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 Marlene Diaz. Her topic, from Precalculus: graphing rational functions.

How can this topic be used in your students’ future courses in mathematics or science?
When graphing rational functions, we are able to see the different asymptotes a function has. A rational function has horizontal, vertical and sometimes slant asymptotes. Knowing how to find the asymptotes and knowing how to graph them can help in future classes like Calculus and calculus 2. In those classes you will learn about limits. When finding the limit of a rational function the horizontal asymptote is checked and that’s what the limit is approaching. For example, we have BOTU, which is big on top is undefined, when undefined it can either be to negative or positive infinity and depending on what x is approaching. For example,

in this case we see that x has a higher degree on top therefore the limit is infinity. Another example would be

in this example we have that the degree is higher at the denominator therefore the limit is zero. In both cases we are able to evaluate both the limit and the horizontal asymptote and how they work with each other.

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

A fun activity that can be created to enforce the learning of graphing rational functions is a scavenger hunt. A student can be given a rational function to start the game, they have to find all the pieces that would help them find the graph of the function. The pieces they would have to have include the horizontal and vertical asymptotes. Once they find one piece at the back of the notecard there would be a hint of where the other piece can be. There would be other pieces mixed in with the correct one and the students would have to figure out which one they need. After they are done collecting all their cards, they would show them to the teacher and if it’s correct they get a second equation and if its incorrect they have to try again. This would most likely be played in groups of two and which ever team get the most correct will win a prize.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.

Something I have always used as a review or to better understand a topic is Khan Academy. The reason I think this website helps me is because you are able to watch a video on how to graph a rational function, there are notes based on the video and there are different examples that can be attempted by the student. Furthermore, the link I found to help learn the graphing of rational functions breaks every step down with different videos. The first video is called graphing rational functions according to asymptotes, the next one is with y-intercepts and the last one is with zeros. After seeing all the videos there are practice problems that the students can do. At the end of the link there are more videos but, in these videos, you can ask any questions that the you might still have, and you can also see previous questions asked. The way the website is organized and detailed can be very beneficial for a student to use and it is always good to give students different explanations of the topic. The link to Khan Academy is: https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:rational/x2ec2f6f830c9fb89:rational-graphs/v/horizontal-vertical-asymptotes

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.

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.

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.

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:

If Simplifying Rational Expressions Is Aspirin Then How Do You Create The Headache?

https://en.wikipedia.org/wiki/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 comes from my former student Heidee Nicoll. Her topic, from Precalculus: solving equations with rational functions.

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.

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.

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

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.

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.

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.

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