonal 1 8

Use Figurate number formulas and Successive Differences method to determine the missing values. (Do you notice any patterns?)

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?”

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

Make a table Use inductive reasoning

Guess & Check

Look for a pattern Write relevant equation & solve

Trial and error

Solve a simpler problem

Use formula & solve Use common sense

Draw sketch / graph Work backwards Look for a “catch”

COMBINATION of these strategies

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.

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?

Month # of Pairs @ start of month

# of Offspring Pairs produced

# of Total Pairs @ end of month

1st 1 0 1

2nd 1 1 2

3rd 2

4th

5th

6th

7th

8th

9th

10th

11th

12th

Working Backward – Determine a wager at the track (p.22)

Using Trial & Error – Find DeMorgan’s birth year (p.23)

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

Draw a sketch – Straight 4 line segments puzzle (p.25)

Use common sense – Coin denominations (p.26)

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

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

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 …

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.

Type of Beverage

Tally results Percent of the Total

Angle Measurement

Lemonade 15

Cola 10

Cherry 5

Pepsi 20

Fanta 10

Bar graph or Histogram (vertical or horizontal) X-Y axes show

comparisons Discrete data b/c data is

categorical EX: Animal ages …

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 …

World Motor Vehicle Production

Europe

Other

U.S.A.

Japan

Canada

Chart Wizard Activity Represent the Ohio

Tornado Activity as a circle graph, bar graph and line graph.

Month # of TornadoesJanuary 6

February 14

March 36

April 113

May 157

June 204

July 168

August 86

September 37

October 19

November 38

December 3

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