Saturday, November 14, 2009

M5M1 b - Developing Area Formulas (4)

M5M1. Students will extend their understanding of area of geometric plane figures.
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
* make-it-bigger
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.

Saturday, November 7, 2009

M5M1 - Developing Area Formulas (3)

M5M1. Students will extend their understanding of area of geometric plane figures.

As we discussed in the previous post, the GPS expects students to determine the area of rectangles and squares by counting or calculation. Then, in Grade 5, students are expected to derive and use formulas to determine the area of parallelograms, triangles, and circles. Interestingly, there is nothing about area mentioned in Grade 4. It is listed as one of the "Concepts/Skills to Maintain," but there is no specific standard about the area measurement in Grade 4. Many people might wonder about the feasibility of fifth graders actually deriving the area formulas of parallelograms and triangles on their own. Do they have enough background knowledge? What background knowledge do they need to increase the likelihood of their deriving those formulas?

In a previous post on the idea of teaching through problem solving (April, 2009), how children can learn through problem solving new mathematical ideas. Those mathematical ideas are the ones that will serve as the bridge between M3M4 (area of rectangles and squares) and M5M1 (area of parallelograms, triangles, and circles). As we will see shortly, those specific understandings will be used over and over to derive the formulas. So, in Grade 3, finding the area of L-shapes may be simply a complex application of what they learned, but, in Grade 5, the focus should be on ways of thinking involved in calculating the area. If those understandings are made explicit, students are much more likely to be successful in deriving the area formulas. So, I encourage you to read that post again (or for the first time, if you have not read it before).

By the way, element (f) of this standard says, "Find the area of a polygon (regular and irregular) by dividing it into squares, rectangles, and/or triangles and find the sum of the areas of those shapes." Actually, this element is simply one of the strategies developed in the L-shape lesson, that is, sub-dividing the given unfamiliar shape into a collection of familiar shapes. The only difference is what shapes are available to students as familiar shapes. When students work on the L-shape problem, they only knew how to calculate the area of rectangles and squares. However, after students have learned the formulas for the area of parallelograms and triangles, students can also use those figures. So, in case you are wondering if you can afford to spend an extra time to discuss something that is not explicitly mentioned in the GPS, the L-shape lesson does address the GPS directly, too.

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.

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Elaboration of Georgia Performance Standards by Tad Watanabe is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.