Perimeter and Area Dice Game and Function to find dimensions of equal perimeter and area.

Many careers like architecture, aeronautical and graphic design, engineering, farming, sports, painting, construction, manufacturing, military applications, boating, the coast guard, and many others include the use of area and perimeter on a regular basis.  Perimeter is the linear  (one dimensional) distance around the outline of the rectangle. Either add the two lengths plus the two widths, or first add the length and width and then double that sum. P = L + L + W + W or  P = 2 L + 2 W or P = 2 (L + W). Area is the two dimensional space inside the rectangle. Simply, multiply length times width or count the square units inside the rectangle.

So I created the Perimeter and Area Dice Game:

1. Roll 2 Dice to determine the dimensions of a rectangle, draw the rectangle, and label the dimensions below the middle of the base and to the middle right of the right height. 

2. Calculate the Perimeter and Area of that rectangle and record it as P=_____u  and A= ____u^2. 

3. After you fill the page with rectangles, separately, find the sum of all of your perimeters and areas at the bottom of the page.

I challenged the older students to create a function that shows the dimensions of a rectangle when its perimeter and area are the same number of units. They first did some guess and check and determined that a 4x4 unit rectangle, also a square would yield a perimeter of 16 units and an area of 16 square units. After some more testing, they determined that a 3x6 or 6x3 rectangle would yield a perimeter of 18 units and an area of 18 square units. They became uncomfortable when they tried to use a side length of 1 or 2, seeing that they could never create rectangle with equal perimeter units and area of square units and any other dimension value. But for instance, they did determine that they could find a side of 10 x 2.5 with a perimeter of 25 units and an area of 25 square units. So we decided to create a function to find all combinations of two dimensions to satisfy this challenge.

1. Set the perimeter formula equal to the area formula. Perimeter = 2L + 2W and Area =  LW, so P = A or 2L + 2W =  LW or LW = 2L + 2W.

2. Use the subtraction and division property of equality to isolate length. Subtract 2L from each side of the equation

LW -- 2L = 2W

3. Factor out L from the left side of the equation.

L(W -- 2) = 2W

4. Divide both sides of the equation by (W — 2)

So, we get L = (2W)/(W — 2)

5. Substitute the whole number widths to 

record the corresponding lengths. To make this easier, create a T chart showing the coordinates of whole number unit widths and corresponding lengths of rectangles when its perimeter and area are the same number of units.                                              

6. Graph these coordinates on a coordinate plane. I provided a T chart and graph in the pdf for this purpose.