Chapter 12: Area of Shapes

In this chapter, we continue our study of measurement by focusing on area of shapes.  We examine how students in elementary school determine areas of shapes: progressing from primitive methods to more sophisticated methods and culminating in the use  of area formulas. All methods for determining areas of shapes rely on what area means,  and all use basic principles about area to decompose and recompose shapes to work  with the pieces. These methods function in much the same way as calculation methods  in arithmetic rely on what the operations mean and use the basic properties of arithmetic  to decompose and recompose calculations. We review some familiar area formulas and  discuss how these formulas are derived from the meaning of area and basic principles  about area. We also examine the distinction between area and perimeter, which is a common source of confusion for students in elementary school. Finally, we consider how the  Pythagorean theorem can be viewed as a fact about areas, and we see how the theorem  can be derived from decomposing squares in different ways and equating areas. 

Standards for Mathematical Content in the CCSSM  

In the domain of Measurement and Data (Kindergarten–Grade 5), students learn the  concept of area. They measure areas by counting squares, and they connect area to  multiplication. Students recognize area as additive, and they use additivity to determine areas of shapes. They distinguish linear and area measures, and they consider rectangles  with the same perimeter and different areas (and vice versa). In the domain of Number and  Operations—Fractions (Grades 3–5), students find areas of rectangles of fractional side  lengths. In the domain of Geometry (Kindergarten–Grade 8), students find areas of triangles and quadrilaterals by composing and decomposing shapes, and they understand  and apply the Pythagorean theorem.  

Standards for Mathematical Practice in the CCSSM  

Opportunities to engage in all eight of the Standards for Mathematical Practice described  in the CCSSM occur throughout the study of area, although the following standards may  be especially appropriate for emphasis:  

• 1 Make sense of problems and persevere in solving them. Students engage in  this practice when they look for multiple ways to determine areas by composing  and decomposing shapes.  

• 2 Reason abstractly and quantitatively. Students engage in this practice when  they connect the reasoning for deriving an area of a shape with an area formula for  that shape, thus developing both a geometric and an algebraic perspective on area.  

• 4 Model with mathematics. Students engage in this practice when they use area and  perimeter to find quantities of interest. For example, animals might cluster in a circular  pack to keep warm. The circle’s circumference and area might give information about  the number of animals in the cluster and how many are cold on the boundary.  (From Common Core Standards for Mathematical Practice. Published by Common  Core Standards Initiative.) 

Chapter Summary