Fences for Grazing
Task
Create a fenced-in space with the maximum area for your cow Bessie, given 100 feet of fencing. How many poles would you have for this area if you need 1 pole every 5 feet? How do you know it is the maximum area? What is that area? Explain.
Now, instead of having the grazing area in the middle of a field you decide to use a side of your barn. With the same amount of fence and the side of the barn being 50 feet, find the maximum area of this alternative grazing pasture. How do you know this is the maximum area? Explain. What is the area? Explain.
Compare the 2 answers and reasoning behind them. What do they suggest? What conjectures can you make? Extend your thoughts.
Alternate Versions of Task
| More Accessible Version:
Create a fenced-in space with the maximum area for your cow, Bessie, given 100 feet of fencing. How many poles would you have for this area if you need one every 5 feet? How do you know it is the maximum area? What is that area? Explain.
More Challenging Version:
Create a fenced-in space with the maximum area for your cow, Bessie, given 100 feet of fencing. How many poles would you have for this area if you need one every 5 feet? How do you know it is the maximum area? What is that area? Explain.
Now, instead of having the grazing area in the middle of a field you decide to use a side of your barn. With the same amount of fence and the side of the barn being 50 feet, find the maximum area of this alternative grazing pasture. How do you know this is the maximum area? Explain. What is the area?
Compare the 2 answers and reasoning behind them. What do they suggest? What conjectures can you make? Extend your thoughts.
Without changing the amount of fencing or barn side used, how else could you increase the area on which the cow can graze?
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Context
We had been studying area and perimeter. I wanted a problem that would use the two concepts, and hopefully students would come to some conclusions about squares and maximizing area.
What This Task Accomplishes
This task will tell me which students are comfortable with working with area. It will tell me which students have grasped the concept of area and perimeter and can work with and express these concepts.
What the Student Will Do
Most students started by drawing a diagram. Some played around with a number of possible rectangles as they tried to get the largest area.
Time Required for Task
50 minutes
Interdisciplinary Links
This type of problem can be used with any planning activity students might be doing, designing playgrounds, for example.
Teaching Tips
Encourage students to test other rectangles. Question students as to whether they could find a rectangle with a larger area.
Suggested Materials
Possible Solutions
Maximum area is actually a circle (though no one in my class thought of that) and using Pi as 3.14 the area is about 796.18 square feet. The more sides the polygon has the bigger the area will be with a given perimeter. The rectangle with the largest area is 25' x 25' with an area of 625 sq ft. A square maximizes the area of rectangles with a given perimeter. You will need 20 fence posts five feet apart.
The rectangle with the largest area using the side of the barn is 25' x 50' with an area of 1,250 sq ft. But, again, if a student used polygons with increasing sides, drew a diagram to scale and accurately approximated the area, that would be an indication of a deep understanding of the problem. The maximum area using the side of the barn would be to use the whole side of the barn and curve the 100' of rope into half an oval. Some students may draw the diagram to scale and approximate the area.
| More Accessible Version Solution:
Maximum area is actually a circle and using pi as 3.14 the area is about 796.18 square feet. The more sides the polygon has the bigger the area will be with a given perimeter. The rectangle with the largest area is 25 feet by 25 feet with an area of 625 sq. ft. A square maximizes the area of rectangles with a given perimeter. You will need 20 fence posts 5 feet apart.
More Challenging Version Solution:
The original solution and, by changing the surface area, you can increase the grazing area. For instance, if the fence is placed around a small hill, the number of square feet of grazing area will be increased.
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Task Specific Assessment Notes
Novice:
This student is applying inappropriate concepts of area and perimeter. His/her strategy does not help to solve the problem. There is little explanation and inaccurate use of a diagram.
Apprentice:
The student uses a strategy that is partially useful. S/he uses all the fencing in the second part, but not the first part. There is some evidence of reasoning, but they could not completely carry out the mathematical procedures. The explanation is incomplete. There is some use of mathematical representation.
Practitioner:
This student shows a broad understanding of the problem. They use a strategy that will lead to a solution of the problem for rectangles. They use effective mathematical reasoning with a clear explanation and appropriate use of mathematical representation and terminology.
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Expert:
The Expert has a deep understanding of the problem and identified the appropriate mathematical concept (maximizing the area by keeping the three sides equal). S/he uses a sophisticated strategy (use of Pythagorean Theorem) that uses refined and complex reasoning. There is an effective explanation (using mathematical equations to communicate). Mathematical representation is used to help communicate ideas.
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