M5M1 d - Developing Area Formulas (7)

M5M1. Students will extend their understanding of area of geometric plane figures.
d. Find the areas of triangles and parallelograms using formulae.

The most common difficulty I observe when children (or adults) are asked to find the area of triangles or parallelograms isn't that they can't remember the formulas, but rather they don't seem to have a clear understanding of what "b" (base) and "h" (height) are supposed to be. This is particularly telling when students are asked to find the area of parallelogram like the one shown below:

Although there are sputtering of errors such as adding all measurements, the most common error is multiplying the lengths of two adjacent sides, which is an overgeneralization of the area formula for rectangles. Some students may even write, "a = bh," on the paper, yet still multiply two adjacent sides. Even if we help students derive the formulas, it will be a good idea to provide students to deepen their understanding of the formulas. When you give problems, you may want to include extra information like the one above. Another potentially useful activity is to have students find all triangles with the area of 2 square units on a regular 5 by 5 geoboard.

Many students will find the following triangles:

Note that the one on the bottom right is really the same as the one in the top middle, just in different orientation.

If you ask them what strategy they used, many will say that the product of the base and the height must be 4. So, the possible combination is either 2 and 2 or 1 and 4 (or 4 and 1). If students are having difficult time going beyond those found above, you may want to focus on the two triangles on the left and ask if there is any other triangle you can make with the base of 2 units and the height of 2 units. You may even want to draw both triangles on the same base:

Eventually, someone will notice that you can "slant" the triangle a bit further:

It is important to discuss here if the height is still 2 units for this triangle. Once they agree that the height of this new triangle is 2 units, others may find another one.

If you put all of these triangles (and a mirror image of one of them) on the same base, you can see a picture like this:

Students may notice that the "top" vertex is moving along a line, and if the geoboard was larger, we could slant the triangle even further by moving the vertex further and further to the right along the same line. If you ask them to describe the relationship between the line and the base of the triangle, students will notice that they are parallel. From here, we can generalize that the area of triangle does not change if we move a vertex along the line that is parallel to the opposite side. We can then re-emphasize that this is indeed the case because all of these triangles have the same height, and the height is really the distance between the base and the parallel line containing the opposite (from the base) vertex. The distance between the two parallel lines is measure by the length of perpendicular segments from one to the other, so the height of a triangle can be measured anywhere although some are more commonly used in textbook drawings than others:

You can do a similar activity with parallelograms, as well. Through these questions, you can deepen students understanding of the formulas - in particular the understanding of the height in the triangle (and parallelogram) formula.