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.