Fencing
Task
I have a miniature collection of farm animals and I would like to fence in a rectangular area to put them in. I have only 20 centimeters of fencing.
Find all the different rectangles that you can that have a perimeter of 20 centimeters.
Write or tell about any patterns you see about your rectangles.
Write or tell why you think you found all the ways to make the different rectangles.
Alternate Versions of Task
| More Accessible Version:
I have a miniature collection of farm animals and 20 pieces of plastic fencing. Show all the different rectangular fences I can build around my animals using every piece of fence.
Write or tell me about any patterns you see about your rectangles.
Write or tell me why you think you found all of the ways to make the different rectangles.
More Challenging Version:
I have a miniature collection of farm animals, and I would like to fence in a rectangular area for them. Each fence piece is 1/2 centimeter long, for a total of 40 fence pieces. Find all of the different rectangles that have a perimeter of 20 centimeters.
Write or tell me about any patterns you see about your rectangles.
Write or tell me why you think you found all of the ways to make the different rectangles.
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Context
The second-grade class I visited just finished working with non-standard and standard measure, finding perimeter and naming different shapes. I thought this problem would allow them to think about perimeter from a different perspective.
What This Task Accomplishes
This task assesses students' knowledge of perimeter in a non-routine way. Instead of measuring different shapes and adding the sides to find the perimeter, students have to think about the whole perimeter and try to break it up into its parts (length of sides). This task will not only assess their knowledge of perimeter, but will also assess their problem-solving and reasoning skills.
What the Student Will Do
Most students started to draw rectangles. Some students asked for and used centimeter rulers even though I gave them centimeter graph paper. Many started with the short side of about one, two or three centimeters and built the rectangle's base and top simultaneously counting as they went along. Most had to verify the perimeter many times by counting the lengths of the sides centimeter by centimeter. Some verified using the addition equation.
Time Required for Task
45 minutes
Interdisciplinary Links
This task could be used with a social studies unit or science unit on farming.
Teaching Tips
I gave out centimeter graph paper so students could easily draw their rectangles. Review with your students that a square is a rectangle.
Suggested Materials- Centimeter graph paper
- Centimeter rulers (available if asked for)
Possible Solutions
You can make five different rectangles (if you use whole number lengths for sides).
1 x 9
2 x 8
3 x 7
4 x 6
5 x 5
| More Accessible Version Solution:
The solution is the same as that of the original version.
More Challenging Version Solution:
.5 x 9.5
1 x 9
1.5 x 8.5
2 x 8
2.5 x 7.5
3 x 7
3.5 x 6.5
4 x 6
4.5 x 5.5
5 x 5
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Task Specific Assessment Notes
Novice:
This student was finding area and not perimeter. S/he also has 10 different rectangles each with an area of two square centimeters instead of one rectangle with a perimeter of 20 centimeters.
Apprentice:
This student has found some of the rectangles, but not all of them. S/he used a strategy that is partially useful, but because the strategy is guess and check the student did not see any patterns and, therefore, did not find more solutions. There is some use of mathematical representation and mathematical terminology.
Practitioner:
This student has a broad understanding of the problem. S/he uses a strategy that leads to a solution. The mathematical equations and representations help make a clear presentation.
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Expert:
This student has a deep understanding of the problem, including the ability to identify the appropriate mathematical concepts necessary to explain the solution. The student used an efficient strategy, first drawing a rectangle with the short side of one centimeter, then two, three, four and five centimeters. When asked about any patterns s/he made the comment about the short and long side adding to 10.
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