In my capstone class for future secondary math teachers, I ask my students to come up with ideas for *engaging* their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Jason Trejo. His topic, from Precalculus: using Pascal’s triangle.

How could you as a teacher create an activity or project that involves your topic?

After some research and interesting observations I came across while examining Pascal’s Triangle, I feel like I could create some sort of riddle worksheet that involves the Triangle. Once I have taught my students how to create Pascal’s Triangle, I could give my students riddles such as:

- Once you go and strive in prime, belittling your neighbors isn’t a crime.
- Students might notice that each number (other than 1) in a prime number row is divisible by that prime number:
- Row 7= 1, 7, 21, 35, 35, 21, 7, 1
- Row 11= 1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1

- Naturally shallow slides aren’t much fun, but with a fib of raunchy, it is this one.
- Given that I have gone over the Fibonacci sequence with my students prior to these riddles, I could include this one. The students should eventually see that if you take shallow diagonals on Pascal’s Triangle, the sum of those diagonals are the consecutive numbers in the Fibonacci sequence.

- In a game on blades, you can’t be a schmuck with a puck. Be nimble and quick to look for the stick.
- This one is a little more straightforward compared to the last two so hopefully the students will make the connection to notice the hockey stick pattern on the diagonals of Pascal’s Triangle. When adding the numbers down a diagonal, then the number to the side and below will be the sum, thus looking like a hockey stick.

- What else is there? What else is in store? What patterns can you find when you know who to root four?
- The “typo” is intentional to give a hint at another pattern the students might notice on Pascal’s Triangle. Now I am challenging the students to find more patterns within the Triangle such as:
- Sum of rows are the powers of 2
- Rows relate to the powers of 11 (get murky after the 4
^{th}row) - Counting numbers, triangular numbers, etc.

- The “typo” is intentional to give a hint at another pattern the students might notice on Pascal’s Triangle. Now I am challenging the students to find more patterns within the Triangle such as:

- Students might notice that each number (other than 1) in a prime number row is divisible by that prime number:

The purpose of this activity would extend the use of Pascal’s triangle from what they already know. I could assign this at the beginning of the lesson and if no one understands what the riddles meant, we could come back as a class and figure them out together once the lesson was done. These riddles could be an assignment of their own if I introduce them after they are very familiar with Pascal’s Triangle.

How can this topic be used in students’ future courses in mathematics and science?

I would say the primary use most students will get from Pascal’s Triangle would be to find the coefficients of binomials since it is much easier when working on binomial expansions, but there are also other ways they can use the Triangle as well. For one, it can be of great use in many courses that involve since it is a visual in seeing the number of combinations there are based on the number of items used. For example, say there are 6 different pieces of candy in a bowl and you need to know how many different ways can you choose 3 candies? Using Pascal’s Triangle, we look at the 6^{th} row and the 3^{rd} entry in that row (remembering the top row is Row 0 and the first 1 in each row is Entry 0), we can see that there are 20 possible combinations of 3 different pieces of candy. Other than that, even based on the riddle activity from above, students can use Pascal’s Triangle and its various patterns to help remember such things as triangular numbers, powers of 11, etc.

How has this topic appeared in high culture?

Within the past few years, the Shanghai-based design company, Super Nature Design, created the interactive art piece “Lost in Pascal’s Triangle”. This structure takes inspiration from Pascal’s Triangle and allows people to “explore the concept and magnification of the Pascal’s Triangle mathematics formula.” The following link takes you to the website that gives a bit more information behind the piece and shows how people can interact with the structure through a xylophone-type console: http://www.supernaturedesign.com/work/pascaltriangle#8

Another quick application that can be done through Pascal’s Triangle is by seeing the relationship between the Triangle and Sierpinski’s triangle (as shown below):

The pattern is by shading in every odd number on Pascal’s Triangle, you start creating Sierpinski’s triangle which is found in many works of art like these:

It might actually be a small but fun project to have the students create something like this at the beginning of the lesson and then explain the relation of the two special triangles.

__References__:

Pascal Triangle Information: http://jwilson.coe.uga.edu/EMAT6680Su12/Berryman/6690/BerrymanK-Pascals/BerrymanK-Pascals.html

Image of Pascal’s Triangle: http://mathforum.org/workshops/usi/pascal/images/pascal.hex2.gif

Lost in Pascal’s Triangle: http://www.designboom.com/weblog/images/images_2/andrea/super_nature_design/pascaltriangle01.jpg

Super Nature Design: http://www.supernaturedesign.com/work/pascaltriangle#2

Pascal and Sierpinski Triangle : http://mathforum.org/workshops/usi/pascal/images/sierpinski.pascalfrac.gif

Sierpinski Pyramid: http://www.sierpinskitetrahedron.com/images/sierpinski-tetrahedron-breckenridge.JPG

Sierpinski Art Project: http://fractalfoundation.org/wp-content/uploads/2009/03/sierpkids1.jpg