Engaging students: Pascal’s triangle

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 Roderick Motes. His topic, from Precalculus: Pascal’s triangle.

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History – What are the contributions of various cultures to this topic?

Through doing this project I learned that the history of Pascal’s triangle is actually pretty fascinating, and could be an excellent talking point for students.

Pascal’s Triangle was named after Blaise Pascal, who published the right angled version of the triangle, the binomial theorem, and the proof that n choose k corresponds to the kth element of the nth row of the triangle. But this wasn’t the first time interesting results about the triangle had been published, not even in the west.

The triangle was actually independently developed and worked on as early as the 11’th century in both China and modern day Iran. In China two mathematicians, Chia Hsien and Yang Hui, worked on the triangle and it’s applications to solving polynomials. Hsien used the triangle to aid in solving for cubic roots. Hui built upon the work of Hsien and actually gave us the first visual model of the triangle and used the triangle to aid in solving higher degree roots.

Pascals_triangle_China

Independently Omar Khayyam in Persia (modern Iran) used the triangle and binomial theorem (which was known to Arabic mathematicians at the time) to solve nth roots of polynomials.

In addition the triangle was used before Pascal to solve cubic equations, and in Europe in particular we get to the old controversy of Cardano and Del Ferro of ‘who found the general formula for cubic roots’ because another Italian man by the name of Niccolo Tartaglia claimed to have used the triangle to solve cubics and dervice the formula before Cardano published his formula.

So there were a variety of cultures who all independently recognized the significance of the triangle and used it well before Pascal. Consequently the triangle is called many things in many cultures. In China it is referred to as Yang Hui’s triangle, in Iran it is still called the Khayyam-Pascal triangle. All this goes to show that the history we think we know of mathematics may not be quite so true, and that mathematical understanding is the product of many cultures over many years.

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Technology- How can you effectively use technology to engage students on this subject?

There are a variety of technological resources you could use to craft a lesson. In particular I’m fond of the Texas Instruments exploration lessons. The lessons are available for free at education.ti.com and come with a slew of materials and handouts prepared for you. I’ve used the TI Nspire to teach the Law of Sines and the activity went tremendously well.

For Pascal’s Triangle and Binomial Theorem there are equivalent lessons with the TI Nspire and TI 84. The links are included at the end of this. The lessons allow the students to see Pascal’s triangle side by side with the triangle of coefficients which they are generating on the calculator. This could be backed up with having the students physically create the triangles on paper and see that they match up. The lesson then has the students conjecture what they believe the binomial theorem is.

This could be a powerful lesson for engaging learners of various strengths. Kinetic learners will love the physical action of the calculator, visual learners will love seeing the triangles update in real time.

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Curriculum- How can this topic be extended to your students future math courses?

Pascal’s triangle has a large relationship to probability and statistics. There are a variety of ways you can tie statistics lessons back to Pascal’s triangle and the binomial theorem. In particular we can examine how we might game a Pachinko machine in order to maximize our winnings.

Pachinko (or Plinko or a variety of other things depending on where you are) is fairly simple in idea.

You have a rectangular grid of pegs in which each row is slightly offset from the row above it. You drop a disc or puck of some kind down and attempt to get it into one of the small bins at the bottom. Sometimes prizes will be attached to certain bins (this is a popular carnival game) and sometimes money will (this is also a popular gambling game.)

The bin in which the puck will land follows a normal distribution based on the starting position. This is unsurprising and can be introduced very easily in a Statistics class when you’re teaching about probability distributions and normal distributions. What is more interesting is that this is very deeply related to Pascal’s Triangle.

Overlaying the triangle on top of the machine yields a triangle which shows the number of possible paths to get to each point. You can use this to make a statistical analysis and actually assign values to the probability of landing in a given spot. Using this knowledge you can game the machine and maximize your odds of getting the giant teddy bear or the fat stack of cash.

This application of Pascal’s triangle and its relationship to elementary combinatorics (which should hearken back to Middle School mathematics in addition to being extendable into Statistics,) is looked at in depth in a paper by Katie Asplund of Iowa State University. I have included this paper below. In addition to this suggestions she also relates a specific activity useful in the exploration where the students look at the various options of n choose k and relate the possibilities back to Pascal’s Triangle. I could not get the link for that specific activity as it requires access to Mathematics Teacher which I was unable to find using the UNT Library Resources.

plinko

References and Other Such Things

http://www.math.iastate.edu/thesisarchive/MSM/AsplundCCSS09.pdf

–          This paper is written by Katie Asplund. In it she explores a variety of patterns and connections between Pascal’s Triangle and various parts of the high school math curriculum. In particular she is interested in seeing how she can relate the patterns to her own high school pre calculus class. I recommend reading this entirely because it is simply illuminating and has quite a few suggestions you could implement.

http://pages.csam.montclair.edu/~kazimir/history.html

–          This website has a quick history of Pascal’s triangle as well as several applications. Using this and Wikipedia I was able to learn about the histories and cultures which led to our modern understanding of the triangle. In particular Omar Khayyam is a very interesting person to talk about if you feel like injecting some history of the Islamic Golden Age and the history of Mathematics after the fall of Rome. Khayyam was a Poet as well as a mathematician, and was one of the first to openly question Euclid’s use of the Parallel Postulate.

http://education.ti.com/calculators/downloads/US/Activities/Detail?id=11139&ref=%2fcalculators%2fdownloads%2fUS%2fActivities%2fSearch%2fKeywords%3fk%3dPascal

–          This is the TI Nspire activity on the Binomial Theorem and Pascal’s Triangle. It’s fairly straightforward but, like many of the TI Activities, it has some nice tricks that it uses the calculator to accomplish.

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