Creating good models

Teaching Pre K-8, Jan 1999 by May, Lola

Sometimes it helps to see a math problem while you think about it.

Providing good models in the minds of students is endemic to the art of teaching, but especially so in math. Teachers use various types of materials and create pictures to form the models.

Let's look at teaching the concepts of area and perimeter. These are frequently taught close to each other with the result that students are sometimes confused about when to multiply and when to add.

Area models. Area, which involves multiplication, can be taught before perimeter. The meaning of multiplication facts can be modeled by using square tiles or graph paper.

In the example shown here, the model is for an area 3 x 5. If you ask, "What is the area of this large rectangle?" students can easily see that the answer is 15. Have students continue to show various models of the multiplication facts, each time asking, "What is the area?"

What's happening here is that students are being programmed to think multiplying when they hear the word area. Area is a two-dimensional figure and is found by multiplying the length by the width.

Teaching perimeter. Later in the curriculum, the concept of perimeter is taught. The idea that perimeter is the distance around an object is frequently taught by using graph paper, geoboards and pictures in text books. With a different model, students can easily see that perimeter is a linear measure.

Rectangles can be created out of strips of heavy paper, connected in the corners with brads. This example is a rectangle that is 6 units long and 4 units wide. Ask the students to remove one of the four brads.

The perimeter is now linear, one dimension. The solution is 6 4 6 4 or (2x6) (2x4).

A commercial material called GeoStrips can be used for many topics in geometry, but is very good at showing perimeter. The strips are in most mathematics catalogs.

Another model. Compound multiplication can be shown by a different model than the one usually used in textbooks. This model shows binomial times a binomial.

Improper fractions. A model is needed for changing an improper fraction to a mixed number. Using two pieces of paper the same size, the students write the number 1 on one side of each paper. Each paper is then folded into fourths. Shade fourfourths of one of the papers on the side opposite the number 1. Shade three fourths of the other paper on the side opposite the number 1.

When the shaded side of both papers are face up, the students see the model for the improper fraction: seven-fourths. Turn over the paper with four-fourths shaded and the students see the model for the mixed number: one and three-fourths.

How about a model that will show the reverse activity, changing a mixed number to an improper fraction? Let's change 2 1/2 to an improper fraction. The model is using an oval to represent 1 (see illustration.)

Good models are needed so students can reconstruct or visualize ways to find a solution.

Next month: Follow the Clues.

Lola May is a mathematics consultant, a former Math Consultant for the Winnetka (IL) Public Schools and a Teaching Editor of Teaching K-8.