# 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.

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

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