Unit 4 - Measurement in Two and Three Dimensions Overview

This unit deepens students’ understanding of measurement by expanding the concept of measurement to two dimensions through the areas of planar figures and to three dimensions through volumes of solid figures. Students likely have prior experience with calculating the areas of conventional figures and composites of those figures. One reason for studying area is that it often represents quantities that are otherwise difficult to compute. For example, the area under the graph of an object’s speed corresponds to its total distance traveled. Therefore, techniques for calculating area can be adapted to find other quantities. Students likely have prior experience with calculating the areas of conventional figures and composites of those figures. Students may also have experience calculating the volumes of conventional solids. The unit introduces students to Cavalieri’s principle, which relates the area of a figure to its cross-sectional lengths and the volume of a solid to its cross-sectional areas. The focus of the unit is on justifying area and volume formulas with which students are already familiar and using area and volume to model real-world physical scenarios. 

ENDURING UNDERSTANDINGS
Students will understand that …

• The area of a figure depends on its height and its cross-sectional widths.
• The volume of a solid depends on its height and its cross-sectional areas.
• The geometry of a sphere is completely determined by its radius.

KEY CONCEPTS

•  4.1: Area as a two-dimensional measurement – Connecting one- and two-dimensional measurements to develop an understanding of area as a measurement of flat coverage
• 4.2: Volume as a three-dimensional measurement – Connecting two- and three-dimensional measurements to develop an understanding of volume as a measurement of space occupied
• 4.3: Measurements of spheres – Measuring areas and volumes of spheres to make sense of round objects in the physical world