a. Suppose your 6th grade students are investigating how many different rectangles can be formed that have a perimeter of 20 and integer lengths and widths.

Question

a. Suppose your 6th grade students are investigating how many different rectangles can be formed that have a perimeter of 20 and integer lengths and widths. They organize all the possibilities into a table, with columns for length, width and area. Create the table for yourself and graph the data using length as independent variable and area as the dependent variable.

Answer 1

Hi Iesha,

 

Perimeter = 2L+2W. So in this case, 2L+2W=20, and simplifying, L+W=10. As L varies, W=10-L; and Area = L*W= L*(10-L).

To get a complete picture of the situation, you should consider ALL nine possible integer value combinations of length and width that add up to 10: (1,9); (2,8); (3,7); (4,6); (5,5); (6,4); (7,3); (8,2); and (9,1). Yes, (5,5) is a square, but remember, all squares are rectangles, so it still counts.

Yes, it’s true (0,10) and (10,0) are not rectangles, but you could add them to the graph anyway to show those areas are zero.

You will get an inverted parabolic curve. It will start at the origin, go up until x=5 and then come back down again to (10,0). To see what this looks like, you can use your graphing calculator or favorite website to graph the area: y=x*(10-x). [Length=x; Width=10-x] My favorite graphing site is Desmos.

 

Sue

Answer 2

Hi Iesha!

 

The possibilities are L (length) =6 and w (width) =4, L=7 and w=3, L=8 and w=2, and L=9 and w=1. All of these will give you rectangles with a p (perimeter) of 20 so you have 4 rectangles that are options with integers.  You multiply the L x w to get the area for each one.  E.g., multiply 6 x 4 to get the first area, so it will be 24.  Make a table by having three columns: one with L, one with W, and one with A.  Graph it by making your x-axis be L (Length) since that's your independent variable and have your y-axis be A (area) since that's your dependent variable.  You will see that when you graph it there will be an inverse relationship between the two variables (L and w), which means that when one increases the other decreases.  The line will go down from left to right.  Hope this helps!

 

Eric