# Mid-America Transportation Center

## Teacher Resource Center

Lesson Overview
 Lesson Title: Applications of Parabolas--Vertical Crest Road Design Teachers: Mr. John Catlett (Primary) Brief Description: Students will come up with the equations of the parabolas used in the design of Highway overpasses based on the entrance grade, exit grade and initial elevation Topics Introduced: Parabolas, Length of Curves Transportation, Distribution, and Logistics Curriculum Framework Components Addressed: Transportation Operations Transportation Systems/ Infrastructure Planning, Management and Regulation Suggested Grade Levels: 10th Grade 11th Grade 12th Grade Subjects: Mathematics Standards Taught: 12.4.2 Math 2006 12.4.4 Math 2006

Lesson Information
Learning Expectations:

Students will learn how parabolas are used in roadway design and be able to use technology to find the equations of the parabolas.

Plan Of Action:

1. Review Equations of Parabolas in standard form: ax^2+bx+c
2. Show video/simulation (drive over) of viaduct. Have student look for what happens as they go over the viaduct. What shape does their line of travel seem to take?
3. Use Slide show to show formulas for calculating coefficients used in the equation of the parabola used in the overpass design. Pay special attention to slides 5 and 6. point out to students that they will be working with Type 1 curves.
4. Have students work through examples of creating the equations on geogebra.
5. In groups, with given parameters, have students develop the equations for the overpass design and graph the parabolas.
6. Have students check their equations and graphs using geogebra.
7. End with a discussion of other factors that might influence the curve design: Speed limits, stopping distances, line of sight.

Data Set Used:

Photos of existing over passes were used and placed into geogebra files to fit into a coordinate plane.

Materials Needed:

LCD Projector, Computer lab/Mobile lab with internet access.

Preparation Period:

1-2 hrs...if you need to install Geogebra on all the computers in the Computer Lab.

Implementation Period:

Lesson: 1 class period in computer lab. Students: 20-30 minutes additional outside of class.

Science, Math, Engineering and / or Technology Implications:

Helping students understand the math (equations of parabolas) behind the engineering ( design of the overpass).

Unexpected Results:

None were experienced.

Considerations for Diversity in Education:

No special diversity considerations were made in this lesson.

Lesson Files
ParabolasInVerticalCurveDesign
PowerPoint of lesson presentation on Overpass Design
[size: 3608576] [date uploaded: Mar 03, 2011, 4:39 pm ]

Parabola Exploration Worksheet
Exploring parabolas in y=ax^2+bx+c. Used with Parabola Exploration Activity in geogebra
[size: 35328] [date uploaded: Mar 03, 2011, 4:44 pm ]

Parabola Exploration
Geogebra activity using sliders to show how the constants a, b, c affect the shape of a parabola in y=ax^2+bx+c
[size: 3715] [date uploaded: Mar 03, 2011, 4:47 pm ]

Overpass Equation Worksheet
Worksheet which works through finding equation overpass vertical curve (parabola). Goes with Overpass Example, Overpass Problem 1 and Overpass Problem 2 geogebra files.
[size: 162304] [date uploaded: Mar 03, 2011, 4:52 pm ]

Overpass Example
Geogebra file accompanying example problem of Overpass Equation Worksheet
[size: 187455] [date uploaded: Mar 03, 2011, 4:54 pm ]

Overpass problem 1
Geogebra file accompanying problem 1 on Overpass Equation Worksheet
[size: 130183] [date uploaded: Mar 03, 2011, 4:55 pm ]

Overpass Problem 2
Geobra file used with problem 2 on Overpass Equation Worksheet
[size: 156225] [date uploaded: Mar 03, 2011, 4:57 pm ]

Overpass Equation Checker
Geogebra file used by students to check their work on equations developed by hand using given input data.
[size: 3821] [date uploaded: Mar 03, 2011, 4:58 pm ]

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