Problem Solving
Ch 1
Shake Hands with
Everyone
•
Some things to think about:
How many handshakes occurred?
How did you keep track that you didn’t shake
someone’s hand twice?
How do you know you shook everyone’s hand?
Is Your Friend a
Cheater?
You’re friend invited you to play a game
You get 2 darts where there are six sections
scored 1, 2, 3, 4, 5, and 6.
The rule is you throw both darts. If the second
score is greater than the first, you win. Otherwise
you loose.
Is the game fair?
Review
•
There are many different approaches to a
problem. And there is not one that is better than
another. This chapter will focus on trying
different methods.
•
The first part of your homework will focus on
solving problems similar to the handshake
problem.
•
The second part of you homework will focus on
solving problems with a graphical model.
Day 2 - Geometric
Modeling
•
Let’s look at the handshake problem again.
•
Can anyone think of how to draw a geometric
figure to represent our class size and the number
of hand shakes?
draw a shape with the # of
sides = to the # of students.
vertices = ppl
total # of diag + the # of sides =
# of handshakes
Practice - on your own
1. How many diagonals are there in an octogon?
2. An icosagon (20 sides) has how many diagonals?
3. If there were 35 people in this class, how many
handshakes would be required if everyone were
to shake hands?
One of the most important
mathematical tools:
Always
draw
a picture!
Finding a Pattern
We’re in Algebra now right?!
So let’s look at the handshake problem
algebraically...
Who knows
what sigma
notation is?
Sigma Notation
Let’s find a pattern for the total number of
handshakes for different class sizes.
2 people = 1 handshake
3 people = 3 handshakes
4 people = 6 handshakes
5 people = 10 handshakes ...
•What
is the pattern in the total number of
handshakes?
Pattern:
•
Below is the sequence representing the total
number of handshakes:
•
1, 3, 6, 10, ...
Now here is the series of how the handshakes are
increasing when another person is added to the
class...
1 + 2 + 3 + 4 + ...
Sigma Notation
•
This needs to represent a sum = the total number of handshakes.
•
Let n = the number of people in the class.
•
i.e. when n = 2, there was 1 handshake.
when n = 3, there was 2 handshakes added to the previous amount.
•
Is there an expression representing the number of people and the
number of handshakes added to the total?
n-1
What would the sigma notation look like for 8
people in the class? Then evaluate it.
•
Remember:
8
 n 1
n2
(2-1) + (3-1) + (4-1) + (5-1) +
(6-1) + (7-1) + (8-1) = 28
Day 3 - A Recursive
Solution
•
Recursive - a rule in which determining a
certain term in a sequence relies on the previous
term(s).
Handshake Problem
(again)
Yesterday, we looked at how the total number of
handshakes were increasing with each person
added to the room:
•1,
2, 3, 4, ...
•Let’s
think of it a different way:
Handshake Problem
(again)
With 2 people, there is 1 handshake
With 3 people, 2 more handshakes were added
With 4 people, 3 more handshakes were added
•
Therefore:
H2 1
H3  H2  2  3
H4  H3  3 6
Handshake Problem
(again)
•
This is a recursive sequence.
•
Can you think of a way to write it algebraically?
H n  H n 1  n  1 
Something
important is
missing!
Checkpoint
1. Generate the first five terms of the sequence that
has the recursive definition T1  2, T n  3T n 1
2. Find a recursive formula that will generate the
sequence 1, 2, 6, 24, 120, 720
Demonstrations
•
1st - Think of an expression that represents an
even number
•
2nd - Think of an expression that represents an
odd number
Even and Odd Integers
•
The even and odd integers can be divided into
two sets with no intersection, called disjoint sets.
•
a) Draw a picture to represent this situation
•
Keep your answer on a whiteboard and wait to
share with other groups.
Even and Odd Integers
•
Prove it!
•
I will give each group one of the problems (b - g)
to prove algebraically.
•
Hint: even integers are represented as 2n and
odd integers are represented as 2n - 1
Even and Odd Integers
•
Which set of integers contain the answer to each of the following
operations:
•
b) Add two even integers
•
c) Add two odd integers
•
d) Add an even and an odd integer
•
e) Multiply two even integers
•
f) Multiply two odd integers
•
g) Multiply an odd and an even integer
•
Keep your answer on one of your whiteboards