The Exploring Area and Perimeter investigation uses a simple, relatively open problem (creating a play area with 320 feet of fence) as a means to focus students' attention on the relationships between area and perimeter for various shapes. Interactive sketches help students visualize change. Because the computer does the area calculations, students are free to focus on the effect of changing side lengths on the enclosed areas.
Assessment of student understanding should be ongoing as students work with each sketch. Questions are included before and/or after each sketch in order to encourage students to reflect on their thinking and make connections among sketches and shapes. In addition, information about perimeter and area relationships, and specific suggestions for teachers, are listed below. You may wish to investigate the sketches and answer the questions yourself prior to reading the information below.
Students may be asked to keep a written record of their findings for each sketch. Prompts such as "describe more about that idea" or "explain what you mean to another student and then try to write as if you were speaking" can be used to encourage students to write more. Helping students to describe their thinking thoroughly and in detail here will communicate high expectations for reflection throughout the sketches.
The maximum triangle area is approximately 4926 square feet. The maximized triangle is equilateral; the sketch approximates this triangle when its side lengths are between 106 and 107 feet. High school and advanced middle school students could use the Pythagorean Theorem to find the triangle height for an equilateral triangle with perimeter 320 feet and then find the exact area for the maximized triangle. All students should realize that the area would likely be increased slightly if the side lengths were each 106 2/3 feet long (i.e., 320 divided by 3).
Note that both endpoints of the segment used to control the sketch are labeled "A" - although this notation is non-standard, it has been used throughout the sketches so that the segment endpoints match the labels on the shapes.
The maximum area is obtained when the play area is a square. The square is 80 feet on each side and has an area of 6400 square feet. The graph of length (x) versus area (y) is a parabola with maximum at (80, 6400). High school students should be encouraged to find the function that describes the relationship between the lengths of the sides and the area of the shape, i.e., y=x(160-x). Middle school students could create a table and graph to determine that the function is not linear.
Pentagon, Hexagon, and Octagon Sketches
For each of these sketches, students can change the lengths of the sides by moving the points along the segments at the top of each sketch. To change the angle measures without changing the side lengths, student can drag the corner points on the shapes themselves. Although not all corner points can be dragged, students can create approximate regular polygons by a combination of dragging points on the segments and corners of the shapes.
The questions included with these sketches encourage students to reflect, generalize, and predict. Students may notice that (1) the shape areas increase as more sides are added, (2) maximized shapes have equal length sides, (3) maximized shapes are equiangular, and (4) as the areas increase, the shapes get rounder. Students should be encouraged to re-examine the previous sketches and their own writing, and compare their ideas with other students' work.
Students with sufficient knowledge of angle measure relationships in polygons can be asked to find the angle measures for the interior angles of regular pentagons, hexagons, and octagons. Students with this knowledge can also determine the exact maximum area for the regular hexagon. High school students with an understanding of basic trigonometry can find the exact areas for the regular pentagon and octagon. Once their investigations are completed, students at all levels can be asked to write a paragraph summarizing their findings. This encourages students to reflect on and communicate their thinking.