Rational Roller Coasters

Author:

Marie Argo

Unit Title:

Rational Roller Coasters

Grade:

8th

Subject:

Honors Algebra I

Estimated Duration:

8 days – 65 minute classes 

Unit Activities:

Activity 1: Roller Coaster Energy Placement & Designs

Activity 2: Rational Roller Coaster Design & Simulator

Activity 3: Building 3-D Model

Activity 4: Testing, Refining, and Communicating

Date:

June 2015

Unit Academic Standard

F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

F.IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

F-IF.1Understand that a function from one set (domain) to another set (range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of is the graph of the equation y = f(x)

F-IF. A.Int.1 Understand the concept of a functions and use function notation.

The Big Idea (including global relevance)

Engineers use knowledge of functions and energy to design roller coasters that are safe and entertaining. While safety and thrill tend to oppose each other, it is an engineer’s job to find the maximum amount of thrill and safety for each ride.

Global Relevance

Local: After 6 injuries upon opening the Son of Beast, the ride had to take out its 118 foot loop. Eventually, the ride was taken down in 2012 due to too many injuries (Story Link) (WKYC News, Kings Island: Son of Beat completely demolished).

State: Ohio averages almost 4,000 injuries per year related to Roller Coaster and Amusement Park Incidents.

Country: From 1990 – 2005, there were 52 deaths caused by thrill rides. The probability that you will get injured on a thrill ride is more likely than getting attacked by a shark. 

The Essential Question

What makes Roller Coasters functional and entertaining?

The Hook

Students will be asked to participate in the instant challenge, Loop the Loop. They will construct the initial hill and loop of a roller coaster using foam insulation and perform multiple trials to find the ideal height to successfully loop the loop. They will test five different heights and slopes while keeping the loop the same in order to see how the initial height of a roller coaster affects its speed and ability to loop the loop. This activity will then lead into the big idea, essential questions, and initial research regarding roller coasters and potential and kinetic energy. 

The Challenge

Congratulations Engineer!  You have been selected to submit a proposal for a design of King’s Islands newest roller coaster!  As with any engineer you will need to consider what makes a roller coaster safe and what makes it exciting. You will use your knowledge of algebraic functions to design your coaster!

Coaster Model Requirements & Constraints:

  • Minimum Function Components (thrill)
    • 1 Linear Function
    • 2 Non-linear functions
    • 1 Non function
  • Stopping Component (safety)
    • Stop on 1ft landing strip
  • Roller Coaster must be free-standing
  • Materials:
    • 10 feet of ½’’ vinyl tubing
    • 1/4’’ ball bearing
    • 10 twist ties
    • 5 zip ties
    • 1 foam board
    • 3 ft. Tape
    • 10 popsicle sticks

Blueprint Design:

  • You will need to draw a 2-D side view of your coaster on graph paper.  You will need to label and include the following:
    • Each function and non-function, it’s characteristics (increasing, decreasing, positive, negative) and what its purpose is (linear = speed, etc. This should be in complete sentences)  
    • An average rate of change for each function and non-function and all functions must be named in function notation.
    • You must write a function rule for at least 2 functions on your roller coaster.
    • Get this approved by your teacher! 
    • Use a highlighter and function notation to color code the different parts (ex: linear pieces = pink, etc)
    • You will have the opportunity to do some research on roller coaster designs and will be able to test your design in a roller coaster simulation app to see if it is functional!
  • Presentation of Results to the Kings Island Rep (Your roller coaster model and your 2D blueprint displayed – see rubric)

Good Luck!  Also, the winner of the challenge will be determined by:

  • The “fastest” coaster with all of the required parts (speed at any given point)
  • The coaster that will make the best addition to Kings Island as evidenced by your final presentation to KI Rep.

Description of Challenge (Either Product or Process is clearly explained below):

List the Constraints Applied

Create a 3D roller coaster that is functional (the ball bearing moves continuously through the track), meets the functional requirements above, and includes a safe landing. (see above)

Time & Materials (see above)



Anticipated Guiding Questions (that apply to the Challenge and may change with student input.)
  • Does initial height have anything to do with speed?  
  • Can we represent different parts/pieces of the track to functions? (linear function, non-linear function, not a function?)
  • Can we model a function for the coaster’s initial height?
  • Can we prove that pieces of the track are a function or non-function by providing a table as proof?
  • Can we determine the importance of having linear tracks (speed) and non-linear tracks (to slow down the speed, and non-functions - loops for fun)?
  • 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.)
Engineering Design Process (EDP)

Checkpoint #1: Pre-Assessment will be given to determine the student’s initial level of understanding (summative)

Lesson 1: Researching and Brainstorming – The students will be given the challenge (identify and define). They will research (gather information) roller coaster to find real life designs.  They will then brainstorm (identify alternatives) by designing 2-D blueprints for a coaster. Checkpoint #2: Teams will give an oral presentation to the teacher about their 2-D design of functions and their relationship with roller coasters (formative). This will help them select their best idea for functionality (select solutions).  (Activity 1 and 2)

Lesson 2: Building and Testing – Students will explore a roller coaster interactive applet that will allow them to test the functionality of their coaster (implement solution).  If it fails (evaluate solution) and their roller coaster can’t continuously move, they can make modifications to their design (refine).  Students will then build a roller coaster and test it for functionality (re-implement solution).  And, they will be allowed to make improvements after they test it (refine). (Activity 1 and 2)

Checkpoint #3 (formative): Present Results – Students will make a PowerPoint on the back of their 3-D roller coaster. The display will include their 2-D drawing with the labels of the different types of functions that their coaster design includes.  They will also be required to take a picture of their final coaster design.

Checkpoint #4: Post-Assessment will be given to determine the student’s final level of understanding (summative).

Misconceptions

Oftentimes, students do not realize that there are other types of functions besides linear.  They are unaware of non-linear functions and non-functions.  They also struggle with being able to see distance and time graphs and describe the scene as distance being a function of time.  They typically see the graph as the path the person walked (up a hill down a hill). 

Unit Activities

Prior to Unit: Students will take pre-assessment (1.0.0a.) and be assigned teams and roles.

Lesson 1: Unit Introduction, Research, and Brainstorming (4 days)
Day 1 (Lesson 1 – Activity 1)

Hook (1.1.01b) (40 mins)

Students will participate in the foam pipe marble coaster lab and instant challenge: How much height to loop the loop?

Big Idea & Essential Questions (25 mins) (CBL PowerPoint) (1.1.01c.)

  • Teacher will begin the Challenge Based Learning (CBL) process to generate the big idea and communicate the challenge.
  • By the end of class, a big idea should be generated and the teacher will have a challenge ready for the students tomorrow.

Day 2 (Lesson 1 - Activity1)

Challenge (1.1.01d) (1.1.01f) (15 mins)

  • Part 1: Students will be given the challenge
  • Part 2: Teams of 3-4 will be reminded of jobs and refinements made:

§  Project Manager – when the team has a question this person comes to the teacher

§  Time Manager – Keeps teammates on task and keeps the group on the timeline path.  Also checks in with the teacher to get tasks approved.

§  Materials Supervisor – tries to handle any disagreements and hears all sides of the issues and will determine the best solution.

§  Communication Specialist – Takes down any notes, observations, questions that arise during testing and building

§  All team members will design, build, and test! 

§  Students will fill out Step 1 of the Engineering Design Process Team WS (1.1.01f)

Research Web Quest (1.1.01e) (30 mins)

  • Part 3: Teams will visit the three sites below in order to learn about roller coasters designs, potential and kinetic energy, and how they work.

i. Roller Coaster Energy Placements (Potential & Kinetic) Tutorial

1. Link (Website: www.eduplace.com, Title: Energy & Motion)

ii. Roller Coasters – How they work! Tutorial

1. Link (Website: www.learner.org, Title: Amusement Park Physics)

iii. Roller  Coaster Design Research

1. Link (Website: www.coastergallery.com, Title: Coaster Gallery)

iv. Teams will discuss their findings and note them on the Engineering Design Process WS – Step 2.

Day 3 (Lesson 1 – Activity 2)

Brainstorm (1.1.02g) (65 mins)

  • Part 1: Students will individually draw a very basic 2D blueprint of their coaster from a side view on the graph paper provided.  The coaster will need to include:

§  Minimum Function Components (thrill)

- 1 Linear Function

- 2 Non-Linear Functions

- 1 Non-Function

§  Stopping Component

§  Roller Coaster must be free standing

- Ball bearing must come to a complete stop on a 1 ft. landing strip.

§  Students will share their ideas with their group and fill out Step 3 of the EDP WS.

§  Teams will select the design they want to proceed with (Step 4 of EDP WS).

  • Part 2: Teams will then use a roller coaster “builder” to educate themselves on the reasons for the different parts of the roller coaster (hills, straight aways, etc.) and to get a safety rating.

§  Note: This will not directly relate to their individual design, however, the knowledge gained can be used to make sure their coaster is safe.  

§  Link (Website: www.learner.org, Title: Design a Roller Coaster)

  • Part 3: Students will then use the roller coaster simulator to verify the functionality of their drawing.

§  Link (Website: www.funderstanding.com, Title: Roller Coaster Game)

  • Part 4: Students will make necessary changes before drawing final design.
Day 4 (Lesson 1 – Activity 2) 

Sketch (30 mins)

  • You will need to draw a 2-D side view of your coaster on graph paper.  You will need to label and include the following:
    • Each function and non-function, it’s characteristics (increasing, decreasing, positive, negative) and what its purpose is (linear = speed, etc. This should be in complete sentences)
    • Ordered pairs of points for each function or non-function (ex: (1,2) ). You will need a minimum of 5 points from each function or non-function (line).  
    • An x/y chart table (function table) for each function and non-function with the ordered pairs organized.  You must also have a sentence describing what makes the function linear, non-linear or not a function. This will be your proof. (I know you can tell by looking at the line/graph, but how can you tell by looking at the table?)
    • An average rate of change for each function and non-function and all functions must be named in function notation.
    • You must write a function rule for at least 2 functions on your roller coaster.
    • Get this approved by your teacher! 
    • Use a highlighter and function notation to color code the different parts (ex: linear pieces = pink, etc)
    • You will have the opportunity to do some research on roller coaster designs and will be able to test your design in a roller coaster simulation app to see if it is functional!
    • Students will begin building tomorrow.

Lesson 2: Building, Testing, and Communicating (4 days)

Day 5 & 6 (Lesson 2, Activity 1)

Building (130 minutes)

  • Students will begin building the roller coaster based on their drawing.
  • They should be aware that they only have TWO days to build and then the design is finished. 
  • They should use the following materials to build:
  • Materials

§  10 feet of ½’’ vinyl tubing

§  1/4’’ ball bearing

§  15 twist ties

§  1 foam board

§  3 ft. Tape

  • If students begin to finish, they are permitted to test their design with a ball bearing. 

Day 7 (Lesson 2, Activity 2)

Testing and Redesign (65 minutes)

  • Students will begin testing their tracks with a ball bearing.
  • Students should make any modifications that are necessary to enhance the functionality of the design.
  • When they are finished with testing and modifying, students should alert the teacher to time their design.  Records will be kept here:  Roller Coaster Time Stats
  • Students can then begin to work on their PowerPoint. 

Day 8 (Lesson 2 - Activity 2)

Communicating Results (40 mins) (1.2.02h.)

  • Students will make a PowerPoint and share their findings. The context is that they are presenting their design firms design to the KI representative. The display will include their 2-D drawing with the labels of the different types of functions that their coaster design includes.  They will also be required to take a picture of their final coaster design.

Post-Assessment (25 mins) (1.2.02i.)

Additional Resources

Pre-Test (1.0.0a.)
Day 1 (Lesson 1 – Activity 1)
1.1.01b. – Hook WS
         Hook Video (on WS)
                 Link (Youtube.com, Title: Banshee Roller Coaster REAL POV Kings Island Ohio 2014)
1.1.01c. – Challenge Based Learning PowerPoint
Big Idea/Societal Impact Video
                    Link (WKYC News, Kings Island: Son of Beat completely demolished).
Day 2 (Lesson 1 - Activity1)
1.1.01d. – Challenge WS
1.1.01e. – Research WS
         How Roller Coasters Work (on WS):
                     Link (Website: www.learner.org, Title: Amusement Park Physics)
         Potential and Kinetic Energy Simulation (on WS):
                   Link (Website: www.eduplace.com, Title: Energy & Motion
         Database for Roller Coasters (on WS):
                   Link (Website: www.coastergallery.com, Title: Coaster Gallery)
1.1.01f. – Engineering Design Process WS
Day 3 & 4 (Lesson 1 – Activity 2)
1.1.01g. – Brainstorm WS
         Safety Rating Test (on WS)
                   Link (Website: www.learner.org, Title: Design a Roller Coaster)
         Roller Coaster Timer & Simulator (on WS):
                   Link (Website: www.content3.jason.org, Title: The Jason Project – Coaster Creator)
Day 5 & 6 (Lesson 2, Activity 1)
Day 7 (Lesson 2, Activity 2)
1.2.02h. – Rubric
Day 8 (Lesson 2 - Activity 2)
Post-Test (below) 1.1.01i

Pre-Unit Post-Unit Assessment Instrument

Pre-Test (1.0.0a.)

Post-Test  (1.2.02i.)

Results: Evidence of Growth in Student Learning

Below is the Pre-Test and Post-Test scores (out of 30 points total). Every student showed growth with an average increase of 24.4 points. The highest student grew 29.5 points and the lowest student grew 13 points. Students noted that created the roller coasters helped to understand the importance of different types of functions and it allowed them to have a visual example.

Results
Reflection

As evidenced by the data above, the unit was definitely a success in terms of student achievement. Students were highly engaged during the 8 day unit and were highly motivated to do well. Students struggled creating 2-D scale models that included detail. In addition, students struggled to have the roller coaster be free-standing. In the future, I would create a lesson on making 2-D scale models in addition to have more strict requirements and examples. Finally, I would take away the free-standing requirement as students spent a large majority of the time at the end on this requirement. This took away from the overall purpose and learning objectives (besides engineering/design). 

Overall, this unit went really well and I would recommend any Algebra or 8th grade math teacher to complete this on functions, slope, or equations!