When I am learning something new, whether it's color theory, weaving, or adult education curriculum planning, doing something visual or manipulative is important to me. I have found that the same is true for my adult learners of mathematics. Traditional paper/pencil/problem/practice methods of teaching mathematics have a place, but there must be something more for the ideas to sink in.

Perimeter and Area
The concept of perimeter and area was exceedingly difficult for my students. We reviewed the formulas for rectangles (A=s², A=lw, P=4s, P=2l+2w) and worked some standard problems. I found that my students weren't really grasping the concept that perimeter is the distance around an object; area is the object's footprint. There was great confusion around when to use the perimeter formulas .vs. the area formulas.

I decided to explore the footprint concept. Each student received two sheets of centimeter (cm) square graph paper, a length of string, and a ruler with both English and Metric measures. Colored pencils, scissors and tape were in the center of the table. Each of us taped two sheets of graph paper together end-to-end. We then placed our foot on the graph paper and traced around it. (Keeping shoes on provides for a smoother line and, consequently, an easier estimation task.)

Tracing Footprints
The first task was to find the perimeter of their footprint (in cm.) using the available tools. (I made it clear that the measure they were finding was an estimate.) The first reaction was to try measuring the distance around the footprint with the ruler. Some students puzzled over the formulas. Someone soon saw that the string might be helpful. Before long they were all fitting the string around the outline of their foot, marking the distance on the string, and measuring this distance. We recorded these measurements.

The next task was to find the area of their footprint (in square cm.). Two methods were suggested:

1. Try to find a length and width for the footprint and apply the formula.
2. Count each block inside the tracing.

We discussed the advantages and disadvantages of each. For the first method, if you find a length and width, you can use the formula. But, do you draw the rectangle on the outside or the inside of the tracing? We agreed that you would probably do both and take the average of the two as the estimate of the area.

The second method would yield a pretty accurate answer, but would take forever. Through this discussion we came up with a third method: "rectangle up" the inside of the footprint, count the length and width of each rectangle, use the formula to find the area of each rectangle, and sum the areas. We decided that partial squares in the tracing would be ignored since this was an estimate.

Each student chose their favorite method of finding the area, and we recorded these next to the perimeters. We wound up with a table that looked like this:

We compared the perimeter and area for each foot. We had a grand time looking at our feet and comparing their shapes (short, fat feet; long, skinny feet; etc.) and the resulting perimeter and area of each. Someone even noted that her total footprint (both feet) would be double the area of her single foot.

After this activity, we went on to find the perimeter and area of odd-shaped rectangular figures (those with rectangular cut-outs and add-ons), with much greater success.

The class was lively, participatory, and, hopefully, when they get to the GED exam and encounter a problem dealing with perimeter or area, they will look to their feet and get to work.

Veronica Kell is an Adult Secondary Education (ASE) instructor and computer literacy instructor at the MWCC/Devens Learning Center. She can be reached by e-mail at vkell@ma.ultranet.com