Patterns and Functions

Extensions: 

• Fixed Area
• How many people can be seated if we have 24 unit tables?  
• Or, what is the perimeter of a rectangular table whose area is 24 square units?
• Discussion. Allow the students to use the square tiles to form tables and to list all rectangles with an area of 24 square units in a chart. Stress that we are trying to find all rectangles with a given length. After the students have explored this problem, organize the data.
• Ask students to observe the patterns.  
• Are they the same as the problem with fixed perimeter? 
• As the length increases, the width decreases. But the perimeter decreases to a certain point and then starts to increase, but the decrease and increase are not as symmetric as in the pattern with fixed perimeter. Graphing the problem displays this pattern visually. The maximum perimeter is 50 units for the long, skinny 1 × 24 rectangle. The minimum perimeter is 20 units for the fat, "almost square" 4 × 6 rectangle. 
• Ask, "Could we have a larger perimeter?" Students might suggest cutting the tables in half to form a rectangle with dimensions of 1/2 × 48. Other students will claim that if we allow rational (or real) numbers, we could go on forever.  
• "What about the minimum perimeter? Could we get a smaller perimeter?" [Yes, if we try lengths between 4 and 6.] Try 5 for one side, then the other side will have to be 24/5 (the area is fixed at 24 square units). The perimeter is 19.6 units, which is less than 20 units. This process can go on until we reach a number that when squared is 24. Students can use a calculator to compute the square root of 24, or they can estimate the square root of 24 as 4.9: 4.92 = 24.01, or approximately 24.
• Call attention to the fact that this graph is not a parabola (its shape is called a hyperbola). 
• Ask the students to read information from the graph: If the perimeter is 30 units, what are the dimensions of the rectangle?  
• If the length is 7 units, what is the perimeter of the rectangle?  
• If the length is 24 units, what is the perimeter?
• For older students, this process can be generalized as
• P = 2(l + w) and A = lw, or w = A/l. Thus P = 2(l + A/l).
• Fixed Area for Plane Figures
• For a fixed perimeter, which plane figures will have the greatest area?
• Discussion. If we extend our investigations to other figures in the plane, such as polygons and circles, students can discover that the circle is the plane figure that has the most area for a fixed perimeter.