In Topic C, Students continue work with geometry as they use equations and expressions to study area, perimeter, surface area, and volume. This final topic begins by modeling a circle with a bicycle tire and comparing its perimeter (one rotation of the tire) to the length across (measured with a string) to allow students to discover the most famous ratio of all, pi. Activities in comparing circumference to diameter are staged precisely for students to recognize that this symbol has a distinct value and can be approximated by 22/7 or 3.14 to give students an intuitive sense of the relationship that exists. In addition to representing this value with the pi symbol, the fraction and decimal approximations allow for students to continue to practice their work with rational number operations. All problems are crafted in such a way to allow students to practice skills in reducing within a problem, such as using 22/7 for finding circumference with a given diameter length of 14 cm, and recognize what value would be best to approximate a solution. This understanding allows students to accurately assess work for reasonableness of answers. After discovering and understanding the value of this special ratio, students will continue to use pi as they solve problems of area and circumference (7.G.B.4).
In this topic, students derive the formula for area of a circle by dividing a circle of radius r into pieces of pi and rearranging the pieces so that they are lined up, alternating direction, and form a shape that resembles a rectangle. This “rectangle” has a length that is 1/2 the circumference and a width of r. Students determine that the area of this rectangle (reconfigured from a circle of the same area) is the product of its length and its width: 1/2(C)(r) = 1/2 2(pi)(r)(r) = pi(r)2 (7.G.B.4). The precise definitions for diameter, circumference, pi, and circular region or disk will be developed during this topic with significant time being devoted to student understanding of each term.
Students build upon their work in Grade 6 with surface area and nets to understand that surface area is simply the sum of the area of the lateral faces and the base(s) (6.G.A.4). In Grade 7, they continue to solve real-life and mathematical problems involving area of two-dimensional shapes and surface area and volume of prisms, e.g., rectangular, triangular, focusing on problems that involve fractional values for length (7.G.B.6). Additional work (examples) with surface area will occur in Module 6 after a formal definition of rectangular pyramid is established.