The students will use varying numbers of tiles to form shapes, and then find the minimum and maximum perimeter for each.
Before the Activity
As an introduction, put some square tiles in the shape of a letter C on the overhead or draw it on the board. Ask students how many sides the figure has, what its area is, and what its perimeter is. Then ask the students to make the following figures (having only right angles if using a geoboard):
1) 4 sides having an area of 12
2) 8 sides having an area of 8
3) 12 sides having an area of 10
During the Activity
Working in groups, assign each group different areas to build. They may use either square tiles or a geoboard, but they should record their shapes on dot paper or graph paper. Group 1 could do areas 1, 5, 9, 13, and 17
Group 2 could do areas 2, 6, 10, 14, and 18
Group 3 could do areas 3, 7, 11, 15, and 19
Group 4 could do areas 4, 8, 12, 16, and 20
Demonstrate on the overhead taking 5 tiles and making a "plus sign" shape would have a perimeter of 12 while an arrangement with a 2 by 2 square with an extra tile on top has a perimeter of 10. Each group would have a calculator connected to the TI-Navigator™.
The students should put the area in L1 on their calculators, the maximum perimeter in L2, and the minimum perimeter in L3 for each number of tiles.
When each group is finished, they should send the results using TI-Navigator. The lists would be combined and sent back to the students. Then make a scatter plot using L1 and L2. Try to find a formula for the maximum perimeter if you know the area. (max per = 2n+2)
Area vs. Maximum Perimeter Area vs. Minimum Perimeter
Now make a scatter plot of L1 and L3. Do you notice a pattern? (It looks like steps. The steps keep getting longer. There are two of each length of step.)
After the Activity
Discuss why the numbers appear to all be even. Ask students to predict the maximum and minimum perimeters for different areas beyond those done in the groups.
Also talk about strategies used for finding maximum and minimum perimeters. Discuss what shapes gave a maximum or minimum. Explain why square and rectangular numbers are important in figuring the minimum perimeter.