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University of Nebraska–Lincoln

Transportation Systems Engineering

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Lesson Overview
Lesson Title:
Approximating the Area Under a Curve

Mrs. Sara Bonn (Primary)

Brief Description:
This lesson is introducing the concept of using rectangles to estimate the area under a curve.

Topics Introduced:

Transportation, Distribution, and Logistics Curriculum Framework Components Addressed:
Transportation Operations
Health, Safety and Environmental Management
Suggested Grade Levels:
11th Grade
12th Grade

Standards Taught:
12.1.3 Math 2009
12.1.2 Math 2009
12.1.4 Math 2009
12.2.1 Math 2009
12.2.2 Math 2009
12.2.4 Math 2009
12.4.1 Math 2009

Lesson Information
Learning Expectations:

Students are expected to know the difference between LRAM, RRAM and MRAM and how to calculate each. In addition, they are expected to know the impact the increasing and decreasing nature of a curve has on the accuracy of each. Finally they should know how increasing the number of rectangles improves the accuracy of approximation.

Plan Of Action:

Students will review how to find distance, given a rate and time, by multiplying speed (rate) by time to obtain distance. This only works if the rate is constant over a period of time. We will review this concept graphically, having speed/rate along the y-axis and time along the x-axis. Students will see, typically for the first time, that this distance is represented as the area under the “curve” (i.e.; horizontal line), which is also the area of a rectangle. Using this concept, students will be asked to find the area, using rectangles, given an increasing rate of speed. Students will need to use multiple rectangles and will need to decide how to determine the height of each rectangle. The placement of the rectangle will determine the height and will also determine the type of rectangular approximation they are using (RRAM, LRAM or MRAM.) Students will then perform a similar exercise given decreasing rates of speed. They will be asked to determine what impact the increasing and decreasing curves have on the accuracy of the 3 methods. Finally, we will discuss the impact of increasing the number of rectangles under the curve.

Data Set Used:

Students will be given a couple of different scenarios, where train speeds are captured over time intervals. They will use these speeds and times to create graphs and to calculate the areas under the curve. The specific data is contained within the Powerpoint presentation.

Materials Needed:

Students will need paper (graph paper may be preferred), a pencil and calculator.

Preparation Period:

No prep time is needed.

Implementation Period:

This lesson can be taught in one class period.

Science, Math, Engineering and / or Technology Implications:

In this lesson the students will be finding and using distance, rates and time. These are all used when roads and freeways are being built. This is a great engineering lesson.

Unexpected Results:

Some students did not use the base of the rectangle even though they used the height. Instead they used the total from the start to the end of the interval so they had an inflated base for every rectangle after the first. Once we figured this out, they easily adjusted.

Considerations for Diversity in Education:

This is a math lesson, and cultural diversity is not used. Area and different geographic formations could easily be used during this lesson.

Lesson Files
Estimating with Finite Sums
This is the presentation used during the lesson. In the "Notes" section, I have included many of the questions I asked on each slide
[size: 812032] [date uploaded: Jun 22, 2011, 11:11 am ]

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