Problem Solving (Lecture)

1. Today, problem solving,...

2. Problem Solving …
Problems have always existed
We don’t generally need a
computer to solve them
But, the computer often makes
problem solving easier
and faster
This is because
the computer is a
useful tool for solving problems

3. Problem Solving (cont’d)
There are many ways to classify
problems
For example:
Easy vs. Difficult
Mathematical vs. non-
mathematical
Another way to classify problems is
by their answers or solutions…

4. The Various
Types of
Problems

5. 1. Problems with well-
defined solutions
The problem – “How do
you bake a cake?”
The solution (of
course) – “Follow
a recipe”

6. 2. Problems with many
solutions
The problem – “What should I
do tonight?”
Some possible solutions –
Do my homework
Go to the library
Go fishing
Watch TV
Visit Mr. Wachs at home?”

7. 3. Problems with an
optimal (or best) solution
The problem – “How much tin
should be used for a can to hold
apple juice?”
There are many solutions – The
best solution – “Use a
can that uses the
smallest amount of tin
while holding the most juice

8. 4. Problems with no
solution
The problem – “How many
ancestors do you have?”
The solution – ??????
A “No” solution will mean a
solution that today we cannot
verify as true or false

9. What happens when we
solve problems?
1. We learn new ideas about
what the problem asks
2. We practice things we know
3. We transfer ideas from one
area to another
4. Our curiosity is stimulated

10. Watch out for Problems
When following an outline, watch out for things
like Einstrillings (a mind-set or mind-fix)
Read this..
PARIS
IN THE
THE SPRING

11. Say out loud, the COLOR of each word
Yellow Red Green Black
Black Green Red Yellow
Red Yellow Black Green
Green Black Yellow Red
Yellow Red Green Black
Another example…

12. Can you read this?
Your mind might be a pitfall in
solving a problem
Aoccdrnig to a rscheearch at Cmabrigde
Uinervtisy, it deosn't mttaer in waht oredr the
ltteers in a wrod are, the olny iprmoetnt tihng is
taht the frist and lsat ltteer be at the rghit pclae.
The rset can be a total mses and you can sitll raed
it wouthit porbelm. Tihs is bcuseae the huamn
mnid deos not raed ervey lteter by istlef, but the
wrod as a wlohe....... amzanig huh?

13. The 5 Steps
to Problem
Solving

14. The 5 Steps to Problem Solving
There are 5 steps you can use
to solve any problem
1.What is the problem?
2.Make a model of the problem
3.Analyse the model
4.Find the solution
5.Check the solution

15. Let’s Try it…
(1.) The problem:
A salmon swims 3 km upstream
and the current brings her back
2 km each day
How long does it
take her to
swim 100 km?

16. Solution…
1. What is the problem?
Does the salmon swim the 3 km
during the day and then at night drift
back 2km when resting?
Or, does she swim continuously all
day, slowed down by the current, so
that she swims only 1 km during the
day?
Does “day” mean daylight or 24
hours?

17. We’ll define the problem as …
How many 24 h. days does it
take for a salmon to swim 100
km if in the 24 h. period she
swims upstream and the
current brings her back 2 km.
while she is resting?
What is the problem (cont’d)

18. 2. Make a model of the problem
For this problem, the model will be a
“picture” of what happens
Start 3 km Upstream 24 h.
2 km back
Etc.
This gap is here so
that we can easily
see her path. It is
not counted

19. 3. Analyse the model
Make a table to keep track of her path:
Day
Number
traveled
forward
Maximum number
reached (end of
day)
Number
pushed
back
Total distance
travelled by next
day
1
2
3
•
•
•
97
98
3 km
3 km
3 km
•
•
•
3 km
3 km
3 km
4 km
5 km
•
•
•
99 km
100 km
2 km
2 km
2 km
•
•
•
2 km
2 km
1 km
2 km
3 km
•
•
•
97 km
98 km

20. 4. Find the solution
While you are making up the table, look at
the model
In this case the “picture” to check the
numbers
When the table headings are written down,
fill in the table to get the solution
We can see by the table that by the end of
day 98 she will reach the 100 km point
 The salmon will take 98 days to
swim 100 km

21. 5. Check the solution
Check the table for errors
Sometimes, ask others to check
your solutions
Even if errors are not
found, however, this
does not prove the
solution is correct!

22. Let’s Try another one …
(2.) The problem:
One day, a cabdriver picked up three young
couples and took them to a dance
One girl was dressed in red, one in green,
and one in blue
The boys all wore the same three different
colours (red, green, blue)
When the three couples were dancing, the
boy in red danced with his female partner
over to the girl in green and said…

23. The problem (cont’d)
“Isn’t it weird, Mary? Not one of us is
dancing with a partner dressed in the
same colour”
Given this
information, can
you deduce the
colour of the
partner the girl in
red is wearing?

24. Solution… (5 steps)
1. What is the problem?
To use the information given
to find out what
colour the
partner of the
girl in red is
wearing

25. 2. Make a model of the problem
For this problem, the model
will incorporate another type
of model called a logic chart
Drawn as follows…
Red Girl Green Girl Blue Girl
Red Boy
Green Boy
Blue Boy

26. 3. Analyse the model
Make sure you have all the boys
and all the girls?
And all the colours?
Red Girl Green Girl Blue Girl
Red Boy
Green Boy
Blue Boy

27. 4. Find the solution
We know the Red boy is not dancing with
the Green girl (because he talked to her)
Red Girl Green Girl Blue Girl
Red Boy
Green Boy
Blue Boy
X
The Red boy says “Not one of us is
dancing with a partner of the same
colour” (this applies to all colours) 
X
X
X

28. Find the solution (cont’d)
 The Red boy must be
dancing with the Blue girl
Red Girl Green Girl Blue Girl
Red Boy
Green Boy
Blue Boy
X
X
X
X
Yes

29. Find the solution (cont’d)
Because the Blue girl is already
dancing with someone, she cannot be
dancing with the Green boy
Red Girl Green Girl Blue Girl
Red Boy
Green Boy
Blue Boy
X
X
X
X
Yes
X
 The Green girl must be dancing
with the Blue boy
Yes

30. Find the solution (cont’d)
Because the Blue boy is already
dancing with someone, he cannot be
dancing with the Red girl
Red Girl Green Girl Blue Girl
Red Boy
Green Boy
Blue Boy
X
X
X
X
Yes
X
 The Green boy must be dancing
with the Red girl
Yes
X
Yes
The Solution

31. 5. Check the solution
Check for errors
Ask others to check your solutions
Even if errors are not found,
however, this does not prove the
solution is correct!
For Example: We are assuming
that the boys are always dancing
with the girls!

32. Let’s Try one more …
(3.) The problem:
How many different games are
possible in a best of three
volleyball playoff
between the
“Huskies” and
the “Bunnies”

33. Solution… (5 steps)
1. What is the problem?
How many different games are
possible if a team must win two
games in a best-of-three playoff
This means that a team
must win two games
to win the playoffs

34. 2. Make a model of the
problem
For this problem, the
model will incorporate
another type of model
called a tree diagram
Drawn as follows…

35. Make a model of the problem (cont’d)
H B
H B
H B B
H
H H
B B
H B
Huskies win Bunnies win
Game 3
Game 2
Game 1

36. 3. Analyse the model
Does the model give a
clear picture of what
happens as each game
is played?

37. 4. Find the solution
Use the tree diagram to
count up the number of
different possibilities
The answer is 6

38. 5. Check the solution
Check over the solution for errors
Write down the possibilities…
1. Huskies win, Huskies win
2. Huskies win, Bunnies win, Huskies win
3. Huskies win, Bunnies win, Bunnies win
4. Bunnies win, Huskies win, Huskies win
5. Bunnies win, Huskies win, Bunnies win
6. Bunnies win, Bunnies win

39. Using Analogy as Your Model
Sue is a good basketball player
Sue is 6 ‘ 2 ‘’ tall and slim
Will Sue be a good gymnast?
Ingrid is 6 ‘ 2 ‘’ tall, slim, a good
basketball player, and a good gymnast
Using analogy we observe similarities
and conclude that Sue will be a good
gymnast
However, analogy does not guarantee
our conclusions are correct

40. Using Analogy as Your Model (cont’d)
For instance…
What if I told you that even though
Sue is a good basketball player
She is a wheelchair athlete
So does this change her ability in
gymnastics?
You decide if your analogy still
stands…
You must make a judgment on the
validity of your analogy

41. Let’s Recap
the 5 steps to
problem
solving

42. 1. What Is the Problem
What am I trying to find?
What do I know?
What information is given?
State in your own words
What information do I need?
Take out vague words
Interpret the wording
Restate the problem

43. 2. Make a Model of the Problem
Get what you need (e.g. Paper)
Draw (e.g. logic chart, picture, tree
diagram)
Build it
Is there a pattern?
Use analogy (where two things are
alike)
Look for something familiar
Use top-down design (next time)

44. 3. Analyze The Model
Look at the model to see if it does what
you want
Could try trial and error
If it doesn’t work … make it again
For number problems, use a chart or table
If you have time, “sleep on it”
If it seems to difficult to solve as is,
change it to a slightly different problem;
now solve the new problem
Then see if the new solution can help
solve the original problem

45. 4. Find The Solution
Fill out the chart
Do what the
problem states
Use the model

46. 5. Check The Solution
Is it reasonable?
Do the problem
again
Use a team
approach

47. Conclusion:
 There is definitely an advantage to using a
structured approach to problem solving
 This applies to the various types of problems
encountered
 You can always incorporate the use of the 5
steps to problem solving