7.3 GeoGebra Illustrative Mathematics - Measuring Circles

In this unit, students apply their knowledge of proportional relationships to the context of measuring circles. They learn the relationships between radius, diameter, circumference, and area of circles and use these relationships to solve problems. This builds on students’ work from previous grades with perimeter and area of polygons. Students will build on this work in grade 8 when they study the volume of spheres, cylinders, and cones.

The unit begins with activities designed to help build up students’ vocabulary for describing circles more precisely. The terms "center," "radius," "diameter," and "circumference" are introduced. Then students investigate the relationship between circumference and diameter and see that it is a proportional relationship. They apply this relationship to solve problems.

Next, students explore the area of circular regions. They see an informal derivation that shows where the formula  comes from and then use this formula to solve problems. Finally, students solve problems that require deciding whether the situation relates to the circumference or area of a circle.

The first section of this unit, in which students recognize and apply proportional relationships involving circumference, serves as a bridge between the foundational work with proportional relationships in the previous unit and the more advanced applications in the following unit. The remaining sections of this unit, which deal with the area of circles, are preparation for the continued geometry work students will do later in this course.

A picture of three different circular objects. The leftmost object is a wagon wheel with a measuring tool starting from one point on the wheel, goes through the wheel center to a point on the other side of the wheel. The center object is a plane propellor with three identical propellor blades. A measuring tool starts from the center of the propellor and goes to the end of the blade. The third object is of a sliced orange. A measuring tool goes around the entire circular region of the orange.

A note on using the term "circle":

Strictly speaking, a circle is one-dimensional. It is the boundary of a two-dimensional region, rather than the region itself. The circular region is called a “disk.” Because students are not yet expected to make this distinction, these materials refer to both disks and the boundaries of disks as “circles,” using illustrations to eliminate ambiguity.