Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

 A B C

Solutions for Session 10, Grades 6-8, Part B

See solutions for Problems: B1 | B2 | B3 | B4 | B5 | B6

Problem B1

Rectangular Prisms
With a Volume of 72 cm3

Dimensions

Surface Area

 1 by 1 by 72 290 cm3 1 by 2 by 36 220 cm3 1 by 8 by 9 178 cm3 2 by 2 by 18 152 cm3 2 by 4 by 9 124 cm3 3 by 4 by 6 108 cm3 4 by 4 by 4.5 104 cm3 4.16 by 4.16 by 4.16 103.8336 cm3

Volume does not uniquely determine the size of a rectangular prism. In terms of size, as the dimensions of the rectangular prism become more similar, the surface area decreases. Said another way, the surface area of a rectangular prism is minimized as its shape approaches a cube.

 Problem B2 Since 72 has many factors (1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72), Mr. Cellucci's choosing 72 cm3 for the rectangular prism's volume allowed for a range of rectangular prisms to be constructed.

Problem B3

 a. Some students noticed that rectangular prisms that looked more like cubes had smaller surface areas. Other students focused on the dimensions and saw that as the dimensions "got closer," the surface area "got smaller." One group hypothesized that a perfect cube would minimize surface area, and then set out to determine the necessary dimensions and to construct that cube. b. Drawing the two-dimensional net and three-dimensional solid helped students assign appropriate dimensions and visualize what the rectangular prism would look like before they constructed it. Drawing the net of the three-dimensional solid was also helpful in understanding and determining surface area. Students saw that the total area of the two-dimensional net was the surface area of the rectangular prism and that the net is, in effect, "wrapped around" the solid. By working with the nets, students also focused on the number of identical faces and started to realize they could find the area of certain faces and then multiply to get the total surface area.

 Problem B4 One of the standards listed for grades 6-8 is "Develop strategies to determine the surface area and volume of selected prisms,..." which this problem asks students to do (repeatedly). They must determine surface areas, and they must, in a sense, determine volumes in reverse -- using a volume and coming up with dimensions of a prism that yield it. In addition, this problem directly addresses the common misconception pointed out in the standards and cited in the previous part, "Exploring Standards."

Problem B5

The mathematical purpose of the lesson is clear. Students understand that they can create a range of rectangular prisms with the same volume. Some of the factors that make the lesson successful include the following:

 • The lesson is hands-on. Students get to measure, construct, and fill the prisms with rice. • Students organized data in a table and looked for patterns. Some students will more readily notice number patterns than geometric patterns. • Students moved from making two-dimensional nets to three-dimensional solids. Creating nets for the rectangular prisms helped students to visualize them first.

The lesson allowed students to enter into the task at a variety of points. Mr. Cellucci also underscored the practical applications by suggesting situations in which you would want to be "economical" and minimize packaging materials.

Problem B6

In the summary discussion, Mr. Cellucci helped students focus on the dimensions, as well as the surface area, of their rectangular prism by prompting them for the dimensions and recording them in a chart. This allowed students to focus on relationships among the dimensions and between dimensions and surface area. The resulting generalizations were the following:

 • As the dimensions get closer or the "lengths more similar," the surface area gets smaller. • A cube will have the smallest surface area. • You can find the dimensions of a cube by taking the cube root of the volume. (This was generalized from the students' understanding of dimensions and area of a square and of taking a square root.)