Changing Perimeter

Students should experience changing perimeter as the area remains constant, look for patterns, and make conjectures about the minimum and maximum area.

This activity helps students look patterns as they collect data about the minimum and maximum perimeter formed by a set of tiles.

Procedures:

Divide the class into groups of 3-4 students. Ask students the difference between area and perimeter.

Put some tiles in the shape of a letter U 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

Instruct the groups of students that we will be using cubes in the lesson today, but we are only looking at the face of each so we are only thinking of them as squares. With the cubes, ask the groups of students to make the following figures (having only right angles).

4 sides having an area of 12

8 sides having an area of 8

12 sides having an area of 10

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.

Assign each group different areas to build. They will use the tiles to build their shape and then 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

As the students determine the maximum and minimum perimeter they can form for each area they should send someone to the overhead to record their results. If an overhead graphing calculator is available, enter the data in the graphing calculator:

The area in L1

The maximum perimeter in L2

The minimum perimeter in L3

After students have recorded all the results view the area (x-axis) vs. minimum perimeter (y-axis) numerically and graphically. Ask students what patterns they see. Relate the pattern to the cubes the students built.

After students have recorded all the results view the area vs. maximum perimeter numerically and graphically. Ask students what patterns they see. Relate the pattern to the cubes the students built.

Monitor the groups of students as they are forming the figures. Assess if they see any patterns taking place during their investigation.

Closure: Ask students to write down 1-2 ideas they learned about the relationship of area and perimeter from today’s lesson

Homework Assignment: Take 2 grids home tonight and draw 4 shapes: One with an area of 21 with the smallest possible perimeter, one with an area of 21 with the largest perimeter possible, one with an area of 22 with the smallest possible perimeter, and finally one with an area of 22 with the largest possible perimeter. Describe how these four drawings fit the patterns we say in class today.