Talkin' Trash
Leo Paveglio, Barb Steinbrunner,
Rich Van Gilst, Carolyn Wahl
Class (Materials): General Math(graph paper, eyeball estimation)
Algebra I(graph paper, point/slope form)
Algebra II(calculator/computer, linear regression)
Goals:Students should learn to fit a linear equation to data, use the equation to predict future outcomes and compare these to Monroe County's actual predictions.
Time Required: One class period with unfinished work assigned as homework.
Extensions may also be assigned as homework.
Background: Students should be able to write the equation of a line of best fit or a line of regression (Algebra II), graph on a coordinate plane or calculator (Algebra II) and interpret the graph. Algebra II students should have some
experience with arithmetic sums.
Setting: This project addresses the growing waste problem by trying to predict the number of tons of waste Monroe County, Indiana, will produce in the future. Listed are Monroe County's actual figures for the years 1990-1994 with their projections for the years 1995-1999.
Problem: (See also attached student sheets.)
The United States is the largest waste producer in the world with most of the waste being dumped into the ground. As with most states, Indiana's landfills are filling at an alarming rate. To make adequate and financially feasible plans for the future, the County Solid Waste Management needs a way to predict the amount of waste that will need placement. Listed are the tons of waste generated for the years 1990-1994 (and the predictions for 1995-1999). These numbers include solid waste and material that may be recycled.
Tons of Waste in Monroe County, Indiana
1990132,956
1991135,686
1992136,102
1993137,793
1994139,417
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1995140,739
1996142,158
1997143,430
1998144,521
1999145,920
2000
Funded in part by the National Science Foundation and Indiana University 1995
Algebra II
Problem:
The United States is the largest waste producer in the world with most of the waste being dumped into the ground. As with most states, Indiana's landfills are filling at an alarming rate. To make adequate and financially feasible plans for the future, the County Solid Waste Management needs a way to predict the amount of waste that will need placement. Listed are the tons of waste generated for the years 1990-1994. These numbers include solid waste and material that may be recycled.
Tons of Waste in Monroe County, Indiana
1990132,956
1991135,686
1992136,102
1993137,793
1994139,417
1.Using the above data and the appropriate technology, create a linear model that will
predict the total waste in Monroe County for the years 1996 to 2000. State the linear
equation and explain the meaning of each of your constants and variables.
2.The Monroe County Waste Management has made the following predictions:
Future Tons of Waste in Monroe County, Indiana
1995140,739
1996142,158
1997143,430
1998144,521
1999145,920
Using the equation from problem 1, compare your predictions to these.
3.Look at your scatterplot from problem 1. Is there any point(s) which does(do) not appear to fit the same pattern as the rest? If so, eliminate it (them) and create a linear model which fits the remaining points.
4.Compare your two models. How are they different? How many tons of waste does each predict for 2005, 2010 and 2020? Which do you think is better? Why?
Funded in part by the National Science Foundation and Indiana University 1995
Select the better of your two models in problem 4 to use for questions 5 - 7.
5.At the end of 1994, Monroe County landfill still had room for three million tons of waste. In what year will the landfill become full?
6.In 1994, 46,736 tons of the total waste was recycled and not put into the landfill. If recycling patterns remain the same, how long will it take for the landfill to be at its capacity?
7.At what rate must the people of Monroe County recycle in order for the current landfill to be usable until the year 2050? Do you think that it would be realistic to recycle at this rate?
Funded in part by the National Science Foundation and Indiana University 1995
Algebra I
Problem:
The United States is the largest waste producer in the world with most of the waste being dumped into the ground. As with most states, Indiana's landfills are filling at an alarming rate. To make adequate and financially feasible plans for the future, the County Solid Waste Management needs a way to predict the amount of waste that will need placement. Listed are the tons of waste generated for the years 1990-1994. These numbers include solid waste and material that may be recycled.
Tons of Waste in Monroe County, Indiana
1990132,956
1991135,686
1992136,102
1993137,793
1994139,417
1.Plot the information given above and draw the line of best fit. Using two points, write the equation of that line. Please show your work and graph.
2.Using the graph and line from problem 1, predict the number of tons of waste for the years 1995, 1997 and 1999. Please show your work.
3.In 1994, 46,736 tons of the waste was recycled. If this rate remains the same, how many tons of waste will be recycled in 1995?
4.If 15% of the waste is non-burnable and one incinerator burns 350,000 tons of waste per year, how many incinerators will be needed in 1997? Explain your answer in sentences.
5.Assume 15% of the waste can be recycled. If the remainder is deposited in a landfill, what should its capacity be as you plan for the year 1997?
6.Describe the assumptions you have made as you did these problems. How could these assumptions change your results?
Funded in part by the National Science Foundation and Indiana University 1995
General Math
Problem:
The United States is the largest waste producer in the world with most of the waste being dumped into the ground. As with most states, Indiana's landfills are filling at an alarming rate. To make adequate and financially feasible plans for the future, the County Solid Waste Management needs a way to predict the amount of waste that will need placement. Listed are the tons of waste generated for the years 1990-1994. These numbers include solid waste and material that may be recycled.
Tons of Waste in Monroe County, Indiana
1990132,956
1991135,686
1992136,102
1993137,793
1994139,417
1.Plot the information given above and draw the line of best fit. Explain any patterns or trends
2.Using the graph and line from problem 1, predict the number of tons of waste for 1995, 1996, and 1997.
3.One-third of the waste was recycled in 1994. If this trend continues, how many tons of waste will be dumped into the landfill in 1996? Please show all work.
4.What could influence or alter the amount of waste produced on a yearly basis?
Funded in part by the National Science Foundation and Indiana University 1995
Sample Solutions:
General Math:Problem one can be graphed on a coordinate plane. Some patterns students may observe are a positive slope of the line and an increasing amount of waste. The amount of waste in tons for 95, 96 and 97 will vary, but should increase by approximately 1000-1500 tons per year. The amount of waste depends on the student's assumptions, such as the amount of waste recycled, public awareness or the manufacturer's response to the recycling effort.
Algebra I:Answers will vary for problem one. One solution uses the data points (93, 137 793) and (94, 139 417) resulting in the equation: Tons = 1624(year) - 13,239
(Note: x represents the last two digits of the year and y the tons of waste.)
This formula yields the following for problem two:
YearTons
95141,041
97144,289
99147,537
For problem 3, using a proportion yields 47,280 tons of recycled waste.
For problem 4, one incinerator will be more than adequate.
Problem 5 is deliberately vague. Students= answers will depend on their assumptions. These may include assumptions such as the county opening a new landfill or more/less waste being recycled.
Algebra II:Using a line of regression program for problem one, the equation is: y = 1503x + 133385. (Note: x represents 0 for 1990, 1 for 1991, etc.) Answers will vary for problems 2 - 7. By eliminating the second point and using the remaining four, problem 3 yields the new equation:
y = 1617x + 132929. Assuming this is the better model, answers for problems 5-7 are 2013, 2022, and 70.9%, respectively.
Evaluation:The evaluation method will depend on the course. Considerations may be given to the clear statement of the problem, assumptions/constraints, model design, analysis (strengths/weaknesses of the model), and conclusions.
Extensions:One extension may be to collect data from your county or state solid waste management and compare models. Also, students could explore the internet for other sources of data. Following are two other extension ideas:
General Math:Give the students the Monroe County predictions for 1995-1999 and have them compare their results to those of Monroe County.
Algebra II:Using the points from problem 3, create an exponential model for the total waste. What does this model predict for 2005, 2010, and 2020? Compare the model with your linear model, answer questions 5 - 7 with this new model.
Funded in part by the National Science Foundation and Indiana University 1995
Bibliography:
Swetz & Hartzler, 1991. Mathematical Modeling in the Secondary School Curriculum.
"Time to Waste." National Council of Teachers of Mathematics.
Resource:
Monroe County Solid Waste Management, Bloomington, Indiana.
Funded in part by the National Science Foundation and Indiana University 1995