Ahoy again! What can you see now? Building Rational Functions
Lesson 4 of 12
Objective: SWBAT to develop a non-linear model and formally define and apply inverse and direct variation.
Today’s activity definitely requires some prep work on behalf of the teacher well before the lesson (and probably some money too!). To complete today’s investigation students are going to need at least 8 viewing tubes of various sizes. The tubes should range in length from 10-70cm, but all have the same diameter. Paper towel, toilet paper, and wrapping paper tubes could be used and pieced together to make the various lengths. However, I plan to just buy some PVC piping and cut it with a hacksaw to various lengths. I figure this will last better from hour to hour and year to year. I am going to shoot to cut about 20 different tubes of varying lengths so that my 13 teams can share these and there will be plenty to rotate through. I also am going to label the tube with its length in centimeters to save students some class time.
I am purposely skipping a warm-up type activity today to allow more time for data collection. I also did not assign homework last night so we should be starting the activity right away.
To introduce today’s activity, I am going to take a moment to have students’ recall what they found with Arthur the Pirate’s spyglass problem from unit 1. In particular, I want students to recall that the further they stood from the wall the more they could see through their view tube and the closer they were the less they could see. At this point, I am just going to mention that this is an example of direct variation. As the distance from the wall increased/decreased the diameter of the viewing window increased/decreased by the same factor. We will talk in more detail about this later this class period.
In this activity students should be collecting data from a fixed distance from the wall. You may want to assign a distance to all teams if you want all students to come up with the same function, or you may want to allow students to select their own distance (and keep it the same from measurement to measurement) to get a variety of functions.
I think the variety of functions would definitely lead to more interesting conversations in the end. However, I don’t know that my students are quite ready to discuss the different aspects of a rational function that this will raise. I am also concerned about whether this will require too much class time. However, I am going to allow each group of students to choose a distance. I think it will provide a great context down the road for stretch factors and evaluating k in the rational form y=k/x even if we don’t discuss that today.
Once students collect their 8 data points, I ask them to make a scatter plot of the data using their calculator. I expect that the scatter plot will help students come up with an equation to model their data. My idea is that students should pick up on the fact that this situation models a function similar to the parent graph y=1/x. The should understand transformations well enough to appreciate that the parent graph has been stretched. Clearly, this part of the activity screams Math Practice 4. But, other mathematical habits of mind are also being developed in this part of the activity. In order to complete the task, students are paying attention to the structure of a rational functions and how a stretch factor affects a plot (Math Practice 7). They will also need to identify patterns make inferences to create a good model for the data (Math Practice 8). Of course, as students use their calculator to test their models they are also engaging in Math Practice 5.
For students who are not ready to reason from a general model, I will encourage a guess and check approach to finding the stretch factor in Question 9. I will have them pick a function they think works, graph it on the same plane as their scatter plot, and then make an educated guess as to whether that stretch factor needs to increase or decrease. They can keep tweaking their equation until it is close. I think this will really help students to see how that constant value affects the graphs.
If possible, I will ask these students to also solve for the stretch factor algebraically. In order to support this task, I will likely help students to select a point (x,y) and guide them as the solve for the constant.
Today’s closure question is going to be a bit tricky for students.
I believe it’s the language that trips up my students on a question like this. It is important to help students decode the phrase "varies directly with the square of x." To do this I will ask things like:
What does it mean to vary directly?
When y varies directly with x, what do we know? Look back at the definition.
If y varies directly with x squared what does that mean?
Today, I may go as far as providing students with a mathematical hint: y=kx^2.
Tonight, I will assign Homework 3 - Rational Functions from this unit. This assignment helps students to continue practicing solving rational equations and simplify rational expressions. It also prompts students to recall transformation rules of basic functions to help students be more successful in upcoming lessons on transformation of rational functions. Students also practice skills and concepts developed in class today like sketching the graphs of rational functions and applying definitions of inverse and direct variations.
Teaching Note: I will make sure to remind students that their about why dividing by zero causes a function to be undefined needs to be complete prior to tomorrow's class. See the homework section of Lesson 1 from this unit for more information about this assignment.