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.

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.

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.

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.

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.

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.

Tic tac toe

Source: http://www.xkcd.com/832_large/

39 Ways to Love Math

From Math with Bad Drawings:

Last week, 6,000 mathematicians met in Baltimore. They crowded in conference rooms, swapped gossip over beers, and wherever free food appeared, they lined up like ants.

On a table in the hallway of the convention center, I stationed paper, markers, and the following invitation:

I got 39 replies, 39 tributes to math’s power—in short, 39 ways to love mathematics.

See the results here: http://mathwithbaddrawings.com/2014/01/22/39-ways-to-love-math/

“Or” / “and”

One of the formulas typically taught in mathematics is

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

In ordinary English, the probability that either event $A$ or $B$ happens is the probability of event $A$ plus the probability of event $B$ minus the probability that the both occur.

For example, when rolling two fair six-sided dice, the probability that at least one three appears is

$P(A \cup B) = \displaystyle \frac{1}{6} + \frac{1}{6} - \frac{1}{36} = \displaystyle \frac{11}{36}$ .

It’s necessary to subtract something off at the end because it’s possible for the first die to be a four and simultaneously the second die to be a four.

This can be a conceptual barrier for students if it’s not directly addressed. In mathematics, the word “or” means “one or the other… or maybe both.” In the previous example, event $A$ was “first die is a four” and event $B$ was “second die is a four,” and it’s possible that both events could occur simultaneously.

Of course, this is different than the way we typically use “or” is spoken English. For example, in the final episode of each season of “The Bachelor,” the guy has to choose one woman or the other… and there’s no possibility of him choosing both! When a student says, “Next semester, my morning class will be history or physics,” we don’t think that there’s a possibility that the student will choose both classes… the student will choose one or the other, but not both.

In terms of mathematical logic, the word “or” in ordinary speech really is an “exclusive or.”

As I said, this isn’t a big deal for students to see, but in my opinion it’s best to directly address this subtlety rather than have students confused about which meaning of the word “or” they should be using when doing their homework.

P.S. The good news is that the word “and” means the same thing in the language of probability/logic as its meaning in ordinary speech.

My “history” of solving cubic, quartic and quintic equations

When I teach Algebra II or Precalculus (or train my future high school teachers to teach these subjects), we eventually land on the Rational Root Test and Descartes’ Rule of Signs as an aid for finding the roots of cubic equations or higher. Before I get too deep into this subject, however, I like to give a 10-15 minute pseudohistory about the discovery of how polynomial equations can be solved. Historians of mathematics will certain take issue with some of this “history.” However, the main purpose of the story is not complete accuracy but engaging students with the history of mathematics. I think the story I tell engages students while remaining reasonably accurate… and I always refer students to various resources if they want to get the real history.

To begin, I write down the easiest two equations to solve (in all cases, $a \ne 0$ :

$ax + b = 0 \qquad$ and $\qquad ax^2 + bx + c = 0$

These are pretty easy to solve, with solutions well known to students:

$x = -\displaystyle \frac{b}{a} \qquad$ and $\qquad x = \displaystyle \frac{-b \pm \sqrt{b^2-4ac}}{2a}$

In other words, there are formulas that you can just stick in the coefficients and get the answer out without thinking too hard. Sure, there are alternate ways of solving for $x$ that could be easier, like factoring, but the worst-case scenario is just plugging into the formula.

These formulas were known to Babylonian mathematicians around 2000 B.C. (When I teach this in class, I write the date, and all other dates and discoverers, next to the equations for dramatic pedagogical effect.) Though not written in these modern terms, basically every ancient culture on the globe that did mathematics had some version of these formulas: for example, the ancient Egyptians, Greeks, Chinese, and Mayans.

Naturally, this leads to a simple question: is there a formula for the cubic:

$ax^3 + bx^2 + cx + d = 0$

Is there some formula that we can just plug $a$ , $b$ , $c$ , and $d$ to just get the answer? The answer is, Yes, there is a formula. But it’s nasty. The formula was not discovered until 1535 A.D., and it was discovered by a man named Tartaglia. During the 1500s, the study of mathematics was less about the dispassionate pursuit of truth and more about exercising machismo. One mathematician would challenge another: “Here’s my cubic equation; I bet you can’t solve it. Nyah-nyah-nyah-nyah-nyah.” Then the second mathematician would solve it and challenge the first: “Here’s my cubic equation; I bet you can’t solve it. Nyah-nyah-nyah-nyah-nyah.” And so on. Well, Tartaglia came up with a formula that would solve every cubic equation. By plugging in $a$ , $b$ , $c$ , and $d$ , you get the answer out.

Tartaglia’s discovery was arguably the first triumph of the European Renaissance. The solution of the cubic was perhaps the first thing known to European mathematicians in the Middle Ages that was unknown to the ancient Greeks.

In 1535, Tartaglia was a relatively unknown mathematician, and so he told a more famous mathematician, Cardano, about his formula. Cardano told Tartaglia, why yes, that is very interesting, and then published the formula under his own name, taking credit without mention of Tartaglia. To this day, the formula is called Cardano’s formula.

So there is a formula. But it would take an entire chalkboard to write down the formula. That’s why we typically don’t make students learn this formula in high school; it’s out there, but it’s simply too complicated to expect students to memorize and use.

This leads to the next natural question: what about quartic equations?

$ax^4 + bx^3 + cx^2 + dx + e = 0$

The solution of the quartic was discovered less than five years later by an Italian mathematician named Ferrari. Ferrari found out that there is a formula that you can just plug in $a$ , $b$ , $c$ , $d$ , and $e$ , turn the crank, and get the answers out. Writing out this formula would take two chalkboards. So there is a formula, but it’s also very, very complicated.

Of course, Ferrari had some famous descendants in the automotive industry.

So now we move onto my favorite equation, the quintic. (If you don’t understand why it’s my favorite, think about my last name.)

$ax^5 + bx^4 + cx^3 + dx^2 + ex + f = 0$

After solving the cubic and quartic in rapid succession, surely there should also be a formula for the quintic. So they tried, and they tried, and they tried, and they got nowhere fast. Finally, the problem was solved nearly 300 years later, in 1832 (for the sake telling a good story, I don’t mention Abel) by a French kid named Evariste Galois. Galois showed that there is no formula. That takes some real moxie. There is no formula. No matter how hard you try, you will not find a formula that can work for every quintic. Sure, there are some quintics that can be solved, like $x^5 = 0$ . But there is no formula that will work for every single quintic.

Galois made this discovery when he was 19 years old… in other words, approximately the same age as my students. In fact, we know when wrote down his discovery, because it happened the night before he died. You see, he was living in France in 1832. What was going on in France in 1832? I ask my class, have they seen Les Miserables?

France was torn upside-down in 1832 in the aftermath of the French Revolution, and young Galois got into a heated argument with someone over politics; Galois was a republican, while the other guy was a royalist. More importantly, both men were competing for the hand of the same young woman. So they decided to settle their differences like honorable Frenchmen, with a duel. So Galois wrote up his mathematical notes one night, and the next day, he fought the duel, he lost the duel, and he died.

Thus giving complete and total proof that tremendous mathematical genius does not prevent somebody from being a complete idiot.

For the present, there are formulas for cubic and quartic equations, but they’re long and impractical. And for quintic equations and higher, there is no formula. So that’s why we teach these indirect methods like the Rational Root Test and Descartes’ Rule of Signs, as they give tools to use to guess at the roots of higher-order polynomials without using something like the quadratic formula.

Real references:

http://mathworld.wolfram.com/QuadraticEquation.html

http://mathworld.wolfram.com/CubicFormula.html

http://mathworld.wolfram.com/QuarticEquation.html

http://mathworld.wolfram.com/AbelsImpossibilityTheorem.html

http://mathworld.wolfram.com/QuinticEquation.html

http://library.wolfram.com/examples/quintic/

http://library.wolfram.com/examples/quintic/timeline.html

MAA Calculus Study: Persistence through Calculus

I just read a recent post by David Bressoud, former president of the Mathematical Association of America, concerning the percentage of college students in Calculus I who ultimately enroll in Calculus II. Some interesting quotes:

[J]ust because a student needs further mathematics for the intended career and has done well in the last mathematics course is no guarantee that he or she will decide to continue the study of mathematics. This loss between courses is a significant contributor to the disappearance from STEM fields of at least half of the students who enter college with the intention of pursuing a degree in science, technology, engineering, or mathematics.

And:

Our study offered students who had chosen to switch out a variety of reasons from which they could select any with which they agreed. Just over half reported that they had changed their major to a field that did not require Calculus II. A third of these students, as well as a third of all switchers, identified their experience in Calculus I as responsible for their decision. It also was a third of all switchers who reported that the reason for switching was that they found calculus to require too much time and effort.

This observation was supported by other data from our study that showed that switchers visit their instructors and tutors more often than persisters and spend more time studying calculus. As stated before, these are students who are doing well, but have decided that continuing would require more effort than they can afford.

And:

[W]e do need to find ways of mitigating the shock that hits so many students when they transition from high school to college. We need to do a better job of preparing students for the demands of college, working on both sides of the transition to equip them with the skills they need to make effective use of their time and effort.

Twenty years ago, I surveyed Calculus I students at Penn State and learned that most had no idea what it means to study mathematics. Their efforts seldom extended beyond trying to match the problems at the back of the section to the templates in the book or the examples that had been explained that day. The result was that studying mathematics had been reduced to the memorization of a large body of specific and seemingly unrelated techniques for solving a vast assortment of problems. No wonder students found it so difficult. I fear that this has not changed.

The full post can be found at http://launchings.blogspot.com/2013/12/maa-calculus-study-persistence-through.html.

Coin problems

Here’s a problem that a friend posed to me a while ago. Apparently this is called the Coin Problem, but I’d never heard of it before.

McNuggets used to come in boxes of 6, 9, or 20. Given that scheme, what is the largest number of nuggets that cannot be ordered exactly?

Here’s a similar problem:

In American football, teams can score points in increments of 3 (field goal) and 7 (touchdown plus extra point). What is the largest number that can’t be a valid football score? (I’ve ignored other possible ways of scoring — 2-point safeties, 6-point touchdowns without the extra point, 8-point touchdowns with a two-point conversion — because the problem is utterly trivial with these extra options.)

I’m not going to give the answers (if you want to cheat, see the above link), but I suggest questions like these as a way of engaging elementary-school students (who have mastered addition and multiplication) with a non-traditional math question.

On derivatives

This note appeared in the October 2012 issue of the American Mathematical Monthly.

On the news: A very literal pie chart

A very literal pie chart can be seen 20 seconds into this news clip.

A Review of WuzzitTrouble: an app for math education

Most apps and computer games that claim to assist with the development of mathematical knowledge only focus on rote memorization. There’s certainly a place for rote memorization, but I’ve been very disappointed with the paucity of games that encourage mathematical creativity beyond, say, immediate recall of the times tables.

Enter WuzzitTrouble, a new app that was developed by Keith Devlin, a professor of mathematics at Stanford and one of the great popularizers of mathematics today. An introduction to WuzzitTrouble can be seen in this promotional video:

One minor complaint about WuzzitTrouble is that the first few levels are so easy that it’s easy for children to low-ball the game… in much the same way that the first few levels of Angry Birds are utterly easy. (My other complaints is that the game only assume one user, so that a parent can’t play the game without affecting a child’s settings.) However, the level of difficulty does eventually increase. Here’s another promotional video showing how to solve Level 1-25:

Here’s a sampling of some of the higher levels. Remember that the wheel has 65 steps along the circumference, as shown in the above picture and videos.

Level 2-5: Using cog wheels of size 5 and 9, pick up keys at 23 and 36 and prizes at 27, 45, and 55.
Level 2-15: Using cog wheels of size 5, 7, and 9, pick up keys at 11, 16, and 21 and prizes at 32 and 42.
Level 2-25: Using cog wheels of size 5, 9, and 16, pick up keys at 24, 48, and 59; prizes at 11 and 37; and avoid a penalty at 64.
Level 3-3: Using cog wheels of size 3, 4, and 5, pick up keys at 7, 17, and 27 and prizes at 12 and 22.

In the words of their promotional materials:

At InnerTube Games, we set out to design and build mobile casual video games and puzzles that can attract and engage a large number of players, yet are built on fundamental mathematical concepts and embed sound mathematics learning principles.

We start with one simple, yet powerful observation. A musical instrument won’t teach you about music. But when you pick up an instrument and start playing – badly at first – you cannot fail to learn about music. And the more you play, the more you learn. In fact, using that one instrument, you can go all the way from stumbling beginner to virtuoso concert performances. It’s the music that changes, not the instrument. In modern parlance, the instrument is a platform. And (well designed) platforms are good for learning because they make the learning meaningful and put the learner in charge.

InnerTube Games does not build video games to “teach mathematics.” Rather, we build instruments which you can play, and we design them so that when you play them, you cannot fail to learn about mathematics. Moreover, each single game can be used to deliver mathematical challenges of increasing sophistication.

Our vision for learning design is to build the game around core mathematical concepts and practice so it looks and plays like the familiar casual games on the market. As a result, you won’t be able to see the difference by playing the first few levels, or by watching someone else play. It’s the educational power under the hood that makes our games different.

We’re not making a secret of the fact that our games are math-based. It’s not “stealth learning;” it’s a form of learning through action that the brain finds natural, having much in common with what educational researchers call embodied learning.

Wuzzit Trouble is our first puzzle to reach the market. It is built around the important mathematical concepts of integer partitions–the expression of a whole number as a sum of other whole numbers–and Diophantine equations. At the easiest levels of the puzzle, these provide engaging practice in basic arithmetic, leading to arithmetical fluency.

But that’s just the start. Integer partitions and Diophantine equations are major areas of mathematics, still being worked on today by leading mathematicians.

Freeing the Wuzzits won’t take you into those dizzy realms—at least in the initial release, which comes loaded with puzzles aimed at the Elementary and Middle School levels. But as you progress, you will face challenges that increasingly require higher-order arithmetical thinking, algebraic thinking, strategy design and modification, optimization, and algorithm design, all crucial abilities in today’s world. Getting three stars can require considerable ingenuity.

As you attempt to free each Wuzzit and maximize your score, you will be developing and applying valuable conceptual, analytic thinking skills that sharpen your mind—all without lifting pencil to paper.

As educators and former educators, all of us at InnerTube are very aware of the importance of learners meeting agreed standards. In its initial release version Wuzzit Trouble provides natural learning in the following areas of the US Common Core Curriculum:

*Grade 2, Operations & Algebraic Thinking #2

*Grade 2, Number & Operations in Base Ten #2, #8

*Grade 3, Operations & Algebraic Thinking #1, #4

*Grade 4, Operations & Algebraic Thinking #5

*Grade 6, Number System #5, #6

But we don’t want anyone to play our game purely to hit those Common Core markers. We want you to play it because it’s fun and challenging. Improvement in those CC areas comes automatically. Just like learning music by playing a musical instrument!

The analogy that I prefer is playing basketball. When young children are first learning to play basketball, there’s a place for learning how to dribble, how to pass, how to shoot free throws, etc. (These are analogous to learning how to add, subtract, multiply, and divide.) But children don’t just learn skills: they also go out and play. That’s where the WuzzitTrouble app fits in: it offers children a chance to just play with mathematics and enjoy it.

More references:

http://profkeithdevlin.org/2013/09/03/the-wuzzits-free-at-last/

Review: Wuzzit Trouble