Problem Solving
Strategies

Look
for a pattern
Example:
Solution:

Find
the sum of the first 100 even positive numbers. 
The
sum of the first 1 even positive numbers is 2 or 1(1+1)
= 1(2).
The sum of the first 2 even positive numbers is 2 + 4
= 6 or 2(2+1) = 2(3).
The sum of the first 3 even positive numbers is 2 + 4
+ 6 = 12 or 3(3+1) = 3(4).
The sum of the first 4 even positive numbers is 2 + 4
+ 6 + 8 = 20 or 4(4+1) = 4(5).

Look for a pattern:
The sum of the first 100 even positive numbers is 2 + 4 +
6 + ... = ? or 100(100+1) = 100(101)
or 10,100.

 Make an organized list
Example:

Find
the median of the following test scores: 73, 65, 82, 78, and
93.

Solution:

Make
a list from smallest to largest:
65
73
78 Since 78 is the middle number, the median is 78.
82
93


Guess
and check
Example:

Which
of the numbers 4, 5, or 6 is a solution to (n + 3)(n  2) =
36? 
Solution:

Substitute
each number for “n” in the equation. Six is the solution
since (6 + 3)(6  2) = 36. 

Make
a table
Example:

How
many diagonals does a 13gon have? 
Solution:

Make
a table:
Number
of sides

Number
of diagonals

3

0

4

2

5

5

6

9

7

14

8

20

Look for a pattern.
Hint: If n is the number of sides, then
n(n3)/2 is the number of diagonals. Explain in words why this
works. A 13gon would have 13(133)/2 = 65 diagonals. 
 Work
backwards
Example:

Fortune
Problem: a man died and left the following instructions for his
fortune, half to his wife; 1/7 of what was left went to his son;
2/3 of what was left went to his butler; the man’s pet pig
got the remaining $2000. How much money did the man leave behind
altogether? 
Solution:

The
pig received $2000.
1/3 of ? = $2000
? = $6000
6/7 of ? = $6000
? = $7000
1/2 of ? = $7000
? = $14,000


Use logical
reasoning
Example:

At the
Keep in Shape Club, 35 people swim, 24 play tennis, and 27 jog.
Of these people, 12 swim and play tennis, 19 play tennis and
jog, and 13 jog and swim. Nine people do all three activities.
How many members are there altogether? 
Solution:

Hint:
Draw a Venn Diagram with 3 intersecting circles.

 Draw a diagram
Example:

Fortune
Problem: a man died and left the following instructions for
his fortune, half to his wife; 1/7 of what was left went to
his son; 2/3 of what was left went to his butler; the man’s
pet pig got the remaining $2000. How much money did the man
leave behind altogether?

 Solve a simpler problem
Example:

In a delicatessen,
it costs $2.49 for a half pound of sliced roast beef. The person
behind the counter slices 0.53 pound. What should it cost? 
Solution:

Try a
simpler problem. How much would you pay if a half pound of sliced
roast beef costs $2 and the person slices 3 pounds? If a half
pound costs $2, then one pound would cost 2 x $2 or $4. Multiply
by the number of pounds needed to get the total:
3 x $4 = 12.
Now try the original problem: If a half pound costs $2.49, then
one pound would cost 2 x $2.49 or $4.98. Multiply by the number
of pounds needed to get the total: .53 x $4.98 = $2.6394 or $2.64. 
 Read the problem carefully
Know the meaning of all words and symbols in the problem.
Example:

List the
ten smallest positive composite numbers. 
Solution:

Since
positive means greater than 0 and a composite number is a number
with more than two whole number factors, the solution is 4, 6,
8, 9, 10, 12, 14, 15, 16, 18. For example, 4 has three factors,
1, 2, and 4. 
Sort out information
that is not needed.
Example:

Last year
the Williams family joined a reading club. Mrs. Williams read
20 books. Their son Jed read 12 books. Their daughter Josie
read 14 books and their daughter Julie read 7 books. How many
books did the children of Mr. and Mrs. Williams read altogether?

Solution:

You do
not need to know how many books Mrs. Williams has read since the
question is focusing on the children. 
Determine if there is
enough information to solve the problem.
Example:

How many
children do the Williams have? 
Solution:

There
is not enough information to solve the problem. You do not know
if Josie, Julie, and Jed are the only children. 
 Create problem solving
journals
Students record written responses to openended items such as those
tested on FCAT in mathematics. Student identifies problem solving strategies.
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State of Florida
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