## Tuesday, November 3, 2009

### M3M4 - Developing Area Formulas (2)

M3M4. Students will understand and measure the area of simple geometric figures (squares and rectangles).
a. Understand the meaning of the square unit and measurement in area.
b. Model (by tiling) the area of a simple geometric figure using square units (square inch, square foot, etc.).
c. Determine the area of squares and rectangles by counting, addition, and multiplication with models.

Once students understand that area is the amount of space inside any geometric figures, we are ready to start thinking about ways to measure the area of various shapes. The next step is to pick a unit and actually "cover" shapes to see how many units will be needed. So, what should we use as a unit? Although we will eventually use squares as units, we may want to think about using anything that can cover the plane without a hole or an overlap. Also, using a familiar objects might be helpful to focus students' attention on the process of area measurement. One such familiar object might be index cards. Students can measure the area of the surface of desks or any other large rectangular regions.

If students have many index cards available to them, they will cover the rectangular region in many different ways. Here are three possibilities.

In this particular example, no matter how you cover the rectangle, it takes 24 small rectangles. So, we can say that the area of the rectangle is 24 units.

After measuring the area by actually covering rectangles with units, many students will realize that some ways of covering the given shape is easier to count than others. For example, the arrangements like the one on the left requires us to actually count all of the units to determine how many units were used. On the other hand, since the other two arrangements will result in equal groups (either rows or columns), we can use multiplication to find the area (either 4x6 or 8x3).

At this point, you might want to give students only 3 or 4 unit pieces to see if they can think about ways of calculating the area. A common error at this stage is to do the following:
and .

So, the area is 4x3=12 units. It is important for students to understand here why they cannot rotate the unit as they measure how many units will fit in each dimension of the rectangle. What we are trying to do when we measure the second dimension is how many rows (in this example) of 4 units there are. If we turn the unit as shown on the right, we are no longer counting the number of rows of 4 units.

You may want to ask students what we can do to avoid this type of confusion. Some students will realize that if we use a square as a unit, then it doesn't matter whether we rotate it since squares have 4 equal sides. You can then introduce that the standard units of area measurement are squares with unit length on each side, e.g., 1 cm, 1 in, 1 ft, etc.. Each unit square is said to have the area of 1 cm2, 1 in2, 1 ft2, etc., respectively. Actually, I am not sure exactly how the GPS wants these standards units of area to be handled. Unlike the units for volume, these area units are not mentioned in the GPS. However, it seems strange not to talk about the units when we are talking about the area of rectangles.

By using unit squares, we can also make it easier to determine the number of units that fit along each dimension of a rectangle by simply measuring their lengths. So, if a rectangle is 5 inches wide and 8 inches long, that means we can fit 5 1-inch squares along one row and there will be 8 rows. Therefore, we can multiply 5 and 8 to get 40 cm2. It is important that students understand that when 2 lengths are multiplied together, the product mysteriously becomes the area measurement. The two lengths we are measuring are simply telling us how many unit squares will fit along each side of the given rectangle.

Also note that students are not introduced to letters as variables until Grade 5, the formula should be written (if it is to be written at all) as, Area of Rectangle = Length x Width. Again, it is important to emphasize that this formula is to calculate the area of rectangles. Some students (and adults, unfortunately) will say that area is "length x width," but it is only a formula for a specific shape. Area is the amount of space inside a shape, no matter what the shape is.