b. Derive the formula for the area of a parallelogram.
As discussed in the previous post, through activities like finding the area of L-shaped region, students can develop the understanding that "when we are given an unfamiliar shape, we may still be able to calculate its area by somehow making a familiar shape (or a collection of familiar shapes)." Moreover, students can develop the following strategies to make a familiar shapes:
* divide the given shape up into several familiar shapes
* cut and re-arrange to make a familiar shape
Now they are ready to tackle this standard.
In many textbooks, students are asked to find the area of parallelograms like the one shown below using what they already know:
Some students will count the number of unit squares, making appropriate adjustments when only a part of a unit square is inside the parallelogram. Other students will try to change the parallelogram to a rectangle, a familiar shape they already know how to calculate the area of. The typical way that this is accomplished by cutting a triangular segment from one end of the parallelogram and moving it to the other side, as shown below:
Since this rectangle is 6 cm wide and 4 cm long, area can be calculated by 6 x 4, or 24 cm2.
In most textbooks, this method is then generalized to derive the formula for calculating the area of parallelograms: Area of Parallelograms = Base x Height. So, is this the end? Have we successfully addressed this particular standard? I argue that the formula at this stage is an overgeneralization. Students at this point may have difficulty calculating the area of parallelograms like the following:
Some students will try to create a new rectangle like before and notice that "the height (in red) stops here!"
Others might try to turn the figure and make a rectangle like this:
Unfortunately, they can't determine the length and the width of this new rectangle other than actually measuring them, which isn't possible if the figure isn't drawn to scale. Even if the figure is drawn to scale, actually measuring the length and the width will introduce measurement errors. So, what can students do? Actually, there are a lot of things they can do using the understanding they developed through the L-shape lesson. Here are some possibilities:
Note that (a), (c) and (d) use the "cut and re-arrange" strategy, (b) uses the "divide up" strategy, and (e) uses the "make-it-bigger" strategy. In (b), (c) and (d), the "familiar" shape students created are parallelograms that can be changed to rectangles by cutting and re-arranging right triangles.
Some of you may be wondering about (e) since students have not learned how to calculate the area of triangles. In this case, instead of calculating the area of each triangle, this student actually pushed together the two triangles that were used to make a bigger rectangle. The two triangles will make a rectangle whose dimensions are 5 cm by 6 cm.
Actually, some students may use this make-it-bigger strategy with the first parallelograms. If they did, then, this "slanted" parallelograms do not pose any challenge to them since they can use exactly the same strategy to this one as well. This strategy could have been used to derive the formula for calculating the area of parallelograms, too. Look at the figure below:
The area of the original parallelogram (un-shaded part in the figure on the left) can be calculated by subtracting the area of shaded rectangle (in the middle figure) from the large rectangle. However, this difference is really the area of the yellow rectangle in the figure on the right. That means that the area of the parallelogram is the same as the area of rectangle you can build on the base whose length is the distance between the base and its opposite side, or more accurately, the distance between the parallel lines containing the base and its opposite side. If we consider this distance between the base and its opposite side as height, we still have the same formula, Area of Parallelogram = Base x Height.
The important idea here, though, is what constitute as the height. The height of a parallelogram is the distance between the base and its opposite side, and the distance between two parallel lines is the length of a perpendicular segment connecting them. It is not the length of the adjacent side to the base. In case of a rectangle, which is a special type of parallelograms, the adjacent side may be used as the height because it is perpendicular to the base. However, that is not generally the case in parallelograms. Thus, understanding what the height of a parallelogram is may be the most important aspect of deriving the formula. Unfortunately, students don't understand this idea because they aren't asked to grapple with parallelograms like the second one we saw above, or derive the formula through the make-it-bigger strategy. I hope you will seriously consider giving your students this challenge as they try to derive the formula for calculating the area of parallelograms.