The West Hancock Booster Club is considering replacing the existing grass football field with a new type that is softer for the benefit of the opposing team. Visiting teams have been complaining about the large number of injuries from our hard-hitting players smashing them into the ground. Local fans have agreed to volunteer labor and equipment. The Booster Club is concerned only with the cost of the sod for the field. They are looking for the best buy for their money. Below are price quotes from various local nurseries: 6' x 2' roll $1.35
6' x 6' roll $4.00
8' x 3' roll $2.75
6' x 3' roll $2.25 The field dimensions are 120 yards x 160 feet. Which is the best buy?
How many rolls of sod will be needed?
What will be the total cost of the sod? Bonus: Make a scale diagram of how the sod will be laid down on the field.
This problem was developed to correspond with a geometry unit. This year we had an outstanding football team and thought this theme would foster student interest in the problem. Along with the use of multiplication and division of whole numbers and decimals, students need to compute the area of each roll of sod and the area of the football field. Students will need to recognize that all units of measure need to be converted to the same unit. Students will need to find a way to compare unit prices, which may include comparing decimal amounts if students determine the cost per square foot of each size sod. Several strategies were used to solve this task, including diagramming and using formulas. Most students knew to convert yards into feet. Some students had difficulty computing the unit price. An extension to the problem would be for students to create a scale diagram of how the sod would be laid. When this problem was first used, this part was included. It created a great deal of difficulty and was dropped from the next field test. Students had difficulty "cutting" the sod into 4' x 6' pieces and then putting the 2' x 6' remainders together. 1-2 hours This task could be linked to a consumer unit on comparison shopping. It can also be linked to other tasks involving working with areas such as shingling, carpeting, wall papering, etc. To get students started, the area of a rectangle and customary unit conversions were reviewed. In one situation, students worked in groups of three for about 15 minutes to brainstorm ideas on how to approach the problem, then separated to do individual work. This format worked well.
The cost per square foot is as follows: 6' x 2' roll $1.35 cost per square foot = $0.1125 6' x 6' roll $4.00 cost per square foot = $0.1111 8' x 3' roll $2.75 cost per square foot = $0.115 6' x 3' roll $2.25 cost per square foot = $0.125 So the least expensive rolls are the 6' x 6'. There does not seem to be a big difference in the costs per square foot, but it really adds up when you buy the quantity needed. In fact, there is an $800 difference when comparing the least expensive, to the most expensive sod. 57,600 square feet of sod is needed. 1,600 6' x 6' rolls would be needed and would cost $6,400.
The Novice will have no solution. The explanation will be unclear and there will be little evidence of mathematical reasoning and/or language. The Novice will not know where to begin and will not have a strategy for even getting started toward a correct solution.
The Apprentice will have some understanding of the task. The Apprentice might be able to convert yards to feet, finding the area of the football field. The Apprentice may be able to determine the number of pieces of sod that are needed, but will have no strategy for determining the least expensive. There will be some evidence of mathematical reasoning and/or language and perhaps an attempt at a mathematical representation.
The Practitioner will have a solid understanding of the problem. The Practitioner will be able to convert yards to feet, finding the area of the football field. The Practitioner will be able to determine the number of pieces of sod that are needed and will have a strategy for determining the least expensive. The Practitioner will use mathematical language and representations to communicate clearly and will use sound mathematical reasoning.
The Expert will have extensive understanding of the problem. The Expert will have an efficient strategy for finding a solution and will compare the four different costs for the reader. The Expert will use sophisticated mathematical language and may make mathematically relevant comments or observations about his/her solution, such as discussing the significance in decimal amounts when determining the cost per square foot.
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