Engaging students: Simplifying 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 Peter Buhler. His topic, from Algebra II/Precalculus: simplifying rational expressions.

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

One activity that could be performed when introducing rational expressions is to demonstrate the reason for simplifying. Before teaching students to simplify, instead ask them to evaluate the expressions given various x values. As they struggle through the painstaking process of taking squares, distributing, multiplying, adding and subtracting as they attempt to evaluate the rational expression, take note of how long it may take the students. Then have several students share their method. Following the student sharing, show your efficient method that allows you to simplify the expression before beginning to evaluate.
This not only shows the students that it is quicker, but it often provides more accurate answers to the process that must be taken to “cancel” the terms and then evaluate. Students should be more willing to participate in the following lesson on simplification due to the desire to do less work. This could also be an opportunity to discuss why it is often helpful to look for “shortcuts” or tools that can be used to simplify long or tricky problems into something manageable, even by high school students.

 

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B2. How does this topic extend what your students should have learned in previous courses?

This topic actually extends several previous topics seen in middle school mathematics. One of these topics is reducing fractions. This actually builds on the topic of finding the greatest common factor (GCF), which students learn in elementary school. To reduce a fraction, students find a GCF from both the top and bottom of the fraction, and then simply eliminate that factor leaving the expression in a simplified form. This could be utilized to introduce the idea of simplifying rational expressions, as students will likely be familiar with reducing fractions to their most simplified form.
This can also be applied to multiplying by fractions, as the GCF can be pulled out of the top and bottom of the fractions and simplified, making the multiplication of the fraction simpler. One last possible application could be in solving proportions, as students are typically taught to simplify the proportions before attempting to solve. The common theme in all of these is simplifying in order to make a problem easier and is a more efficient process for most students.

 

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D2. How was this topic adopted by the mathematical community?

There are many advanced applications of simplifying rational expressions. One such function is the Pade approximant, which is an approximation of a rational function of a given order. It was created by Henri Pade in 1890 and has been used to model certain rational functions. While this is certainly an advanced rational expression, it still holds true as there is a polynomial on the top and the bottom, which can be factored and simplified.
Rational functions have also been commonly used to model certain equations in STEM field such as functions of wave patterns for molecular particles, various forces in physics, and other fields that take mathematical ideas and apply them to a science. As a teacher introducing the topic of simplifying these expressions, one could display various applications of these functions and how they are used in a day-to-day setting. Students should be able to see beyond the cut-and-dry steps of simplifying the expressions and understand the implications beyond what they are doing.

References:

http://blog.mrmeyer.com/2015/if-simplifying-rational-expressions-is-aspirin-then-how-do-you-create-the-headache/
https://en.wikipedia.org/wiki/Rational_function

 

 

Engaging students: Solving Equations with Rational Functions

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 Heidee Nicoll. Her topic, from Precalculus: solving equations with rational functions.

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

To jog the students’ memory of what rational functions look like and what some of their properties are, I would do a relay race with them.  The class would be divided into two groups, and each group would have a different rational function, not anything too difficult, but something for which they could easily compute values, something like f(x)=-2/x and g(x)=3/x.  On the board would be two large papers, each with a table of values and a blank graph.  The x-values would be filled in, but the y-values would be blank.  The students would line up, and the first student in each line has to compute the y-value for the first given x-value, then grab the one marker for his/her team, go up to the board and write that value in the table.  The next student will compute the next value, and so on.  The students would be able to use the calculators on their phones if necessary, but they would not be able to use graphing calculators since they would be able to just plug the function in and look at the table.  Once the teams had all the y-values written down, the next student would have to come up to the board and plot the first point on the graph, and so on, until all the points were plotted.  The very last student would connect the dots to make a curve.  Then we could have a class discussion about vertical asymptotes, and how they show up in the table as an error or undefined value.  We could talk about what they remember of end behavior, horizontal asymptotes, x- and y-intercepts, and that could lead into the rest of the lesson.

 

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

Desmos online graphing calculator is quite nifty.  The functions show up in different colors, and you can graph points as well as lines and curves.  I found a sort of online worksheet on Desmos talking about rational functions, and modified it.  This is the link to the modified version: https://www.desmos.com/calculator/zi62lrxnim It leads the student step by step, as they click on each function to see it on the graph, through looking at the vertical asymptotes, x- and y-intercepts, any holes or slant asymptotes, and at the very end gets them thinking about intersections and solving equations.  The purpose would be to remind the students of all the properties of rational functions that we should think about when solving, and how graphing the functions to get a solution is a viable option.  In the activity, the students are also asked to move a few slides to graph the correct asymptotes.  In this way they are not just taking in information, but are required to provide some answers of their own.  All of this information should be already learned, so it would just be a review for the students as they take what they already know and learn how to apply it to solving equations with rational functions.

 

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What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

This is a paragraph from Encyclopedia Britannica about Apollonius of Perga and his contributions to geometry.

Greek geometry entered its golden age in the 3rd century. This was a period rich with geometric discoveries, particularly in the solution of problems by analysis and other methods, and was dominated by the achievements of two figures: Archimedes of Syracuse(early 3rd century bc) and Apollonius of Perga (late 3rd century bc). Apollonius is best known for his Conics, a treatise in eight books (Books I–IV survive in Greek, V–VII in a medieval Arabic translation; Book VIII is lost). Apollonius presented a comprehensive survey of the properties of these [parabolas, hyperbolas, and ellipses]. A sample of the topics he covered includes the following: the relations satisfied by the diameters and tangents of conics (Book I); how hyperbolas are related to their “asymptotes,” the lines they approach without ever meeting (Book II); how to draw tangents to given conics (Book II); relations of chords intersecting in conics (Book III); the determination of the number of ways in which conics may intersect (Book IV); how to draw “normal” lines to conics (that is, lines meeting them at right angles; Book V); and the congruence and similarity of conics (Book VI).  (Knorr).

We would read it as a class and I would point out that a hyperbola is the parent function for rational functions, y=1/x, and that when we are talking about asymptotes, we are using information that Apollonius worked on and studied.

Works Cited

Biographical Dictionary. n.d. Image. 18 November 2016.

Knorr, Wilbur R. Encyclopedia Britannica: Greek Mathematics. n.d. Website. 18 November 2016.

 

Original Desmos Activity: https://www.desmos.com/calculator/3azkdx4llk

Modified Desmos Activity: https://www.desmos.com/calculator/zi62lrxnim

Clowns and Graphing Rational Functions

I thought I had heard every silly mnemonic device for remembering mathematical formulas, but I recently heard a new one: the clowns BOBO, BOTU, and BETC for remembering how to graph rational functions.

  • BOB0: bigger (exponent) on bottom, x = 0
  • BOTU: bigger on top, undefined
  • BETC: bottom equals top eponent, coefficients (i.e., the ratio of coefficients)

Which naturally leads to this pearl of wisdom:

Engaging students: Graphing rational functions

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 Rory VerNesie. His topic, from Precalculus: graphing rational functions.

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So one question may be how this topic can be used in your students future courses in math or science?  The students start to learn about this concept in high school, and progressively builds on it until they are expected to know it in college.  Courses that require this are: Complex Analysis, Numerical Analysis, Differential Equations, Abstract Algebra, Real Analysis and Meromorphic Functions. These classes deal with understanding what happens as we approach a limit or when the denominator approaches zero.  In Abstract Algebra, they talk about a Field of Rational Expressions while Complex Analysis deals with a ratio of polynomials with complex coefficients. In Differential Equations, Rational Functions are seen in slope fields, Separable Equations, and Exact equations. Also in Real Analysis, the talk about convergence using 1/n.  Also, Laplace Transforms and partial fractions in electronics and physics  may need graphing along with partial fraction decomposition. All in all, graphing Rational Functions is a important part of math because they deal with division over zero or a singularity.

 

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Next, a very important man in the history of rational functions and their graphs was Charles Hermite. This man found a way or algorithm to integrate Rational Functions, while in early 19th century Ostrowski extended this idea and algorithm to Rational Expressions. The neat thing about Hermite is that he helped extend this idea to complex numbers and developed the idea of using interpolation to find the coefficients of rational functions. Without these contributions from Hermite and Ostrowski we would not be able to graph the derivatives of rational functions or the anti derivatives of rational functions. The methods discovered by these men were profound and in some ways led to the discovery of news ideas in math such as partial decomposition and other integration techniques that help integrate Rational Functions. Without these men, Rational Functions and there uses would be known about less.

 

 

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A great activity involving graphing rational functions would be to have the kids get into groups and assign the each group a certain rational function. Every group would have a team leader who would be in charge of making sure everything about the function gets done. These responsibilities would include, graphing the function, finding the zeros of the function, the asymptotes(Horizontal and Vertical), Removable Discontinuities if any, and the y intercept.  The students would then present what they found and would answer any questions for the class. This activity would be a good cooperative learning exercise for students who maybe are not the best at math. This could be a major confidence booster and fun activity for the students. Also they students are learning from each other so they are engaging in discovery learning.

All in all, graphing rational functions is a major part of mathematics and all of these statements mentioned above show how important rational functions are. They deal with division by zero and limits and are a great way to engage students in the novelty of singularities. Graphing polynomials also look really neat when you graph them.

Work Cited

http://integrals.wolfram.com/about/history/

http://www.sciencedirect.com/science/article/pii/0898122176900237

tutorial.math.lamar.edu

Slideplayer.com

 

Engaging students: The quadratic formula

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 former student Chais Price. His topic, from Algebra: the quadratic formula.

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What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

What is the quadratic formula you ask? It is basically a tool used to find roots of the quadratic equation. It all started around 3000 BC,when the Ancient Babylonians needed a method for calculating how much to pay the tax collector. Imagine that you are an Ancient Babylonian farmer with a square field that is placed in the middle of your farm. On this field you plan to plant some crops. After some experimenting you discover that if you double the length of each side of the square field, you end up with 4 times as many crops as before. This observation demonstrated the area of the field and the amount of crops that you can grow and the square of one of the side lengths are all proportional. From here we come up with the first quadratic equation. Let “x” represent the length of a side and “m” be the amount of crops you can grow on a square field of side length 1. Then we have “c” which is the entire area of crop that can be grown. Thus c= mx^2. It is now April 15 in Babylonia and the tax collector comes and says to give him “c” crops to pay your taxes. Now you ask yourself, how big does your square field need to be to grow that amount of crops? Well as it turns out, we just take x= (c/m)^1/2. The Babylonians however, did not have a way to solve square roots accurately. They were just estimations. This square field representation is just a basic representation. Let’s say that your field was not a square but a triangle divided into 2 right triangles where a and b are the amount of crops that you can grow in each field.

triangle

To calculate the amount of crops, you being a very intelligent Babylonian farmer, you come up with the equation

c= ax^2+bx.

The next step is to divide by “a” and then complete the square yielding

a \displaystyle \left(x + \frac{b}{2a} \right)^2 = x^2 + \displaystyle \frac{b}{a}x + \displaystyle \frac{b^2}{4a^2}

Now we substitute into the previous equation. We now have an equation of the form

a \displaystyle\left( x + \frac{b}{2a} \right)^2 = c + \displaystyle \frac{b^2}{4a^2}

Solving again for the tax collector, we need to solve for “x.” This gives us what we know as the quadratic formula:

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

Something worth noting since you are an Ancient Babylonian farmer, is that all the roots you find are positive since negative numbers have not been discovered yet. In addition, the quadratic formula shown above is just an illustration broken down step by step. The Babylonians had no general formula for the quadratic formula, but there method for the quadratic formula can be closely associated with the method of completing the square.

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How has this topic appeared in the news?

There is a pretty interesting article I read which had to do with the quadratic equation. A teacher provided a list of quadratic equations that the students would pick from. This equation was to be to model for a parabolic device that the student was responsible for building. The student was instructed to spend no more than 12 dollars on this project. They would use the equation to calculate a positional focus which suspended a marshmallow which would cook slowly due to the rays of the sun reflected upon it. This lesson stretched using a quadratic equation to form a focus. Then the student would graph the data and calculate the rate of temperature increase. Student were also asked to make predictions on what temperature the marshmallow would get to. Once the experiment is complete, the student is anxious to see how accurate their model is to the actual equation they chose. This is a very good lesson that covers a pretty broad range of topics.

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So the concept of the quadratic formula would develop beyond what the Ancient Babylonians discovered. 1500 years after the Babylonians, the Egyptians came along with a different approach for similar type problems where the quadratic formula was used. The mathematics behind the calculations were known to be very complicated. However, they recorded these calculation on a table ( much like our multiplication tables) so that when a farmer or an engineer required some kind of proportion or dimensional measurement using the quadratic formula, rather than solving the problem again, they would just look it up on the table that had already been calculated. They reproduced this table and made it a master look up guide to follow for such solutions.

Then the Babylonians would reappear in time and discover the method of completing the square. This method still required somewhat of an educated guess. However, since the Babylonians had a 60 base numerical system ( which the Egyptians did not) addition and multiplication were easier to compute and check calculations. Even still there was not a general formula or equation established. Then Pythagoras and Euclid came along and developed a general formula to solve the quadratic equation. Pythagoras argued that the ratio of the area of a square and the length of a side does not always produce an integer and indeed can have a rational solution. Euclid advanced beyond Pythagoras and claimed you could have irrational results. These were logical claims made from a theoretical point of view because at that time there was no way to calculate the square root of a number by hand. Finally in 700 AD, a mathematician named Brahmagupta from India came up with a general solution to the quadratic equation using numbers. He also was on board with Euclid and his irrationals as well as an equation producing 2 roots. His work more than likely inspired another Hindu Mathematician named Baskhara who around 1100 AD formulated the complete solution we are familiar with today. He was also the first to acknowledge that any number greater than 0 has 2 square roots. It was around this same time in history that a Jewish mathematician Abraham bar Hiyya derived the quadratic formula and brought it to Europe. It wouldn’t be until another 500 years that the quadratic formula would be adopted into the formula we know today.

 

Works Cited

 

Budd, Chris, and Chris Sangwin. “101 Uses of the Quadratic Equation.” Plus Math. Plus Magazine… Living Mathematics , 01 Mar 2004. Web. 10 Sep 2014. http://plus.maths.org/content/101-uses-quadratic-equation

Tracey, Wong Briggs. “Students use quadratic equations to cook marshmallows.” USA Today. USA Today, 05 Mar 2007. Web. 10 Sep 2014. http://usatoday30.usatoday.com/news/education/2007-03-04-teacher-parabola-side_N.htm?csp=34

Hell, Dr. . “The History Behind the Quadratic Formula .” . BBC H2G2, 13 October 2004. Web. 10 Sep 2014. http://news.bbc.co.uk/dna/place-lancashire/plain/A2982567

 

 

 

Engaging students: Finding the asymptotes of a rational function

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 former student Belle Duran. Her topic, from Algebra: finding the asymptotes of a rational function.

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How has this topic appeared in high culture?

Although the topic itself has not appeared in high culture, idea of asymptotes brings me the idea of the myth of Tantalus. In a nutshell, Tantalus was always committing crimes against the Gods of Olympus but always going unpunished. One day, he invites the Gods to his home for a feast in which he serves the Gods a rather vile dish. This ultimately angered the Gods to the point of punishing Tantalus by hanging him from a fruit tree amidst a lake, sentencing him to suffer eternal hunger and thirst. Tantalus was always so close to the water and fruits, yet they stayed beyond his reach. In the same way, when a graph has an asymptote then a part of the graph will approach that asymptote without ever touching it or being equal to it.

 

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What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

The word, “asymptote” derives from the Greek word, “asumptotos” which translates to “not falling together.” The term was first introduced by Apollonius of Perga in his work on conic sections, but used the term to represent a line that will not meet the curve in any finite point. Other achievements by Apollonius includes the introduction of eccentric and epicyclic motion to explain the motion of the planets as well as the hemicyclium which is a sundial with hour lines drawn on the surface of a conic section to give greater accuracy.

 

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How does this topic be used in your students’ future courses in mathematics or science?

One way finding asymptotes can be used in students’ future courses are to understand finding the limits of a function. When it comes to limits, it can be shown that vertical asymptotes are concerned with objectives in which the function is not usually defined and near which the function becomes large positively or negatively, or if a line x=a is called a vertical asymptote for the graph of a function of either the limit to positive infinity as x approaches positive a or negative a. Likewise, horizontal asymptotes are concerned with finite values approached by the function as the independent variable grows large positively or negatively. In other words, a line y=b is a horizontal asymptote for the graph is either the limit of the function is b as x approaches positive infinity or negative infinity.

References

http://www.greekmyths-greekmythology.com/the-myth-of-tantalus/

http://www-history.mcs.st-and.ac.uk/Biographies/Apollonius.html

http://jwilson.coe.uga.edu/emat6680/greene/emat6000/greek%20geom/Apollonius/apollonius.html

http://www.education.com/study-help/article/horizontal-vertical-asymptotes/

http://oregonstate.edu/instruct/mth251/cq/Stage3/Lesson/asymptotes.html

 

 

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.

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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.

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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.

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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&gt;.

“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.