- Author:
- Carrie Robledo, James O'Neal
- Subject:
- Algebra, Geometry
- Material Type:
- Activity/Lab
- Level:
- Middle School, High School
- Tags:

- License:
- Creative Commons Attribution Non-Commercial Share Alike
- Language:
- English

# Building a Light Rail

## Overview

**This task explores the real world topic of building light rails. Throughout the implementation of this task the students will learn about the cost of building railways and how to implement them within a budget. This task explores such mathematical concepts of using coordinates to find the distance between points, using coordinates to build polygons and find the area and length of sides, and writing equations of parallel lines.**

# Instructor Directions

*Building A Light Rail*

*Submitted by James O’Neal*

*Charlotte-Mecklenburg Schools*

Driving Question / Scenario | How can you maximize the distance between points to cover the most area? |

Project Summary | This task explores the real world topic of building light rails. Throughout the implementation of this task the students will learn about the cost of building railways and how to implement them within a budget. This task explores such mathematical concepts of using coordinates to find the distance between points, using coordinates to build polygons and find the area and length of sides, and writing equations of parallel lines. |

Estimated Time | 3-4 days |

Materials / Resources | Rulers, protractors, Desmos.Light rail video explaining the benefits how they run:https://www.youtube.com/watch?v=knqt3dXZJT4 |

Grade | 8-10th |

Subject(s) | Algebra 1, Geometry |

Educational Standards | NC.M1.G-GPE.4Use coordinates to prove simple geometric theorems algebraically.Use coordinates to solve geometric problems involving polygons algebraically• Use coordinates to compute perimeters of polygons and areas of triangles and rectangles.• Use coordinates to verify algebraically that a given set of points produces a particular type of triangle or quadrilateral.NC.M1.F-LE.5Interpret expressions for functions in terms of the situation they model.Interpret the parameters 𝑎 and 𝑏 in a linear function 𝑓(𝑥)=𝑎x+𝑏 or an exponential function 𝑔(𝑥)=𝑎𝑏𝑥 in terms of a context.NC.M1.G-GPE.5Use coordinates to prove simple geometric theorems algebraically.Use coordinates to prove the slope criteria for parallel and perpendicular lines and use them to solve problems.• Determine if two lines are parallel, perpendicular, or neither.• Find the equation of a line parallel or perpendicular to a given line that passes through a given point. |

Project Outline | |

Ask | Task 1:1. What are some ways we can find the distance between two points?2. What is the shortest distance between two points?Task 2:1. How can I make the quadrilateral into an easier quadrilateral to work with?2. What shapes are formed when you inscribe the image with a square?Task 3a:1. What does connecting the points B&E form?2. Every line has an equation, can you find the equation of the line?Task 3b:1. If Keisha’s light rail runs alongside the connecting railway BE, what does that mean mathematically?2. What do you need from the line connecting BE that will help you find the equation of the line forming Keisha’s stop HD? |

Imagine | Imagine that Raleigh is considering adding a light rail system to connect together different parts of the city. In the map to the right, each point represents a different part of the city that could become part of the light rail system. You are in charge of recommending which points the city should connect. Here are the guidelines from the city: The city only has enough money to create 4 light rail lines, so you can only make 4 recommendations The city wants to connect together the points that are furthest apart so travel will be easier between those parts of the city. |

Plan | Tell the students that they will be provided with rulers and to use desmos scientific calculator for computing costs and distances. Have them write out and draw their design for their railways. |

Create | Have students use straight edge rulers or protractors and have erasers and extra copies handy. The process of drawing lines and erasing them can become really messy. Have students create the connections and also the polygons needed to find the area of the railway system. |

Improve | Along the way students may look at lines and think they are longer than others. Have students prove their thinking by testing all lines to confirm their spatial reasoning. Also on task 3A, you could improve Sam’s route by putting the stop at (0,6). It would be shorter than the straight shot starting from (0, 2.25). |

Closure / Student Reflections | This could be a good time to talk about equitable access for people who may not own a car. Students can reflect on why transportation and equitable access to it is so important. You could go into the economics of why certain projects are voted on during elections on why taxes may be increased for certain projects that some may see as unprofitable to them. |

Possible Modifications / Extensions | You could modify which sections of the project to give at certain times. I would modify the 45,000,000 because the number may not be as realistic to a budget for certain cities. You could extend this into the classroom where students find the shortest distance between certain classes and how they could change the layout of the school to make their period change the shortest distance for them. To add more technology to this all of this task, students can use desmos to graph points and test their lines. This will be a good way to bring the project to a digital environment. Points can even be preset by teachers in Desmos as well. |