Designing a Mathematical Rollercoaster

thingiverse

: In this activity, students use their knowledge of different types of functions and what their graphs look like in order to design a rollercoaster. Students are given certain specifications they must follow. First they will hand design the coaster, then they will use a graph plotter to digitally plot their graphs. Finally, students will use modeling software to 3D print their rollercoaster and test it out with a marble or small ball. Team: 3Dears Team Members: Alice Nudelman, Elana Reiser, Laura Levin Print Settings Printer Brand: MakerBot Printer: MakerBot Replicator (5th Generation) Rafts: Yes Supports: No Resolution: Standard Infill: use a low infill for prototyping Notes: Note: Use a low infill for prototyping then a higher infill percentage for the final version. How I Designed This How I designed this: NOTE: The full design description is included in the lesson plan. Tips for designing the 3D image: If you are making an inner groove, be sure to put the groove on the correct side of the rollercoaster (i.e. the inside). It’s easy to get disoriented when you are zoomed into the object. On one of our prototypes, the ball got stuck at the bottom and was not able to complete the track because we made the curves too steep. (see image) If using a 3D modeling software program such as 123d Design, be careful not to use a Filet inside of the groove since it doesn't print properly since the inner strands have no support. (see image) You might want to add a support if your rollercoaster begins at a high point. Use trial and error to get just the right design. So much is learned when redesigning projects. Students should ask each other for assistance (peer tutoring and peer mentoring) when designing their formulas. Here the angles were too steep. The filament didn't have enough of a support when using a Filet. Standards CCSS Overview and Background This project was designed in the spirit of the common core standards. Students apply mathematical knowledge to a real-world example. They work in groups to make sense of their task of designing a rollercoaster and then persevere in solving it. Students then analyze their models and attend to precision as they explain to other group members why to choose a certain type of function and label their graphs. Students also use appropriate tools, such as graphing and modeling software. Finally, students present their work. Objectives After completing this project, students will learn • How rollercoasters work, including some basic physics. • How to apply their knowledge of functions and transformations to build a model. • How to create a 3D model of their rollercoaster design. Subjects Math & Physics Audiences This project is intended for high school or college students who have studied functions. Skills Learned (Standards) Students get practice graphing various types of functions such as sine and cosine curves, polynomials, logarithmic functions, and linear functions. They will have to make transformations of these graphs and also analyze them. This includes finding relative maximum and minimum values, finding intervals of increasing and decreasing, and finding intersection points. The following common core standards are satisfied: CCSS.Math.Content.HSF.IF.C.7.a Graph linear and quadratic functions and show intercepts, maxima, and minima. CCSS.Math.Content.HSF.IF.C.7.b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. CCSS.Math.Content.HSF.IF.C.7.c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. CCSS.Math.Content.HSF.IF.C.7.e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. CCSS.Math.Content.HSF.BF.A.1.b Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. CCSS.Math.Content.HSF.BF.B.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Next Generation Science Standards: MS-PS3-2 Develop a model to describe that when the arrangement of objects interacting at a distance changes, different amounts of potential energy are stored in the system. Lesson Plan and Activity Review/Introduction (if necessary) Review how to graph linear, polynomial, logarithmic, trigonometric, and exponential functions, as well as doing transformations of them. Day 1: Background and Start Design First ask students to share anything they know about rollercoasters. Then you can show pictures of rollercoasters such as the ones in picture 1. If it hasn’t already been mentioned, students can see the loops, the dips and hills, and the banked turns. Students may notice that rollercoasters don’t have circular loops. This is because when they used to be built like that passengers got neck injuries. Making the loop tear-shaped has fixed this problem (optional physics explanation can be found here: http://gizmodo.com/why-roller-coaster-loops-are-never-circular-1549063718). Explain how a rollercoaster works: A chain pulls the car to the top of the first hill. After that, it coasts through the rest of the track. For this to work, the first hill has to be the biggest one. Ask students to read through the following page: (http://www.physicsclassroom.com/mmedia/energy/ce.cfm) where they can learn about the law of conservation of energy. They will learn that potential energy is highest at the top of a hill and kinetic energy is highest at the bottom of the hill when the rollercoaster is moving fastest. There is an animation that shows both types of energy levels at various points on the rollercoaster (see picture 2). Now onto the fun part! Ask students to design their own rollercoaster given the knowledge they now possess. Since we will be modeling with a marble or ball, ask students to start their coaster at the top of a hill. Imagine that there is a set of stairs that lead to the top of the hill and that is where the ride starts. You may want to not allow loops as we have found them difficult to print. If you are able to figure out how to make loops work, please share in comments. Give students the lesson handout/worksheet to complete in groups. Heterogeneous grouping works best so that students of varying levels can learn from each other[1], but it has been found that this tends to occur when groups are assigned at random[2]. If you would like to ensure individual accountability, you can assign each group member a role, such as one person in charge of the final sketch, one person in charge of plotting the graphs, and one person in charge of using the 3D modeling software. On the handout, they will be sketching a design of their rollercoaster, explaining how they got each piece of their graph and transformations used, finding where their graph is increasing and decreasing, points of relative extrema, and where potential and kinetic energy are at their maximum. They will then be asked to plot their graph using a free browser-based grapher located at desmos.com. Days 2 & 3: Finish Worksheet Once students have their graph plotted (see pic 3), ask them to make sure that all pieces are drawn in black and remove the gridlines. They can then grab a screenshot of the graph (see pic 4). One method to do this is by creating a Desmos account then saving the graph. On the top right an option will appear to share your graph. Clicking on Share and then Image will create a good quality screenshot they can save. Day 4 & 5: 3D Modeling The next step is to create a 3D file to print the rollercoasters. The method we used is to first convert their file into an .svg file. This can be done with a free online converter such as http://image.online-convert.com/convert-to-svg. We then imported this file into 123d Design, http://www.123dapp.com/design(see pic 5). Then we drew a Spline that resembled the original curve. Splines are curves that connect several points. If needed, a tutorial can be seen at https://www.lynda.com/123D-Design-tutorials/Drawing-splines/371320/415659-4.html. Then delete the original curve. Change the grid linear snap as needed (see pic 6). Next, we scaled the spline to the maximum printer size. This can be done by adding lines in order to make the sketch an enclosed region. We then scaled the closed sketch by measuring the length and height and then scaling it according to the printer bed size (see pic 7 & 8). After scaling we deleted the added lines to get the curve back. Next, we drew the profile shape of the groove we wanted to put on our curve to make a track on it. We chose half of a hexagon for printability purposes (see pic 9 & 10). We originally used a half-circle but this printed incorrectly because there was no support for the top half as it was printing. We then rotated the sketch to position the profile shape perpendicular to the curve plane (see pic 11). Make sure the groove is positioned correctly. Then we Swept this profile through the curve path and saved our file (see pic 12). We added a support to the front so that our model would stand up (see pic 13). This was then exported as an .stl file and brought to the MakerBot desktop software. We positioned the curve to lay flat and then exported as a MakerBot file to print. You can decide how much support and tutorials to give your students on this process but our advice is to allow them to make mistakes as they will learn best by figuring out how to fix them. Print each group’s rollercoaster outside of class. Day 6: Wrap-up Have each group try rolling a marble or ball down their rollercoaster. Then they will each present to the class, sharing their answers to worksheet questions and how they made their model. Duration of Lesson It is estimated that this project will take 6 days, but may take longer if students need to review functions or need longer to design their rollercoasters. Preparation Students should know what linear, polynomial, trigonometric, logarithmic, and exponential functions are, how to graph them, and how to perform transformations on them. Rubric and Assessment By the end of this project, students will have used their knowledge of various types of functions to build a rollercoaster model. First they will sketch a design, then they will use technology to plot their graph. Finally, they will build a 3D model of their design. A possible rubric for grading is below: (10 points) Students followed the rules for constructing their graph. (50 points) The equations are correctly labeled and the domains make sense. (20 points) Students answered worksheet questions correctly. (20 points) Student participated in group work and class discussions. Handouts & Assets: The student worksheet is attached. This includes a description of their assignment and a place to record answers. References Optional physics explanation of why rollercoasters need tear shaped loops and not circular loops:http://gizmodo.com/why-roller-coaster-loops-are-never-circular-1549063718 Website for students to read to learn how rollercoasters follow the law of conservation of energy:http://www.physicsclassroom.com/mmedia/energy/ce.cfm Video for students to watch to learn how to label their graphshttps://www.educreations.com/lesson/view/rollercoaster-demo/39271878/ Webpage for browser-based graphing software students use to plot their design: desmos.com Converter we used to convert from image file to .svg file:http://image.online-convert.com/convert-to-svg Modeling software to design the 3D image:http://www.123dapp.com/design Spline tutorial:https://www.lynda.com/123D-Design-tutorials/Drawing-splines/371320/415659-4.html Notes: Kagan, S. (1994). Cooperative Learning. San Clamente, CA: Kagan Publishing. Stahl, R.J. (1994). The essential elements of cooperative learning in the classroom. (ERIC Document Reproduction Service No. ED 370881).