Experience First

Today students dive into the world of rational functions by looking at the concentration of anesthesia in a patient’s body. First students make sense of the shape of a rational function by sharing noticings and wonderings about the graph of C(t). Students find times at which the concentration of anesthesia reaches a certain value and tie this in to solving a rational function. A large focus of the activity is thinking about what happens to the concentration of anesthesia after a long, long time, arriving at the idea of a horizontal asymptote. It is interesting to note that while the concentration does not actually reach 0 according to this model, a drug is considered out of the patient’s system if the concentration is below 0.7%.

As you are monitoring students, continue to use the language of “after a long, long time” or “after a very long time” to get at the idea of a horizontal asymptote.

Students may get stuck in solving rational functions algebraically. Usually they are hesitant to try something they are not sure will work. Encourage students to test out their ideas instead of steering them to the proper way of solving the rational function immediately. When students are stuck, a helpful question can be “If you knew what to do, what would you do?” This seems silly to us because obviously they’re stuck because they don’t know what to do, but in my experience this question has worked 100% of the time and students just need your permission to try out what they were already thinking of doing in their head. In most cases they do have an idea and some strategies they can pull from!

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Formalize Later

The idea of comparing the degree of the numerator and denominator for end behavior is a critical one moving forward into AP Calculus. Even though we are not using formal limit notation in Precalculus, the idea of a function approaching a certain value as x goes to infinity is an important one! Have students make their own conjectures about whether there will or will not be a horizontal asymptote and why. Encourage them to think about what they know about fractions and what will make a fraction small versus large.

You may find that students need more than the Important Ideas notes to get a feel for slant asymptotes. We strongly recommend doing question 2 on the Check Your Understanding together and looking at the graph to identify what a slant asymptote looks like. Have students explain to each other why the equation of the slant asymptote will always be linear.