Sunday, December 6, 2009

M5M1 c - Developing Area Formulas (5)

M5M1. Students will extend their understanding of area of geometric plane figures.
c. Derive the formula for the area of a triangle.
In some textbooks, the area of triangles is studied before the area of parallelograms. In those cases, students typically examine how to calculate the area of right triangles. They notice that the area of a right triangle is a half of the rectangle you can make by using two copies of the right triangle.

They will then investigate how the area of more general triangles may be calculated, for example a triangle like this one:

Typically, students will split the triangle into two right triangles and apply the same method as before:

From here, those textbooks often generalize the way to calculate the area of triangles as:

However, sometimes students develop a misconception that the base must be the side of triangle that can be split to form 2 right triangles. In particular, they may not be able to calculate the area of the following triangle:

They want to split the triangle into two right triangles as shown, but they cannot determine the length of "base" and "height" in this case:

What they could do is to use the make-it-bigger approach and form a larger right triangle by adding on a smaller right triangle as shown below:

You can not only calculate the area of the given triangle by using this method, this method can be generalized to support the derivation of the formula. However, the challenge for students to derive (or verify the previously developed) formula is that they have to apply the distributive property as shown below:

Students may be used to applying the distributive property over addition, but they may not have had much experiences with the distributive property of multiplication over subtraction. Moreover, they have to treat the expression 4 ÷ 2 as a quantity.

Although it is possible to discuss the area of triangles before the area of parallelograms, my preference is to discuss parallelograms first. If parallelograms are "familiar" shapes, students can use a variety of methods to find the area of triangles. Here are just a couple students have come up on their own:

From the method on the left, we can see that the area of the triangle is the half of the area of the parallelogram, and the area of the parallelogram may be calculated by multiplying the "base," which is one side of the triangle, and the "height," which is the distance between the base and the parallel line containing the third vertex. From the method on the right, we can see that the area of the triangle is equal to the area of the new parallelogram. The area of the new parallelogram may be calculated by multiplying the "base," which is a side of the given triangle, and a half of the "height" of the triangle.

Either way, we can generate the formula, Area = Base x Height ÷ 2. Just as was the case with parallelograms, it is important that students understand that any of the three sides of the triangle may serve as the base and for each base, there is a corresponding height. The height is the distance between the base and the third vertex, which is the same thing as the length of perpendicular segment from the third vertex to the base.

By the way, it is probably important to write the formula with "÷ 2" since students only learn to model fraction multiplication in Grade 5 according to the GPS.

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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.