The role of "area" in a Common-Core aligned high school classroom

Question

Some background: I recall becoming much more adept at the concepts of area and measurement during high school geometry. However, as I scour the Common Core standards, "area" only shows up in high school standards when talking about the sectors of a circle. I could also argue that concepts of area and surface area must be understood in order to perform the majority of modeling tasks, but it is not explicitly mentioned.

Area is supposed to be covered in the pre-high school standards; however, experience shows that students entering the high school geometry classroom have forgotten the majority of formulas they've learned and, though they relearn the formulas reasonably quickly, they appear to have never or rarely applied these formulas to compound shapes. Only one or two had ever applied a decomposition argument to justify the area of a triangle, much less a parallelogram or trapezoid (which are 6th-8th grade standards)!

I have a few related questions that should be taken in this context, in addition with the knowledge that learning from the middle grades is often forgotten:

• Presuming a student has actually experienced their math education in a CC-aligned manner, what role should the idea of area play in his or her geometry education at the high school level?

• Presuming a student's math education was hastily shoehorned into CC alignment sometime around the 6th or 7th grade, what role should the idea of area play into his or her geometry education at the high school level?

This question is in the context of common core alignment, but can also be answered in the context of college preparedness, developmental need, etc.

Answer 1

One of the drawbacks to the Common Core Standards is the implied assumption that students will not need revision on previous material. All classes need reteaching on certain topics at times, and teachers will need extra time for such remedial instruction. Clearly if students are lacking foundational material that was supposedly taught in previous years, the teacher will have to fill in these gaps before moving forward. This is even more true as you work with students who are lacking knowledge because of a change in curriculum.

Area concepts are extremely important. Americans tend to overemphasize logic and proofs in their geometry courses, sometimes to the detriment of geometry content. (Other countries spread the study of proofs more evenly throughout the math curriculum.) Most geometry students will not go on to do college level math and they should clearly leave the course with sufficient knowledge of geometry to handle basic work requirements and needs from daily life. Clearly, measurement concepts are among the few areas of geometry that most people will need in life. I think this is why the Common Core emphasizes area concepts so much in grades 3 through 7.

I agree that the Common Core does not address area as thoroughly as it should at the high school level. However, one of the main stated aims of the Common Core was to trim curricula, so the standards never repeat topics supposedly already covered. The Common Core does, however, go into a few area concepts besides circle sectors at the high school level. For example:

G-GPE 7: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.★

G-GMD 1: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.

G-MG 2:Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).★

For teachers looking to increase their students' familiarity with measurement concepts, the two starred standards (indicating a connection with modeling) would be a good opportunity to allow students to practice their understanding of area through practical exercises or even project-based learning.