What’s so special about this
little triangle and its pattern of
numbers?
In this investigation, you’ll
discover the answer to that
question along with some
amazing mathematics!
Blaise Pascal was a natural genius who lived
from 1623 to 1662. He was born in Clermont,
France.
His father oversaw his son’s studies at home.
He confined Pascal to the study of languages
and directed that his studies should not include
any mathematics.
This only excited his curiosity more and at
the age of twelve, he began to explore geometry
and made some great discoveries! His father
gave him a copy of Euclid’s book, called
Elements, which he mastered quickly!
As an adult he abandoned the study of math for awhile and became
curious about religion. But in 1653, he invented the arithmetical
triangle that we are about to begin studying.
Throughout this presentation, you’ll learn about and explore this
amazing triangle and its many mathematical connections!
 In this view of the triangle, you get a
first look at Pascal’s discovery. He wrote
“figurate” numbers (more on this later!)
On the first line, are written numbers of
the first order. The second line are the
natural numbers or numbers of the 2nd
order.
On the third line are numbers of the
third order and so on. Do you remember
anything special about these numbers. If
not, you’ll be reminded soon!
This triangle had so many patterns and
mathematical connections to algebra,
geometry, probability that it amazes all
who take time to explore it!
ROWS…
Numbers in a horizontal line make up
rows. Rows are numbered like this….
To be able to discuss this triangle
intelligently, let’s get some basic
vocabulary down pat!
0th row
1st row
2nd row
3rd row
ELEMENTS…
Any number in the triangle is
referred to as an element.
1 is the 0th element on each row
4 is the 1st element in the 4th
row
10 is the 2nd element in the 5th
row. It is also the 3rd element.
1st: Can you figure out the pattern
that helps to create each new row?
How would you describe it?
Click here when you are ready to
check your answer
The sum of two adjacent
numbers is equal to the number
directly below it and between
them! (I bet you said it just like
this! You are a genius!)
B
91 14 1
A
What you see are
portions of Pascal’s
Triangle. Can you
use what you know
already to be able
to fill in the missing
numbers?
C
15
35 35
56
E
120 210
462
D
715 286
495
3003 1365
105
120 560 1820
153
171
190 1140
1716
3003
G
3003
F
In this next set of activities, you’ll be searching for the patterns
found when finding sums in Pascal’s triangle!
Recall: what is a row? How are they numbered?
Directions: Fill in the table below by finding the sum of the
elements in the first few rows of Pascal’s triangle. See if you
can answer the questions using the pattern you may discover!
Row
0 1 2 3 4 5 6
Row Sum
1 2
•What is the pattern of the sums?
•How could you relate the row number
to the sum of that row?
•How would you express the sum of
the elements in the 20th row? In the
100th row?, in the nth row?
Shift your eyes to looking at the diagonals now, instead of the rows!
Where is the element that will give the sum
of the first 4 elements of the first diagonal
(1+2+3+4)?
Where is the element that will give the sum
of the first 4 elements of the second diagonal
(1+3+6+10)?
What is the pattern that will give the sum
of any number of elements in any diagonal?
The sum of the first 4 elements
of the 2nd diagonal (1+2+3+4)
can be found in the 4th (how many elements you were
adding) position of the 3rd diagonal ( adjacent to the
2nd diagonal where you were adding)
What is the pattern that will give the sum of any
number of elements in any diagonal? Answer: The
sum of any number of elements in any diagonal can
be found by looking for the element in the next
adjacent diagonal which has the same position
number as the number of elements you were
adding!
The triangular sum of the
elements up to and including
row 2 is…
1+1+1+1+2+1=7
Find the sum of all the elements in
Pascal’s Triangle down to and including
the first 6 rows. Fill in the table below….
Row
0
Triangular Sum 1
1
3
Do you see a pattern? See if you can fill
in the table below without adding all the
elements….
Row
6 7 8 9 10 11
Triangular Sum
2
7
3
4 5
Get a copy of a filled in Pascal Triangle.




Work in groups of 4.
Have one member of your group color in all the multiples of 3.
Have another member color in all the multiples of 5
Another member fills in all the multiples of 7
The last member of the group fills in all the multiples of 9
Do you see a pattern? What is different about the pattern with the
multiples of 9? Can you figure out what it is about 3, 5 and 7 that
causes this pattern and why it doesn’t continue at 9? Does it continue
with other numbers larger than 9?
Check out an interesting web site at :
http://www.cs.washington.edu/homes/jbaer/classes/blaise/blaise.html




Download a copy of a Pascal worksheet from
http://mathforum.com/workshops/usi/pascal/mid.color_pascal.html
Open a Pascal worksheet in the ClarisWorks paint program.
Choose you favorite color from the paint palette and using the
divisibility rule for 3, paint all the cells that contain a multiple of 3.
Choose another color and paint all the cells that contain one less
than a multiple of 3 with this color.
Choose another color and paint all the cells that contain two less
than a multiple of 3 with this color.
Do you see a pattern?
Can you see any “symmetry” in the design?
Copy your design into a drawing program, duplicate it 6 times and link it
together to create a hexagon.
 Look at the center point where they are joined. Can you see the optical
illusion of a cube?
Notice the seven elements grouped
together. This group is known as a
Pascal flower. The middle cell of 15 is
surrounded by 6 cells called petals.
Starting with the petal above and
to the left of 15, the alternating
petals are highlighted in red and
are numbered 5,20, and 21.
The three remaining petals are
highlighted in yellow and contain
the numbers 10, 35, and 6
•Find the product of the red petals. Find the product of the yellow petals.
What do you notice? Can you explain this?
•Write the prime factorization of the numbers in the red petals. Write the
prime factorization of the numbers in the yellow petals. Compare the two
prime factorizations. Share your discovery!
•See if it works with any other Pascal flower!
What is 11 to the 0th power?
What is 11 to the 1st power?
What is 11 to the 2nd power?
What is 11 to the 3rd power?
Where can you find these answers in
Pascal’s triangle? Click here to check
What is 11 to the 0th power?
What is 11 to the 1st power?
What is 11 to the 2nd power?
What is 11 to the 3rd power?
Where can you find these answers in
Pascal’s triangle?
11 to any power is
found by using the
digits in the row that
has the same power
number!
Do you remember the Fibonacci Sequence,
(1,1,2,3,5,8,13,….) where each term is the sum
of the two previous terms? This sequence can
be found in Pascal’s triangle by following
“shadow diagonals” See below for the
illustration
Do you remember the triangular numbers?
These numbers are created by thinking of the
number of dots you need to make a triangle.
These numbers are found in the third diagonal
of Pascal’s triangle!
Pascal’s triangle is found with many
more interesting number patterns such
as …..
To explore these kinds of numbers in Pascal’s triangle, visit the
website found at :
http://forum.swarthmore.edu/workshops/usi/pascal/pascal_numberpatterns.html
Remember solving problems involving combinations….they were
like this…..
If I have 3 different socks,how many different ways could I choose 2
of them. Using a list where a, b, and c are the different socks, I see I
have three possibilities… ab, ac, and ac. This answer and answers to
all types of “combination” problems are found in Pascal’s triangle!
Watch!
Since there are 3 items to
choose from, go to row 3!
Since you are choosing only 2
items, go to the 2nd element
in that row and find your
answer!
Use the information from
the previous slide, and
Pascal’s triangle to help
you answer this question!
What is the probability of choosing
one combination of two colors if
you have 5 colors to choose from?
Use the triangle to help you, give
your probability answer as a
fraction!
When ready, click on the arrow
below to check!
The total number of combinations of 2 items
that can be made from 5 items is found in the
2nd element of the 5th row! Therefore the
probability of choosing one combination is
1/10!
This great problem was found on the www at
http://forum.swarthmore.edu/workshops/usi/pascal/pizza_pascal.html
Use what you learned from the previous slides to answer the questions following the story of Antonio’s Pizza
Palace.
It’s Friday night and the Pizza Palace is more crowded than
usual. At the counter the Pascalini’s are trying to order a
large pizza, but can’t agree on what toppings to select.
Antonio, behind the counter, says, “I only have 8 different
toppings. It can’t be that hard to make up your mind. How
many different pizzas can that be?”
“Well, we could get a plain pizza with no toppings,” says Mr..
Pascalini.
“Or we could get a pizza with all 8 toppings,” says Mrs..
Pascalini.
“What about a pizza with extra cheese and
green peppers?” asks Pepe.
“You’re not helping!” Antonio yells at Pepe.
“Get back to work!”
As Pepe starts to clear off the nearest table,
he mumbles to himself, “or a pizza with
anchovies, extra cheese, mushrooms and
olives.”
Antonio hands an order pad to Mr.. Pascalini
and says, “When you decide, write it down
and I’ll make it.” Then he helps the next
people in line , who know what they want: a
large pizza with mushrooms, green peppers
and tomatoes.
How many different pizzas can be ordered at the Pizza
Palace if a pizza can be selected with any combination of the
following toppings: anchovies, extra cheese, green peppers,
mushrooms, olives, pepperoni, sausage, and tomatoes?
As you solve the problem, look for patterns and answer the following questions:
1. How many different pizza can you order with only one topping?
2. How many different pizzas can you order each with seven toppings?
3. Are the number of one-topping pizzas and the number of seven-topping
pizzas related? (Why or why not?)
4. How many different pizzas can you order with two toppings?
5. How many different pizzas can you order with six toppings?
6. Are the number of two-topping pizzas and the number of six-topping pizzas
related? (Why or why not?)
That’s all, folks!!!!
I hope you enjoyed
investigating my
special little triangle
and that you made
many mathematical
discoveries. But for
now…...
B
13 1
91 14 1
105 15
A
C
20 15
35 35 21
70 56
E
D
120 210
165 330 462
715 286
2002 1001 364
3003 1365
105 455
120 560 1820
680 2380
18
19
153
171 969
190 1140
1716 1716
G
495 792
3003 3432 3003
6435 6435
F
Row
Row Sum
0 1 2 3 4 5 6
1 2 4 8 16 32 64
Did you discover….?
•The pattern of the sums forms the
sequence of the powers of 2.
• The sum of any row is equal to 2 to the
row power
20
•The sum of the 20th row is 2
100
•The sum of the 100th row would be 2
n
•The sum of the nth row would be 2
Find the sum of all the elements in
Pascal’s Triangle down to and including
the first 6 rows. Fill in the table below….
Row
0
Triangular Sum 1
1
3
2
7
3 4 5
15 31 63
Do you see a pattern? See if you can fill in
the table below without adding all the
elements….
Row
6 7 8 9 10 11
Triangular Sum 127 254 511 1023 2047 4095
The triangular sum of any number of rows is equal to
(2
(Row Number +1)
- 1)
What did you discover? Share your
discovery with the class! Click below to
see how Pascal’s triangle can solve this
problem!
Pascal’s Triangle
solves this
problem by
adding all of the
elements in the
8th row...
1+8+28+56+70+56+28+8+1= 256
possible pizzas!