Assignment: Penny's Pet Pens
Penny, the owner of Penny's Pet Enclosures, makes enclosures used by boarding kennels for dogs, cats, and other pets. Penny creates these enclosures in modules. She has all the walls pre-constructed and has them put together on the site they are to be built. All the walls have gates in them. The first enclosure has four walls as shown in figure 1.
Figure 1
The construction of two enclosure and three enclosure pens are shown in figure 2. Adding walls to the first enclosure makes second and third enclosures.
Figure 2
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Enclosures
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Number of Walls
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4
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5
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1.) Use the chart to the right to sketch a picture or diagram of four and five enclosures put with walls around them.
2.) How many walls does Penny need for 4 enclosures?
3.) How many walls does Penny need for 5 enclosures?
4.) Describe as completely as possible any pattern that you noticed in drawing or acting out the situation.
5.) Without drawing a diagram, predict the # of walls for 6 enclosures. walls.
You answered the previous questions by acting out what Penny would do to put the Walls and Enclosures together. You also drew a diagram of what the walls and enclosures would look like.
Another way of analyzing this situation involving enclosures and walls is to think of the number of enclosures as an input and the number of walls as an output. We can use a table to represent the relationship between the number of enclosures as an input and the number of walls as an output.
6.) Complete the following table by entering the # of walls corresponding to each # of enclosures.
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Enclosures (Input)
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1
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2
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3
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4
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5
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6
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7
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8
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9
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Walls (Output)
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7.) Describe as completely as possible any pattern that you notice in the table you made.
8.) Use the pattern you noticed to predict the number of walls needed for:
a. 10 enclosures: walls.
b. 11 enclosures: walls.
c. Explain how you got these numbers for parts a and b.
You can graph the # of walls corresponding to a # of enclosures.
9.) Plot the values from your table
10.) Should you draw a line through the points? Why or why not?
11.) Use the graph to predict the number of walls needed for 12 enclosures.
12 enclosures: walls.
The problem solving strategies used above are helpful in solving the problems posed. However, if we need many enclosures, the strategies used so far become somewhat tedious. If we can create a formula by which Penny can calculate the # of walls she would have for a given # of enclosures that would be helpful.
12.) a. Write a sentence that expresses how the number of walls is related to the number of enclosures put together.
b. Test your relation by using it to find the # of walls at 5, 6 and 8 enclosures.
5 enclosures: walls 6 enclosures: walls 8 enclosures: walls (Check the table on page 1 to see if you got the correct number of walls.)
13.) (a) Write an equation that expresses the relationship between the # of walls and the # of enclosures. (Let "E" stand for the # of enclosures, and "W" for the # of walls.).
(b) Test your equation by using it to find the number of walls at 5, 6 and 8 enclosures. Show your substitution.
5 enclosures:
6 enclosures:
8 enclosures:
(Check the table on page 1 to see if you got the correct number of walls.)
14.) Use your equation from question 13 to find the number of walls for the following number of enclosures put together: 13, 25, 30, and 52. Show all your work.
13 enclosures:
25 enclosures:
30 enclosures:
52 enclosures: