Transcript Problem Solving Strategies

Problem Solving
Strategies
Algebra/Use a Variable
Look for a Pattern
Make a List
Solve a Simpler Problem
Using Algebra/Variables

How to use this strategy:
– Look for a pattern that can be described in
words and/or a general formula
– Finding Patterns:
1.
2.
3.
4.
Look at the sequences of numbers
Look at input/output table for rule
Start with simpler case first
Use finite differences
Using Algebra/Variables

When to use this strategy:
– The phrase ‘any number’ is used
– There is a list of numbers that can be
generated from the problem
– There is an input/output that can be
described as a rule
– A large number of cases are involved
Example 1

1.
2.
Determine the sum of the first 50 odd
counting numbers.
Step 1: odd counting numbers are 1, 3,
5, 7, 9 … and so on. I am adding
together the first 50 of these odd
numbers starting with 1+3+5…
Step 2: The plan will be to start with
simpler problems and look for a pattern
in the smaller/simpler sums
Example 1
Step 3: Use an
input/output table to
keep track of the
simpler sums and
look for a pattern.
Input-how many odd
#’s am I adding?
Output-what is the
sum?
Do you see the pattern?
3.
In
What odd
#’s I am
adding
Out
1
1
1
2
1+3
4
3
1+3+5
9
4
1+3+5+7
16
5
1+3+5+7+9
25
Example 1

The pattern that related the input to the
output is:
– Sum of n number of odd numbers is n x n
(the sum of the first 3 odd numbers was 3 x
3)
– So the sum of the first 50 0dd numbers is 50
x 50, or 2500

Step 4: Look back and verify answer
Example 1

Imagine each odd number represented by
tiles in the following shapes (note the
number of tiles used represents the odd
number, show below are 1, 3 and 5)
Example 1

Now add the shapes:
1+3
1+3+5
forms a square of how
many tiles?
Pattern Strategy

When to use:
– A list of data is given
– A list or sequence of numbers is involved
– Trying to predict or generalize
Make a List Strategy

When to use:
– When looking for a pattern/rule
– Data can be easily generated and organized
– Listing results from ‘Guess and Test’
Solve a Simpler Problem

When to use:
– Problem involves complex or complicated
computation
– Involves very large or small numbers
– Looking for a pattern/rule
Example 2

Carl Friedrich Gauss (1777-1885) was a mathematician
that made many contributions to math, physics,
astronomy and related sciences. A story of his
knowledge of math at a young age is attributed to his
exasperated third grade teacher.
The teacher was constantly trying to find things for Gauss to do
since he always finished his math lessons way before his
classmates. The teacher gave him the assignment of adding the
counting numbers 1 through 100 figuring that would keep Gauss
busy for awhile. The teacher was astounded when Gauss was
able to complete the task in only a few minutes. How did he do
this so quickly? (this is before calculators remember)
Example 2
Step 1: we are adding the counting
numbers 1,2,3,4… and so on, up to 100
 Step 2: combine strategies of making a
list (table), solving simpler problems,
looking for a pattern, and using a variable
 Input will be how many numbers I am
adding
 Output is the sum of the numbers

Example 2
Input
What #’s are added
Output
1
1
1
2
1+2
3
3
1+2+3
6
4
1+2+3+4
10
5
1+2+3+4+5
15
Example 2
There seems to be no operation that clearly
relates the input to the output (input times 2 for
example)
 The finite difference does not give much insight
 Gauss tried a new approach as follows:

– Write the numbers you are adding in reverse order
– Compare each pair top # paired to bottom# and
notice they make the same sum in every pair
Input
What #’s are added
Output
1
2
1
1
3
3
1+2+3
3+2+1
4
1+2+3+4
4+3+2+1
5
1+2+3+4+5
5+4+3+2+1
1+2
2+1
have 2 pairs of 3’s
have 3 pairs of 4’s
have 4 pairs of 5’s
have 5 pairs of 6’s
6
10
15
Input
What #’s are added
Output
1
2
1
1
3
3
1+2+3 have 3 pairs of 4’s
3+2+1 3 x 4 = 12
6
4
1+2+3+4 have 4 pairs of 5’s
4+3+2+1 4 x 5 = 20
10
5
1+2+3+4+5 have 5 pairs of 6’s
5+4+3+2+1 5 x 6=30
15
1+2
2+1
have 2 pairs of 3’s
2x3=6
Example 2

What is the relationship now?
– If you are adding 2 pairs of 3’s you have
exactly twice the output for the sum of the
first two numbers
– If adding 3 pairs of 4’s, you get 12 which is
double the output for the sum of the first 3
numbers (1+2+3)
– If adding 4 pairs of 5’s, you get 20, which is
double the sum of the first 4 numbers
(1+2+3+4)
Example 2

So if I want to add the numbers 1,2,…,10 using
this pattern I would take the list:
1+2+3+4+5+6+7+8+9+10
10+9+8+7+6+5+4+3+2+1
Reversing the list gives you 10 pairs of 11’s which is
=10 x 11 or 110.
But this is double what I need, so divide 110 by 2 to get
55. 55 is the sum of 1+2+…+10
How do you write this using variables?
What is the sum of 1+2+…+100?
Example 2


Adding n, number of
the counting numbers
can be found by
(when n was 10, n+1
was 11)
Sum of first 100
numbers is:
n  n  1  2
100  101  2  5,050