Author: John Quintanilla

I'm a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT's program for preparing secondary teachers of mathematics and science.

Gravity wells

xkcdgravity_wells

This is one of most creative diagrams that I’ve ever seen: the depth of various solar system gravity wells. A large version of this image can be accessed at http://xkcd.com/681_large/.

From the fine print:

Each well is scaled so that rising out of a physical well of that depth — in constant Earth surface gravity — would take the same energy as escaping from that planet’s gravity in reality.

Depth = \displaystyle \frac{G M}{g r}

It takes the same amount of energy to launch something on an escape trajectory away from Earth as it would to launch is 6,000 km upward under a constant 9.81 \hbox{m}/\hbox{s}^2 Earth gravity. Hence, Earth’s well is 6,000 km deep.

Here’s some more details about the above formula.

Step 1. The escape velocity from the surface of a spherical planet is

v = \displaystyle \sqrt{ \frac{2GM}{r} },

where G is the universal gravitational constant, M is the mass of the planet, and r is the radius of the planet. Therefore, the kinetic energy needed for a rocket with mass m to achieve this velocity is

E = \displaystyle \frac{1}{2} m v^2 = \frac{GMm}{r}

Step 2. Suppose that a rocket moves at constant velocity upward near the surface of the earth. Then the force exerted by the rocket exactly cancel the force of gravity, so that

F = mg,

where g is the acceleration due to gravity near Earth’s surface. Also, work equals force times distance. Therefore, if the rocket travels a distance d against this (hypothetically) constant gravity, then

E = mgd

The depth formula used in the comic is then found by equating these two expressions and solving for d.

Poorly worded homework problems

A personal pet peeve of mine are grade-school homework problems that are extremely poorly worded, thus leading to unnecessary confusion and bewilderment in students who (sadly) are already confused and bewildered more often than they (or we) would like. Here are two examples that I’ve seen recently.

(1) A worksheet gives the numbers 144 and 300 with the instructions “Find all of the ways to multiply to make each product. First, find the ways with two factors, and then find ways to multiply with more than two factors.”

The second half of the instructions can easily be interpreted by a child to mean “Find all of the ways to write 144 and 300 as a product with more than two factors.” This reading of the question (probably not intended by the author) will take even a gifted child a really, really long time to complete. Furthermore, I’m a professional mathematician, and even I have no idea off the top of my head if there’s an easy formula for the number of ways that a number can be expressed with an arbitrary number of factors greater than 1.

(2) A rocket blasts off. At 10.0 seconds after blast off, it is at 10,000 feet, traveling at 3600 mph. Assuming the direction is up, calculate the acceleration.

I assume that the author was trying to be cute by adding the “it is at 10,000 feet” part of the problem. Or the author wants the student to develop skill at weeding out unnecessary information (like the height) and identifying just the important information (the final velocity and the time) to calculate the quantity of interest.

But it’s aggravating that the information in the problem is not consistent, so there is no solution. In other words, it’s impossible for a rocket to travel with constant acceleration at travel 10000 feet at 3600 mph 10 seconds later.

To begin,

3600 \displaystyle \frac{\hbox{mile}}{\hbox{hour}} = 3600 \displaystyle \frac{\hbox{mile}}{\hbox{hour}} \times \displaystyle \frac{\hbox{5280 feet}}{\hbox{1 mile}} \times \displaystyle \frac{\hbox{1 hour}}{\hbox{3600 seconds}} = 5280\displaystyle \frac{\hbox{feet}}{\hbox{second}}.

Therefore, the (presumably constant) acceleration is

\displaystyle \frac{5280 \hbox{~feet/second}}{10 \hbox{~seconds}} = 528 \hbox{~feet/second}^2.

However, using calculus, we can compute the height of the rocket by integrating twice:

v(t) = \int 528 \, dt = 528t + v_0 = 528t

y(t) = \int 528t \, dt = 264t^2 + y_0 = 264t^2

Therefore, the height of the rocket after 10 seconds is y(10) = 26,400, not the 10,000 feet given in the problem.

Engaging students: Square Roots

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 again comes from my student Allison Metzler. Her topic, from Pre-Algebra: square roots.

green line

A2. How could you as a teacher create an activity or project that involves your topic?

The following activity, http://ispeakmath.org/2012/05/03/square-roots-with-cheez-its-and-a-graphic-organizer/, effectively engages students because it’s hands-on and allows the students to work together. The students would start with their own cheez-its, creating the smaller squares (1, 4,9). Then, they would work in groups by combining their cheez-its to make bigger squares. Eventually, they would come together as a class to see how big of a square they could create. This involves square roots because each time the student would create a square (assuming they know the properties of a square), they would see that the square root would equal the base of the square. Also, they would see that the base of a square could be any of its four sides because they are all congruent or equal. Thus, the reasoning behind the name, “square root”, would become more apparent. Because they wouldn’t have a calculator as a resource, this visual method of teaching would give the students a more efficient way of calculating square roots. This activity is an effective way to get the students to remember the concept of square roots because it involves food, it’s hands-on, and they’ll learn a visual method of calculating square roots.

green line

D4. What are the contributions of various cultures to this topic?

Many cultures have contributed to the concept of square roots. From 1800 BC to 1600 BC, the Babylonians created a clay tablet proving 2^1/2 and 30*2^1/2 using a square crossed by two diagonals. Within that time (1650 BC), a copy of an earlier work showed how the Egyptians extracted square roots. From 202 BC to 186 BC, the Chinese text Writings on Reckoning described a means to approximate the square roots of two and three. In the 9th century, the Indian mathematician Mahāvīra stated that square roots of negative numbers do not exist. Then, in 1546, Cantaneo introduced the idea of square roots to Europeans. The last major contribution to the concept of square roots was in 1528 when the German mathematician, Christoph Rudolff, introduced the modern root symbol in print for the first time.

To present this to the students, I would use the following timeline and proceed to briefly mention what each culture contributed to the topic of square roots.

Square Root Timelinegreen line

E1. How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

The video, https://www.youtube.com/watch?v=AfBQGLowyKU, uses Elvis’s (You’re So Square) Baby I Don’t Care and recreates it with lyrics relating to square roots. This video not only accurately describes the main components of square roots, but also includes actual examples of perfect squares and square roots. It points out that the square root is the inverse of the square of a number. It also describes the base and the exponent which are directly related to the square root. Because the video is based off an actual song, it should effectively engage students and help them remember it since it’s catchy. Also, it is a great way to introduce the topic to the students where they want to know more, but aren’t overwhelmed with the amount of new information.

green lineReferences:

Banta, Willy, prod. Think I’m a Square, Baby I Don’t Care. Perf. Elvis Presley. YouTube, 2011. Web. <http://www.youtube.com/watch?v=AfBQGLowyKU&gt;.

Reulbach, Julie. “Square Roots with Cheez-Its and a Graphic Organizer.” I Speak Math., 3 May 2012. Web. <http://ispeakmath.org/2012/05/03/square-roots-with-cheez-its-and-a-graphic-organizer/&gt;.

“Square Root.” Wikipedia. Wikipedia Foundation Inc., 10 Jan. 2014. Web. <http://en.wikipedia.org/wiki/Square_root#History&gt;.

For your enjoyment:

HungerGamesSquareRoot

Am I Going to Die This Year?

Here’s an unexpected application of exponential growth that I only learned about recently: the Gompertz Law of Human Mortality. It dictates that “your probability of dying in a given year doubles every eight years.”

Here’s the article that I read from NPR: http://www.npr.org/blogs/krulwich/2014/01/08/260463710/am-i-going-to-die-this-year-a-mathematical-puzzle?sc=tw&cc=share.

An analysis of subtraction algorithms from the 18th and 19th centuries

Today I happily link to this wonderful article about how elementary school students “should” subtract two numbers, as it challenges the commonly held notion that there is only one way that subtraction should be implemented.

The common algorithm taught in schools today is the Decomposition Algorithm.

http://www.youtube.com/watch?v=3itmfsP6HoM

But there’s also the Equal Additions Algorithm.

http://www.youtube.com/watch?v=AN8XN_MSucI

And the Complement Algorithm.

http://www.youtube.com/watch?v=krNVuaIwi-o

And the Austrian Algorithm.

The author concludes:

The teaching and learning of subtraction is just as important today as it was in the past. Innovations in technology and mathematics curriculum have certainly occurred since the 1700s and 1800s, but the need for the teaching and learning of subtraction has not changed. Today, in many classrooms, subtraction is often taught through student-invented algorithms. Looking to the past may give teachers insight into invented algorithms or other algorithms students may use. Additionally, many teachers who do not encourage students to invent strategies teach only the “standard subtraction algorithm” presented in nearly every textbook across the United States, the decomposition algorithm. This research and analysis provides the modern teacher with an opportunity to reflect on the algorithms being taught in his or her classroom and allows the teacher to begin to think about why decomposition became the dominant algorithm in the United States. Teachers can ask their students to reflect on whether they agree with this historical turn of events. Incorporating the history of subtraction algorithms into modern elementary school mathematics invites robust mathematical discussion of subtraction and also of how, for many mathematical operations, there isn’t just one algorithm, but rather many algorithms from which to choose.

Exploring the history of subtraction in past school mathematics may provide us with insight into students’ mathematical struggles as they attempt to conceptualize not only subtraction, but also negative numbers and other notoriously challenging mathematical concepts. As educators and researchers, we need to devote more attention to issues in mathematics education such as the development of specific algorithms in elementary mathematics.

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.

green line

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.

green line

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.

green line

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.