Pascal’s Triangle

A big thank you to Turning Point for hosting us this month and raising the bar for all future hosts. We started with a tour of TP’s building and saw their classrooms, offices, and rooftop deck (!). Evidence of great student work is everywhere with student posters and presentations on the diverse topics of supply and demand, classified ads for housing, and dice and probability. It was wonderful to see such a beautiful, well-established community-based education program with full-time staff. And they provided refreshments!

For this meeting, we looked at Pascal’s Triangle, since it came up in discussion at the end of the October craps meeting in relation to calculating combinations, as well as the Pascal and the gambler story.

Wow, there is so much amazing, overlapping math in this one figure…

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Part One

Tyler started us out by giving us a worksheet with the beginning numbers of the triangle completed, but with lots of holes. Our first task was to work independently on a partially completed version of Pascal’s Triangle to extend the patterns and complete the triangle up to the 12th row.Tyler asked us to take note of the different patterns we saw. As we worked, people started to talk about the patterns, starting with the basic rule for building the triangle: Each number is formed by adding the two numbers centered above it.

If this is your first time seeing Pascal’s triangle, take a moment to copy the triangle above and see if you can create the next row. 1  5  10  10  5  1 ?   ?   ?   ?   ?   ?   ?

Tyler walked around to different tables, asked us to explain what we had found so far and planted some seeds:

– What do you notice about the rows in the triangle?

– Is there anything interesting happening in the different diagonals?

– Did anyone find the triangular numbers?

– Did anyone find the Fibonacci numbers?

– There’s an interesting pattern in which sums of consecutive numbers make the perfect squares. Can you find it?

– 5 goes into the second number of the row, 10, evenly. Is this true for all odd numbers in the triangle?

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Part Two

Next, the group discussed the patterns we noticed (some of these discussions happened at the end of the meeting after Part Three):

– Sequences found in the diagonals: Looking at the diagonals, the 2nd diagonal is the counting numbers (1, 2, 3, 4…), the 3rd diagonal is the triangular numbers (1, 3, 6, 10…), the 4th diagonal is the tetrahedral numbers (1, 4, 10, 20…) and so on, though we didn’t really know all the names of these different series beyond the tetrahedral numbers.

– Finding each diagonal in the one below it: If you look at the difference between each number in a diagonal series, you’ll find the series just above. For example, in the differences between the numbers in the triangular numbers (1, 3, 6, 10…), you’ll find the counting numbers (2, 3, 4…). And in the difference between the numbers in the tetrahedral numbers (1, 4, 10, 20…), you’ll find the triangular numbers (3, 6, 10…).

– Diagonals and summation: Each diagonal is a summation, or adding up, of the diagonal above it. For example, the summation of (1, 2, 3, 4…) is (1, 1+2, 1+2+3, 1+2+3+4), which could also be written as (1, 3, 6, 10…).

– Binomial coefficients: Each row of the triangle gives the coefficients in binomial expansions. For example,  . The coefficients are 1, 2, 1, which correspond to the 2nd row of Pascal’s triangle (not including the top 1). . The coefficients are 1, 3, 3, 1, found in the 3rd row of the triangle.

Ramon explains the numbers in the binomial expansion

Ramon’s notes

– Pascal Petals: Choose any number surrounded by 6 numbers. Multiply alternating numbers times each other (6x10x35) and compare to the product of the other alternating numbers (5x20x21), something interesting happens. The group talked about how this becomes obvious when we look at the prime factors of the petals, but we were still left with some questions about why it worked out this way.

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Part Three

After the group had discussed a number of different patterns in the triangle, Tyler asked us to put aside Pascal’s triangle and work on the following sheet of questions:

Antonio’s Pizza Palace

At first glance, this scenario seems to have nothing to do with Pascal’s triangle. We worked on the questions and noticed connections with other problems that some of us have done with students. “How many different pizzas can you order with two toppings?” reminded some us of the Handshake Problem. This led us to a visualization and a familiar series of numbers that seemed accessible to students.

The question of “How many different pizzas can you order with six toppings?” seemed pretty challenging. Some of us knew how to calculate combinations to find this number or came across it another way, but we wondered how students might approach this question. A few people recognized that the number of pizzas with 2 different toppings mirrors the number of pizzas with 6 different toppings, since every 2-topping pizza is a pizza without the other 6, and vice-versa. We talked about how a teacher might bring in laminated magnets of the toppings, so that students could move them around to visualize on the board.

Ooh, and we learned how to calculate combinations using the TASC calculator!

In processing this question sheet, we got deep into a discussion of how the pizza combination numbers are found in Pascal’s triangle, during which we tried to understand the connection between combinations and the patterns in Pascal’s triangle. We opened a lot more questions than we answered, and hope to continue the exploration through the discussion list and our meetings. The big takeaway was that we learned how to use Pascal’s Triangle to solve combination problems. For example, if we want to know how many combinations of 2 toppings we could make out of 8 possible choices, we start at the top and move 8 rows down, and then move two cells to the right, to 28.

We tested this with the handshake problem, since most of us remembered the solution. We can treat the handshake problem as a combination problem. There are 9 people, taken two at a time. So we move 9 rows down the triangle, and then 2 to the right, and we get 36–the solution to the handshake problem!

Since were were still buzzing about some of these discoveries, we stuck around after the meeting and brought up some other interesting observations, which you’ll see below. It would be awesome to dig deeper into these in a future meeting!

Exploring many patterns at once

Finding tetrahedral and triangular numbers through combinations

Finding combinations and permutations in the sums of two dice

Comparing an equation for finding triangular numbers with the formula for finding combinations

What the …?

(Try coloring in all the even numbers in Pascal’s triangle.)

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In attendance: Eric, Esther, Joe, Leo, Linda, Maggie, Ramon, Solange, Tyler

Programs represented: CUNY, Turning Point, NYCDOE, York College, CUNY Start, BMCC, Fifth Avenue Committee

Location: Turning Point, 423 39th Street, Brooklyn

Respectfully submitted by Eric