Question

the flower bed problem
anthony chens family operates a full - service pool and landscaping business. sometimes their clients want low decorative
fences around their flower beds.
anthonys father wants a quick way to estimate the amount of fencing needed. mr. chen asked anthony to help him. anthony
realizes that this is just a perimeter question.
he starts his task by analyzing the relationship between the length of the side of a square flower bed and the perimeter of the
flower bed. this will tell him the amount of fence needed to enclose the flower bed. anthony realizes that lengths of sides of
flower beds are not always whole numbers, but he decides to use square tiles to build models of flower beds of various sizes to
help him find a pattern. in his models, 1 tile represents 1 square foot.

1. draw the next two models in anthonys concrete representation next to the models shown.
2. make a table representing the relationship between the length

of the side of a square flower bed and the perimeter of the
flower bed. explain how the pattern in your tabular
representation relates to the concrete models.

Explanation:

Step1: Analyze the pattern of the models

The first model has \(1\) tile (\(1\times1\) square), the second has \(4\) tiles (\(2\times2\) square), the third has \(9\) tiles (\(3\times3\) square). So the next two models will be \(4\times4 = 16\) tiles and \(5\times5=25\) tiles.

Step2: Calculate the perimeter for each model

For a square of side length \(n\) (where \(n\) is the number of tiles on one side), the perimeter \(P = 4n\).

• When \(n = 1\), \(P=4\times1 = 4\)
• When \(n = 2\), \(P = 4\times2=8\)
• When \(n = 3\), \(P=4\times3 = 12\)
• When \(n = 4\), \(P=4\times4=16\)
• When \(n = 5\), \(P=4\times5 = 20\)

Step3: Create the table

Length of side (\(n\))	Perimeter (\(P\))
\(2\)	\(8\)
\(3\)	\(12\)
\(4\)	\(16\)
\(5\)	\(20\)

The pattern in the tabular representation is that the perimeter \(P\) is \(4\) times the length of the side \(n\) of the square flower - bed. In the concrete models, each side of the square made up of tiles has a length \(n\) (number of tiles on the side), and since the perimeter of a square is \(4\times\) side - length, the relationship \(P = 4n\) is represented both in the table and the concrete tile - based models.

Answer:

1. The next two models (for \(n = 4\) and \(n = 5\)) are squares made up of \(16\) (\(4\times4\)) and \(25\) (\(5\times5\)) tiles respectively.
2. The table has a linear relationship \(P = 4n\) where \(n\) is the side - length of the square flower - bed (in terms of number of tiles) and \(P\) is the perimeter. In the concrete models, each side of the square of tiles has length \(n\), and using the formula for the perimeter of a square \(P=4\times\) side - length, we can see the correspondence between the tabular and concrete representations.