Saturday, October 24, 2009

M3M4 & M5M1 - Developing Area Formulas (1)

M3M4. Students will understand and measure the area of simple geometric figures (squares and rectangles).
M5M1. Students will extend their understanding of area of geometric plane figures.

In the past several posts, I discussed important ideas involved in helping students develop multiplication and division algorithms. With today's post, I want to start a new series on how to help students develop various area formulas (M3M1 & M5M1). In today's post, however, I want to focus my attention on the basic ideas about teaching and learning of area measurement.

I have discussed previously (December, 2008) some basic ideas of teaching measurement. As we teach measuring of any attribute (e.g., length, weight, area, angle, etc.), we must first help students understand the particular attribute we are trying to measure. Without understanding the attribute, measuring it will not make any sense. For that purpose, comparison activities are very useful. Typically, we start with direct comparisons, then move on to indirect comparisons. After students understand the concept, we can start thinking about quantifying the amount of the attribute, i.e., measuring the object. Many people suggest that we start with non-standards units first. One reason for this suggestion is that if we start with standard units, students will have to learn the idea of using units to quantify the attribute and the idea of standard units. Moreover, when we try to measure with standard units, we typically use measurement instruments, such as rulers and protractors. So, students will also have to learn how to use measurement instruments. If we start with non-standard units, students can focus on the notion of quantifying, or measuring, first. Once students understand how a particular attribute can be measured using a non-standard unit, they can use that understanding to both measuring with standard units and learning how to use measurement instruments. After all, the notion of "standard" units probably does not make sense unless you have some experiences with "non-standard" units.

So, what does this all mean when we are teaching the area measurement? Obviously, the first focus should be on helping students understand what area is about. To do so, it seems like we should have students engage in some comparison activities. So, for example, let's ask students to compare the following two rectangles (one is actually a square, but we know that squares are rectangles, don't we?).

Note that I am just showing you the drawing of rectangles, but you should give children cut out pieces to compare. Moreover, I included grid lines to show the dimensions of the rectangles, but you may not want to do so when you give these shapes to students.

Anyway, when students are given these two shapes and asked "Which is bigger?" you see generally two different ways students will compare these shapes. Some children will compare the shapes by overlapping them (on the left below). Others will put the shapes next to each other (on the right).

We can ask students if these two ways of comparing are actually comparing the same attribute (we may not want to use this particular word with 3rd graders). Students may not know, but they can understand that if the conclusions we get from these two ways are different, then, they couldn't be comparing the same attribute. So, have them try comparing these two shapes using both ways. For example, if we put the shapes next to each other and rotate one of them around the other shapes, you see that the two shapes are the same size:

When you overlap shapes, we see that the square is actually "bigger" than the rectangle:

Since the results are different, these two ways of comparison are indeed comparing two different attributes. We know that the first comparison was comparing the length around these shapes, or perimeter, while the second comparison is about area, i.e., the amount of space inside the shapes.

I am not saying that children will understand the difference between the perimeter and the area by doing this one activity. However, it is important for children to have a number of comparison activities to compare the length around and comparing the amount of space inside. When two same objects give different results like the one above several times, students might develop a better sense of the difference between these two attributes.

Once students understand what the area as an attribute is about, we can now move into the discussion of measuring it. So, in the next post, we will discuss M3M1, in which students think about the area of rectangles and squares.