Transcript Document 7107883

Ch.1 The Art of Problem Solving
1.
2.
3.
4.
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

“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-WaterJug-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=
You show it works!
[n(n+1)] / 2

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
Use Figurate number formulas and Successive Differences method to
determine the missing values. (Do you notice any patterns?)
Figurate Number
1st
2nd
3rd
4th
5th
6th
Triangular
1
3
6
10
15
21
Square
1
4
9
16
25
Pentagonal
1
5
12
22
Hexagonal
1
6
15
Heptagonal
1
7
Octagonal
1
8
7th
8th
1.3 Strategies for Problem
Solving


Logic Riddles - handout
General 4-step problem solving
developed by George Polya (18881985) 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/dp
p/news/deadly-ohio-tornado-left$100m-in-damage-060810
http://www.cnn.com/2010/US/06/06/mi
dwest.storms/index.html?eref=rss_
topstories&utm_source=feedburne
r&utm_medium=feed&utm_campai
gn=Feed%3A+rss%2Fcnn_topstori
es+%28RSS%3A+Top+Stories%2
9&utm_content=Google+Feedfetch
er

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
Lemonade
15
Cola
10
Cherry
5
Pepsi
20
Fanta
10
Percent of the
Total
Angle
Measurement

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
Japan
U.S.A.
Other
Canada

Chart Wizard
Activity

Represent the Ohio
Tornado Activity as
a circle graph, bar
graph and line
graph.
Month
# of Tornadoes
January
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