“Some Really Cool Things Happening in Pascal’s

Triangle”

Jim Olsen

Western Illinois University

Outline

1. Not your typical teacher workshop – What kind of session will this be?

2. Triangular numbers

3. Ten Cool Things About Pascal’s Triangle

4. Tetrahedral numbers and the Twelve Days of Christmas

5. A Neat Method to find any Figurate Number

1. What kind of session will this be?

• This session will be less like your typical teacher in-service workshop or math class,

• and more like– A play– A musical performance– Sermon– Trip to a Museum

• I’m going to move quickly.• I will continually explain things at various

levels.

Answering non-Math Questions in advance

Which Illinois Learning Standards?– This relates strongly to

• Goal 6: Number Sense

• Goal 8: Algebraic Thinking

• Goal 10: Probability and Counting

– Relates weakly to• Goal 9: Geometry/Spatial Visualization

Answering non-Math Questions in advance

Q: Which Grade Level Is This?

A: Grade 3 to middle school to high school to Math 355 (Combinatorics) in college

Answering non-Math Questions in advance

Q: How do I use this in my classroom?

A1: Understand it first, then get creative.

A2: Get the handouts and websites at the back.

A3: Communicate with me. I’d like to explore more answers to this question.

Answering non-Math Questions in advance

Q: Isn’t this just math trivia?

A: No. These are useful mathematical ideas (that are “deep,” but not hard to grasp) that can be used to solve many problems. In particular, basic number sense problems, probability problems and problems in computer science.

What kind of session will this be?

• I want to look at the beauty of Pascal’s Triangle and improve understanding by making connections using various representations.

2. Triangular numbers

15 10 6 3 1 54321 TTTTT

In General, there are Polygonal Numbers

Or Figurate Numbers

Example: The pentagonal numbers are

1, 5, 12, 22, …

+31+2 +9+8+7+6+5+4

Let’s Build the 9th

Triangular Number

• The Triangular Numbers are the Handshake Numbers

Number of People in the

Room

Number of

Handshakes

2 1

3 3

4 6

5 10

6 15

Interesting facts about Triangular Numbers

• Which are the number of sides and diagonals of an n-gon.

A-B A-C A-D A-E

D-E

B-C B-D B-E

C-D C-E

Why are the handshake numbers Triangular?

Let’s say we have 5 people: A, B, C, D, E.

Here are the handshakes:

It’s a Triangle !

15 10 6 3 1 54321 TTTTT

Q: Is there some easy way to get these numbers?

A: Yes, take two copies of any triangular number and put them together…..with multi-link cubes.

459 T

9

9+1=10

9x10 = 90

Take half.

Each Triangle has 45.

459 T

n

n+1

n(n+1)

Take half.

Each Triangle

has n(n+1)/2

2

)1(

nnTn

Another Cool Thing about Triangular Numbers

Put any triangular number together with the next bigger (or next smaller).

21 nTT nn

And you get a Square!

819298 TT

Ten Cool Things About Pascal’s Triangle

Characterization #1

• First Definition: Get each number in a row by adding the two numbers diagonally above it (and begin and end each row with 1).

Example: To get the 5th element in row #7, you add the 4th and 5th element in row #6.

Characterization #2

• Second Definition: A Table of Combinations or Numbers of Subsets

• But why would the number of combinations be the same as the number of subsets?

etc.

Five Choose Two

2

5

etc.

2

5

{A, B} {A, B}

{A, C} {A, C}

{A, D} {A, D}

etc.

etc.

Form subsets of size Two Five Choose Two

{A, B, C, D, E}

• Therefore, the number of combinations of a certain size is the same as the number of subsets of that size.

subsets 10

102

5 2 choose 5

subsets 120

1207

10 7 choose 10

Characterization #1 and characterization #2 are equivalent, because

r

n

r

n

r

n 1

1

1

Show me the proof

Characterization #3

Symmetry or“Now you have it, now you don’t.”

7

9

2

9

rn

n

r

n

Characterization #4

The total of row n

= the Total Number of Subsets (from a set of size n)

=2n

32215101051 5

n

n

nnnn2

210

Why?

Characterization #5

The Hockey Stick Principle

The Hockey Stick Principle

Characterization #6

The first diagonal are the “stick” numbers.

…boring, but a lead-in to…

Characterization #7

The second diagonal are the triangular numbers.

Why?

Because of summing up stick numbers and the Hockey Stick Principle

Now let’s add up triangular numbers (use the hockey stick principle)….

And we get, the 12 Days of Christmas.

A Tetrahedron.

Characterization #8

The third diagonal are the tetrahedral numbers.

Why?

Because of summing up triangular numbers and the Hockey Stick Principle

Tetrahedral Numbers are Cool Like Triangular Numbers

• Do the same things.– Find a general formula.– Add up consecutive Tetrahedral Numbers.

Find a general formula.

Use Six copies of the tetrahedron !

35 20 10 4 1 54321 TeTeTeTeTe

Find nTe

Combine Two Consecutive Tetrahedrals

• You get a pyramid!– Wow, which is the sum of squares.– (left for you to investigate)

Characterization #9

This is actually a table of permutations.

Permutations with repetitions. Two types of objects that need to be arranged.

For Example, let’s say we have 2 Red tiles and 3 Blue tiles and we want to arrange all 5 tiles. How many permutations (arrangements) are there?

For Example, let’s say we have 2 Red tiles and 3 Blue tiles and we want

to arrange all 5 tiles.

There are 10 permutations.Note that this is also 5 choose 2.Why?Because to arrange the tiles, you

need to choose 2 places for the red tiles (and fill in the rest).

Or, by symmetry?…

Characterization #10

Imagine a ball being dropped from the top. At each pin the ball will go left or right. **

The numbers in row n are the number of different ways a ball being dropped from the top can get to that location.

Row 7 >> 1 7 21 35 35 21 7 1

Imagine a pin at each location in the first n rows of Pascal’s Triangle (row #0 to #n-1).

Ball droppingThere are 21 different ways for the ball to

drop through 7 rows of pins and end up in position 2.

Why?

2

7Because position 2 is

And the dropping ball got to that position by choosing to go right 2 times (and the rest left).

Ball dropping demonstrations

Physical – with actual balls.

Virtual at http://www.subtangent.com/maths/flash/quincunx.swf

Equivalently

To get to Wal-Mart you have to go North 2 blocks and East 5 blocks through a grid of square blocks.

Wal-Mart

Home

There are 7 choose 2 (or 7 choose 5) ways to get to Wal-Mart.

Pascal’s Triangle gives it to you for any size grid!

A Neat Method to Find Any Figurate Number

Number example:

Let’s find the 6th pentagonal number.

The 6th Pentagonal Number is:

• Polygonal numbers always begin with 1.

1 + 5x4 + T4x3 1+20+30 = 51

• Now look at the “Sticks.”

– There are 4 sticks

– and they are 5 long.

• Now look at the triangles!

– There are 3 triangles.

– and they are 4 high.

The kth n-gonal Number is:

• Polygonal numbers always begin with 1.

1 + (k-1)x(n-1) + Tk-2x(n-2)

• Now look at the “Sticks.”

– There are n-1 sticks

– and they are k-1 long.

• Now look at the triangles!

– There are n-2 triangles.

– and they are k-2 high.

Jim Olsen

Western Illinois University

jr-olsen@wiu.edu

faculty.wiu.edu/JR-Olsen/wiu/

•Resources,

•Evaluation,

•Thank you.