After viewing “Snowboarding,” the online tutorial simulation from Unit 3 of NROC’s Algebra 1—An Open Course, that reviews the properties of functions and relations, students draw a graph representing the hills and loops of a roller coaster. The students use coordinates from their graph to discuss different types of relations (functions, linear functions, etc.). Finally, they create a scale model of a rollercoaster, thereby applying their learning to the use of a proportional function in a real world situation. This blog post builds on the “Design a Roller Coaster,” also from Unit 3.
• Identify linear functions, relations, and non-linear functions from graphs.
• Create a linear equation from two points on a graph.
• Use a proportional function in a real world endeavor.
This capstone project allows students to showcase their understanding of functions and relations as well as apply them to a 3-D construction project.
Applying the abstract concepts of functions and relations to real life can be difficult for a student. Projects and real life examples can help. The information below is an overall outline of how such a project can be introduced. Also provided is a link to a handout to help guide students through the project.
1. Warm Up: Have your students respond freely in a journal to the following prompt. “How do Roller Coasters work? What makes them fun? What (if anything) do they have to do with functions?” (Example answers: The work on gravity. Getting pulled around and going fast are fun! How fast a coaster goes is a function of how far it’s fallen. How pulled you feel when going around a corner is a function of how tight a turn is and how fast you’re going.)
2. Explain that today you are going to begin a project that will begin with using functions and relations to design a track and end with building scale models of roller coasters. First, you’re going to see a tutorial simulation that uses snowboarding tracks to show you how to represent a track or path as a graphed relation between height and distance. It will help you identify functions and non-functions using graphs and tables also.
3. Show the simulation, having students discuss and vote on correct answers. When finished ask: If shown a graph of another track, a roller coaster track, could you do what we did here?
4. Students work in pairs at computers or individually with graph provided as part of the “Design a Roller Coaster,” project from Unit 3 of Algebra 1—An open Course. They should answer questions one and two either from the website or from the handout I created to go with the project. Note that the students are being asked to do the same thing as in the snowboarding tutoring simulation they just viewed: Identify sections of the track as representing a linear function, non linear function, or not a function (it does not pass the vertical line test). They are NOT being asked to generate more advanced functions from the graph provided. You can have them write an equation for the portion of the graph that represents a linear function.
5. Introduce them to the project idea of building a scale model of a roller coaster. When one scales something, one uses a linear, proportional function, so this provides awesome real life use of the math they’ve been learning. Essentially, they will need to generate a scale factor using the ratio between the height their coaster would be in real life, and how tall the maximum height they plan for their model. This “factor”—the ratio between the true height and the height of their model--will determine all other heights in their model.
This project is based on the NROC Algebra 1--An open Course Rollercoaster Project, but a bit more fleshed out. To me, hands on building and real life applications are what make projects awesome, so I’ve added it in! See the handouts I’ve made to guide this project. WARNING: They are not complete, but I’d love your help to make them that way! I’ve posted what I have in a Shared Google Doc. Please feel free to edit and improve, and let me know what you’ve done! You’re welcome to edit the original if you feel confident or to send me written up ideas for future inclusions.
• Fun and innovation should be encouraged. If students would rather make a scale model of their house, the school, an airplane, or most anything, have them go for it! No kits, though.
• Many other functions apply to roller coasters than scaling functions for models. For example, if friction is ignored, the speed of a rollercoaster at any point along the track is determined by how far it has fallen from rest at the top of the coaster’s highest hill. As an extension project, have some students try to find that relation, determine whether it is a function, and if so can they use it to predict speed? (for the instructor: It’s a non-linear function sqrt(2gh) is what you’re looking for, where “g” is acceleration on earth from gravity and “h” is the height fallen.
• For students struggling with finding the equation of a linear portion of a graph invite them to review the NROC presentations from Unit 3 and perhaps redo the tutorial Snowboarding simulation from Unit 3 on functions and linear proportionality.
• In a classroom with limited space resources, students can create a scale illustration on large butcher paper of their planed coaster. Remember, they should still report the scale factor between their original graph and their poster as well as the scale factor between their poster and real life.
• This is a very scalable activity (pun intended). If you don't want to do the full project, you can still watch the snowboarding tutorial sim and discover the equation of a line on the NROC Algebra 1 Roller Coaster Project website. Then, instead of having students make coasters, print out pictures of coasters. Have students find out how tall the coaster is in real life and have them figure out: the proportionality between the image and the actual coaster, 2. The slope (or entire equation) of some linear portion of the coaster if it is projected onto an X,Y coordinate system.
Please refer to the Google document handout.