# PROBLEM SOLVING

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```					PROBLEM
SOLVING

Chapter 1
   My web page
   A Fable
   Information Sheet
   Syllabus

   History of Math Education
Math Anxiety
   Lets go to Taco Bell
   Burrito \$1.79
   \$2
   Change: \$18.21
Math Anxiety
   Timed Tests

   Overcoming Math Anxiety by Sheila
Tobias
   MATH – An American Phobia by Marilyn
Burns
Three approaches to problem solving:

   Experimentation

   Intuition

   Deduction
Try to remember as little as possible!

Try to figure out as much as possible!
Real learning takes place when we
connect social knowledge (facts that
have to be told to you) to what’s inside
the learner.

In the Math classroom we will be
creating situations that cause you to
interact with ideas and by that, connect
something new to what you already
know.
Water Jug Problem
You are stranded on a desert island. You
have an unlimited amount of water but
only 2 containers: a 5 gallon jug and a 3
gallon jug. You need exactly 4 gallons
and you will be miraculously transported
off the island. Both jugs are irregularly
shaped and no markings can be made on
them. How can you use these 2
containers and your unlimited source of
water to obtain exactly 4 gallons of
water?
Marilyn Burns Video
DAY 2
Cannibals and Missionaries
In a far away place there are 3 cannibals and 3 missionaries who
wish to cross a river. They have learned to live together
peacefully without any threat to their lives as long as they abide
by one rule. The cannibals can never outnumber the
missionaries. As long as there are more missionaries than
cannibals or an equal number of each, everyone is safe. If the
cannibals ever outnumber the missionaries, they will eat them!

There is a boat that will hold 2 people that will be used to get all
six across the river. No one swims. The boat cannot travel
unmanned. If there is a missionary on one side by himself and 2
cannibals come to that side, there is no rule that keeps one in
the boat. He will get out and eat that missionary.

How can we get all six people safely across the river in the least
number of moves?
Problem Solving
is the Cornerstone
of School Mathematics.

The NCTM Principles and
Standards
page 2
You have a tremendous
responsibility and a great
privilege!

Page 375
   SPOT

   Fixation

   The Book of Think by Marilyn Burns
   Pick a number between 1 and 100
   Double it
   Take half of what you have now
   Subtract 5 from that
What did you get?
   Pick a number between 1 and 100
   Double the result
   Take half of the result
   Subtract the number you started with
What did you get?
   Choose any two odd numbers or any two
even numbers.
   Add the two numbers together.
   Divide the sum by 2.
   Subtract the smaller of the two original
numbers from the larger.
   Divide the difference by 2.
What did you get?
   Pick a 2 digit number
   Multiply by 6
   Subtract 3
   Divide by 3
   Subtract 6 less than the original number
   Subtract one more than the original number
   Divide by 2
What did you get?
Three digit fun!
   Write down the number of the month
you were born in.
   Double it.
   Multiply by 50
   Subtract the year of your birth
   Circle the last 2 digits.
What do you notice about the result?
Looking for Patterns
1x9=
21 x 9 =
321 x 9 =
4321 x 9 =
54321 x 9 =

87654321 x 9 =
67 x 67 =
667 x 667 =
6667 x 6667 =

6,666,667 x 6,666,667 =
9x1=
9x2=
9x3=
9x4=
9x5=
9x6=
9x7=
9x8=
9x9=
11 x 4 =    11 x 9 =

11 x 26 =   11 x 72 =

11 x 89 =   11 x 2345 =
“Fifteen”
Fill in the box below with the Counting
Numbers 1-9. Use each number only
once and make each row, column, and
diagonal have a sum of 15.
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45

45 ÷ 3 = 15

1 + 5 + 9 = 15               2+   6+   7 = 15
1 + 6 + 8 = 15               3+   4+   8 = 15
2 + 4 + 9 = 15               3+   5+   7 = 15
2 + 5 + 8 = 15               4+   5+   6 = 15
Standard 3 X 3 Magic Square
2 X 2 Magic Square
Old McDonald had a total of 37
chickens and pigs. All together they
How many chickens were there and how
many pigs were there?
Old McDonald had a total of 37 chickens and
pigs. All together they had 98 feet.
How many chickens and how many pigs?
Old McDonald had a total of 37 chickens and
pigs on his farm. All together they had 98 feet.
How many chickens were there and how many
pigs?
Toni is thinking of a number.
If you double the number and add 11, the
result is 39.
What number is Toni thinking of?
Guess and Check
Place the digits 1,2,3,4, and 5 in these circles so that
the sums across and vertically are the same.
10 Commandments for
Teachers
Introduction to Problem Solving
   Two engineers were standing on a street
corner. The first engineer was the second
engineer’s father but the second engineer was
not the first engineer’s son. How could this
be?

   If an electric train is traveling 40 miles an
hour due west and a wind of 30 miles per hour
is blowing due east, which way is the smoke
from the train blowing?
Introduction to Problem Solving
   At noon a rope ladder with rungs 1 foot
apart is hanging over the side of a ship,
and the twelfth rung down is even with
the water surface. Later, after the tide
has risen 3 feet, which rung of the
ladder is just even with the surface of
the water?
DAY 3
Homework Questions
Page 6
#2
Introduction to Problem Solving
Census Taker Problem
During a recent census, a census taker went to a
man's house who had three children. The father of
the children told the census taker that the product
of his children's' ages is 72. The census taker waited
the sum of their ages is the same as his house
number. The census taker looked at the house
number and then replied "I still don't know how old
they are." The father said "Oh, I forgot to tell you
that my oldest child likes chocolate pudding!" The
census taker was then able to write down the
children's ages and go on to the next house. How old
are the children. How do you know?
Polya’s Problem Solving
Principles
page 9
   Understand the problem

   Devise a plan

   Carry out the plan

   Look back
The process is more
Guess and Check

8                  16

2        6

14       12       18   11        15
8                       13

14       18             12        9

22

13        17
Make an Orderly List/Table
How many different scores could you make if
you hit the dartboard with three darts?

1
5
10
10’s   5’s   1’s   Score
Explain how you could determine the
number of ways you can make change
for a dollar using quarters, nickels,
dimes and pennies.
242
ways to make change for \$1
Draw a Diagram
Example1.5 page 16
In a stock car race the first five finishers in
some order were a Ford, a Pontiac, a
Chevrolet, a Buick, and a Dodge.
Problem Solving Strategies
   Guess and Check
   Make an orderly list
   Draw a diagram
Polygon Perimeter Lab
DAY 4
Homework Questions
Page 18
Polygon Perimeter Lab
Polygon Perimeter Lab
Polygon Perimeter Lab
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Number Sequence Lab
Number Sequence Example
1st   2nd   3rd   4th   5th   6th

4      7    10    13    16    19
20th Figure
• Three horizontal rows of dots. The top
row has 20 dots, the middle row has 20
dots, and the bottom row has 21 dots.
nth Figure
• Three horizontal rows of dots. The top
row has n dots, the middle row has n dots,
and the bottom row has n + 1 dots.
DAY 5
Number Sequence Lab
1st   2nd   3rd   4th       5th
Number Sequence Lab
1st    2nd    3rd
Number Sequence Lab
1st     2nd   3rd
Number Sequence Lab
1st    2nd   3rd
Continue these numerical
sequences.
 15, 30, 45, 60, ___, ___, ___
75, 90, 105, . . .
Arithmetic Sequence
common difference = 15
 15, 30, 45, 75, ___, ___, ___, . . .

120, 195, 315, . . .
Fibonacci “type” sequence
1, 1, 2, 3, 5, ___, ___, ___, . . .
8, 13, 21, . . .
The Fibonacci sequence
Leonardo of Pisa
AKA Fibonacci
   Fibonacci Fun, Fascinating Activities
with Intriguing Numbers by Trudi
Hammel Garland

   Fascinating Fibonaccis, Mystery and
Magic in Numbers by Trudi Hammel
Garland

   The Famous Rabbit Problem
 15, 30, 60, 120, ___, ___, ___, . . .
240, 480, 960, . . .
Geometric sequence
common ratio = 2
 15, 30, 90, 360, ___, ___, ___, . . .

1,800, 10,800, 75,600, . . .

   15, 30, 120, 960, ___, ___, ___, . . .
15,360, 491,520, 31,457,280, . . .
   15, 20, 26, 33, ___, ___, ___, . . .
41, 50, 60, . . .

 1, 4, 7, 10, 13, ___, ___, ___, . . .
16, 19, 22, . . .
Arithmetic sequence
common difference = 3
 19, 20, 22, 25, 29, ___, ___, ___, . . .

34, 40, 47, . . .
• 1, 4, 9, 16, 25, ___, ___, ___, . . .

• 1, 8, 27, 64, 125, ___, ___, ___, . . .
• A, E, F, H, I, K, ___, ___, ___, . . .

• 8, 5, 4, 9, 1, ___, ___, ___, . . .
Carl Gauss
   Page 27
   Fractal, Googols, and Other
Mathematical Tales by Theoni Pappas

1 + 2 + 3 + 4 + . . . + 98 + 99 + 100 = 5050
1+2+3+4+5+6+7+8+9=
1 + 2 + 3 + 4 + . . . + 98 + 99 + 100 =
34, 39, 44, 49, . . .
   What are the most likely choices for
the next two numbers in the sequence?

   What number will be the 46th term?
(Make a table – establish a pattern)
34, 39, 44, 49, . . .
• What number will be the 46th term?
34, 39, 44, 49, . . .
• What number will be the 46th term?
Term #   Term
1     34                    = 34 + 5(0)
2     39 = 34 + 5           = 34 + 5(1)
3     44 = 34 + 5 + 5       = 34 + 5(2)
4     49 = 34 + 5 + 5 + 5   = 34 + 5(3)
34, 39, 44, 49, . . .
• What number will be the nth term?
Term #   Term
1     34 = 34 + 5(0)
2     39 = 34 + 5(1)
3     44 = 34 + 5(2)
4     49 = 34 + 5(3)

46     259 = 34 + 5(45)
34, 39, 44, 49, . . .
• Which term is 359?
Term #   Term
1     34 = 34 + 5(0)
2     39 = 34 + 5(1)
3     44 = 34 + 5(2)
4     49 = 34 + 5(3)

n         34 + 5(n – 1) = 5n + 29
34, 39, 44, 49, . . .
• Which term is 359?
Term #   Term
1     34 = 34 + 5(0)
2     39 = 34 + 5(1)
3     44 = 34 + 5(2)
4     49 = 34 + 5(3)

66    359 = 34 + 5(65)
Find the sum (S).
34 + 39 + 44 + 49 + . . . + 359 = S
Find the sum (S).
34 + 39 + 44 + 49 + . . . + 359 = S
359 + 354 + . . . . . . . . + 34 = S
393 + 393 + 393 + . . . . . + 393 = 2S

393 shows up in this list 66 times.
(66)(393) = 25,938
If two of the sums = 25,938
The sum is 29,938 ÷ 2 = 12,969
27, 30, 33, 36, . . . , 363
• Find the next two missing terms.
27, 30, 33, 36, . . . , 363
• Find the 84th term.
Term # Term
27, 30, 33, 36, . . . , 363
• How many terms are there?
Term #   Term
1     27 = 27 + 3(0)
2     30 = 27 + 3(1)
3     33 = 27 + 3(2)
4     36 = 27 + 3(3)

84     276 = 27 + 3(83)
27, 30, 33, 36, . . . , 363
• How many terms are there?
Term #   Term
1     27 = 27 + 3(0)
2     30 = 27 + 3(1)
3     33 = 27 + 3(2)
4     36 = 27 + 3(3)

84      276 = 27 + 3(83)
113     363 = 27 + 3(112)
Find the sum (S).
27 + 30 + 33 + . . . + 363 = S
Find the sum (S).
27 + 30 + 33 + . . . + 363 = S
363 + 360 + . . . . . + 27 = S
390 + 390 + 390 + . . .+ 390 = 2S

(113)(390) = 44,070

44070 ÷ 2 = 22,035
52, 59, 66, 73, . . . ,815
• Find the 48th term.
52, 59, 66, 73, . . . ,815
• How many terms are there?

Term #    Term
1      52 = 52 + 7(0)
2      59 = 52 + 7(1)
3      66 = 52 + 7(2)
4      73 = 52 + 7(3)

48      381 = 52 + 7(47)
52, 59, 66, 73, . . . ,815
• How many terms are there?

Term #    Term
1      52 = 52 + 7(0)
2      59 = 52 + 7(1)
3      66 = 52 + 7(2)
4      73 = 52 + 7(3)

110     815 = 52 + 7(109)
Find the sum (S).
52 + 59 + 66 + 73 + . . . + 815 = S
Find the sum (S).
52 + 59 + 66 + . . . + 815 = S
815 + 808 + . . . . . . . + 52 = S
867 + 867 + . . . . . . + 867 = 2S

(110)(867) = 95,370

95,370 ÷ 2 = 47,685
Pascal’s Triangle
Example 1.10 Page 31
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
Problem Solving Strategies
   Look for a pattern
   Make a Table
   Use a variable
   Solve an equivalent problem
   Solve an easier, similar problem
Did You Know?
Page 35
A lecture hall has 40 rows of seats.
There are 10 seats in the first row, 12
in the second row, and so on, with two
more seats in each row than the
previous row. How many seats are in
the lecture hall?
DAY 6
NIM
(The Gold Coin Game)
Place 15 gold coins (or toothpicks) on a
desktop. The players play in turn and,
on each play, can remove one, two, or
three coins from the desktop. The
player who takes the last coin wins the
game.
Work backwards to come up
with a strategy to win!

   You pick from 1, 2 or 3 coins – pick all and win!
   You go first, pick 3 – AND WIN!

   He picks from 12 (take 3 - leave 9; take 2 - leave 10; take 1 -
leave 11)

   You pick from 9 (take 1 - leave 8) or 10 (take 2 – leave 8) or 11
(take 3 – leave 8)

   He picks from 8 (take 3 - leave 5; take 2 - leave 6; take 1 - leave
7)

   You pick from 5 (take 1 - leave 4) or 6(take 2 – leave 4) or 7
(take 3 – leave 4)

   He picks from 4 (take 3 - leave 1; take 2 - leave 2; take 1 - leave
3)

   You pick from 1, 2 or 3 coins – pick all and win!
Pigeon Hole Principle
If you have more pigeons than pigeon
holes, at least one hole has more than
one pigeon in it.
3 pigeons, 2 holes
The sock problem
John owns 20 blue and 20 black socks
which he keeps in complete disorder.
What is the minimum number of socks
that he must pull from the drawer on a
dark morning to be sure he has a
matching pair?
Jar of Marbles
   8 Red
   12 Yellow
   16 Blue

What is the minimum to be drawn to be
sure you have at least 2 of the same
color?
Make sure you have more pigeons than
pigeon holes.
Jar of Marbles
   8 Red
   12 Yellow
   16 Blue
   20 Green

What is the minimum to be drawn to be
sure you have at least 2 of the same
color?
Jar of Marbles
   8 Red
   12 Yellow
   16 Blue
   20 Green

What is the minimum to be drawn to be
sure you have at least 3 of the same
color?
Jar – student names
   24 students
   14 girls
   10 boys

How many must you draw to insure 2 of
the same sex?
Jar – student names
   24 students
   14 girls
   10 boys

How many must you draw to insure 3 of
the same sex?
Jar – student names
   24 students
   14 girls
   10 boys

How many must you draw to insure 5 of
the same sex?
Jar – student names
   24 students
   14 girls
   10 boys

How many must you draw to insure 2 of
opposite sex?
Eliminating Possibilities
   Either Jim, John or Yuri are in the
shower.
   You can recognize John’s voice and
Yuri’s voice.
   You do not recognize the voice of the
person singing in the shower.

Who is in the shower?
Beth, Jane and Mitzi play on the basketball team. Their positions
are forward, center and guard. Given the following information,
determine who plays each position.
(a)   Beth and the guard bought a milkshake for Mitzi.
(b)   Beth is not a forward.

forward             center            guard
Beth
Jane
Mitzi
   Beth plays center
   Jane plays guard
   Mitzi plays forward
Classic Handshake Problem
If everyone in class today were to
shake hands with each person in the
class today, how many handshakes would
take place?

*Make an easier problem, Use a model,
Make a table, Look for a pattern.
Number of People   Number of Handshakes
Problem Solving Strategies
   Working backwards
   Pigeon Hole Principle
   Eliminating Possibilities
Homework Questions
Page 35
#7
1=1
1+2+1=4
1+2+3+2+1=9
1 + 2 + 3 + 4 + 3 + 2 + 1 = 16

1+2+3+…+ 99 + 100 + 99 +…+3+2+1=
1+2+3+…+ (n-1) + n + (n-1) +…+3+2+1=
Page 35 #9
2n
n=1        n=2       n=3     n=4

n2 – n + 2
n=1        n=2       n=3     n=4

n3 – 5n2 + 10n - 4
n=1         n=2      n=3     n=4
Page 35, #13
A lecture hall has 40 rows of seats.
There are 10 seats in the first row, 12
in the second row, and so on, with two
more seats in each row than the
previous row. How many seats are in
the lecture hall?
10, 12, 14, 16, . . .
40th row?
10, 12, 14, 16, . . .
40th row?

1         10 = 10 + 2(0)
2         12 = 10 + 2(1)
3         14 = 10 + 2(2)

40        88 = 10 + 2(39)
10, 12, 14, 16, . . . , 88
10 + 12 + 14 + … + 88 = S
10, 12, 14, 16, . . . , 88
10 + 12 + 14 + … + 88 = S
88 + 86 + . . . . + 10 = S
98 + 98 + 98 + . . .+ 98 = 2S

40 rows,      (40)(98) = 3920
Divide by 2
1960
BAGELS – no digit is correct
PICO – digit is correct, wrong place
FERME – correct digit, correct place
1.5 Reasoning Mathematically
   We use inductive reasoning to draw a
general conclusion based on information
obtained from specific examples.

   Think about all the bears you have seen
(or seen pictures of). You could draw
the conclusion that all bears are black,
brown or white.
Now, think about all the square whole
numbers:

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, . .

Notice that they are all either multiples of 4
or one number larger than a multiple of 4.

Inductive reasoning leads us to generalize
that the square of any whole number is either
a multiple of 4 or 1 more than a multiple of 4.
Inductive Reasoning is a powerful way to
create and organize information. It
leads us to what seems to be true. In
order to state the observation for fact
we would have to prove our conjecture.

*There is in fact a rare type of bear in
Alaska whose fur in dark blue, called the
blue bear.
Example 1.14
Page 52
Look at several examples of 3 consecutive
integers.
8, 9, 10             25, 26, 27

33, 34, 35            100, 101, 102

121, 122, 123         600, 601, 602
It appears that exactly one of the three will
always be a multiple of 3.
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
• Place n dots on a circle.
• Join each pair of dots with a line segment making
sure that no more than 2 line segments intersect in a
point.
• Observe the number of regions inside each circle.
Use Inductive Reasoning
   Observe a property that holds in
several examples.
   Check to see if the property holds in
more examples. Try to find an example
in which it does not hold.
   If it holds in every example, state a
generalization that the property is
probably true.
Conjecture
   A generalization that seems to be true
but has not yet been proven.
Theorem
   A conjecture that has been proved.
Representational Reasoning
A representation is an object that
captures the essential information
needed for understanding and
communicating the properties and
relationships.

Often it is displayed visually, as in a
diagram, a graph, a map, or a table.
NCTM Standard
page 54
Page 36
Homework problem #7
1=1
1+2+1=4
1+2+3+2+1=9
1+2+3+4+3+2+1=16
1+2+3+4+5+4+3+2+1=25
1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1 = 25
Test – one week from today!
   Have ALL homework completed before
next class period.
   I will answer questions on sections 4 and
5 first then any left over questions you
might have from previous sections.
   Print a copy of the review list from my
web page.
   Elimination lab next time.
DAY 7
Homework questions
Page 48
Page 48, #3
•   Jumps
•   \$ Doubles
•   Pays \$32
•   Jumps
•   \$ Doubles
•   Pays \$32
•   Jumps
•   \$ Doubles
•   Pays \$32
Page 48, #5
•   1st card                  0
•   2nd card                  1
•   3rd card                  2
•   4th card                  3
•   5th card                  4
•   6th card                  5
•   7th card                  6
•   8th card                  7
•   9th card                  8
•   10th card                 9
# 19 - 20 people, at least 2 have same
number of friends at party.
Friendship is mutual.
• Case 1: Everyone has at least one friend.

• Case 2: Exactly one person has 0 friends.

• Case 3: 2 or more people have 0 friends.
Homework questions
Page 60
Review List
Canoe Problem (frog problem)
Elimination Lab
1. The Browns were giving a small dinner party and
Candy had mailed out invitations to three other
couples.
2. Don called to say that he would be late but his
wife would come with the Carters.
3. The Adams were the first to arrive, followed by
Abe, Ann, and Betty.
4. When Mr. Jones arrived, Bill and his host
greeted him at the door.
5. Abe’s wife and Ann helped Candy get the dinner
were planning the up-coming weekend of golf.
Ann Betty   Candy   Doris Adams Brown   Carter   Jones
Abe
Bill
Carl
Don
Brown
Carter
Jones
SUDOKU
4   5                   9
6   2       7               5   3
5       4   3   1       8
9   8                   1
5   3                       9   4
2                   4   7
4       1   9   2       6
7   6               5       1   8
1                   8   2

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