Transcript Pascal’s Arithmetic Triangle

Kelly Shattuck
MAT 2009
Pascal’s Triangle
Triangle Terminology
Elements
Rows
Diagonals
Patterns in the Rows
 Sum of the Rows
 The sum of the numbers in each row is equal to a power
of 2 where n is the row number.
20 = 1
21 = 1+1 = 2
22 = 1+2+1 = 4
23 = 1+3+3+1 = 8
24 = 1+4+6+4+1 = 16
 Powers of 11’s
 If a row is made into a single number by using each
element as a digit, the number is equal to a power of 11
where the power is the row number.
Patterns in the Diagonals
 Triangular Numbers
 Triangular numbers can be found on the diagonal
starting with row 3.
where stands for the
term and
.
1, 1+2, 1+2+3, 1+2+3+4, 1+2+3+4+5, etc
Hockey Stick Pattern
 The diagonal of numbers of any length starting with
any of the 1s bordering the side of the triangle and
ending on any element inside the triangle is equal to
the number below the last element of the diagonal not
on the diagonal
Now…
Let’s Color!!
Coloring Multiples
 Even Numbers
Coloring Multiples
 Multiples of 3
Coloring Multiples
 Multiples of 4
Coloring Multiples
 Multiples of 7
What is the probability of
tossing 2 Heads if you toss
4 fair coins?
Applications
 It shows you the results of heads and tails when a fair, 2-sided
coin is tossed
Example: Toss a fair coin 4 times.
0H
TTTT
1
1H
HTTT
THTT
TTHT
TTTH
2H
HHTT
HTHT
HTTH
THHT
THTH
TTHH
4
6
3H
THHH
HTHH
HHTH
HHHT
4
4H
HHHH
1
Applications
 Pascal’s Triangle saves the trouble of using this tedious
formula
Example:
 Pascal’s Triangle Video
1
4
6
4
1
Applications
 The numbers in each row of the triangle are precisely
the same numbers that are the coefficients of binomial
expansions.
Example: Expand
1
4
6
4
1
Lessons and Activities
 Pattern Exploration
 Middle School level exploration of the triangle
 Coloring Multiples Exploration
 Coin Tossing Activity
 Exploring theoretical and experimental probability
 Pizza Problem
 Discovering the number of combinations of pizza topping
 Binomial Coefficients
 Relates the triangle to the Binomial Theorem