Problem Solving

1
Problem Solving
2
Problem Solving
Teachers of problem solving need to DO the
problems
Models good behavior
Increases your problem solving skills
Provides insight
Stimulates excitement
3
Problem Solving
Consider the introduction of a new concept.
Give a formula.
Do a few examples.
Practice
Problems all similar with different numbers
4
Problem Solving
Mathematical history
Information about problem solving
Source of ideas
Connections between disciplines
Show applications
Link concepts

Provide challenges for individuals
5
Problem Solving
Gauss 1 2 3 . . . 98
99 100
100 99 98 . . . 3 2
1
101 101 101 . . . 101 101 101
6
Problem Solving
What is problem solving?
Process
Challenge
Synthesizes information
7
Problem Solving
Polya problem-solving steps
Understand problem
Make a plan
Carry out plan
Look back

8
Problem Solving
Four CDs are purchased at 13.98 each. How much
money was spent?
Forget sales tax
Is this a problem solving problem?
For first grader?
For you?

9
Problem Solving
How many squares do you see?
10
Problem Solving
How many squares do you see?
11
Problem Solving
You are giving a party. You are not a guest
First doorbell ring brings one guest.
Second doorbell ring brings 3 new guests.
Third doorbell ring brings 5 new guests.
Each new doorbell ring brings 2 more new guests
than last one.
How many guests enter on the 20th ring?
What is the grand total number of guests after
the 20th ring?

12
Problem Solving
Fold a piece of paper in half
How do you know it is folded in half?
Are there other ways to fold a paper in half?


13
Problem Solving
Four essentials of a problem-solving problem
Non-routine solution
Challenge offered
Challenge accepted
Positive attitude fostered

14
Problem Solving
What is the question?
15
Problem Solving
Tangrams (can be related to last slide)
http//enchantedmind.com/puzzles/tangram/tangram.
html
16
Problem Solving
Classic calculus problem - - maximum volume
17
Problem Solving
What is the fewest number of weighings for 8
blocks, knowing one is light and 7 are the same?
You may place as many blocks on each pan as you
want.
18
Problem Solving
Teaching problem solving
DO the problems yourself
Avoid specialization
Use appropriate problems
Pace the use of problem solving problems

19
Problem Solving
Problem solving resources
http//www.whitehouse.gov/kids/math/index.html
http//mathforum.org/
Mathematics Teacher (NCTM)
Teaching Middle School Mathematics (NCTM)
Teaching Children Mathematics (NCTM)

20
Problem Solving
Tower of Hanoi
http//illuminations.nctm.org/tools/tool_detail.a
spx?id40
21
Problem Solving
Generalizing the Tower of Hanoi
?
?
?
22
Problem Solving
Traits of good problem solvers
Thinking
Reasoning
Educated guesses with follow-up
Generalization building
Flexible thought

Stating the obvious
Skipping steps
Talking to self
23
Problem Solving
Strawberry Ice Cream
There are three children
Each child has a counting-number age
The product of their ages is 72
The sum of their ages is the same as this house
number
Guest looks at the house number and asks for more
information

Guest is told, The oldest likes strawberry ice
cream.
Guest gives ages of the children
24
Problem Solving
25
Problem Solving
Buffons principle
Parallel line segments
Toothpick length apart
Drop toothpicks
On or Off segment
http//www.mste.uiuc.edu/reese/buffon/buffon.html
26
Problem Solving
How many squares are on a checkerboard?
27
Problem Solving
28
Problem Solving
29
Problem Solving
8x8 --gt 1
7x7 --gt 4
6x6 --gt 9
?
?
?
1x1 --gt64
TOTAL 204
30
Problem Solving
Suppose the earth is a sphere and the
circumference at the equator is 25,000 miles. A
band is placed around the earth, concentric with
the equator. The circumference of the band is
25,000 miles 10 feet. What is the thickest
thing that could fit between the band and the
equator?
31
Problem Solving
CB 2?(RE x)
CB 2?RE 2?x
CB CE 10 feet
CB 2?RE 10 feet
So, 2?RE 2?x 2?RE 10 feet
2?x 10 feet Subtracting 2?RE from both sides

32
Problem Solving
A mile-long, horizontal bridge is built with no
expansion joints. Neither end will move. The
bridge expands to a length of 1 mile 2 feet due
to temperature changes. The expansion causes the
bridge to bow up in the middle. What is the
distance between the center of the bridge in its
normal and in its expanded state?
33
Problem Solving
ABC is a right triangle - - Pythagorean theorem
AC2 (2641 feet)2 - (2640 feet)2
AC ? 72.67 feet
34
Problem Solving
School has 1000 lockers -- all closed
1000 kids enter school and open or close specific
lockers
Kid 1 opens every locker
Kid 2 closes all multiples of 2
Kid 3 changes (opens or closes, depending) all
multiples of 3
Kid 4 changes all multiples of 4
Etc.,
After all students enter school,
How many lockers closed?
Which ones?
35
Problem Solving
Locker 1 2 3 4 5 6 7 8 ...
Kid 1 O O O O O O O O ...
Kid 2 O C O C O C O C ...
Kid 3 O C C C O C O C ...
Kid 4 O C C O O C O O ...
Kid 5 O C C O C C O O ...
Kid 6 O C C O C C O O ...
Kid 7 O C C O C C C O ...
Kid 8 O C C O C C C C ...
36
Problem Solving
Look at factors of each locker number
Primes and composites (non-square) have even
number of factors
Implication -- each ends up as things started
Squares have odd number of factors
Implication -- each of these opposite of as
things started

37
Problem Solving
A sheet of paper is folded in half, and then in
half again, and again, and so on, until a total
of 50 folds is made. The question is, How high
is the stack of paper? The assumption here is
that the 50 folds can be made. Typical paper is
0.003 inches thick.
Stack of paper is 250 thick x 0.003 ? 3.3776 x
1012 inches thick.

38
Problem Solving
What is the largest number you can write using
only three digits?
999
999
999
Worked from top down
Equals 9387,420,000
About 500 miles long if typed
39
Problem Solving
For a month, you know
One-third of income for housing
One-third income for transportation and education
25 income for food and entertainment
200 left for saving, giving, investing, and
shopping?
How much does the individual make a month?

40
Problem Solving
Algebraic solution
x 2400
41
Problem Solving
Geometric solution
Big rectangle divided
Thirds horizontally
Fourths vertically
12 squares rectangle
1 square 200

12 X 200 2400
42
Problem Solving
Your job is to write numerals on pages in a book.
You are paid by the number of digits you write.
You wrote 642 digits. How many pages did you
number?
1 through 9 - - 9 pages
10 through 99 - - 180 digits - - 90 pages
189 digits used
642 189 453
9 90 151 250 pages
43
Problem Solving
Potential errors when adding decimals
Ragged decimals
Regrouping

44
Problem Solving
9 2.7 done like 800 372
9 8.9 0.1
8.9 - 2.7 7.2

Add 0.1
45
Problem Solving
Whole number times a tenth
46
Resources
Brumbaugh, D. K., Ortiz, E., Gresham, G.
(2006) Teaching Middle School Mathematics Mahwah,
NJ Lawrence Erlbaum Associates Brumbaugh, D.,
Rock, D. (2006 (3rd Ed.) Teaching Secondary
Mathematics Mahwah, NJ Lawrence Erlbaum
Associates. Brumbaugh, D., Rock, D.
(2001) Scratch Your Brain C1 Pacific Grove, CA
Critical Thinking Books and Software.