### PROBLEM SOLVING

```PROBLEM SOLVING
Polya’s 4 Steps to Problem Solving
1.
2.
3.
4.
Understand the problem
Devise a plan. Strategies to use?
Carry out the plan
Look back!
Guess and Test
• Clues suggesting Guess and Test strategy
–
–
–
–
Limited number of possible answers to check
Have a good idea what the answer is
Can systematically try all possibilities
No other obvious strategy to try
Use a Variable
Variable: A letter used in place of numbers.
• Clues suggesting Use a Variable strategy
–
–
–
–
Problem suggests an equation
A proof or a general solution is required
An unknown quantity is related to know ones
Problem says things like “for any #”,
“consecutive”, “odd”, “even”.
Draw a Picture
• Clues suggesting Draw a Picture strategy
–
–
–
–
Problem involves a physical situation
Involves geometric figures or measurements
To gain a better understanding of the problem
Visual representation of the problem is possible
Is the sum of five consecutive
numbers a multiple of 5?
____ _____ _____ _____ _____
_x__ _x+1_ _x+2_ _x+3_ _x+4_
= 5x + 10
5x + 10 = 5 ( x + 2)
A rectangular field has a perimeter of
220 meters and an area of 2,856 square
meters. What are its dimensions?
220 m = l + w + l + w = 2 l + 2 w
l
l + w = 110
50 x 60 = 3000
40 x 70 = 2800
42 x 68 = 2856
w
w
l
Ans: 42 m by 68 m
It takes 64 cubes to fill a cubical box that
has no top. How many cubes are NOT
touching a side or the bottom?
16 
 16
It takes 64 cubes to fill a cubical box that
has no top. How many cubes are NOT
touching a side or the bottom?
16 
 16
It takes 64 cubes to fill a cubical box that
has no top. How many cubes are NOT
touching a side or the bottom?
8
16 
 16
8
It takes 64 cubes to fill a cubical box that
has no top. How many cubes are NOT
touching a side or the bottom?
8
16 + 16 + 8 + 8 + 4 = 52
16 
 16
8 4 
64 – 52 = 12
After the first two terms in the sequence,
each number is the sum of the preceding
two terms. Find the missing numbers.
4 ___ ___ ___ ___ 67
Tom and Barb have only nickels and
dimes. Barb has 75 cents and Tom has
90 cents. Both have the same number of
coins and Tom has the same number of
dimes as Barb has nickels. How many
nickels does Barb have?
The surface of Big Lake is 31 feet
above the surface of Long Lake.
Long Lake is half as deep as Big
Lake and the bottom of Long Lake is
8 feet below the bottom of Big Lake.
How deep is each lake?
Will the sum of any n consecutive
integers have a factor of n?
```