# Engaging students: Polynomials and non-linear 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 Jessica Martinez. Her topic, from Algebra II: polynomials and non-linear functions.

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

There is a Ted Talk video showing the math behind professional basketball player Michael Jordan’s hang time. The video connects a popular sport and player with mathematics by using quadratic equations to explain how MJ stays in the air as long as he does. You can see that the video is aimed at a younger audience since it’s done with cartoon animation, and it’s fairly easy to follow along as it explains the math. The video explains how they derived the formula for MJ’s jump shot by using his initial velocity and the force of gravity along with the variable of time. It also provides a great visual representation of how jumping into the air resembles a parabola of a quadratic function when they place MJ jumping against a graph. The video shows how applicable quadratics are by explaining that the roots of the parabola of MJ’s jump shot are the spots where he jumps and where he lands again. We could also calculate the maximum height of MJ’s jump by finding the vertex of the parabola and I could modify the equation as a problem for my students to solve. For example, we could look up the world record for highest jump and I could ask my students to calculate what the initial velocity would be for that person to get the highest jump using MJ’s hang time.

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

As a student, the first couple of times I looked at the graphs of polynomials I always thought, “Huh, those kind of look like rollercoasters”. I did some researching and I found a project where students are asked to use polynomials to analyze and design rollercoasters. As a teacher, I could introduce this project with a short video or advertisement of a popular theme park (like Six Flags) to get their interest and show some of the cool rollercoasters in action. Then I would have the students answer word problems about rollercoasters and their polynomial functions to find the local max/min of the coasters, where the function is increasing/decreasing (riding down or up on the rollercoaster), and what type of function best models certain parts of the coaster (quadratic, cubic or quartic). After my students have worked some polynomial function problems, I would have them pair up or work in groups to design their own rollercoaster using polynomials. I would also like to collaborate with a physics teacher as well; by using physics, my students could test the equations of their coasters with velocity, force and acceleration and see if they are realistic or not (and they could also see how this topic extends to other courses).

What interesting (i.e., uncontrived) word problems using this topic can your students do now?

Whenever a new infectious disease begins to spread rapidly (like Ebola or the Zika virus), there is coverage of the spread all over the news, making this topic highly relevant for my students. The spread of infectious disease can be modeled through non-linear functions such as exponential functions. I could create multiple word problems about the Ebola outbreak in Africa; for example, I could have my students pretend that scientists have developed a vaccine for the Ebola virus but now the problem is distributing the vaccine to all of the infected people. I would have my students pretend they were a disease control team trying to race against the spread of this disease in order to vaccinate the people before it was too late. By using actual date on reported cases in a specific country in Africa (like Liberia or Sierra Leone), my students could find the exponential function that best represents their data. They could then use that function to estimate the time it would take for all of the population in their country to be infected and compare that to the rate and time it would take to distribute all of the vaccines to the people (making estimates based on research of the country and how it has handled disease spread in the past). Since actual data won’t always match precisely with a mathematical function, I would have my students discuss what other variables and factors could affect their calculations as well.

References

Dawdy, T. (n.d.). Roller Coaster Polynomials. Retrieved September 23, 2016, from http://betterlesson.com/lesson/435674/roller-coaster-polynomials

Honner, P. (2014, November 05). Exponential Outbreaks: The Mathematics of Epidemics. Retrieved September 23, 2016, from http://learning.blogs.nytimes.com/2014/11/05/exponential-outbreaks-the-mathematics-of-epidemics/?_r=0

TEDEd. (2015, June 04). The math behind Michael Jordan’s legendary hang time – Andy Peterson and Zack Patterson. Retrieved September 23, 2016, from https://www.youtube.com/watch?v=sDbmcPnzwy4

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