**Repaving the School Parking Lot**

Gregory Graue

**Class:** This project is appropriate for geometry, college level algebra, and developmental mathematics classes.

**Materials:** Calculator, paper, pens, pencils, graph paper, and measuring tape.

**Goals:** This project will reinforce the use of area and volume to help estimate the amount of asphalt present in the parking lot. The use of formula manipulations will be emphasized in the calculation of mass from the density and volume. The project will also encourage teamwork in problem-solving and presenting/explaining solutions to other students.

**Time Required:** One option is to introduce the project on Thursday, allow the students to gather information and formulate solutions on Friday, and have the group presentations the following Monday. You may wish to allow one week for the students to prepare written reports. For college students, after introducing the project in class, allow one weekend before the group presentations to the class. Allow one hour to explain the problem, assign project teams, and explain requirements for the written report and oral presentation and one hour for group presentations. You may wish to allow two weeks for the students to prepare a written report.

**Background:** Students will need to know how to determine the areas of triangles, rectangles, circles, and sectors from composite figures. They will need to know how to perform unit conversions and how to find the volume of rectangular prisms. Students also need to have an understanding of density and how to use it and volume to find mass.

**Setting:** Your school needs to have the parking lot repaved due to the erosion of the existing asphalt. First the old pavement needs to be broken up and hauled away. The school district has a dump truck with a 25-ton hauling capacity with which to remove the old asphalt.

**Problem:** The school district needs an estimate of the amount of asphalt present to determine the approximate number of trips the district's dump truck will need to haul away the old asphalt.

**Report Requirements:** Students will need to list assumptions used to arrive at a conclusion, give justification of the conclusions reached, make oral presentations of their findings, and turn in a research paper detailing and explaining how the group arrived at a conclusion.

**Sample Solution:** A group of students found the area of the school parking lot to be 36,000 sq. ft. and the average thickness of the blacktop to be 9 inches (0.75 ft.). They concluded that the shape of the asphalt to be removed was a rectangular prism. Thus, the volume was

(area of base) X height = (36,000 sq. ft.) X (0.75 ft.) = 27,000 cu. ft.

The students determined the density of the asphalt to be 40 lb./cu. ft.

Mass = density X volume = (40 lb./cu ft.) X (27,000 cu. ft.) = 1,080,000 lbs.

To determine the number of truck loads required to haul away the asphalt, the students needed to convert the pounds to tons using 1 ton = 2,000 lbs.

1,080,000 lbs. X 1 ton/2000 lbs. = 540 tons

To determine the number of trips required to haul away 540 tons of asphalt, the students used the fact that each dump truck load is 25 tons.

540 tons X 1 load/25 tons = 21.6 loads of asphalt

Thus, the truck must make 22 trips to haul away the asphalt.

**Evaluation:** Give emphasis to the assumptions the students make to arrive at a mathematical conclusion in both the written report and group presentation. Require each student to speak at some point during the oral presentation to the class. For the written report, emphasize clarity of expression so that someone not in the class could read the report and understand the problem and the solution.

**Extensions:**

1. Students could create a cost estimate of removing the old asphalt and repaving the lot with new asphalt.

2. Students could compare an estimate in U.S. units and an estimate in metric units and determine which estimate is the better buy.

3. Students could decide how to repaint the lines on the parking lot in order to have the maximum number of cars possible park on the lot.

**Teacher Notes:** Problem solutions should have a list of assumptions detailing the shape and density of the asphalt in the parking lot. The calculations of area(s), volume, density, and unit conversions should be clearly explained in the written report and oral presentations.

Students may have difficulty determining the density of the asphalt in the parking lot. You may want to consult a science teacher in your building if you are not familiar with experimental methods to determine density. The determination of density could be a shared project with a science teacher if you and the science teacher have the same students. The resources in the school library could also be helpful in determining the density of asphalt.

A paving contractor would make an excellent outside expert for a class speaker after the project is concluded. An excellent topic for discussion would be steps involved in preparing written estimates for potential jobs.

Students should be placed in groups of three or four with at least one high-achieving and one low-achieving student in each group. Extra supervision may be necessary for high school classes if you take the class outside to gather data for the project.

**Repaving the School Parking Lot**

**Student Problem Statement**

**Group Members** _______________________________________________________________

**Setting:** The parking lot of your school needs to be completely repaved from the ground level due to erosion of the existing asphalt. The school district central office needs to estimate the amount of asphalt present to determine the approximate number of trips the district's dump truck with 25 tons of hauling capacity will need to haul away the old asphalt.

**Problem: **Your job is to determine, along with your group members, the amount of asphalt present in the parking lot. You will then need to estimate the density of the asphalt. Using this information, you will be able to determine the number of trips necessary for the dump truck to haul away all of the old asphalt.

**Report Requirements:** You will need to list assumptions used to arrive at a conclusion, give justification of the conclusions reached, make oral presentations of these findings, and turn in a research paper detailing and explaining how the group arrived at a conclusion.

*Funded in part by the National Science Foundation and Indiana University 1995*