Engaging students: Multiplying and dividing rational expressions

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 Kayla (Koenig) Lambert. Her topic, from Pre-Algebra: solving proportions.

B) Curriculum: How does this topic extend what your students have learned in previous courses?

Multiplying and dividing rational expressions extends so many topics because the students have to use what they have learned up to multiplying and dividing the rational expressions. For example, this topic extends multiplying and dividing fractions. For multiplying and dividing fractions the students need to multiply across the numerators and multiply across the denominators and then simplify when possible (Multiplying Rational Expressions). Students also use factoring, which they should have learned before getting to this topic. When factoring, the students should remember different ways to factor. Some different ways are finding the greatest common factor, factoring by grouping, and finding the perfect square. They should also remember how to factor polynomials of different degrees.

The students also need to remember how to divide numerical fractions because they use the same method when dividing rational expressions; multiplying by the reciprocal. Another topic students should have previously learned is how to simplify rational expressions and how to multiply polynomials. Lastly, the students should also remember what a term, coefficient, constant, degree of a term, degree of a polynomial and should remember different types of polynomials (monomial, binomial, etc.). I could keep going with what topics are used when multiplying and dividing rational expressions all the way down to counting, addition, and subtraction. There are obviously so many different topics students have learned in the past that are extended when multiplying and dividing rational expressions.

D) History: What are the contributions of various cultures to this topic?

We can break multiplying and dividing rational expressions into many different mathematical subjects. In order to accomplish multiplying and dividing rational expressions, basic algebra and other basic mathematics had to come first. Methods of multiplication were documented by ancient Egyptian, Greek, and Chinese civilizations (Multiplication-Wikipedia). Around 1800 BC, Egyptians were the first known to use fractions. In 1600 BC, the Babylonians already knew solutions to quadratic equations and also solutions to equations to the third and fourth degree (Mathematics History). Egyptians used papyrus to make papers and used these to “calculate fractions” (Mathematics History).

The word polynomial comes from the Greek work “poly” meaning “many” and from the Latin word “binomium” meaning “binomial” and was introduced in Latin by a French mathematician, Franciscus Vieta (Polynomial-Wikipedia). The history of algebra goes back to ancient Egypt and Babylon where people learned to solve linear and quadratic equations. Also, Islamic mathematicians were able to multiply, divide and find the square root of polynomials. The Hindu-Arabic numerical system was first described by Brahmagupta who gave rules for addition, subtraction, multiplication and division. In orient mathematics, algebra “ultimately evolved from arithmetic” (Mathematics History). Nicole Oresme, from Normandy, was the first person to use fraction and exponents. Many cultures have contributed to multiplying and dividing rational expressions, but I would have to say that the Egyptians, Babylonians, Chinese, and Arabic have contributed the most.

E) Technology: How can technology (YouTube, Geometers Sketchpad, graphing calculator, etc.) be used to efficiently engage students with this topic?

Rational functions are used for many things including:

Fields and forces in physics
Spectroscopy in chemistry
Enzyme kinetics in biochemistry
Electronic circuitry
Aerodynamics
Medicine concentration
Wave functions for atoms and molecules
Optics and photography to improve image resolution
Acoustics and sound

Since the above topics are a little too advanced, I could show the student a video on YouTube to introduce the topic and to show them what multiplying and dividing rational functions are used for in the real world. After this, I would explain to the students that many other careers use rational functions like architects, foresters, and chemists. After talking about the topic, I could them give them a problem like the one below and ask them to graph the rational function with their calculator and can use their calculator to set up tables of values for their rational function. This will make it easy for them to see the maximum and minimum of the function and to see how the function behaves.

Example 9 from PreCalculus:

A rectangular page is designed to contain 48 square inches of print. The margins at the top and bottom of the page are 1 inch deep. The margins on each side are 1 ½ inches wide. What should the dimensions of the page be so the least amount of paper is used?

Works Cited

Larson, Ron, and David C. Falvo. “Precalculus – Ron Larson, David C. Falvo – Google Books.” 7 Feb. 2012. http://books.google.com/books?id=JRzhE6yqeFcC&pg=PA125&dq=what+are+rational+functions+used+for&hl=en&sa=X&ei=1lo1T9zDN-GusQLcrpyuAg&ved=0CFwQ6AEwBQ#v=onepage&q=what%20are%20rational%20functions%20used%20for&f=false.

“Mathematics History.” ThinkQuest : Library. 7 Feb. 2012. http://library.thinkquest.org/22584.

“Multiplication – Wikipedia, the free encyclopedia.” Wikipedia, the free encyclopedia. 7 Feb. 2012. <http://en.wikipedia.org/wiki/Multiplication>.

“Multiplying Rational Expressions.” Purplemath. 7 Feb. 2012. http://purplemath.com/modules/rtnlmult.htm.

“Polynomial – Wikipedia, the free encyclopedia.” Wikipedia, the free encyclopedia. 7 Feb. 2012. http://en.wikipedia.org/wiki/Polynomial_Functions#Polynomial_functions.

“Who Created Fractions | Ask Kids Answers.” AskKids Answers | AskKids.com. 7 Feb. 2012. http://answers.askkids.com/Math/who_created_fractions.

Volume of solid of revolution

In Calculus I, we teach two different techniques for finding the volume of a solid of revolution:

Disks (or washers), in which the cross-section is perpendicular to the axis of revolution, and
Cylindrical shells, in which the cross-section is parallel to the axis of revolution.

Both of these could be expressed as either an integral with respect to x or as an integral with respect to y, depending on the axis of revolution. I won’t go into a full treatment of the procedure here; this can be found in places like http://www.cliffsnotes.com/math/calculus/calculus/applications-of-the-definite-integral/volumes-of-solids-of-revolution or http://mathworld.wolfram.com/SolidofRevolution.html or http://en.wikipedia.org/wiki/Disk_integration or http://en.wikipedia.org/wiki/Shell_integration.

A natural question asked by students is, “If I have the choice, should I use disks or shells?” The correct answer, of course, is “Pick the method that gives you the easier integral to compute.” But that’s not a very satisfying answer for novice students who’ve just been exposed to integral calculus. So, over the years, I developed a standard reply to this query:

That’s an excellent question, and it’s one of the classic conundrums faced by mankind over the years.

Should I choose Coke… or Pepsi?

McDonald’s… or Burger King?

Ginger… or Mary Ann?

Disks… or shells?

The answer is, it just takes a little practice and experience to determine which technique gives you the easier integral.

If you don’t get the cultural reference, here’s a reminder. As of 10 years ago, I could still tell this joke to college students and still get smiles of acknowledgement. But, given the passage of time, I’m not sure if this same joke would fly college students now.

The Law of Averages

Colloquially, the Law of Averages dictates that what ought to happen does happen if it happens long enough. If a gambler plays a casino for a very long time, he is almost certainly guaranteed to lose. If a weak sports team plays a stronger team in a multiple-game series, then it is almost certain to lose the series.

However, if the gambler plays in the casino for only a little while, then there is a realistic (though less than 50%) chance of coming out ahead. And a weak sports team may defeat a stronger one if only one game is played… hence the appeal of the NCAA basketball tournament and, on a larger scale, the knockout stages of the World Cup.

In my statistics class, I use a simple simple spreadsheet to illustrate that $\hbox{SD}(K) = \sqrt{n p(1-p})$ for a sample count, but $\hbox{SD}(\hat{p}) = \displaystyle \sqrt{ \frac{p(1-p)}{n} }$ for a sample proportion. Here is one image from the spreadsheet:

The user can change the bright green cell to be any positive integer up to 5000. This number represents the number of simulated coins that are flipped. In the above example, ten coins are flipped. Column B shows the results of the simulated coins, while column C shows a running count of the number of heads that have appeared. In the above example, 7 of the 10 flips are heads, for an observed error of +2 (two more heads than the expected number of 5) and a percentage error of 20%.

In class, I would run the spreadsheet several times, and students will see that the observed error usually is in the range of -2 to +2, and the percentage error is usually in the range of -20% to +20%.

By contrast, look what happens when the number of flips increases to a large number, like 5000.

There is now a larger absolute error — in this case, -28. Of course, an absolute error of that size is impossible with 10 coin flips or even 50 coin flips. However, at the same time, the percentage error is now significantly smaller (only -0.56%).

This example gives evidence for the counter-intuitive result that the absolute error grows like $\sqrt{n}$ while the relative error decreases like $\sqrt{n}$ .

The Very First Unteachable Students

This witty essay about teachers complaining about their students made me laugh. Hope you enjoy it.

http://www.insidehighered.com/blogs/education-oronte-churm/very-first-unteachable-students

Growth Rate of Calculus Textbooks

Source: https://www.facebook.com/photo.php?fbid=589563657759289&set=a.250425975006394.53155.241224542593204&type=1&theater

A Capella science

I wish I had the talent to do something like this.

Day One of my Calculus I class: Part 6

In this series of posts, I’d like to describe what I tell my students on the very first day of Calculus I. On this first day, I try to set the table for the topics that will be discussed throughout the semester. I should emphasize that I don’t hold students immediately responsible for the content of this lecture. Instead, this introduction, which usually takes 30-45 minutes, depending on the questions I get, is meant to help my students see the forest for all of the trees. For example, when we start discussing somewhat dry topics like the definition of a continuous function and the Mean Value Theorem, I can always refer back to this initial lecture for why these concepts are ultimately important.

I’ve told students that the topics in Calculus I build upon each other (unlike the topics of Precalculus), but that there are going to be two themes that run throughout the course:

Approximating curved things by straight things, and
Passing to limits

I’ve then quickly used these themes to solve two completely different problems: (1) finding the speed of a falling object at impact and (2) finding the area under a parabola. I can usually cover these topics in less than 50 minutes, sometimes in 35 minutes. Again, because I’m not immediately holding my students responsible for the contents of this introduction, I feel freer to move a little quicker than I would otherwise in the hopes of showing the forest for all of the trees.

I then ask the obvious question: what do these two questions have to do with each other. One involves the distance-rate-time formula. The other involves the areas of rectangles. At first blush, these two questions seem completely unrelated. And at second blush. And at third blush.

I tell my class that these two apparently unrelated questions are indeed related by something called the Fundamental Theorem of Calculus. Somehow, the process of finding the area under a curve is intimately related to finding an instantaneous rate of change. I then make a bold, eye-catching statement: The Fundamental Theorem of Calculus is one of the greatest discoveries in the history of mankind, period. And, at the ripe old age of 17, 18, or 19 years old, my students are now privileged to understand this great accomplishment.

This ends my introduction to Calculus I. I’ll then begin the more mundane development of limits on the way to formally defining a derivative.

Day One of my Calculus I class: Part 5

I’ve told students that the topics in Calculus I build upon each other (unlike the topics of Precalculus), but that there are going to be two themes that run throughout the course:

Approximating curved things by straight things, and
Passing to limits

We are now trying to answer the following problem.

Problem #2. Find the area under the parabola $f(x) = x^2$ between $x=0$ and $x=1$ .

Using five rectangles with right endpoints, we find the approximate answer of $0.44$ . With ten rectangles, the approximation is $0.385$ . With one hundred rectangles (and Microsoft Excel), the approximation is $0.33835$ . This last expression was found by evaluating

$0.01[ (0.01)^2 + (0.02)^2 + \dots + (0.99)^2 + 1^2]$

At this juncture, what I’ll do depends on my students’ background. For many years, I had the same group of students for both Precalculus and Calculus I, and so I knew full well that they had seen the formula for $\displaystyle \sum_{k=1}^n k^2$ . And so I’d feel comfortable showing my students the contents of this post. However, if I didn’t know for sure that my students had at least seen this formula, I probably would just ask them to guess the limiting answer without doing any of the algebra to follow.

Assuming students have the necessary prerequisite knowledge, I’ll ask, “What happens if we have $n$ rectangles?” Without much difficulty, they’ll see that the rectangles have a common width of $1/n$ . The heights of the rectangles take a little more work to determine. I’ll usually work left to right. The left-most rectangle has right-most $x-$ coordinate of $1/n$ , and so the height of the leftmost rectangle is $(1/n)^2$ . The next rectangle has a height of $(2/n)^2$ , and so we must evaluate

$\displaystyle \frac{1}{n} \left[ \frac{1^2}{n^2} + \frac{2^2}{n^2} + \dots + \frac{n^2}{n^2} \right]$ , or

$\displaystyle \frac{1}{n^3} \left[ 1^2 + 2^2 + \dots + n^2 \right]$

I then ask my class, what’s the formula for this sum? Invariably, they’ve forgotten the formula in the five or six weeks between the end of Precalculus and the start of Calculus I, and I’ll tease them about this a bit. Eventually, I’ll give them the answer (or someone volunteers an answer that’s either correct or partially correct):

$\displaystyle \frac{1}{n^3} \times \frac{n(n+1)(2n+1)}{6}$ , or $\frac{(n+1)(2n+1)}{6n^2}$ .

I’ll then directly verify that our previous numerical work matches this expression by plugging in $n=5$ , $n= 10$ , and $n = 100$ .

I then ask, “What limit do we need to take this time?” Occasionally, I’ll get the incorrect answer of sending $n$ to zero, as students sometimes get mixed up thinking about the width of the rectangles instead of the number of rectangles. Eventually, the class will agree that we should send $n$ to plus infinity. Fortunately, the answer $\displaystyle \frac{(n+1)(2n+1)}{6n^2}$ is an example of a rational function, and so the horizontal asymptote can be immediately determined by dividing the leading coefficients of the numerator and denominator (since both have degree 2). We conclude that the limit is $2/6 = 1/3$ , and so that’s the area under the parabola.

Day One of my Calculus I class: Part 4

I’ve told students that the topics in Calculus I build upon each other (unlike the topics of Precalculus), but that there are going to be two themes that run throughout the course:

Approximating curved things by straight things, and
Passing to limits

I then applied these two themes to find the speed of a falling object at impact.

I now switch to a second, completely unrelated (or at least it seems completely unrelated) problem.

Problem #2. Find the area under the parabola $f(x) = x^2$ between $x=0$ and $x=1$ .

I draw the picture and ask, “OK, what formula from geometry can we use for this one?” Stunned silence.

I say, “Of course you can’t do this yet. This is a curved thing. Back in high school geometry, you learned (with one exception) the areas of straight things. What straight things had area formulas in high school geometry?” I’ll always get rectangles and triangles as responses. Occasionally, someone will volunteer parallelogram or rhombus or kite.

So I ask the leading question, which of these shapes is easiest? Students always answer, “Rectangles.” Which then leads me to the next question: How can we approximate the area under a parabola with a bunch of rectangles?

Again, stunned silence. I let my students think about it for at least a minute, sometimes two minutes. Hopefully, one student will volunteer the answer that I want, though occasionally I’ll have to coax it out of them.

Eventually either a student volunteers (or else I tell the class) that we ought to use a bunch of thin rectangles. For starters, I’ll use five rectangles and a very rough sketch on the board.

I’ll start with the right-most rectangle… what is its area? Students immediately see that the width is $1/5$ , but the length takes a little bit more thought. And I make my students figure it out without me giving them the answer. Eventually, someone notices that the height is simply $f(1) = 1$ , so that the rightmost rectangle has an area of $0.2$ .

I then move to the rectangle that’s second to the right. This also has a width of $1/5$ , but the height is $(0.8)^2 = 0.64$ . So the area is $0.128$ .

Eventually, we get that the sum of the areas is $0.008 + 0.032 + 0.072 + 0.128 + 0.2 = 0.44$ . Students can easily see that this is a decent approximation to the area under the parabola, but it’s a bit too large.

I then ask the same question that I had before: how can we get a better approximation? Students will usually volunteer either “More rectangles” or “Thinner rectangles,” which of course are logically equivalent. I then proceed with 10 equal-width rectangles. Occasionally, a student volunteers that perhaps we should use thinner rectangles only on the right side of the figure, which of course is a very astute observation. However, I tell my class that, for the sake of simplicity, we’ll stick with rectangles of equal width.

With ten rectangles (and I redraw the picture with ten thin rectangles), the approximation is quickly found to be

$0.1 [ (0.1)^2 + (0.2)^2 + \dots + (0.9)^2 + 1^2] = 0.385$

I like using ten rectangles, as that’s probably the largest number that can be handled in class without a calculator (until the very last step of adding up the areas).

By now, the class sees what the next steps are: take more and more rectangles. At this point, I’ll resort to classroom technology to make the process a little quicker. I personally prefer Microsoft Excel, though other software packages can be used for this purpose. For $100$ rectangles, the class quickly sees that the sum of the rectangles is

$0.01 [ (0.01)^2 + (0.02)^2 + \dots + (0.99)^2 +( 1.00)^2] = 0.33835$

My class can see that the answer is still too large, but it’s certainly closer to the correct answer.

I’ll then tell the class that this is another example of passing to limits, the second theme of calculus. I’ll describe this more fully in the next post.

Day One of my Calculus I class: Part 3

I’ve just told students that the topics in Calculus I build upon each other (unlike the topics of Precalculus), but that there are going to be two themes that run throughout the course:

Approximating curved things by straight things, and
Passing to limits

We are now studying the following problem.

Problem #1. A building on campus is 144 feet tall. A professor takes a particularly annoying student to the top of the building, and throws him (or her) off to his (or her) certain demise. (Usually I pick a student that I know and like as the one to throw off the building. This became a badge of honor over the years.) The distance that the student travels (in feet) after $t$ seconds is $f(t) = 16t^2$ . How fast is the student going when he (or she) hits the concrete sidewalk?

At this point in the lecture, we have done some experimental numerical work with successfully smaller time intervals to find better and better approximations to the speed at impact.

With a time interval of length $3$ seconds, the approximation is $48$ ft/s.
With a time interval of length $1$ seconds, the approximation is $80$ ft/s.
With a time interval of length $0.5$ seconds, the approximation is $88$ ft/s.
With a time interval of length $0.1$ seconds, the approximation is $94.4$ ft/s.
With a time interval of length $0.01$ seconds, the approximation is $95.84$ ft/s.

I’ll then tell the class that this is an example of passing to limits, the second theme of calculus. By making the time intervals smaller and smaller, we get better and better approximations to the true speed at impact.

By this point, students realize that we’re getting better and better approximations… however, we’re probably not going to get the correct answer by just plugging in numbers. And we certainly can’t just take a time interval of $0$ seconds since dividing by zero is a no-no.

Depending on my read of the class — on whether or not they’re ready for a little more abstraction — I’ll then ask the class, “How can we make these fractions without plugging in all of these numbers?” Usually students are at a loss at first. Perhaps someone will volunteer that we ought to introduce a variable… but, in my experience, even bright students at the start of calculus do not have this step of abstraction at the tips of their fingers. So I’ll lay out the fractions that we’ve studied so far, like

$\displaystyle \frac{f(3) - f(2.9)}{0.1} \qquad$ and $\qquad \displaystyle \frac{f(3)-f(2.99)}{0.01}$ ,

and ask, “How could we do this more systematically? Does anyone see a pattern in these fractions?” Hopefully someone will notice that the input of the second function call is 3 minus the denominator; if not, I’ll volunteer this observation to the class. So both of these fractions can be written as

$\displaystyle \frac{f(3) - f(3-h)}{h}$ ,

where $h$ is a small positive number. Let’s now simplify this fraction:

$\displaystyle \frac{f(3) - f(3-h)}{h} = \displaystyle \frac{16(3)^2 - 16(3-h)^2}{h}$

$= \displaystyle \frac{144 - 16(9-6h+h^2)}{h}$

$= \displaystyle \frac{144 - 144 + 96h - 16h^2}{h}$

$= \displaystyle \frac{96h - 16h^2}{h}$

$= \displaystyle 96 - 16h$ .

The last step is permitted because $h$ is assumed to be a nonzero number. I then check to see if the previous work matches this algebraic expression:

If $h=1$ , then $96 - 16h = 80$ , matching the previous answer.
If $h=0.1$ , then $96-16h = 94.8$ , matching the previous answer.

I then ask the class, what’s the ultimate goal with $h$ ? The answer: send $h$ to zero. So we conclude that the velocity at impact is $96 - 16(0) = 96$ ft/s, which is the final answer.

Reviewing, the curved thing was the changing speed of the falling object, which was approximated by the straight thing, the ordinary distance-rate-time formula. Finally, we passed to limits to find the real velocity at impact.

All of the above is eventually done more systematically later in the semester after the properties of derivatives have been more fully developed. However, I think that doing this calculation on the very first day of class gives my students a taste of what’s going to be happening in the days and weeks to come. Again, I emphasize that I probably cover this material in maybe 15-20 minutes, and that I don’t hold students immediately responsible for repeating such a calculation on their own. (I do hold them responsible for this, of course, after they know how to differentiate $f(t) = 16 t^2$ .